
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00024530/00001
Material Information
 Title:
 Linear prediction of continuoustime bandlimited processes, with applications to fading mobile radio
 Creator:
 Lyman, Raphael J
 Publication Date:
 2000
 Language:
 English
 Physical Description:
 x, 94 leaves : ill. ; 29 cm.
Subjects
 Subjects / Keywords:
 Error rates ( jstor )
Fourier transformations ( jstor ) Linear prediction ( jstor ) Mobile radios ( jstor ) Predictability ( jstor ) Signals ( jstor ) Spectral energy distribution ( jstor ) Stochastic processes ( jstor ) Wave functions ( jstor ) White noise ( jstor ) Mobile communication systems ( fast ) Radio  Transmitters and transmission ( fast )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 2000.
 Bibliography:
 Includes bibliographical references (leaves 8893).
 General Note:
 Printout.
 General Note:
 Vita.
 Statement of Responsibility:
 by Raphael J. Lyman.
Record Information
 Source Institution:
 University of Florida
 Rights Management:
 The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. This item may be protected by copyright but is made available here under a claim of fair use (17 U.S.C. Â§107) for nonprofit research and educational purposes. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact the RDS coordinator (ufdissertations@uflib.ufl.edu) with any additional information they can provide.
 Resource Identifier:
 45125797 ( OCLC )
ocm45125797

Downloads 
This item has the following downloads:

Full Text 
LINEAR PREDICTION OF CONTINUOUSTIME,
BANDLIMITED PROCESSES, WITH APPLICATIONS
TO FADING IN MOBILE RADIO
By
RAPHAEL J. LYMAN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2000
Copyright 2000 by
Raphael J. Lyman
This work is dedicated to Miss Lena Margaret Lyman.
I had a little nut tree, nothing would it bear
But a silver nutmeg and a golden pear.
The King of Spain's daughter came to visit me,
And all for the sake of my little nut tree.
I skipped over water, I danced over sea,
And all the birds in the air couldn't catch me.
Mother Goose Rhyme
ACKNOWLEDGMENTS
I have rarely paid much attention to acknowledgments in the books and
articles that I have read, viewing them mostly as formalities, but now that I am assuming the role of presenter, I would not dream of submitting this dissertation without offering my deepest gratitude to those who have made this work possible. The research described here is the result of a team effort. In addition to myself, the team includes my advisor, Dr. William Edmonson, as well as two faculty members, Dr. Scott McCullough and Dr. Murali Rao, from the Department of Mathematics. I am also grateful to Dr. John M. M. Anderson, Dr. Tan F. Wong, and Dr. Haniph A. Latchman, for graciously agreeing to serve on my supervisory committee. Generous support for this project, and especially for my participation in it, was provided by the Motorola LandMobile Products Sector.
At various times during my work I have received assistance from many people. A few of them simply must be mentioned. Dr. Scott Miller and Dr. Tan Wong have been invaluable sources of information regarding communication theory in general and wireless technology in particular. Both have offered many helpful comments and suggestions. Conversations that I have had with Dr. Bert Nelin have yielded interesting insights into the phenomenon of frequencyselective fading. I also thank Dr. M. V. Ramana, who expressed enthusiasm for the problem at an early stage, and participated with me in a memorable brainstorming session. In a more general way, Dr. Leon W. Couch II has aided me greatly by making the benefit of his experience available whenever I have needed it, and I am much obliged.
A word is in order here regarding my relationship with my advisor. Early on, Dr. Edmonson took a considerable risk by providing me with not only the
iv
utmost freedom in seeking a solution for our problem, but also considerable voice in defining the problem itself. His approach throughout has been one of unbounded enthusiasm and support, and I have always sought to make his risk pay off. It would seem that in the academic world a dose of sobriety, when needed, is always available. But in the moments of greatest doubt, when intuition is your only guide, it is probably safe to say that encouragement is what is needed most, and Dr. Edmonson provided it.
Of course, not all the challenges in producing a doctoral dissertation are
academic ones, but thanks to the efforts of Ron Smith in creating the ITEX class ufthesis, the mechanics of manuscript preparation have been considerably eased for me, and for many other graduate students at the University of Florida.
I am grateful to the Institute of Electrical and Electronic Engineers for granting permission to reuse copyrighted material [41, 40]. Thanks to the administrative staff of the Department of Electrical and Computer Engineering, especially Sharon Bosarge, Janet Burke, Janet Holman, Linda Kahila, Chris Reardon, Alice Riegel, Greta Sbrocco, and Wes Shamblin, who have made my life easier in an uncountable infinity of ways. Thanks also to Frances Smith, who has been scrupulous in her efforts to keep our working environment tidy, an underappreciated task, to be sure.
Finally, I owe a very special debt of gratitude to my wife, Chulalak, for her patience and support, and beg the pardon of my twoyearold daughter, Lena, who has had to put up with so much absence on my part.
Raphael J. Lyman
v
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS ............................. iv
LIST OF FIGURES ................................ viii
ABSTRACT .................................... ix
CHAPTERS
1 PREDICTION OF BANDLIMITED PROCESSES ......... 1
1.1 Project Thesis .......................... 2
1.2 Research Contributions . . . . . . . . . . . . . . . . . . . . . 4
1.3 Literature Survey ........................ 6
2 PROBLEM FORMULATION AND ANALYSIS . . . . . . . . . . . 16
2.1 Problem Formulation ...................... 16
2.2 A nalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Dealing with Estimation Errors . . . . . . . . . . . . . . . . 24
2.4 Conclusions ........................... 27
3 PROCESSES WITH FLAT SPECTRAL DENSITIES . . . . . . . . 29
3.1 Basis Functions ......................... 30
3.2 Solution of the Linear Predictor . . . . . . . . . . . . . . . . 34
3.3 Minimum Mean Squared Prediction Error . . . . . . . . . . . 38
3.4 A Bandlimited Process in White Noise . . . . . . . . . . . . 42
3.5 Conclusions ........................... 45
4 APPLICATIONS TO FADING IN MOBILE RADIO . . . . . . . . 47
4.1 Multipath Fading in Mobile Radio . . . . . . . . . . . . . . . 47
4.2 Adaptive Channel Estimation . . . . . . . . . . . . . . . . . 53
4.3 Addressing the Model Mismatch . . . . . . . . . . . . . . . . 74
5 CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . 80
5.1 Conclusions ........................... 80
5.2 Future W ork ........................... 83
APPENDIX EVALUATION OF BASIS FUNCTIONS . . . . . . . . . . . . 86
vi
REFERENCES ................................... 88
BIOGRAPHICAL SKETCH ............................ 94
vii
LIST OF FIGURES
Figure page
3.1 Minimum mean squared prediction error of a bandlimited process... 43 4.1 Model of a fiatfading mobile radio channel . . . . . . . . . . . . . . 49
4.2 Autocorrelation function of a fading parameter . . . . . . . . . . . . 51
4.3 Power spectral density of a fading parameter . . . . . . . . . . . . . . 51
4.4 Example of a complex fading envelope . . . . . . . . . . . . . . . . . 52
4.5 Adaptive channel estimation for a fiatfading channel . . . . . . . . . 54
4.6 Predictive method for maintaining correct channel tracking . . . . . 56 4.7 Linear prediction of a fading parameter . . . . . . . . . . . . . . . . 57
4.8 Predictability analysis for recovery of correct channel tracking .... . 74 4.9 Comparison of fadingenvelope spectra . . . . . . . . . . . . . . . . . 77
4.10 A piecewise approximation of the fadingenvelope spectrum . . . . . 77 4.11 Application of the fiat spectral density to nonadaptive prediction. . . 78
viii
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
LINEAR PREDICTION OF CONTINUOUSTIME,
BANDLIMITED PROCESSES, WITH APPLICATIONS TO FADING IN MOBILE RADIO
By
Raphael J. Lyman
May 2000
Chairman: William W. Edmonson Major Department: Electrical and Computer Engineering
In digital mobile radio, many techniques aimed at compensating for the distorting effects of multipath fading could benefit from a prediction of the fading envelope, a complex time function often modeled as a bandlimited random process. We consider a continuoustime linear predictor applied to a bandlimited process. We show that if the past values of the process are known over an interval of arbitrary positive length, then the mean squared prediction error may be made arbitrarily small, regardless of how far in the future we wish to make the prediction. We also show that this is no longer true when an energy constraint is applied to the predictor, and we discuss what this means for the case in which the prediction is based on past values that are corrupted by estimation errors.
We then go on to solve explicitly for the optimal, energyconstrained predictor when the process spectral density is flat within the band limits. As basis
ix
functions, we use timeshifted versions of the prolate spheroidal wave functions, leading to a simple algebraic optimization problem which may be solved using a Lagrange multiplier. We show how to use the solution to compute the minimum mean squared prediction error under the energy constraint. Then we discuss the case of a bandlimited process emebedded in white noise, showing how to determine if a specified mean squared prediction error can be attained.
Finally, we apply these prediction techniques to a problem in decisiondirected adaptive channel estimation. We show how an adaptive estimator may lose track of the channel when the mobile receiver enters a deep fade. We consider the use of a predicted value of the fading envelope to restart the adaptation after the fade. Our analysis determines the conditions under which we may expect to recover correct carrier tracking using the predicted value. The primary factor is the maximum Doppler frequency, which is proportional to the speed of the receiver.
X
CHAPTER 1
PREDICTION OF BANDLIMITED PROCESSES
Random processes are often classified as being either predictable or regular [52, p. 420]. The future values of a predictable process may be estimated with a mean squared error of zero if the past values of the process are known. It can be shown that the spectrum of such a process can consist only of lines. Regular processes, which contain no spectral lines, are not predictable in this sense.
Processes which contain no spectral lines, but whose spectrum vanishes
outside of a certain band limit, stand between these two extremes. The future of these bandlimited processes cannot be predicted with zero error, but under certain conditions, if one has sufficient knowledge of past values, the prediction error can be made arbitrarily small [52, p. 380].
The aim of this research is to extend these findings and to discover how they may be applied in the analysis of specific signal processing problems. The motivating application was decisiondirected channel estimation in a fading, mobileradio environment. A simple version of the problem is described and analyzed in Section 4.2. Though we wish to maintain our focus on the issue of fading, it is hoped that our results will show a broader potential for bandlimited modeling than has been previously realized.
In the next section we summarize the fundamental ideas that have directed our research, and discuss the significance of our key findings. Section 1.2 contains a list of our research contributions. Finally, Section 1.3 is a detailed survey of the relevant literature.
1
2
1.1 Project Thesis
In communications and signal processing, one sometimes encounters problems which are conveniently formulated in terms of bandlimited random processes. For example, in wireless communications, multipath fading may be viewed as a modulation of the transmitted signal by a complex time function called the fading envelope [54, Ch. 14]. For mobile radio, the fading envelope may be modeled as a bandlimited random process, with the band limits depending upon the speed of the mobile receiver [11].
Some techniques designed to compensate for the effects of fading require an estimate of the current value of the fading envelope. An example is adaptive transmission, which uses this estimate to make appropriate adjustments to the power or rate of transmission. Unfortunately, adaptive transmission is very sensitive to delays which inevitably occur in feeding back estimates of the fading envelope from receiver to transmitter [21]. In such a case we would naturally consider whether the current value could be estimated based upon our knowledge of the past. So adaptive transmission is one example of an application that could benefit from the prediction of a bandlimited process.
Prediction problems are often addressed using the techniques of Wiener filtering, but the solution of the Wiener prediction problem requires that the power spectral density be factorable into minimumphase and maximumphase finiteenergy parts. Unfortunately, this factorization is impossible when the spectral density is zero over a set of positive measure, so the Wiener approach cannot be used to solve the prediction problem for bandlimited processes (see, e.g., Papoulis [52, pp. 402, 493]).
Previous approaches to the problem have focused on the prediction of future values by taking linear combinations of past sample values [72, 5, 8, 9, 59, 62, 50, 45, 68]. The questions that have been dealt with include the conditions under
3
which the prediction error may be made to approach zero, as well as procedures for calculating the predictor coefficients. All of these works assume that samples of known values may be taken arbitrarily far in the past. None addresses the question of how good a prediction is possible if the known past values are confined to a given interval.
In this work, we consider the predictability of a bandlimited random process using a continuoustime linear predictor, rather than a linear combination of past samples. We suppose that a sample function of the process is known over an interval of arbitrary positive length, and we show that future values of that sample function may be predicted with a mean squared error that is arbitrarily small, regardless of how far in the future we wish to make the prediction.
We also show that this is no longer true when we apply an energy constraint to the impulse response of the linear predictor. The constrainedenergy problem is used to analyze the case in which linear prediction is to be based on past values which have been corrupted by estimation errors. If such errors are modeled as white noise, we can then show that they impose a fundamental limitation on the predictablility of the process.
An important contribution of the work, as described in Chapter 3, is the
solution of the bandlimited linear prediction problem in the case where the power spectral density is constant within the band limits. The parameters of the problem are the band limits, the length of the interval of known past values, how far in the future the prediction is to be made, and the energy constraint. With these parameters given, we show how to obtain an expansion for the optimal predictor in terms of a set of basis functions that are timeshifted versions of the prolate spheroidal wave functions [60]. We include a procedure for computing the minimum mean squared error as a function of the energy applied in the constraint. Then we
4
consider the process to be corrupted by white noise, and show how to determine if a specified mean squared prediction error can be attained.
Finally, we apply these prediction techniques to a problem in decisiondirected adaptive channel estimation. We show how an adaptive estimator may lose track of the channel when the mobile receiver enters a deep fade. We consider the use of a predicted value of the fading envelope to restart the adaptation after the fade. Our analysis determines the conditions under which we may expect to recover correct channel tracking using the predicted value. The primary factor is the maximum Doppler frequency, which is proportional to the speed of the receiver. Our conclusion is that the tools we have developed for linear prediction of bandlimited processes can be useful in the analysis of many approaches to fading compensation.
1.2 Research Contributions
We now offer a summary of our contributions. Repeating for emphasis, we consider a continuoustime, bandlimited process, a sample function of which is known on an interval of positive length. We wish to estimate some future value using a predictor which is linear with respects to the known interval. Our findings include the following:
1. The mean squared prediction error of a bandlimited process may be made
arbitrarily small.
2. No linear predictor can make the error zero.
3. If an energy constraint is placed on the impulse response of the predictor,
then the mean squared prediction error has a minimum which is greater
than zero.
5
4. If the prediction is based on noisy data, the mean squared prediction error has a minimum, greater than zero, even without an energy constraint. In the case of a bandlimited process whose power spectral density is flat within the band limits, our contributions include:
5. A method for constructing the optimal linear predictor in the energyconstrained case.
6. A procedure for computing the minimum mean squared prediction error
under the energy constraint.
7. Assuming the process to be corrupted by white noise, a method for
determining whether a specified mean squared prediction error can be
attained.
Regarding the issue of mobile radio fading, our contributions include:
8. An approach for applying the techniques of items 57 above to a problem
in adaptive channel estimation.
9. The finding that, for the problem in item 8, success of the proposed
approach depends primarily on the maximum Doppler frequency, which is
proportional to the speed of the mobile receiver.
10. A method of computing the greatest value of maximum Doppler frequency for which the proposed approach will work.
Chapter 2 contains proofs of items 14. The techniques in items 57 are
developed in Chapter 3. The problem in adaptive channel estimation is analyzed in Section 4.2. Finally, some topics for future research are listed in Section 5.2.
6
1.3 Literature Survey
In this section we shall review the literature that has proved relevant to
our research. The survey is in five parts. Section 1.3.1 includes previous material dealing directly with the prediction and extrapolation of bandlimited signals. In Section 1.3.2 we discuss sources for the various mathematical techniques that we have employed in seeking a solution to this problem. Then, in Section 1.3.3, we consider treatments of various issues in mobileradio fading, the problem which motivated our interest in bandlimited processes. Section 1.3.4 includes sources on equalization and carrier recovery, including blind techniques and the issue of decisiondirected convergence. Finally, in Section 1.3.5, we discuss some general references.
1.3.1 Bandlimited Prediction and Extrapolation
Up until now, investigations of bandlimited prediction have focused on the following question: Suppose you have a stationary random process whose spectral density vanishes outside of some finite interval, and suppose you may obtain sample values of this process arbitrarily far in the past. You wish to predict a future value as a linear combination of the past samples. Under what conditions is it possible to make the prediction error approach zero, and how can the predictor coefficients be calculated, knowing only the band limits and not the exact spectral shape? Note that such coefficients may not be optimal for a given spectral density and set of sample values.
An early treatment of this problem can be found in a 1962 text by Wainstein and Zubakov [72, p. 70]. They suppose that the past values of the process are uniformly sampled at a rate higher than three time the Nyquist rate, which is six times the spectral band limit. They show that the mean squared prediction error may be made arbitrarily small by taking linear combinations of values that extend further and further into the past. The proof is constructive, showing how the
7
coefficients may be calculated. The problem is not central to their development, but the book is referred to in many of the later entries.
In 1966, Beutler [5] addressed the problem in the general context of the
recovery of bandlimited signals from irregularly spaced past samples. His treatment includes both stochastic and nonstochastic signals, and uniform sampling may be considered a special case. Using the gap and density theorems of Levinson [36], he showed that it is theoretically possible to drive the prediction error to zero if the past samples are taken at any rate higher than the Nyquist rate. This rate improves on that of Wainstein and Zubakov by a factor of three, though the book by those authors is not listed among Beutler's references. Beutler's proof, however, does not give a procedure for calculating the predictor coefficients.
In a 1972 correspondence, Brown recognized the works of Wainstein and Zubakov, and of Beutler, and then offered a procedure for calculating the coefficients in the case of uniform sampling at only twice the Nyquist rate [8]. Fjallbrandt contributed a letter in 1975 dealing with certain interpolation and extrapolation problems [15]. As an extreme case, he finds coefficients for extrapolation with uniform sampling at any rate higher than the Nyquist rate. He offers this as a constructive proof of Beutler's results for the uniformsampling case, though it seems unclear whether the derivation is good for finiteenergy or finitepower signals, and the stochastic case is not referred to explicitly.
Slepian discusses the problem, without reference to the previous authors, in a 1978 article on the discrete prolate spheroidal sequences [59]. In this case, he considers the optimal predictor for N uniformlyspaced samples of a bandlimited process whose spectral density is flat within the band limits. He shows that if the sampling rate exceeds the Nyquist rate, the mean squared error of the prediction approaches zero exponentially in N.
8
The next entry is a 1981 correspondence by Knab who, again, makes no reference to any of the above work [32]. His interest is in the interpolation of a finitepower bandlimited signal when the sample times are not symmetric with respects to the point to be estimated. Knab's approach is interesting because he imposes no necessary stochastic interpretation on the problem, and for this reason, his contribution is difficult to place. He deals with endpoint extrapolation as an extreme case, in the manner of Fjallbrandt. Another interesting point is that he uses a timeshifted approximation of the prolate spheroidal wave functions [31], with an approach that bears some similarity to ours in Chapter 3.
In a 1982 paper, Spletstosser [62] reviews the work of Brown, as well as Wainstein and Zubakov, and then shows how to calculate the coefficients for sampling at 1.5 time the Nyquist rate. The problem was addressed again in 1985, this time in a letter by Papoulis [50]. He shows the possibility of prediction with arbitrarily small error using an approach substantially different from that of Beutler. His treatment is quite brief, though it is clearly laid out, but of the authors we have discussed, he mentions only Wainstein and Zubakov in his references.
This led to a number of followup letters in the same journal, each making
some attempt to set the record straight, and some making additional contributions. Marvasti claimed that all of Papoulis's findings had been proved by others previously [45]. He then offered his own brief proof based on a theoretical framework described in an article by Requicha [56]. In the same issue, Brown reiterated his earlier proof [9], and reminded readers of Beutler's work [5]. Finally, Vaidyanathan draws attention to some past literature and, in an explicitly stochastic framework, shows how to calculate predictor coefficients that work for any sampling rate above the Nyquist rate [68]. This entry by Vaidyanathan, published in 1987, is the last that we have found to deal directly with this problem.
9
This survey shows that contributions dealing with bandlimited prediction have been infrequent, often marked by brevity, and characterized by a lack of discussion concerning specific engineering applications. Often, the topic has been introduced as an adjunct to a more general discussion. In addition, there seems to be some suspicion of bandlimited modeling in general. For example, Wainstein and Zubakov [72] note that the addition of even a small amount of white noise to the bandlimited signal destroys its predictability properties. They add,
Of course, as the intensity of the noise approaches zero, we obtain the
formal possibility of predicting m(t) arbitrarily far ahead. However, to substantially increase the time interval for which the prediction is
possible, we must enormously increase the signaltonoise ratio. (p. 73)
Also, see Slepian [58] for a more general discussion of bandlimited modeling. In our research, we address the issue of noise sensitivity by regularizing the problem with an energy constraint (see Section 2.3).
In contrast to the work on bandlimited prediction, the related but distinct problem of bandlimited extrapolation has been much more extensively studied. This problem in its basic form may be stated as follows: Suppose you have a segment of a finiteenergy signal whose Fourier transform vanishes outside of some finite interval. How can you use that segment to calculate the values of the function that are not on the known segment?
We mention only a few examples that have come to our attention in the
course of our work. In 1975, Papoulis described an iterative procedure for performing the extrapolation on a continuoustime signal [48]. He showed theoretically that the algorithm converges, but numerical difficulties led Cadzow to develop a new algorithm, which is still widely used [10]. Slepian et al. also had addressed the problem in 1961 as an application of the prolate spheroidal wave functions [60, p. 46].
10
Bandlimited extrapolation continues to be an active research field, and has admitted of a wide variety of problem formulations. In particular, the case of periodic bandlimited functions was of some interest to us (see, e.g., SoltanianZadeh [61]) because we briefly considered this as an alternative to the stochastic model described in Chapter 2.
1.3.2 Mathematical Techniques
In solving the prediction problem of Chapter 3, we made use of the prolate spheroidal wave functions. Interest in these functions increased after Bouwkamp discovered a numerical technique for evaluating them [6, 16]. Their original application was the solution of wave equations with boundary conditions defined on prolate spheroids. Later, Slepian et al. wrote a series of articles describing their properties and pointing out several interesting applications to signal processing problems [60]. Papoulis has summarized many of the important properties very succinctly while avoiding any mathematics above the level of basic calculus (49, p. 205]. Frieden also offers a thorough tutorial [19].
The most extensive and authoritative tabulation of the prolate spheroidal wave functions is that compiled by the Naval Research Laboratory [70, 23]. We have relied upon these even though some software for evaluating the functions is available [38, 69, 74, 34], since we had difficulty adapting the software to our application (see discussion in the appendix).
In Chapter 3 we use these functions to obtain an expansion that transforms a functional optimization into an algebraic one, which may then be solved using a Lagrange multiplier. The approach is hinted at in a related problem described by Slepian et al. [60, p. 53]. The details may be found in a numerical linear algebra text by Golub and Van Loan [22, p. 5821.
The more general treatment of Chapter 2 relies heavily on functional analysis. A good text is provided by Conway [12]. For specific results concerning HP
spaces, our main source is Koosis [33]. These books assume a background in analysis and measure theory. The very popular book by Rudin provides a starting point in gaining this background (57]. An interesting, lesserknown alternative is Sprecher [63]. The distinguishing feature of this book is the manner in which the author constructs the real numbers from Cauchy sequences of rational numbers. It is a difficult approach, but provides useful insight for those interested in engineering applications, since this theory is the basis of most iterative optimization algorithms (see, e.g., Luenberger [39, Ch. 7]). Those interested in getting some background in functional analysis without extensive prerequisites are referred to the very wellwritten text by Kreyszig (35].
Functional analysis is very closely related to the subject of integral equations. We see from Section 3.1, for example, that our basis functions may be defined as eigenfunctions of a linear integral operator. Some knowledge of integral equations is therefore of interest. An excellent text is the one by Hochstadt [26]. It is written at an elementary level and covers many fundamental concepts of functional analysis. Also, the treatment is confined to Hilbert spaces, which simplifies the presentation and is perfectly adequate for our purposes.
Our basic problem formulation involves the minimization of an integral (see Section 2.1). At the early stages of our research, an attempt was made at performing this minimization using variational calculus. Although this was not the approach that we ultimately adopted, it provided useful insight. Our references were texts by Fox and Weinstock [18, 73].
1.3.3 Fading in Mobile Radio
Our research of bandlimited prediction was motivated by the problem of fading in mobile radio. This is because the fading envelope may be modeled as a bandlimited random process, as was mentioned in Section 1.1. A very good treatment of fading on digital channels is offered by Proakis [54, Ch. 14]. The
12
fading environment is particularly harsh in terrestrial mobile communications. In a 1968 paper, Okumura et al. covered many aspects of mobile radio propagation [47], including smallscale fading caused by multipath.
Clarke dealt directly with the multipath problem, developing a mathematical formulation that resulted in the bandlimited model mentioned above [11]. Jakes incorporated the model in his 1974 book, showing how it could be used to simulate a fading signal in the laboratory [27, Ch. 1]. For this reason, the model itself is often attributed to Jakes.
The effects of fading on a given channel are intimately connected with
the modulation scheme, and for this reason a good knowledge of modulation is absolutely essential for understanding the fading problem. The treatment by Proakis is characteristically thorough [54, Ch. 4]. Alternatives include Peebles [53, Ch. 5], Stuber [65, Ch. 4] and a review article by Aghvami [2]. The latter two are 7T
geared more specifically towards mobile radio, and include coverage of QPSK
4
and GMSK modulation, which are incorporated in current mobileradio TDMA standards.
Adaptive transmission was mentioned in Section 1.1 as a possible means of fading compensation. The paper by Goldsmith contains a literature review, as well as an analysis of the effects of delayed channel estimates [21].
When multipath effects cause a significant time dispersion of the transmitted signal, the received signal is said to undergo frequencyselective fading. This problem was our initial research interest. Proakis includes the subject in his chapter on fading [54]. We also mention an interesting early research paper by Bello and Nelin [3], which specifically addresses noncoherent and differentially coherent modulation (see also the followup [4]).
13
1.3.4 Equalization and Carrier Recovery
Although the term equalization derives from linear filter theory, it now applies to any scheme aimed at compensating for the effects of a dispersive channel. For general background on equalization, we again recommend Proakis [54, Ch. 10, 11]. Linear techniques have found widespread use in telecommunications, but they often prove inadequate in wireless channels, where they can lead to inordinate noise amplification [65, p. 264]. One alternative is decisionfeedback equalization, which uses the output of the nonlinear detector to compensate for the dispersion in a manner that reduces the noise amplification. Maximumlikelihood sequence estimation is a technique which is in a sense optimal and may be carried out using the Viterbi algorithm [17], which was originally developed for decoding convolutional codes [71]. The Viterbi algorithm has the disadvantage of imposing a heavy computational burden. It also requires an accurate estimate of the channel impulse response, about which we shall say more shortly.
Both telecommunications and wireless channels are characterized by vari/
ablility, so equalization is normally carried out adaptively. Qureshi is a muchcited reference [55]. At this point a brief explanation is in order. The conventional approach to adaptive equalization is to transmit a prearranged training sequence of symbols, which the receiver attempts to equalize using its initial setting. The error is then measured between the receiver's detector input and the original known sequence, and the equalizer is adjusted accordingly. For linear and decisionfeedback equalizers, the adaptive algorithms may adjust the equalizer tap gains directly. In the case of maximumlikelihood sequence estimation, the channel impulse response is needed. Often, the channel is modeled as a linear finiteimpulse response filter, and the adaptive algorithm is used to find the filter coefficients in a systemidentification mode [54, Sec. 113].
14
In some applications it is desirable to perform the equalization without
employing a training sequence. Approaches for such blind equalizers fall into three broad categories: Bussgang techniques, of which the Godard algorithm is a widely used example [20], methods based on higherorder statistics [24], and methods that exploit the cyclostationarity properties of the transmitted signal [1].
Once the equalizer has converged, there is a high probability that the output of the detector will be correct. Thus, the detected symbols may be used in place of the training sequence so that the adaptive algorithm may track slow variations in the channel impulse response while data are being transmitted. This is referred to as decision direction. If a receiver operating in this mode enters a deep fade, however, the detected symbols will no longer be reliable and the adaptive algorithm may lose track of the channel. The conditions under which a decisiondirected equalizer will converge is therefore of interest. Mazo provides an analysis of a simplified but still interesting case [46]. Macchi and Eweda establish some sufficient conditions for convergence in the case of linear equalizers [43], and Kennedy et al. discuss decision direction for decisionfeedback equalizers [30].
Overcoming the difficulties associated with decisiondirected equalization in a fading channel was the primary motivation for pursuing the prediction problem. But first we wish to address the similar, but in some ways simpler problem of carrier tracking. Interesting treatments of decisionaided carrier tracking are included in the books by Proakis [54, Sec. 624] and by Macchi [42, Ch. 12]. Our form of the problem is described in Section 4.2.
1.3.5 General References
We now discuss some references which have been of use throughout the
project. The broad field that encompasses our prediction problem is estimation theory. Kay offers a strong theoretical and wellordered treatment of the subject [29]. A strength of the text is the manner in which it carries a few examples
15
through the entire development, which helps to firm ones grasp on the relationships among the various estimation techniques. It assumes a considerable degree of commitment on the part of the student, however, since the theoretical approach means that some of the more popular estimation methods are dealt with in later chapters.
As was mentioned in Section 1.1, Wiener filter theory plays a central role in many discussions about prediction. This subject is covered in texts on adaptive filter theory, as well as spectral estimation [25, 44, 28, 64]. In particular, Kay's book is again very strong in theory, especially in his emphasis on vector space concepts [28, Sec. 6.3.3]. The organization, however, does not seem as good as his previously mentioned text [29], leading, perhaps, to unnecessary repetition and too great a reliance on illustrative simulations. The more recent book by Stoica [64] offers the virtue of succintness as well as coverage of more recent approaches, including filterbank methods, some of which employ the discrete prolate spheroidal sequences (see also Thomson [66]).
Papoulis is our source for almost all matters concerning stochastic processes [52]. The famous book by Doob is more complete but, again, requires substantial background in higher mathematics [14]. Also, Papoulis and Slepian are two authors that have shown sustained interest in bandlimited modeling [52, 51, 49, 58].
For issues related to digital communications our primary source is the wellknown book by Proakis [54]. In its coverage, depth and clarity it is hard to beat. The book by Stuber [65] is also excellent, and is geared more toward mobile radio.
CHAPTER 2
PROBLEM FORMULATION AND ANALYSIS
In this chapter we study the continuoustime bandlimited prediction problem in its general form1 . Our main objective is to justify the introduction of an energy constraint, since the unconstrained problem fails to have an optimal solution. We argue that the energy constraint is an intuitively satisfying one for the case in which the prediction is to be based on past values that have been corrupted by estimation errors.
In Section 2.1, we obtain a frequencydomain expression for the mean
squared prediction error and then formulate the problem as the minimization of an integral. In Section 2.2, we prove our main results concerning the predictablility of bandlimited processes. In Section 2.3, we discuss the case of corrupted past values. Finally in Section 2.4, we summarize our results and motivate the problem addressed in the following chapter.
2.1 Problem Formulation
In this dissertation, Fourier transforms will be denoted by capital letters, as in the following pair:
F(w) = f (t)etdt,
1 /_oFwe tw
f(t) = F(w)ee
27r oo
SThis chapter is based on "The Predictability of ContinuousTime, Bandlimited Processes," by R. J. Lyman, W. W. Edmonson, S. McCullough, and M. Rao. @ 2000 IEEE. Reprinted, with permission, from IEEE Transactions on Signal Processing, vol. 48, no. 2, pp. 311316, February 2000.
16
17
Now consider a real, continuoustime, widesense stationary, zeromean
random process x(t), which is known on the Tlength interval [t  T  T, t  T], with T, T > 0. We would like a predictor i(t), which is linear on the known portion of x(t),
t (2.1) L() x(A)h(t  A)dA,(21 where h(t) is a real and continuous function of t. Note that both .(t) and h(t) may also depend on r and T. We wish to find a function h(t) which minimizes J = E {[x(t)  1(t)]2}, (2.2) where J is the mean squared prediction error. We focus on linear predictors partly because of their simplicity, but also, when the process is Gaussian, no predictor can achieve a lower mean squared error than the optimal linear predictor, if it exists [13, p. 231].
Suppose we constrain h(t) to be zero except for the interval t E [T,7  + T]. Then we may write (2.1) as a convolution, f(t) = x(A)h(t  A)dA (2.3) = x(t). h*t),
where
h(t) = 0, t V [r, r + T]. (2.4) Thus, at every t, i(t) represents a prediction of x(t) based upon the known interval [t  T  T, t  r.
Note that in (2.3), h(t) is viewed as the impulse response of a linear, timeinvariant filter. This is justified by the stationarity of x(t). In our discussion of prediction, however, a possible point of confusion is the Tlength time delay introduced by h(t) in (2.4). As we shall see, this results in a simple frequency
18
domain formulation of the problem, and serves perfectly well to answer our questions about the predictor defined in (2.1).
We now proceed to find a frequencydomain expression for the mean squared prediction error. We start by defining the error, f0 =t x(t)  i(t)
= x(t)  x(t) * h(t)
= x(t) * [6(t)  h(t)]. (2.5) Let us further define the error filter, e(t) = b(t)  h(t) (2.6) E(w) = 1  H(w). (2.7) It is clear that the error is obtained by passing x(t) through a filter with a frequency response given by (2.7). Now, suppose that x(t) is bandlimited to w < Q. Then we may obtain the mean squared error by J = L Ax(w)E(w)2d&, (2.8) = S (w)1  H(w)2dw, (2.9) 2= n
where Sx(w) is the power spectral density of x(t). Thus, the desired predictor (2.3) minimizes (2.9) with respect to h(t) under the constraint imposed by (2.4).
A careful inspection of (2.9) will show that we are seeking a time function of finite support, h(t), whose Fourier transform approximates the frequency response of a zerophase allpass filter in the frequency range w E [Q, Q]. The time delay incorporated in h(t) makes this simple form possible, since otherwise it would be necessary to introduce exponentials of the form e". Note also that the frequency response for wIw > Q is arbitrary, because x(t) contains no energy at frequencies
19
outside of the Q band limit. This fact is critical to the discussion in the next section.
2.2 Analysis
In this section, we shall refer to the following spaces of functions: ï¿½1 and ï¿½2 contain functions which are, respectively, integrable and square integrable on the real line. ï¿½2([7r, 7r]) contains functions which are square integrable on the interval [7r, ir]. ï¿½2T contains functions f(t) E ï¿½2 such that f (t) = 0 for t ( [T, 7 + T], and ï¿½2(S) contains functions F(w) such that ff." S(w)IF(w)12dw < 00.
Suppose that F(w) E ï¿½2 and f(t) = 0 for t < 0. Then we say that
F(w) is in 1W2 of the upper half plane, written F(w) E 1j2(UHP). Now suppose G(w) E ï¿½2([r, r]), and suppose that the Fourier coefficients {y} of G(w) are given by
_ r
If ,yn = 0 for n < 0, we say that G(w) is in 1j2 of the unit disc, written G(w) E
72(UD).
From the previous section, we may formulate our prediction problem as follows:
minimize J =  S.(w)1  H(w)2dw w.r.t. h(t)
2r _ (2.10) subject to h(t) = 0, t V [r, T + T],
where J is the mean squared prediction error. Our first question is whether a minimum exists for (2.10). We now show that there is an allowable h(t) which makes J arbitrarily small.
Theorem 1 Let r, T and Q be fixed, real numbers with T, Q > 0, and let S(w) E ï¿½1, with S(w) > 0, w E [, Q] and S(w) = 0, w V [, ]. Then for every
20
F(w) E ï¿½2(S) and every E > 0, there is an h(t) E ï¿½, such that JS(w)IF(w)  H(w)I12dw < E. Proof: Consider the space T of functions H(w) for which h(t) E ï¿½T,T. To prove the theorem it is sufficient to show that if G(w) E ï¿½2(S), then for every H(w) E T if
f S(w)H*(w)G(w)dw = 0, then
SS(w)IG(w) 2dw = 0. Now, suppose Jf. S(w)H*(w)G(w)dw = 0. Then
0 = f h(t)etdt *S(w)G(w)dw f L h(t) [f S(w)G(w)etdw] dt.
OO OO
In order to justify the interchange of integration, we must show that
S(w)G(w)e"t is absolutely integrable,
/ S(w)G(w)Jdw < oo. (2.11) That this is so follows from CauchySchwarz, since f S(w)G(w)dw
o
< 0 S(w)d) S(w)G(w)dw
OO OO Both of the integrals on the right hand side are finite by hypothesis. Thus (2.11) is true and the interchange is justified.
21
Now let F(w) = S(w)G(w), then
0 = J h(t) [J0 F(w)etwedw dt
= 2r h h(t)7(t)dt.
Because y(t) is a bounded function which integrates against each h(t) e L7,Tn ï¿½1 to give zero, we conclude that 7(t) = 0 for t E [7, r + T]. Further, since 7(t) is bandlimited, it is an entire function, which implies that if y(t) = 0 on any interval of positive measure, it must be zero everywhere. So this must be true of F(w) also. Thus,
P(w) =S(w)G(w) =0 V w.
This shows that G(w) = 0, w E {A E R: S(A) > 0}. Accordingly, G(w) I2 = 0 for w in this set. Therefore, fo. S(w)IG(w) i2dw = 0, and the theorem is proved.O
The expression for the mean squared error in (2.9) is obtained simply by
letting F(w) = 1 and scaling appropriately. Thus, we have shown that the infimum of the mean squared prediction error is zero for bandlimited processes. Note that this result holds for all  and positive T, i.e., it holds regardless of the length of the known interval or how far in the future we wish to make the prediction. The same argument holds even if we allow h(t) to be nonzero on the interval [r, oo).
Next, we shall show that under certain restrictions imposed on S(w), no linear predictor attains this infimum.
Theorem 2 Let T, T and 0 be fixed real numbers with T > 0 and T, Q > 0, and let S(w) E L'1, with S(w) > 0 V w, and S(w) > 0 forw E M, where M C [, Q] is a set of positive measure. Then there is no h(t) E ï¿½,T such that fnS()11  H(w) I2dw = 0. (2.12)
22
Proof: Suppose such an h(t) exists. Then there is some H(w) such that (2.12)
iz
holds. Let s = Z, and let G(z) = H(s). Since H(w) E ji2(UHP), we have
 Z
G(w) E 12(UD) [33, p. 158]. Due to (2.12), H(w) and thus G(w) is equal to 1 on a set of positive measure. But since G(w) E 72(UD), this implies that G(w) and thus H(w) is identically 1. But this contradicts H(w) E ï¿½2. Therefore no such H(w) or h(t) exists.O
Since no allowable h(t) attains the infimum of J, by definition (2.10) has no minimum. This theorem applies even when T = 0, in which case the value x(t) is included on the known interval and may be obtained without error by sampling at that instant. But in this case, h(t) must be interpreted as a delta function, which is not in ï¿½2. For similar reasons, this theorem does not apply when a random process consists of a finite sum of complex exponentials. It is known that such a process may be predicted with a mean squared error of zero using a finite sum of past values [52, p. 497]. But in this case, S(w) must be interpreted as a finite sum of delta functions, which are not in ï¿½1.
Our next question is what happens to h(t) as J approaches zero. The next
theorem shows that, when h(t) is constrained in energy, the infimum mean squared error of the resulting space of functions is attained. Theorem 3 Let r, T, Q and E be fixed real numbers with T, Q, E > 0, and let S(w) E ï¿½1, with S(w) > 0 V w. Now, consider the space BE of functions h = h(t) E ï¿½L,T such that
+Th2 (t)dt < E. (2.13) Let
K = inf 1 S(w)I1  H(w)2dw.
hEBE 2V n
Then there is an h E BE such that
 f" S(w)1  H(w)12d .
27r _l
23
Proof: Define J: BE + R by J(h) = S(w)I1  H(w)2dw.
27r _.
Since BE is weakly compact, we may prove that J(h) attains its infimum by showing that J(h) is weakly continuous. This can be shown by first observing that if h E BE, then, by CauchySchwarz, IH(w)12 = f + h(t)eIwtdt 2 (jr+T h 2(t) dt) f+T dt < TE. (2.14) Now suppose {hn} is a sequence from BE which converges to he weakly. Then {Ha} converges to Ho pointwise, since H,(w) = h,(t)ee'tdt.
Thus, S(w)1  Ha(w) 2 converges pointwise to S(w)1  Ho(w) 12. From (2.14), we also have
S(w)I1  H,(w)2 < S(w)(1 + TE)2. We can thus apply the Lebesgue Dominated Convergence Theorem to conclude that
J(h,) + J(ho).
Thus, J is weakly continuous and the theorem is proved.0
Theorem 3 implies that there is a function Jmin(E). This function is the minimum mean squared error obtainable when h(t) is constrained to have an energy no greater than E. Note that Jmin(E) is a nonincreasing function of E.
24
2.3 Dealing with Estimation Errors We shall discuss the findings of the previous section as they relate to the prediction of a bandlimited process whose values on the known interval are corrupted by estimation errors. Consider the random process x(t) of Section 2.1, and suppose that we have an estimate of that process, i(A) = x(A) + w(A), A E [t  7  T, t  T], (2.15) where w(t) represents the estimation error. We consider w(t) to be a real, zeromean, widesense stationary process, uncorrelated with x(t), and having an autocorrelation function given by R~.w(t) = a26(t), ea2 > 0. (2.16) This expression indicates that w(t) is being modeled as white noise. Of course, the white noise assumption may not be valid in some applications. Nevertheless, by considering the case of white noise, we can simplify the mathematical treatment and still gain insight into the effects of estimation errors. Also, by using this approach, we can see why E, an upper bound on energy, is a natural choice for a constraint on h(t).
Let us consider the linear predictor from (2.3) with input 2(t) instead of x(t). Then the error becomes
f(t) = x(t)x(t)
= x(t)  i(t) * h(t)
= x(t)  [x(t) + w(t)] * h(t)
= x(t)  z(t) * h(t)  w(t) * h(t) = x(t) * [6(t)  h(t)]  w(t) * h(t). (2.17)
25
Note that this is equivalent to (2.5) with the added term w(t) * h(t). Using a line of reasoning similar to that leading to (2.9), and noting that x(t) and w(t) are uncorrelated, the mean squared error is given by
SS(w)1  H(w)2dw S.ww(w)IH(w)2dw.
27r oo 27r foo SO.2 o
2 oo
12
= J + u h2. (2.18) where h112  foo h2(t)dt. Since I and J both depend on h(t) we write J(h) = J(h) + 2llhl12. (2.19) We know from Theorem 2 that J(h) has no minimum. We shall show, however, that J(h) does have a minimum and that this minimum is greater than zero for any nontrivial process.
Theorem 4 Let fT be as defined at the beginning of Section 2.2, and let k= inf J(h).
hEeC2
hfr,T
Then
1. There is an ho E ,4,T such that k = J(ho)
2. If fJna Sxx(w) dw > 0, we also have k > 0.
Proof: To prove (1.), choose a sequence {h,}, with h ï¿½E L V n, such that J(h,) + k. Then
lim J(hn) + u2Ih 112 = .
n +oo
This means that, for every c > 0, there is an m such that J(h) + o,211IIhl2 < k + E V n> m. (2.20)
26
But since J(hs) > 0, we have &2 2
ua2 11h. 11 < R + E.
Thus,
IIh.l < VW , n> m.
Since {hn} is norm bounded, we may choose a subsequence {hnk } which converges to some weak limit ho. Then there is a sequence of finite sums {gk} with k+Mk
9k E Cl,kh1,, (2.21) l=k
where the coefficients cl,k satisfy k+Mk
CI,k > 0 and E clk = 1, (2.22) l=k
such that {gk} converges strongly to ho. A function gk(t) satisfying (2.21) and (2.22) is called a convex combination of {hn,}. Since a convex combination is a particular type of linear combination, we have 9k E 2,T V k. Now because of (2.20), for every E > 0, there is a p such that, for k > p we have
< J(gk)
1 12 ,2 0
= j S.(w)1  Gk()12dw + IGk(W)12dw
1 a] 1 w  c1,kHl (!),( w ï¿½]] + c,H(w) dw
2 27r 2 oo=k
2 k2 n Sk+Mk 2 +Mk
= S.. (W) 1 E ce,H, (w) d + kf cIkHa (W) Idw
27 J l=k 27 Joo _=k
k+Mk 22 o k+Mk
= (W) c ck[1 H(W ,k + E c(hkH)) d
2xr J =k 27r Jm =k
k+Mk 1 W 2 k+Mk f 0
< E cl,k  S..(w41  Hm() d + E C1,k  "IHnm(U) 12d
l=k 27 J =k 27rJk+Mk, l=k
< kr+e, (2.23)
27
since the cl,k'S sum to 1. Thus, J(gk)  .
Now {gk} converges strongly to ho. Strong convergence implies convergence of the norm so IIgkIl 4 11ho11. And since any strongly convergent sequence also satisfies the definition of a weakly convergent sequence, we may use the argument in the proof of Theorem 3, substituting {gk} for {h} to show that J(gk) + J(ho). We thus have
J(9k) = J(g9k) + 9k12  J(ho) + 2h11 = j(ho). And since {J(gk)} converges to R, we have S= J(ho).
To prove (2.), we note from (2.19) that, if R = J(ho) = 0, then lholl = 0. But then Ho(w) = 0 almost everywhere. Thus, from (2.9) we have J(0) = S (w)dw = 0.
Note that the key result of Theorem 4 is that j has a minimum even if we do not impose a constraint on h(t).
2.4 Conclusions
The results presented in this chapter show that there is no theoretical limit to the predictability of a bandlimited random process, even if knowledge of the process is restricted to a short interval. The usefulness of these findings depends on a number of factors. One of these is the validity of modeling a physical phenomenon as bandlimited [58]. Also, as we have discussed, there is the problem of estimation errors on the known interval.
If the bandlimited model is invalid, then Theorem 1 does not apply, and it is possible that the predictability of the process is limited even if we have perfect knowledge of the infinite past (see, e.g., Papoulis [52, Sec. 14.2]). If, on the other
28
hand, the bandlimited model is valid, then the estimation errors on the known interval become the primary factor which limits predictability, as was discussed in Section 2.3. In fact, the choice of an appropriate model may be influenced by whether the estimation errors on the known interval are expected to be the dominant source of prediction errors.
If the estimation errors can be modeled as white noise with known positive
variance, then J1, the mean squared prediction error, has a minimum that is greater than zero. We recall that the energy of h(t) was central in the analysis of this case. Thus, in the case where values of x(t) on the known interval are error free, an upper bound on this energy is a natural choice for a constraint on h(t), since J would not have a minimum without some constraint.
We now have two objectives. One is to construct a function h(t) which
attains Jmin(E). The other is the computation of Jmin(E) itself. This latter goal would be useful in the case where a linear predictor is desired, but realizing the impulse response h(t) is not possible or practical, making it necessary to resort to some suboptimal approach. Knowledge of Jmin(E) would be useful in determining how closely this suboptimal predictor approaches optimal performance.
As of yet, neither h(t) nor Jmin(E) has been found for the general case.
Solutions have been found for a special case, however, which we discuss in the next chapter.
CHAPTER 3
PROCESSES WITH FLAT SPECTRAL DENSITIES
In this chapter we shall solve the linear prediction problem for bandlimited processes with flat power spectral densities1 . For such processes we have S(w) = 1 for wl < Q and S(w) = 0 elsewhere. In solving this problem, we shall make use of a set of basis functions, {#k(t)}, which are timeshifted versions of the prolate spheroidal wave functions {#k(t)}. As we shall see, these are eigenfunctions of a particular linear integral operator. They have many orthogonality, symmetry and extremal properties that make them very well suited to certain problems in Fourier analysis, especially ones in which a specific time and frequency interval are of interest simultaneously.
In Section 3.1, we discuss the key properties of the prolate spheroidal wave functions and the timeshifted basis functions based upon them. In Section 3.2, we solve the bandlimited linear prediction problem for the flat spectral density case. In Section 3.3, we focus on the computation of the minimum mean squared prediction error. A discussion of the prediction problem for a bandlimited process embedded in white noise follows in Section 3.4. Finally, we offer some conclusions in Section 3.5.
1 This chapter is based on "Linear Prediction of Bandlimited Processes with Flat Spectral Densities," by R. J. Lyman and W. W. Edmonson. @ 2000 IEEE. Used by permission. This paper is currently under consideration for publication in IEEE Transactions on Signal Processing.
29
30
3.1 Basis Functions Discussions of the prolate spheroidal wave functions, as well as proofs of many of their properties, are contained in treatments by Slepian et al. [60], Papoulis [49], and Frieden [19]. We shall summarize some key properties without restating proofs from these references.
3.1.1 Prolate Spheroidal Wave Functions
The prolate spheroidal wave functions, {#n(t)}, are solutions of the following integral eigenvalue problem: Jli( sin O(t  s)
(s) si(t:s) ds = Ag(t). (3.1)
This equation has nontrivial solutions for only a countable set of eigenvalues {A,}. Each An is real and positive, such that the set {An} may be ordered as
1 > A0 > At > A2 > ... > 0. If so ordered we also have lim An = 0.
n+oo
To each An there corresponds only one function on(t) within a constant factor. With a proper choice of this factor, the functions {#,(t)} form a set with the following properties:
T1. Each function 9n(t) is real valued.
q2. The set {#n(t)} is orthonormal on [o, o]; that is,
loo 1 m=n 0( 0 me n.
T3. The set {On(t)} is complete in Obandlimited, finiteenergy functions.
This means that if F(w) = 0 for wl > Q and f" f2(t)dt < co, then there
31
are coefficients {an} such that f(t) may be written
00
n=O
T4. The set {I'(t)} is also orthogonal on [, ], with T JAn n L' VCm(t)1bn (t)dt =
2 0 m n.
T5. The set {n(t)} is complete in finiteenergy functions on [, ]. This is
true even if such a function is not a segment of a bandlimited function. T6. Each function on(t) is Qbandlimited. This means that each Fourier
transform has the property Wa(w) = 0 for IwI > 0.
W7. The set { W(w) is orthonormal on [Q, Q]. This may be seen
by applying Parseval's identity to property T2 and then applying
property 6.
% 8. The set {Wn(w)} is complete in finiteenergy functions on [Q, Q]. This
is true because a finiteenergy function with the property F(w) = 0 for
IwI > Q has an inverse Fourier transform f(t) which is Qbandlimited. By
property T3 we may write
f~t W an ï¿½ (t),
n=o
and taking the Fourier transform we have
oo
F(w) = Zan'n(w).
n=O
99. Let us define On\ I t  2
Ok,trunc(t) 2
0 elsewhere.
32
If W'k,trunc(W) is the Fourier transform of k,trunc(t), then
Tk,trunc(W) = An'I'(w), Iwi * Q. Note that qWk,trunc(W) is not zero for wj > Q. This property says nothing
about the behavior of 'k,trun c(W) outside of the interval [Q, Q].
It should be noted that On(t) depends implicitly on C and T, and may be OT
written Onb(Q, T, t), and that An depends implicitly on the parameter c = , and
2
may be written An(c). In keeping with the convention of Slepian et al., we shall normally suppress this additional notation.
3.1.2 TimeShifted Basis Functions
The basis functions we shall use to solve the linear prediction problem in Section 3.2 are timeshifted versions of {n(t)}. We define
On(t) = On (t  7  Z). (3.2) For each n, the function On(t) satisfies the following integral equation: 17+T sin ï¿½ (t  s) _r ( sin (t  )ds = Antn(t), (3.3) ï¿½. ( s
where An is the nth eigenvalue of (3.1). To see this, we apply (3.2) to (3.1) to obtain
li sin(t s)
0 (s +r + ) sQ(LL) ds = An,4 (t +r + .)
2
Now, with the substitutions s' = s + T + T and t' = t + 7 + we obtain
2 2
Ir+T (S)sin Q (t'  s') A (t)
(t') ds' = Ann(t).
Since this equation must hold for all t and thus all t', the prime notation may be dropped and (3.3) results. This shows that (3.1) and (3.3) have the same set of eigenvalues {An}.
33
The functions {qn(t)} have the following properties, many of which follow directly from the properties of {#n(t)}:
#1'. Each function On(t) is real valued. This follows directly from property W 1.
2. The set {ï¿½n(t)} is orthonormal on [co, oo]. Since each ï¿½,(t) is shifted
by the same amount, the orthogonality of property 92 is preserved.
(D3. The set {qn(t)} is complete in Obandlimited, finiteenergy functions. To
see this, suppose f(t) is such a function. Then the timeshifted function
f (t + + I) is also Qbandlimited. We may thus write
O0
(t +r + 7:) = Zan0n(t).
n=o
Substituting t' = t + r + T we have
00 00
(') = Ean, (e  7  )= an.n(W).
n=O n=O
Since this equation must be true for all t', the prime notation may be
dropped.
(44. The set {ï¿½,(t)} is orthogonal on [7, T + T], with T+T Om(t)n(t)dt = A m=
0 m n.
This may be seen by applying the time shift of (3.2) to property T4.
45. The set {n(t)} is complete in finiteenergy functions on [r, r + T]. This
is the interval of property T5 shifted according to (3.2).
46. Each function On(t) is Obandlimited. Thus, each Fourier transform has
the property 4n(w) = 0 for IwI > Q. This follows from property %6, since
a time shift clearly does not affect the band limitation.
34
47. The set { 'bn(w)} is orthonormal on [Q, ]. The reasoning is the
similar to that of property 97.
(D8. The set {(n(w)} is complete in finiteenergy functions on [Q, Q]. See
the argument from property 98.
9. Let us define
Ok,trunc(t) O
0 elsewhere.
If 'k,trunc(W) is the Fourier transform of Ok,trunc(t), then
4k,trunc(W) = Ann(w), IIwI Q. (3.4) To see this, we note
k,trunc(t) k,trunc (t  Taking the Fourier transforms we have
4k,trunc(W) = iW (T 2 ) Tk,trunc(W) = 1 ( 2) n(W)}, I l
On(t). Thus,
and (3.4) follows.
3.2 Solution of the Linear Predictor Having defined our basis functions {jn(t)} and discussed their key properties, we now return our attention to the prediction problem for a bandlimited process. Let us consider the energyconstrained optimal linear predictor of a bandlimited process whose power spectral density is fiat in the frequency band of interest. The
35
problem may be formulated by substituting Sz,(w) = 1 in (2.10) and adding the energy constraint as follows:
minimize J =  1  H(w)j2dw w.r.t. h(t)
subject to 1. h(t) = 0 t [r, + T] (3.5)
00
2. h2(t)dt = E.
Note that we have written constraint 2 as an equality. We shall see later that the solution to this problem is the same as if it were written with the inequality. Our approach shall be to transform the integrals in this problem into algebraic expressions by expanding in terms of the basis functions {qn(t)}. The algebraic minimization may then be carried out by using a Lagrange multiplier.
3.2.1 Problem Transformation
In the expression for J in (3.5), we wish to expand the terms inside the absolute value signs. In order to do this, we define
F(w) II= (3.6)
0 elsewhere.
By properties I7 and 48, the basis function Fourier transforms { 4,I(w)} form a complete orthonormal set in finiteenergy functions on [Q, Q], so we may write F(w) = E Fn(w) = E Tfn 1 (w) , (3.7) n=O n=O
where
lo 1
Yn = 0F(w) n ( ) d
1 oo
= 2 0f (n()dw
= V2rO(0) (3.8)
36
Thus,
F.(w) = qn(0)In(w). (3.9) By property (4)5, the basis functions {#n(t)} form a complete set in finiteenergy functions on [T, T + T], so we may expand h(t) as well,
o oo
h(t) = t h() = E Pn k,trunc(t), (3.10) n=O n=O
where 4k,trunc(t) is the truncated basis function as defined in property 9 , and the coefficients {Pn} are to be determined.
We consider now the conditions under which (3.10) will converge. Using property 44 we have
S , ktrunc(t)dt = + (t)dt = A. Thus, { #7X= k,trunc(t)} is orthonormal on [T, 7 + T]. Convergence of (3.10) is therefore guaranteed if 00
E AnPn2 < oo. (3.11) n=O
Now, letting the Fourier transform of Ok,trunc(t) be denoted by $k,trunc(w), we may write
Hn(w) = Pn4k,t.,nc(W), n = 0, 1, 2,... (3.12) Using property $9 we have
k,trunc(w) = Ann(w), WI < Q. (3.13) We may now rewrite the objective function of (3.5) as
1
J 11  H(w)12dw 27r n
1 oo 00 2 =E Fn (w) H,(w) dw. (3.14) 27r n=0 n=O
37
Applying (3.9), (3.12) and (3.13) yields
J = I f 0(0)n( )  p.A. n(w) dw. (3.15) 2,7 _ J E 0 (3.15)
n=o n=o When we expand the integrand of (3.15), the cross terms vanish under the integral sign because of orthogonality property 47. We are left with J = 2 (0)  Ap]2 ~ (w)2dw n= o 27 n
o00
= [n(0)  AnPn]2 (3.16) n=O
where the last step is made possible, again, by property 47.
The energy constraint on h(t), which is constraint 2 of (3.5), may also be rewritten using (3.10),
E = h2(t)dt
loo
= Pnktrunc (t) dt 00 =o
o , r+T 2 n=o
where the cross terms again vanish under the integral sign because of the orthogonality property 44. Using this same property again we have
o00
E Anp = E. (3.17) n=o
Note that any set {p,} satisfying (3.17) will also satisfy (3.11), thus guaranteeing the convergence of (3.10).
Using (3.16) and (3.17), we may transform (3.5) from an integral minimization to an algebraic one:
00
minimize J = "[,(0) AnPn]2 w.r.t. {pn}
oo n=0 (3.18) subject to Anp = E, n=0
38
where the time limitation on h(t), constraint 1 of (3.5), is implicit in the expansion (3.10).
3.2.2 Lagrange Multiplier
The technique of Lagrange multipliers will be used to solve (3.18). This technique combines the objective function and the constraint into the following unconstrained optimization problem:
minimize g(p,p) = [0n(0) Anpn]2 +2  E , (3.19) n=o (n=o where p is the Lagrange multiplier, a scalar which is to be determined.
We solve first formally for the pn's, setting the appropriate partial derivatives to zero,
p= 2An[n(0)  A.pn] + 2/AnPn =0, n = 0, 1,2, ... The solution of this equation is Pn () = (0) (3.20) An + I
where the dependence on it is explicitly noted. The multiplier i is then chosen so that the energy constraint of (3.18) is met, 00 00~p]2= rA(A\12 E n[Pn()]2 =n A[n = E. (3.21) n=O n  ï¿½
This series converges for all positive I and gives a unique positive solution for every positive E. The nonlinear equation (3.21) is called a secular equation [22, p. 582]. It may be solved by a line search on the positive axis, and then its solution may be substituted into (3.20) to obtain values for the pn's.
3.3 Minimum Mean Squared Prediction Error
In the previous section, we succeeded in solving for the coefficients {pn}, so the expansion (3.10) for h(t) is uniquely determined for any positive E and the
39
problem (3.5) is solved. It will be recalled that, in a predictor of form (2.1), this h(t) attains the minimum mean squared prediction error under the constraint that f f h2(t)dt = E. As was stated in Section 2.2, there exists a function Jmin(E) which maps E to the minimum mean squared error for given values of Q, T and T. Substituting (3.20) into (3.18), we see that we can compute values for Jmin(E) using the following procedure: For E > 0,
STEP 1: Solve E ï¿½(0)A ( = E for p.
n=O A + P )2 (3.22) STEP 2: Then Jmin(E) = 2(0) 1 An .
n=O An +
Note that, in order to carry out the procedure (3.22), one needs only the eigenvalue A, and the single function value 4,(0) for each n. We choose to focus on Jmin(E) because it provides a useful lower bound on the mean squared error performance of linear predictors.
From (3.22), we see that Jmin(E) is a strictly decreasing function of E. Thus, in (3.5), if the value of E is decreased, the minimum value of J must increase. This justifies the use of the equality in constraint 2. Also from (3.22), we note that Jmin(E) + 0 as E + oo, as we expect from the analysis in Chapter 2.
Now let us consider the behavior of Jmin(E) for small E. Looking at (3.22) we see that in this case, p will be much greater than 1, which is an upper bound for An (see Section 3.1.1). From (3.22) we have 1@
Es z An(0) (3.23) 2n=0
and
002
Jmin (E(p)) E 2(0) 1  A n=o A
40
00 A
= E (0) 12)2n n=o A A2I
00 00o
) E An 0 (0) .(3.24) n=O n=O Now we note from (3.18) that, with E = 0, pn = 0 for all n. Substituting
pn = 0 into the expression for J, we see that the sum E,0 0 (0) is simply Jmin(0). To evaluate the sum explicitly, we use (3.6) to compute
L IF(w)2dw = 2Q. (3.25)
We then use (3.7), (3.9) and property 7 to obtain
n 00
n IF(w)2dw = 21r E 0(0). (3.26)
Sn=O
Setting (3.25) equal to (3.26) we have Eï¿½ 2 (3.27 Jmin(0) = ï¿½(0) =  (3.27) n=O 7"
Note that this is true regardless of the value of T or T, so by (3.2) we have E0 ) (t) =  for all t. To evaluate the braced sum in (3.23) and (3.24) we use (3.3) to obtain
00 (0) + T sin s
E An() = E 4n(0) 0,(s) ds
n=o n=o 7rs
n (0) n (s) sinsds. (3.28) n=O o78 From (3.9), we know that EO 0 O,(0)4n(w) = F(w) as defined in (3.6). The braced expression in (3.28) is therefore the inverse Fourier transform of F(w),
00Q
E q,(0)On(t) = sinc Qt. (3.29) n=0 7r
Thus, substituting into (3.28) we have oo Q2 7+T An O'(0) = 72 sinc2Qs ds n=O
41
Q2
= I, (3.30) where
I = I(Q, T,r)  sinc2 Qsds. (3.31) Combining (3.30) and (3.27) with (3.23) and (3.24) yields
1 Q2
E 2 I (3.32) 22 22
Jmin(E(p)) Jmin(0)  I. (3.33) IL7r2
Solving (3.32) for p and substituting into (3.33) we get
Jmin(E) = Jmin(0)  2 VJE. (3.34) We see that, for small E, Jmin is approximately linear in the square root of E. Further, we see that the slope of the graph may be calculated without knowledge of any 0,(0) or A,. If we normalize with respects to Jmin(0), which is also the power of the process x(t), we get (cf. Equation 3.27)
Jmin(E) J(E) E) " a 1  2vIV/E = 1  GViE, (3.35) where G  2vfi is the absolute value of the initial slope of Jmin plotted against
When either Q or T is fixed, G may not be made arbitrarily large. Indeed, looking at (3.31), we see that, for fixed Q, I will be bounded by the case where r + 0 and T 4 o,
OO
Ima(Q) = sinc2Qs ds. (3.36) We note again that the Fourier transform of sinc 2t is F(w) of (3.6). Using Parseval's relation we have f 2 = JIirF2 (w)d sinc20s ds = 2F2 dW 00 27 
42
7[
= . (3.37) Thus, making use of (3.36) and (3.37) we have
Gmax(') = 2 Imax()= 2. (3.38) Note also,
Imax(T) = lim sinc20s ds = T. (3.39) Thus,
Gmax(T) = 2VT. (3.40) In evaluating A, and 0,(0) = 0,(T  1), we used the extensive tables of spheroidal wave functions compiled in the 1970's at the Naval Research Laboratory [70, 23]. The procedure is described in the appendix. Using (A.6) and (3.2) in (3.22), and noting that Of(t) is symmetric in t for all n, we obtain the curves for Jmin(E) shown in Figure 3.1. The series in (3.22) were truncated at n = 49, which is the highest value of n tabulated in the NRL tables. In the figure, the solid curves represent the function Jmin(E) plotted against the square root of E for sample parameter values T = 0.2, r = .04 and Q = 10,20,..., 50. The dashed straight lines represent the initial slopes of the curves, calculated according to (3.35) and (3.31). The small circles indicate the points at which p = 1 in (3.22). As E decreases below this point, the curves quickly approach their linear approximations, as expected.
3.4 A Bandlimited Process in White Noise
As shown in this and the previous chapter, the mean squared prediction error of a bandlimited process may be made arbitrarily small by allowing the energy of the predictor impulse response to increase, but this is true only if a sample function of the process is known without error on an interval of positive length. Consider a
43
II~
0 0 .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .... . . . . . . . . .
1. . . . . . . . . . . . . . . . . . . . . . .
0.6 ......
IV
.,50
0.4 ..... . ."...".......... 40
f1 \ 2 30
IE
I(E
0
' \\" '
0 2 4 6 8 10 El2
Figure 3.1: The minimum mean squared prediction error that is possible using linear prediction on a bandlimited process whose power spectral density is flat within
the band limits.
7r
process x(t), which is like that of Section 3.2 except that Szx(w) = for IwI < Q.
We use a linear predictor of the form (2.1), which yields a mean squared prediction
error 3J. The calligraphic 3J is used because the process x(t) is unity power.
Now suppose that, instead of x(t) itself, we have an estimate,
(7) = x(7) + w(y(7), 7 E [t  7  T, t  T], (3.41) where the estimation error w(t) is considered to be a real, zeromean, widesense stationary, whitenoise process, uncorrelated with x(t), and having an
autocorrelation function given by
R,,w(t) = awb(t), aW > 0. (3.42)
We again use a predictor of form (2.1), resulting in a mean squared prediction error
J. Clearly, both 3J and J depend on the predictor impulse response h(t), which
44
again may be restricted to h(t) = 0, t V [T, r + T]. As in Section 2.3 we may write
1 00 [00 jf(h) S.(w)1  H(w)l2dw + S.(w)IH(w)I12dw.
27r J0
= J(h) + ,2E. (3.43)
where E = fJ h2(t)dt.
Note that, for fixed E, Jf(h) can be minimized by choosing h(t) such that J(h) = Jmin(E). Thus, we may write
Jmin(E) = Jmin(E) + owE. (3.44) Suppose that, for given values of Q, T and r, we wish to attain a mean squared prediction error min(E) 5 o2, 0 < a2 5 1. We then have
Jmin(E) + 2E < o. (3.45) Rearranging this we have
Smin(E) _ o  u2E = 1(E). (3.46) As an example, consider the values of 0, T and T used in Figure 3.1, and suppose a2 = .002 and a2 = 0.2. The resulting function 1(E) is represented by the dashdot curve at the bottom of Figure 3.1. For 0 = 10, we see that condition (3.46) is met for E in the interval 2.6 < VT < 9.9 approximately. For Q = 20, the curve 1(E) is nearly tangent to Jmin(E) at .1 a 7. For Q significantly greater than 20, a2 cannot be attained for the given values of T, T and aw2.
Though Jmin(E) may be driven arbitrarily close to zero, there is no finite E
that minimizes Jmin(E), as we expect from the analysis of Chapter 2. On the other hand, looking again at (3.44), we recall that Jmin(E) > 0 is strictly decreasing in E > 0, and we note that au.E is strictly increasing in E since au? > 0. Thus, there
45
must be some value of E that minimizes Jmin(E). This is equivalent to saying that, for given Q, T, T and o2, j(h) has a minimum even though J(h) does not. This is also in accordance with the findings of Chapter 2.
3.5 Conclusions
We have presented the solution of the linear prediction problem for a bandlimited process whose spectral density is fiat within the band limits. This special case provides a concrete example that illustrates many of the properties shown to hold for the more general bandlimited prediction problem discussed in Chapter 2. In addition, we have shown how to use the solution to compute values of the function Jmin(E), which for given values of O, T and T as defined in Section 2.1, is the minimum mean squared prediction error that may be attained when the predictor impulse response is constrained to have energy E. We then used this function to analyze the case of a bandlimited process which has been corrupted by white noise, showing how to determine if a specified mean squared prediction error can be attained.
In solving the problem, we used a timeshifted version of the prolate
spheroidal wave functions as basis functions. The special orthogonality, symmetry, extremal and completeness properties of these functions allowed for a simple solution, using a Lagrange multiplier. These same properties aided in determining the behavior of the function Jmin(E) for small E. Computation of the wave function values themselves is a complex numerical problem [6, 16]. Fortunately, we were able to rely upon published tabulated values [70, 23], though one might wish that quality software for computing these functions were more widely available (see discussion in the appendix).
This chapter, and the previous one, offer some insight into the predictability of bandlimited processes. Much work remains to be done on this subject, but at this point it will be helpful to discuss how our findings to date could be used in the
46
analysis of a practical signalprocessing problem. This is the topic of the following chapter.
CHAPTER 4
APPLICATIONS TO FADING IN MOBILE RADIO
It is well known that mobile radio is plagued by the distorting effects of
multipath fading. As we shall see, this problem may be characterized by a complex time function called the fading envelope, which is often modeled as a bandlimited random process [11]. In Section 1.1, we mentioned adaptive transmission as an application that could benefit from the prediction of the fading envelope. The theoretical discussions in Chapters 2 and 3 provided us with some tools for predicting bandlimited processes. This chapter is dedicated to a discussion of fading and the possible application of bandlimited prediction to a problem in adaptive channel estimation.
4.1 Multipath Fading in Mobile Radio Our discussion of fading will be very brief. Additional details may be found in Proakis [54, Ch. 14] and in Stuber [65, Ch. 2,5,6].
Ideally, in a wireless communication system, there would always be an unobstructed line of sight between the transmitting and receiving antennas. Unfortunately, this is rarely practical for mobile communication, in which the mobile receiver is often embedded in a clutter of obstructing objects such as buildings, trees, hills and vehicles. The direct signal path is often completely cut off, such that nearly all of the received signal energy is reflected toward the receiver from these surrounding objects. The different reflections may have comparable amplitudes, but each may have undergone a random shift in phase, as well as a time delay that is longer or shorter than other reflections. At the receiver location, these various phases may add constructively or destructively, leading to the 47
48
phenomenon known as multipath fading. A communication channel characterized by multipath fading is called a fading channel.
In describing the fading channel, we shall use complex signal representation. We shall assume that some form of quadrature modulation is being used; e.g., QPSK or QAM. Each signal is represented by a complex time function whose real part represents the inphase part, and whose imaginary part represents the quadrature part. Using this approach the fading channel can be modeled as a timevarying linear filter. The received signal is given by r(t) =J c(A; t)u(t  A)dA + v(t), (4.1)
where u(t) is the complex representation of the transmitted signal, c(A; t) is the timevarying impulse response of the fading channel, and v(t) is Gaussian noise.
For simplicity, we will restrict our discussion to the case in which, for any t,
c(A; t) is nonzero over only a small range of A. In digital signaling, this occurs when the nonzero range is small compared to the symbol interval, a condition otherwise known as flat fading. In this case, the expression for the received signal becomes r(t) = c(t)u(t) + v(t), (4.2) where c(t) is simply a complex time function, c(t) = x(t) + iy(t). (4.3)
The function c(t) is called the complex fading envelope, and x(t) and y(t) are called the fading parameters. The complex envelope may also be represented in phasor form,
c(t) = a(t)eo(t, (4.4) where
a(t) = Jx2(t) + y2(t), (4.5)
49
u~t r(t
WGN
Figure 4.1: Model of a flatfading mobile radio channel. and
0(t) = arctan y(t) (4.6) a x(t)) (
The flatfading channel model is shown in Figure 4.1. A flatfading channel exhibits high amplitude correlation across the entire bandwidth of the timevarying channel frequency response. A channel for which the frequency response is likely to vary significantly in amplitude across its bandwidth is called frequency selective. Such channels cause a time dispersion or "smearing" of the received symbols, leading to the undesirable phenomenon known as intersymbol interference. Although we shall not be focusing on frequencyselective fading, the theory of the flatfading channel may be used in the analysis of the more complicated frequencyselective case [54].
A popular model for a flatfading mobile radio channel was developed by R. H. Clarke [11]. He viewed the received signal as a superposition of a large number of vertically polarized electromagnetic waves arriving at random angles with random phase shifts. Clarke analyzed the spatial correlation of the fields and determined that, if the receiver is moving with a constant velocity, these fields may be modeled as a stationary Gaussian random process. He also determined the autocorrelation function for this process. Using Clarke's approach, we may view x(t) and y(t) as independent Gaussian random processes with a normalized autocorrelation function given by
Rxx(t) = Ry,(t) = Jo(27rfmt). (4.7)
50
In this expression, Jo is the zerothorder Bessel function of the first kind, and fm is the maximum Doppler frequency, which is given by
fm = f, V (4.8)
C
c
where fc is the carrier frequency, V is the speed of the receiver, and c is the speed of light. In the remainder of the discussion, we shall primarily deal with the real component, x(t). The same analysis, however, applies equally well to y(t). A graph of Rxx(t) is shown in Figure 4.2.
The power spectral density of x(t) is represented by the Fourier transform of (4.7), yielding
1
1 ~ IfI :fm
S~~) = 27rfm 1] P f<
s 2rfm1 2 (4.9)
0 otherwise.
A graph of Szz(f) is shown in Figure 4.3. Note that x(t) is bandlimited to the frequencies If 1 5 fm.
As an example, consider a carrier frequency of fc = 400 MHz and a receiver speed of V = 20 m.p.h. The maximum Doppler frequency would be fm = 12 Hz. Figure 4.4 shows a 400 msec segment of a simulated fading envelope generated with these values. Notice from the plot of the magnitude a(t) that, during this interval, several fades are encountered, including two that fall below 20 dB compared with the average signal power. During a deep fade, almost no signal energy reaches the receiver. As a rule of thumb, some kind of fade will be encountered about once every half wavelength.
Suppose that, in the above example, the symbol rate f, is such that f, > fm; or equivalently, the fading parameters change little over a single symbol interval. Such a condition is called slow fading.
51
0.5
0.4
0.3
0.2
M 0.1
0
0.1 .
0.2
0 0.5 1 1.5 2 t (seconds) Figure 4.2: Autocorrelation function of a mobile radio fading parameter, fm= 1.
1.2
1 .. . . . . . . . . . . . . . . . . . . ... . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
0 .8 .. . . . . . . . . . . . . . .. . ... . . . . . . . . . . . . . . . . . . ... . . . . . . . . .. . . . . . . . . ..
.E
0 . .. . . . . . . . . ... . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . ..... . . . . . . ...
0
1.5 1 0.5 0 0.5 1 1.5 f/f
m
Figure 4.3: Power spectral density of a fading parameter.
52
2
.. . ............................. ...................... ... ...
0 ........ .. . . .. . . .. . .
 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . ... . . . .
2
0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2
2
 1 ... . . . ... . . . ... .. .. . ... . .. .. .. . . . .. ... . .. . . . .... . . ... ...
_ . .... ........ ...... .
2
0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2
0
10
0 ... .. ... !. . ... .. .... .. i.. .. ... i... ...... ... . .. .. . .... .
 0 ......... ......... ........ ....... ... ..
 2 0 ...... . . . . . . . ....... .. .. .... "" .... .... ........ ......
30
40 1 ... . . . . .
0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2
4
0 .. . . . , . . . . . . . .
2
0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2
t (sec)
Figure 4.4: Complex fading envelope, fm = 12 Hz. In this figure, x(t) is the real part, y(t) is the imaginary part, a(t) is the magnitude and O(t) is the phase.
53
4.2 Adaptive Channel Estimation If a digital radio signal is to be detected coherently, some means must be
provided for estimating the carrier frequency and phase. This task is called carrier recovery. For some signal constellations, such as 16QAM, we must also estimate the attenuation of the channel, or more commonly its reciprocal, the channel gain.
Suppose you have a digital communication system in a fading channel. We assume a perfect frequency lock, but we still have to estimate and compensate for errors in phase and gain. If we further assume that the primary source of these errors is the fading channel itself, then these quantities are just the phase and amplitude of the complex fading envelope mentioned in the previous section.
Consider the simple channel acquisition and tracking scheme shown in Figure 4.5. Here, we model u(t) by its discrete version u(nT,), where T, is the symbol interval, and pass it through the flatfading channel of Figure 4.1. It is multiplied by c(nT,) = x(nT,) + iy(nT,), and then white Gaussian noise is added. The lower portion of Figure 4.5 shows a adaptive scheme for estimating c(nT,). At the beginning of the transmission, u(nT) is sent through both the channel and a (singleweight) adaptive filter at the receiver. This may be accomplished by use of a prearranged training sequence. The outputs r(n) and y(n) are then compared and the estimate 6(nT) is updated based upon the difference e(n). We shall call Z(nT,) the channel estimate.
Any common adaptive algorithm, such as LMS or RLS, may be employed. The received signal r(n) is then multiplied by the reciprocol of Z(nT) to remove magnitude and phase distortion. Finally, the detector chooses the symbol fi(n) in the signal alphabet that is closest to i(n).
Once the channel has been acquired, the symbols ii(n) will be correct
with high probability and may be used in place of u(nT,) to track slow channel variations in decisiondirected mode. This works well as long as the error rate is
54
Fading Channel
c(nT) v(nT ) e(n) WGN _ Adaptive (Ts)
 Agorithm
C
Acquisition
y(n)
A B
ii(n) Tracking (nT,)
Figure 4.5: Adaptive channel estimation for a flatfading channel (cuts A, B and C are for reference in Figure 4.6).
low. It is not clear how high the error rate can be before the adaptive algorithm loses track, but estimates in the literature range between P, = .1 and P, = .01 (see Haykin [25, p. 37], Macchi and Eweda [43] and Trabelsi [67] for discussions in the case of decisiondirected equalization, with which this problem bears some similarity).
The scheme of Figure 4.5 is a form of decisionaided carrier recovery. Other decisionaided approaches are offered by Proakis [54, Sec. 624] and Macchi [42, Ch. 12]. Often, the tasks of recovering carrier phase and compensating for channel gain are handled separately. Our approach tracks both simultaneously. We do this only to simplify the presentation, since other approaches introduce nonlinearities that complicate the analysis.
4.2.1 Prediction of the Fading Envelope
Now, when the receiver enters a deep fade, the signal will be overcome by
noise, causing frequent errors in ii(n). Since fi(n) does not equal u(nT,), 6(nT.) is
55
adjusted incorrectly. By the time the receiver comes out of the fade, E(nT,) may have wandered far enough from its optimal value that the decisiondirected channel estimator fails to converge.
We propose a predictive method for overcoming this problem, as shown in Figure 4.6. With the switches in the position shown, while a(nT,) = Ic(nT)J is sufficiently large we see the adaptive scheme operating normally in decisiondirected mode, except that past estimates E(nT,) are clocked into a tapped delay line. When &(nT,) = V(nT,) falls below some threshold value, the switch positions change, turning off the adaptation, which is no longer reliable. A regressor vector of reliable past estimates of Z(nT) is latched, and the adaptive estimate is replaced with a prediction e(nT,) = 1(nT,) + ig(nT,) of the current value of c(nT,) based upon the vector of Z(nT,). Note that, in the figure, r represents how much time has passed since the adaptation was turned off.
When 6(nT,) rises above the fading threshold in absolute value, we return to decisiondirected tracking, using the last predicted value as the starting point for the adaptation. If the prediction is close enough to the true value of c(nT,) when the receiver emerges from the fade, then it may be possible to continue tracking the channel without transmitting a new training sequence. This idea is further illustrated in Figure 4.7. Clearly, the performance of the rpredictor in Figure 4.6 is key. For mobile radio we note that, because the real and imaginary parts of c(t) = x(t) + iy(t) are viewed as independent processes, the problem reduces to the two equivalent problems of predicting x(t) and y(t).
We wish to use the analysis techniques of Chapter 3 to determine the conditions under which the method of Figure 4.6 can work. Two theoretical difficulties present themselves. First, since we want a prediction of x(t), the power spectal density S.(w) is important, as was made clear in the analysis of Chapter 2.
56
ii(n) y(n)) Changes switch
( ( position and
A B resets 7 = 0 when &(nT,) = IZ(nT,)l
Ampl. Thresh. falls below threshold.
Detector
'_ Tapped Delay CLK 
I Io
Line*
T,Clock
8(nT,) CLK Prediction 0 * rPredictor (7 incr. by CLK) 8(nT,) Adaptive 1_e(n) )
Adaptation OnT, ~ ven ]Algorithm (
C
Figure 4.6: Predictive method for maintaining correct channel tracking through a deep fade (cuts A, B and C refer to Figure 4.5).
57
    Thresh.
II
I I I I I I DecisionDirected I I
I Adaptati ~Return to
I
I Adaptation
Prediction I I I I ] I
T t+
Figure 4.7: While in a deep fade, decisiondirected channel tracking does not work well. Using prediction in this region may allow the tracker to converge correctly once the signal comes out of the fade.
58
For mobile radio, however, we normally assume the Ushaped spectral density of Figure 4.3, whereas in Chapter 3 we assumed a flat spectral density.
We have not solved the bandlimited linear prediction problem for the Ushaped spectrum. We nevertheless go forward with the analysis under the assumption that the resulting inaccuracies in our results will not be too great. We do this in order to illustrate a technique that may be applied to bandlimited processes with spectral densities that are not flat once the linear prediction problem is solved for those cases.
The second theoretical difficulty is that the continuoustime analysis of Chapters 2 and 3 assumed that a sample function of x(t) was known over the entire continuum of a positivelength interval. Using the adaptive technique of Figure 4.5, however, the sample function will be known only at discrete times, since the adaptive estimate of c(t) = x(t) + iy(t) is updated once every symbol interval. A linear predictor would therefore be a sum of the form T E x(nT)h(kT,  nT,), where nT, takes values of t on the known interval of x(t) and kT = T is how far in the future, with respects to the known interval, we wish to make the prediction.
An analysis of prediction based on discretetime samples will depend upon the symbol interval T,. Thus, if a sufficient predictor performance cannot be attained at a given symbol rate, we may wonder if better performance might result from sampling at a different rate. Our continuoustime analysis was motivated by a desire to find a performance bound that was independent of T,. We assume that an optimal continuoustime predictor of the form (2.1) will not be outperformed by a discretetime predictor. Although we have not proved this, our heuristic reasoning is that a continuoustime predictor makes use of all the information on the known interval, whereas the discretetime predictor uses only a subset of this information. Extending our analysis to dicretetime prediction, and establishing a rigorous connection between the discretetime and continuoustime cases, as well as
59
solving the bandlimited linear prediction problem for a more general case than that of Chapter 3, remain important research objectives for us.
Even bypassing the theoretical objections, our work on this problem is
incomplete. We nevertheless present a simple case to illustrate how the techniques of Chapter 3 may be applied in the analysis of a practical problem.
4.2.2 Problem Statement
To use the results of Chapter 3, we must know something of the error
statistics of the adaptive estimate c(nT,). This requires an explicit choice of an adaptive algorithm in Figure 4.5. Also, our choice of the parameter r depends upon the duration of the fade; that is, the length of time during which the probability of error in ii(n) is high. In order to calculate the probability of error, we must decide upon a modulation scheme.
These steps are carried out in the following sections. In each case the
choices are made with a view toward simplifying the analysis, in the hope that the resulting presentation will provide clearer insight into the fundamental issues. For notational convenience in what follows, we drop the explicit dependence on T, and refer, e.g., to u(n) and c(n). We now offer the following problem statement:
Consider the adaptive/predictive channel estimation method of Figures 4.5 and 4.6. Assume that the input sequence u(n) is uncorrelated QPSK, whose symbols are chosen with equal probability from the set
u(n) e I + i VE, VE  iVE, VE  i VE, V +i ),
where Eb is the transmitted energy per bit. Thus, Ju(n) = for all n. The fading envelope c(n) is a zeromean complex Gaussian process whose real and imaginary parts are independent and identically distributed, and the signaltonoise ratio at the receiver input, time averaged to include the effects of fading, is such that the overall bit error rate is Pb = .0007.
60
We assume that decisiondirected tracking may proceed as long as the
instantaneous symbol error rate does not rise above P, = .01. When this threshold is crossed, we change position in Figure 4.6 from adaptation to prediction. We wish to resume adaptation as soon as the receiver emerges from the fade, so for the parameter r we choose the mean fade duration, to be calculated in Section 4.2.4. We wish to use a predictor that minimizes the mean squared error based on reliable estimates of c(n) since the time the receiver emerged from the last fade, so for the parameter T we choose the mean time above the fade threshold, also to be calculated in Section 4.2.4. We wish to know under what conditions this approach will allow us to recover correct channel tracking with a probability of PR 0.9.
4.2.3 Error Statistics of a Simple Adaptive Algorithm
As a first step in addressing this problem, we choose a simple adaptive
algorithm. Then we will determine the statistical behavior of E(n), the adaptive estimate of c(n) = x(n) + iy(n), as well as the effect of estimation errors on i(n), the input to the detector. Since we only use adaptation when the error rate is low (P, < .01), we may assume in what follows that ii(n) = u(n). Looking at Figure 4.5, we see that
r(n) = u(n)c(n) + v(n), (4.10) and
1 c(n) v(n)
( () = n) = (n) + . (4.11) Now,
e(n) = r(n)  y(n) = [u(n)c(n) + v(n)]  u(n)6(n). (4.12) Thus, solving for E(n), we have
v(n) e(n)
Z(n) = c(n) + () (4.13) uAn) An)'
61
As an algorithm for updating our estimate of Z(n) we choose e(n) + v(n)
Z(n + 1) = 6(n) + u(n) c(n) + n) (4.14) u(n) u(n) This choice simplifies the characterization of estimation errors. To see this, note that if the variation in c(n) is slow enough, then the error is dominated by the v(n) term. Thus,
(u(n) 1)
6(n) = c(n) + E(n) = c(n) + (n  1)(4.15) u(n  1)'
where
e(n) E(n)  c(n) v(n 1) (4.16) u(n  1)
Now since u(n) is an uncorrelated QPSK sequence with Ju(n)l = V24 , it is clear from (4.16) that e(n) is approximately a zeromean white Gaussian noise sequence
2
with a2  where a = E{Ie(n)12} and wh = e{Iv(n)2}.
Substituting (4.15) in (4.11) we have
c(n) v (n) i(n) = (n) u(n) + v(n) (4.17) c(n) + E(n) c(n) + e(n) Since we are considering the case in which the estimation errors e(n) and additive noise v(n) are not large enough to cause a high symbol error rate, we assume Ic(n)[ > le(n) and Ic(n) > Iv(n)j. Thus, c(n) e(n)
a1 (4.18) c(n) + E(n) c(n)' where the firstorder term is retained because the zerothorder term will be subtracted out shortly. Also, v(n) v(n)
,a . (4.19)
c(n) + e(n) c(n)( Substituting (4.18) and (4.19) into (4.17) we obtain
iu(n) n 1 u(n) + . (4.20)
1 c(n) c(n)
62
We define
S(n) v (n)
6(n)  i(n)  u(n) L( u(n) + (n) (4.21) c(n) c(n)(
Then, holding a(n) = jc(n)j constant,
.{16(n) 12}
S Ic(n)l2 [e(n)u(n) + v(n)] [e*(n)u*'(n) + v*(n)]
c() {je(n)u(n)2  E(n)u(n)v*(n)  E*(n)u*(n)v(n) + Iv(n) 12} .(4.22) Now,
ï¿½ {Ie(n)u(n)j'} = Iu(n) '6 {j,(n)12} = 2Ebo,,2. (4.23) Further, since e(n) is a function of past values of c(n), u(n) and v(n), it is uncorrelated with u(n) and v(n). Thus, E{E(n)u(n)v*(n)} = {1E*(n)u*(n)v(n)} = 0. (4.24) Therefore,
2Eb 62
S{6(n)2 E, + (4.25) a2(n)
This is the effective noise power at the input of the detector, taking into account the estimation errors in c(n). We define the received SNR per bit as b (n) = 2(n)Eb (4.26)
Then, accounting again for the errors in E(n), the effective SNR per bit at the input of the detector is
= 2 (n)Eb a2 (n)Eb yb(n) 9b(n) = 2Eba2 + as 2  2 (4.27) 2Ebef2 + ov 2 a2 2 If we let a2 be the timeaverage value of a2(n), then the timeaverage SNRs are
Eb 2&Eb
 = 2 e 2  (4.28)
2 Eb u,2 + o,,, 2 u,2 2
63
The overall bit error rate is given approximately by (65, p. 223]
1
Pb  a .0007, (4.29) 47b
where the effective average SNR was employed. This yields ,z 357 = 4 714. (4.30) Decisiondirected adaptation may continue as long as the symbol error rate is less than P = .01. The value of a(n) that attains this error rate is called the fade threshold, which we may write as a1. Representing the corresponding SNR by b,f, we have
a~E aEb Eb _Yb,f
7bf = 2bfï¿½  2 (4.31) M r2Es + o 2 2
Since we are interested in the symbol error rate only at the instant when a(n) crosses the threshold af, we do not use averaging. If we assume that Yb,f is high enough that the probability of a simultaneous error in both the real and imaginary parts of i(n) is negligibly small, we have [65, p. 223] P, a 2 Pr (singlebit error) = 2Q 2Vb~)
S.01. (4.32) This yields
y,7  3.32 = b 7, m 6.64. (4.33)
4.2.4 Statistics of the Fading Envelope
Decisiondirected adaptation becomes unreliable when a(n), the magnitude of the fading envelope, falls below the fading threshold au. When this happens, as we have discussed, we switch from adaptation to prediction in Figure 4.5. We wish to resume adaptation as soon as a(n) rises above a1 again. If our predicted value 6(n)
64
is close enough to c(n) at that instant, we may be able to recover correct channel tracking.
Recall that the parameter 7 represents how far in the future we wish to
predict c(t), where t = nT,, with respects to the interval of adaptively estimated values. If this interval is considered to end at the instant that a(n) crosses below af, then 7 should be chosen to equal the expected fade duration; that is, the expected time interval during which a(n) remains below af. We call the expected fade duration tf. It may be computed by [27, p. 36]
t = (lb 2  1), (4.34) where
r = (4.35)
V
and
12rfm
b= S~S"(w)w"dw. (4.36)
2 2,fm
We consider x(t) to be a bandlimited process with a flat spectral density. Since we are only interested in the ratio of b0 to b2 in (4.34), the scaling of (4.36) is arbitrary. Thus, we let Szz(w) = K, w E [2rfm, 2irfm]. We have
1 2wfm
bo = 2 l Kdw = 2Kfm, (4.37) 27r J27rfm
and
1 2wfm K [ 1 27rfm 87r2 3 b2  Kw2d= w31 = ~Kfn3 4.
b2 27rfm K2 3 2 2fm 3 fm (4.38) Thus,
b 2Kfm) = (4.39)
 Fb 4fi32= 'r (23
and
2 1).
tf 47Tfm 1 (er 2 ) (4.40)
65
Recall from Section 4.2.2 that we wish to base our prediction on reliable adaptive estimates of c(n) since the receiver emerged from the last fade. We are thus interested in the mean time above the fade threshold, which we shall call t7. To compute t7, we note that [27]
1 b2 b 2
Nr rer (4.41) tN + t7 bo e
is the average number of times per second that a(n) crosses aI in the positive direction. We thus have
1  rbo 1 er2 (4.42) t N + t = = b2 r
Making use of (4.34) and (4.39) we obtain
_ ,7rbo 1 3 1
t b2  4rfm2 . (4.43) Vb2 r 2r.
Note that, taking the ratio of (4.34) and (4.43) we have f er2  1. (4.44) t7
This may be interpreted as the ratio of the mean time in the fade to the mean time out of the fade. From (4.35) we see that this ratio depends on the SNR at the fade threshold 7b,f and the timeaverage SNR 5. It does not depend on the maximum Doppler frequency fm or the shape of the power spectral density Szz(w). These facts will ease our computations later (see Section 4.2.7).
4.2.5 Recovery of Channel Tracking
We now turn our attention to the instant at which the receiver emerges from the fade and ask under what conditions can we ensure recovery of correct channel tracking with a probability PR > .9. Let us consider the last predicted sample e(n) before switching back to adaptation in Figure 4.6. For that sample we define
&(n) = r(n)  y(n) = [u(n)c(n) + v(n)]  u(n)a(n). (4.45)
66
Solving for 6(n) we have
6(n) = c(n) + v(n)+ .(n) (4.46) u(n) u(n)
We use this value instead of Z(n) in the fading compensator immediately preceding the detector in Figure 4.5. Suppose that this results in the detector making the correct decision ii(n). We switch back to adaptation and, using our adaptive algorithm (4.14), we have
(n + 1) = 6(n) + ) + n) + u(n) (4.47) We see that, as long as we restart adaptation with a correct decision ii(n), the effect of the "big" error 6(n) disappears after one symbol, and since a(n) is above the fade threshold af, adaptation may proceed as in Section 4.2.3, and we consider correct channel tracking to be recovered. Thus, a sufficient condition for recovery of correct channel tracking is the correct detection of the last received symbol before switching back from prediction to adaptation.
To quantify this condition we define
i(n) = 6(n)  c(n), (4.48) which is the fadingenvelope prediction error at the instant the receiver emerges from the fade. Now, when 6(n) is used in the fading compensator, then the input to the detector may be written as (n =c()(n) v(n)
i(n) + (n)
= u(n)  u)(n) + v(n) (4.49) 6(n) 6(n)'
because e(n) = c(n) + i(n). We define 6(n) = u(n)  u(n) = (n)u(n) + v(n) (4.50) 6(n) 6(n)
67
The prediction 6(n) is to be based on adaptive estimates E(n), a process which is approximately zeromean Gaussian, as was discussed in Section 4.2.3. Thus, the minimum mean squared error predictor is linear. Since 6(n) is a linear transformation of a Gaussian process, 6(n) is also Gaussian, and since the real and imaginary parts of 6(n) are independent and identically distributed with zero mean, the same holds for 6(n). Therefore, the phase of 6(n) is uniformly distributed on [7r, 7r].
The Gaussian white noise v(n) is uncorrelated with, and hence independent of 6(n), which is a function of past values of E(n). The prediction error i(n) = 6(n)  c(n) is a sum of Gaussian random variables; thus i(n) is Gaussian. From the theory of mean square estimation, c(n) is the mean of 6(n) conditioned on the adaptive estimates 5(n), so i(n) is zero mean. Also, E(n) must be orthogonal to 6(n) [25, p. 200]. Since i(n) is zeromean Gaussian, orthogonality implies independence.
At the instant the receiver emerges from the fade, we have Ija(n) I = af, but the phase of 6(n) is still randomized. Thus, () and '(n) are independent, zeromean, complex Gaussian random variables with variances 4 and respectively. From (4.50) we therefore have, recalling u(n) = 4 ,
2 2 22Eb^n+ (4.51) at the instant the receiver emerges from the fade. The SNR per bit is Eb _ afEb 2 af (4.52) =6    4 . (4.52) or 2Ebf + o.2 2be 2Eb To attain Pc = .9 we must have (54, p. 272] Pc= 1 Q)2 = .9. (4.53)
68
This yields
Yb = 1.3321. (4.54) In the next section, we shall see that it is unnecessary to compute the value of o2 explicitly.
4.2.6 Predictability Analysis
In this section we shall use the predictability analysis techniques developed in Chapter 3 to determine under what conditions the mean squared prediction error can be made less that u2 as given by (4.52). Now let Jmin(E) be the minimum mean squared prediction error of x(t), the real part of c(t). We note that the adaptive estimation errors e(n) and the predictive estimation error i(n) are complex processes with their powers split evenly between the real and imaginary parts. Using the approach of Section 3.4 we want (cf. Eq. 3.45)
2 2
Jmin(E) + 0E . (4.55)
2 2'
Now 2 = Ic(t)12} is two times the average power of x(t). Thus, if Jmin(E) is the minimum mean squared prediction error of a normalized process we have a2 2 2
Jmin(E) + E < ' (4.56)
2 2 2
or
2 2
Jmin(E)  = l(E). (4.57) Now, from (4.28) we have ab Eb a Eb
 2 = = .= , (4.58)
69
and since, from Section 4.2.3, we know that 2Eba, is approximately equal to the noise variance O , we use (4.28) to obtain U2 U2
   "    ( 4 .5 9 ) a2 2Eb 2 2.59) Also, from (4.52) we have aa 1 a22U2
a2 2 2b 2E a2
_ Yb,f 1 1
b 2Yb 2Y= 2 b 1 , (4.60)
where use was made of (4.58), (4.28) and (4.31). Substituting (4.59) and (4.60) into (4.57) we have
1(E) =   1 E . (4.61) Now T, yb,f and Yb were determined in previous sections. We wish to know under what conditions Jmin(E) will be less than or equal to 1(E) as given by (4.61) for at least some value of E.
It will be recalled that Jmin(E) depends implicitly on 0, T and T, where Q is the band limit of the process x(t), T is the length of the interval of known values on which the prediction is to be based, and 7 is how far in the future we wish to make the prediction. In accordance with the discussion in Section 4.1, the fading envelope c(t) = x(t) + iy(t) is bandlimited to the maximum Doppler frequency fi. We thus choose 0 = 21r f,. The prameters T and T are set equal to the mean fade duration t1 and the mean time above the fade threshold ty respectively, as discussed in Section 4.2.4.
Now note from (4.43) that
T=ty =.43 1 cT ,(.2
S 3 1 c= 3r (4.62) t7 =V 7rf 2 r 2 V L'4 r
70
where c is the timebandwidth product and r, given by (4.35), is already determined. Also, using (4.44), we may define d = = e2  1. (4.63) T t7
We see from this that if one of the three parameters, 0, T or T, is known, the other two may be determined from (4.62) or (4.63). Thus, only one of the three may be considered to be a free variable. We choose , and use the notation Jmin(O, E) to explicitly indicate the dependence of the minimum mean squared prediction error on this parameter.
We see from (4.57) that correct channel tracking will be recovered if
Jmin(O, E) !< 1(E) (4.64) for some E > 0. Note that this condition depends only on Q, which is proportional to the maximum Doppler frequency fin.
4.2.7 Computational Issues o
It is important to note that the relationship between Q, T and r indicated
by (4.62) and (4.63) offers a computational advantage. To see this, we consider the 1r
optimization problem (3.5). We scale the objective function by  for normalization wQ
(cf. Eq. 3.35), then substitute w' = to obtain
f= 11  H(Ow') I2dw' (4.65) =2
T aT
Now let c = and d =  be fixed numbers. Then, letting t' = Ot, the time
2 T
constraint may be written
h () = 0 t' ~ [2cd, c(2d + 1)]. (4.66)
71
Also, with the substitution t' = Qt, the energy constraint becomes
 h2  dt' = E. (4.67) Now let G(w) = H(Qw) =a g(t) = h(l). Our optimization problem becomes minimize 3 = 1  G(w)2dW
2
subject to 1. g(t) = 0 t ( [2cd, c(2d + 1)] (4.68)
2. f g2(t)dt
We see from this that
Jmin(0, E) = Jmin(1, ). (4.69) Thus, we need use the procedure (3.22) only once, scaling appropriately to obtain Jmin(1, E), then use (4.69) to compute Jmin(Q, E) for arbitrary Q. We recall from Section 3.1.1 (see page 32) that n, (t) depends implicitly on Q and T. We may therefore write (3.2) as
0(t) = (Q, T, t  T  . (4.70) Now we let Q = 1 and note from (4.62) that T = 2c. Also, from (4.63) we have
T  = c(2d  1). (4.71) Thus, making use of (A.6) from the appendix, we have
2(0) = 0,(1,2c,c(2d1))
 2n+1 [sO)(c, 1)]2 [R (c, 1 + 2d)] 2, (4.72) 7r
where S(i)(c, rl) and RW n(c, ) are the angular and radial functions discussed in the appendix. Note that in (4.72), we use the fact that V(Q, T, t) is even in t. Our
72
computational procedure then becomes 002 STEP 1: Solve 0 (1,2c, c(2d  1))AnAn (= + E for
n= oc STEP 2: Then Jmi(1, E) = 7r n=O (1, 2c, c(2d  1)) (1 An (C) + M
(4.73)
71"
where the leading factor r = in step 2 is necessary because Jmin(1, E) is normalized.
For this problem, we use (4.30), (4.33) and (4.35) to compute r = .0964. Then, using (4.62) and (4.63) we compute c = 15.92 and d = .00934. For our computations, we round c to 16 and d to .01.
4.2.8 Summary of Analysis Procedure and Discussion
A summary of our analysis procedure is given in Table 4.1. Using values of 7, yb,f and 'b computed in (4.30), (4.33) and (4.54), and also using rounded values of c = 16.0 and d = .01, we obtain the graph of Figure 4.8. We see that we may recover correct channel tracking with probability PR .9 for values of Q less that or equal to .52. This is a maximum Doppler frequency fm =  of less than 2ir
or equal to .08 Hz. This does not indicate good performance because, in mobile radio, the value of fm can be much higher, on the order of tens of hertz. As stated previously, though, our work on this problem is incomplete. The case presented
here is intended mainly as an illustration of how the techniques developed in previous chapters can be applied to the analysis of a practical problem.
There are many approaches we might take to improve the performance of our adaptive/predictive channel tracker. For example, instead of restarting the adaptation as soon as our prediction 6(n) crosses the fading threshold, we might wait a few symbol intervals for the signal to strengthen. If the quality of our prediction has not degraded too much, we might have a better chance of correct recovery at the higher SNR. We also note that our techniques only allow us to
73
Table 4.1: Summary of Predictability Analysis Procedure
1. Based on the desired overall bit error rate, use (4.29) and (4.28) to determine T.
2. Based on the maximum bit error rate to maintain correct decisiondirected
channel tracking, use (4.32) and (4.31) to determine 7yb,.
3. Based on the desired probability of recovering correct channel tracking after
a deep fade, PR, use (4.53) to determine Yb.
4. Use (4.35) to determine r. Then use (4.62) and (4.63) to determine c and d.
5. Using (4.61), for E > 0, plot all positive values of 1(E) on a graph.
6. Using (4.73) and (4.72) one time, and then (4.69), find a value Qo such that
Jmin(Co, E) plotted against E is tangent to the curve for 1(E).
7. If the maximum Doppler frequency fm is less than or equal to Go/27r, correct
channel tracking may be recovered with probability PR.
74
0.01
i 0 .0 0 8 ......... .. .......... .........
i ~ 0=0. 0.52 0.8
Cd
S 0 .0 0 6 .................. .... ....... ...................
0.004 ... ... .... . ....
2 1(E)
S.002
0 . S *
0 0.5 1 1.5 2 2.5 3
ElrV
Figure 4.8: Predictability analysis for recovery of correct channel tracking. perform the prediction based on a finite interval of estimated values, an interval which was chosen somewhat arbitrarily for this example. We would like to extend our approach to the case of prediction based on the infinite past. This would give us a clearer indication of the viability of this approach.
Speaking more broadly, an analysis such as the one we have presented can provide a starting point for investigating the viability of decisiondirected fading compensation in general, since to date much of the research literature on this subject has assumed that the detected symbols are error free (see, e.g., Li et al. [37]).
4.3 Addressing the Model Mismatch As was mentioned in Section 4.2.1, the spectral shape of Figure 4.3, which is the most commonly used in modeling the mobile radio fading envelope, does not match the flat spectrum assumed in Chapter 3. In this section, we consider alternatives for addressing the model mismatch.
75
First, let us consider the prospects of extending the solution of Section 3.2 to processes with spectral densities of the form (4.9). For simplicity, let VJi w21 (4.74)
0 otherwise.
Substituting into the objective function of (2.10) and noting that Q = 1 we obtain
1 1 1
J =  l1i_ i  H(w)2dw. (4.75) 27r _ V1 _W2 1
In Section 3.2, we expanded H(w) in terms of the prolate spheroidal wave functions, allowing us to convert the objective function into an algebraic expression. Unfortunately, the prolate functions are not orthogonal with respects to the weighting function S==(w) as given by (4.74).
We may consider expanding H(w) in terms of functions, such as the Tchebycheff polynomials, which have the appropriate orthogonality properties [7, p. 54]. Alternatively, we may substitute cos 0 = 1flw2 and sin 0 = w in (4.75) to obtain
r=  21  H(sin 0)12d0. (4.76)
2
The function G(0) = H(sin 9) may now be expanded in terms of any complete orthogonal set on [, ]. These may be the prolate functions, Legendre functions, or simply sinusoids. The function h(t) will also be expanded in terms of an orthogonal set on [1r, 7 + T], so evaluating the expansion coefficients for G(0) would require a cross product between the two sets of basis functions. This is potentially the most tedious step. Once this is done, the Lagrange multiplier can be applied and a system of linear equations, such as the one preceding (3.20), will result. Unlike that case, however, we expect more than one unknown, in addition to the Lagrange multiplier, to appear in each equation.
76
Though we are optimistic about solving this problem, we also would like to consider how the solution obtained in Chapter 3 may itself be appropriately applied. In order to do this, let us look again at the objective function of (2.10), which we rewrite for convenience:
1
J=  Sx (w)1  H(w)I'dw. (4.77) 27r fn
We notice that the power spectral density Sx(w) serves only as a weighting function. Regardless of the form of SA(w), we still wish to bring H(w) as close as possible to 1 on w E [Q, 0]. The function S4,(w) specifies only the penalty for deviation from 1 at each w. The graph of Figure 4.9 compares a function of form (4.74), in dotted line, with a function that is constant within the band limit, represented by a solid line. For a fair comparison, each function is normalized with respects to the process power. We see that the difference in the weighting is not very large until w approaches the band limit Q, so it is possible that a predictor optimized for a flat spectral density will also perform well for a process whose density is of the form (4.74). For such a predictor, one of our objectives for future research is to find a useful bound on the true minimum mean squared error, given our computation based on the flat spectral density model.
If we wish to obtain a spectral shape that more closely resembles that of
Figure 4.3, we may consider the approach of Figure 4.10. In this case we combine a flat, bandlimited spectrum with two sinusoidal components at w = +Q. The parameters of the sinusoidal components may be estimated using a correlation on the interval of adaptive esitmates, and the result may be subtracted out before applying the bandlimited predictor. Many variations on this approach are possible. For instance, the flat segment need not occupy the the entire Q bandwidth, and the sinusoidal components need not be placed precisely at the band limits. Also,
77
5 '
4 .5 .. . . . . . . . . it . . . . . . . . . i. . . . . . . . . . i. . . . . . . . .. .I . . . . . . . . . .! . . . . . . . . . .
4 .. . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . ... . . . . . . . . .. . . . . . . .
4 :1
3 .5 . .... .............................. .........
I I:
2 .5 . . .... . . . .. .. . .. . .. . . ... . .
t3 2 .. . .... ....... I..:............
12 .5 ........ . ..... ... .......... ... .... .. ....I .
1.5 .
1 . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . .."
0.5 ...... ... ..................... ...... . . .
0
1.5 1 0.5 0 0.5 1 1.5
WID
Figure 4.9: Comparison of fadingenvelope spectra. The dotted line indicates the spectrum most commonly assumed in mobile radio. The solid line indicates the flat, bandlimited spectrum of Chapter 3.
5
5 . .' : . I.
4 .5 ...... .... ..... ..... ......... ................... .........
3 . 5 . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . ! . . . . . . . . . . .
:I
3 . ..........5.  , .. .
. 2 .5 .......... .. . . .................. ......... ........ I . . .......
U ) 2 . . . . . . . . . . .. . . . ... . . . . . . . . . . . . . . . . . :
ï¿½1 \ . .I:
2. . . . . . . . . ." . . . . . . . . . . . . / . . :. . . .
1.5
1 . ....................... . . ".......... .........
0.5
1.5 1 0.5 0 0.5 1 1.5
Figure 4.10: A piecewise approximation of the fadingenvelope spectrum, using a flat segment and two sinusoidal components.
78
5 4
: I I:
3 .5 . . .. . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . ..
. . .... .. . ... ........
.II I:
S ......... .. ........ ... . ...... .........
2 .
0
1.5 1 0.5 0 0.5 1 1.5 0)
Figure 4.11: Application of the flat spectral density to nonadaptive prediction. the magnitude of each component may be fixed by the model, or left as a random variable to be estimated.
Finally, we note that optimal estimators are often realized using adaptive filtering techniques. Suppose, however, that we prefer a nonadaptive predictor. Then we argue that it makes sense to choose a flat spectral density in carrying out the optimization procedure. The reason may be seen in Figure 4.11. We recall that the fadingenvelope spectral density (4.9) depends upon the maximum Doppler frequency fmn, which is proportional to the speed of the receiver. As the receiver speeds up and slows down, the bandwidth of the spectrum will correspondingly increase and decrease. The characteristic U shape of the spectrum may therefore be wider or narrower, as is shown by the dotted curves in the figure. Thus, a predictor optimized to one of these spectra may perform poorly when the receiver changes speed. A flat spectral density of sufficient bandwidth represents a reasonable compromise, since it distributes an even weighting in the mean squared error penalty across the entire band that is likely to be occupied by the fading envelope.
79
In summary, it is likely that the gap between the flat spectral density model and that of Figure 4.3 will be closed soon. But in the interim, the solution of Chapter 3 will still be a useful tool in analyzing realworld fading problems.
CHAPTER 5
CONCLUSIONS AND FUTURE WORK In this dissertation, we have discovered some key facts about the predictability of bandlimited processes, we have solved the bandlimited prediction problem for processes with flat spectral densities, and we have shown how these findings may be applied in the analysis of a problem in multipath fading. We believe that the approach outlined here has potentially a very wide applicability, not only to fading compensation, but to other problems where the bandlimited model applies. Clearly, though, there is much that remains to be done. In this chapter, we offer some conclusions based on our work and then outline some of our plans for future research.
5.1 Conclusions
Mobileradio fading is an extremely difficult problem to overcome. Still, in view of the market's apparently insatiable appetite for mobile communication services, it seems likely that the urgency for dealing squarely with fadingrelated issues will only increase with time.
It is not surprising that many techniques for fading compensation have been proposed in the literature. Often, as we have discussed, these techniques require an estimate of the current value of the fading envelope. We have mentioned adaptive transmission, and dealt at some length with decisiondirected adaptive channel tracking. As we have seen, such approaches often invlove an implicit prediction problem, since performace analyses assuming the availability of perfect channel estimates fail to account for the effects of estimation errors, as well as delays in obtaining the estimates.
80
81
In evaluating the performance of a receiver, we may wonder which effect, estimation errors or delays, is more important, but our work shows that if the parameter to be estimated may be modeled as a bandlimited process, then the two factors are actually related, because if the estimates are error free, then we can overcome the delay with a prediction whose error may be made arbitrarily small.
The flatfading channel model discussed in Section 4.1, including the characteristic Ushaped spectral density, has been widely applied since its introduction in 1968 [11, 27]. Clearly, this is a bandlimited model, and it has been known since at least 1962 that such bandlimited random processes are predictable in the meansquare sense [72], but this result does not seem to be well known among those researching the fading problem.
The predictability of bandlimited processes, as discussed in Chapter 2, would seem to offer hope that the implicit prediction problems mentioned above can be solved, but as we have seen, these findings must be interpreted with care. After all, many common phenomena, including human speech, can reasonably be modeled as bandlimited. Does this mean that such phenomena can really be predicted with arbitrarily small error?
The answer is no, if there is any error in the process values on which the
prediction is to be based, as was made clear in Chapter 2. And such error is always present, whether caused by an estimation procedure, a measurement technique, or simply the quantization error of storing a number in memory. In applying Wiener analysis to nonbandlimited processes, the effects of noise are often ignored, because the spectrum itself limits predictability. But in bandlimited processes the noise, no matter how small, is always the dominant factor in prediction error.
The application of our techniques to adaptive channel estimation in Section 4.2 was intended mainly as an illustration. In that case, the simplicity of the adaptive algorithm chosen allowed us to trace the main source of errors; it is the
82
white Gaussian noise added to the received signal. In electronic communication systems, such noise arises from thermal effects in the receiver front end. Thus, we may consider trying to reduce our prediction error by improving the input signaltonoise ratio, either by increasing the transmit power or by redesigning the receiver using more expensive lownoise devices. Would such expenditure be worth it?
We hesitate to take such a bruteforce approach because an overall signaltonoise ratio is usually targeted to achieve a desired bit error rate. Raising the SNR just to improve the performance of decisiondirected adaptation may result in a BER that is much lower than necessary. Decisiondirected techniques are widely employed, and are successful in situations where the channel conditions vary slowly, and where deep fades are not encountered very frequently. In such cases, decisiondirected adaptation may be thought of as incurring little cost since, e.g., it does not require side information or additional bandwidth. In the case of mobileradio fading, where such assumptions are not justified, our predictability analysis shows how the formerly negligible costs can become significant.
If the costs are too high, then the adaptive/predictive approach of Section 4.2 will hold little advantage. Regardless of this, however, the implicit prediction problem remains, since it is difficult to see how a decisiondirected approach can otherwise be made to work in a fading environment without increasing power, resorting to diversity, or applying some other resourceconsuming solution. Thus, we see that the techniques of linear prediction developed here may be applied not only to develop new approaches to fading compensation, but also to analyze the feasibility of previously proposed methods which involve such an implicit prediction problem.
We shall continue to pursue these matters, as we are sure others will continue to pursue the general problem of fading. Whether these efforts will eventually
83
succeed in opening up terrestrial mobile channels to the kinds of information rates that are now common in other types of communication is not yet certain. Still, if our contribution can be used to clarify issues and avoid some wasted effort, then our work will have been worthwhile.
5.2 Future Work
Our immediate research objective is to extend the solution of the bandlimited prediction problem to processes with spectral densities of the form (4.9), as was discussed in Section 4.3. In this way, our findings would be more directly applicable to mobile radio fading. As was mentioned in the same section, we would also like to find a useful bound on the minimum mean squared prediction error of such a process, given a computation based on the flat spectral density model.
Also, the techniques that we have developed allow us to carry out a prediction based on a finitelength interval of known values. We may wonder whether such a prediction might be improved if we considered all past values to be known. Thus, we also plan to extend our solution to this case as well.
We pursued a continuoustime analysis in order to find a predictor performance bound that is independent of the symbol interval. In digital modulation, however, the use of adaptive techniques will yield estimates of the fading envelope at discrete times, and these will form the basis of our prediction. This will become especially important as we explore the problem of adaptive equalization. For this reason, an extension of the solution to the discretetime case is of interest. We particularly want to know if the discretetime solution converges to the continuoustime solution in some sense as the symbol interval is shrunk to zero.
Our analysis assumes that the prediction is to be based on a sample function that has been corrupted by white noise. Many estimation techniques result in errors that are not well modeled as white noise. Thus, one of our research goals is
84
to determine how to apply the solution of the bandlimited prediction problem when the corrupting noise is colored.
In addition to considering extensions of the solution in Chapter 3, we also wish to address questions raised in the analysis of Section 4.2, dealing with adaptive channel estimation in a fading environment. At the end of the section, we listed some steps we may take to improve the performance of our adaptive/predictive approach to carrier tracking. Also, it will be recalled that we assumed throughout that a symbol error probability of less than .01 is a sufficient condition for maintaining correct, decisiondirected tracking. Although various figures are cited in the literature (see discussion on page 54), there seems to be little experimental or theoretical justification for them. We wish to address this issue more rigorously, since clearly the greater the symbol error rate that is acceptable, the easier our prediction will be.
Adaptive channel tracking is not the only fading compensation approach that could make use of a prediction of the fading envelope. Many techniques require a current estimate of the channel impulse response, and performance analyses of these techniques often assume that a perfect channel estimate is available. In Section 1.1 we discussed the case of adaptive transmission. Unfortunately, estimation necessarily involves some delay. As we discussed, we would like to explore the application of our predictive techniques to the problem of overcoming this delay, in adaptive transmission as well as other approaches requiring channel information.
A final, broad, and very important question is how the prediction of a
bandlimited process is actually to be carried out. Although we do not wish to pursue a detailed discussion of this subject here, it is not clear that a direct realization of the optimal predictor impulse response is always possible or desirable. Also, we would like to know how to make such a predictor adaptive. The door is
85
wide open here, and we have considered nonlinear approaches, as well as alternative models.
APPENDIX
EVALUATION OF BASIS FUNCTIONS This appendix describes how to compute values of An and 4n(t), as described by Slepian et al. [60], using the tables of spheroidal wave functions published by the Naval Research Laboratory [70, 23]. These include the angular functions, S(n(c, iq) [70], and the radial functions, R )n(c, ) [23]. We adhere to the notational OT
conventions of Slepian et al., with c = 2. In the NRL tables, the symbols I and h are used in place of n and c respectively. The eigenvalues An(c) are appended to the tables of S() (c, q) [70, p. 404].
The function #n( (, T, t) may be found using [70, p. xv] [2An(c)/T]2 So" (c, 2t/T)
On(Q, T, t) =n (A.1) SoT) (c, s)12 ds
In the NRL tables, the angular functions are normalized such that [70, p. xi] S(c) 2 (A.2) 1'1 [SO (c, ) ds 2n ï¿½ 1'
so the denominator of (A.1) is easily handled. Unfortunately, for our prediction problem we are interested in values of 4n(t) for It > T, but So) (c, 7) is only tabulated for q = It < 1 = t < T. This may be handled by using the relation [16, p. 32]
So()(c, z) = ron (c) R (c, z), (A.3) where the angular and radial functions have been extended to entire functions of the complex variable z. The function ,on (c) is called a joining factor, and may be
86
87
evaluated by
S()(c' 1)
on(c) = On (A.4) n (c, 1)
We also make use of [70, p. xv]
2C[~)('1)]
An(c) = ((c, 1)]2. (A.5) Solving (A.5) for 4(c, 1), and using (A.4), (A.3) with z = t, (A.2) and (A.1) we obtain
_ (2n + 1cl
O,(Q, T, t)= (2n+ 1)CSo()(c, 1) 4 (c, 't). (A.6) The function R (c, ) is tabulated for = t > 1 ==  t > T, which is the interval of interest.
Working with tabulated values can be tedious, and it should be noted that some computer programs for evaluating the prolate spheroidal wave functions are available [38, 69, 74, 34], but many of these are written in outdated languages, are not portable, or have not been tested thoroughly.
REFERENCES
[1] K. AbedMeraim, W. Qiu, and Y. Hua. Blind system identification. Proceedings of the IEEE, 85(8):13101322, August 1997.
[2] A. H. Aghvami. Digital modulation techniques for mobile and personal
communication systems. Electronics and Communication Engineering Journal,
5(3):125132, June 1993.
[3] P. A. Bello and B. D. Nelin. The effect of frequency selective fading on the
binary error probabilities of incoherent and differentially coherent matched
filter receivers. IEEE Transactions on Communication Systems, 11:170186,
June 1963.
[4] P. A. Bello and B. D. Nelin. Corrections to "The effect of frequency selective
fading on the binary error probabilities of incoherent and differentially
coherent matched filter receivers." IEEE Transactions on Communication
Technology, 12:230231, December 1964.
[5] F. G. Beutler. Errorfree recovery of signals from irregularly spaced samples.
SIAM Review, 8(3):328335, July 1966.
[6] C. J. Bouwkamp. On spheroidal wave functions of order zero. Journal of Mathematics and Physics, 26:7992, 1947.
[7] A. Broman. An Introduction to Partial Differential Equations, from Fourier Series to Boundaryvalue Problems. New York: Dover, 1989.
[8] J. L. Brown. Uniform linear prediction of bandlimited processes from past samples. IEEE Transactions on Information Theory, 18(5):662664, September
1972.
[91 J. L. Brown. On the prediction of a bandlimited signal from past samples.
Proceedings of the IEEE, 74(11):15961598, November 1986.
[10] J. A. Cadzow. An extrapolation procedure for bandlimited signals. IEEE
Transactions on Acoustics, Speech, and Signal Processing, 27(1):412, February
1979.
[11] R. H. Clarke. A statistical theory of mobile radio reception. Bell System
Technical Journal, 47(6):9571000, JulyAugust 1968.
88
89
[12] J. B. Conway. A Course in Functional Analysis, 2nd ed. New York: SpringerVerlag, 1990.
[13] W. B. Davenport and W. L. Root. An Introduction to the Theory of Random
Signals and Noise. New York: McGrawHill, 1958.
[14] J. L. Doob. Stochastic Processes. New York: Wiley, 1953.
[15] T. T. Fjallbrandt. Interpolation and extrapolation in nonuniform sampling
sequences with average sampling rates below the Nyquist rate. Electronics
Letters, 11(12):264266, June 12, 1975.
[16] C. Flammer. Spheroidal Wave Functions. Stanford, CA: Stanford University
Press, 1957.
[17] G. D. Forney. Maximumlikelihood sequence estimation of digital sequences in
the presence of intersymbol interference. IEEE Transactions on Information
Theory, 18(3):363378, May 1972.
[18] C. Fox. An Introduction to the Calculus of Variations. London: Oxford
University Press, 1950.
[19] B. R. Frieden. Evaluation, design and extrapolation methods for optical
signals, based on use of the prolate functions. In Progress in Optics 9, edited
by E. Wolf, pages 311407. Amsterdam: NorthHolland, 1971.
[20] D. N. Godard. Selfrecovering equalization and carrier tracking in twodimensional data communication systems. IEEE Transactions on Communications, 28(11):18671875, November 1980.
[21] A. J. Goldsmith and S. G. Chua. Variablerate variablepower MQAM for
fading channels. IEEE Transactions on Communications, 45(10):12181230,
October 1997.
[22] G. H. Golub and C. F. Van Loan. Matrix Computations, 3rd ed. Baltimore,
MD: Johns Hopkins University Press, 1996.
[23] S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King. Tables of radial
spheroidal wave functions, volume 1: Prolate, m = 0. Technical Report NRL 7088. Washington, DC: Naval Research Laboratory, 1970. Available through
NTIS, AN: AD723836XAB.
[24] S. Haykin, editor. Blind Deconvolution. Englewood Cliffs, NJ: PrenticeHall,
1994.
[25] S. Haykin. Adaptive Filter Theory, 3rd ed. Upper Saddle River, NJ: PrenticeHall, 1996.
[26] H. Hochstadt. Integral Equations. New York: Wiley, 1973.
90
[27] W. C. Jakes, editor. Microwave Mobile Communications. New York: Wiley,
1974.
[28] S. M. Kay. Modern Spectral Estimation: Theory and Application. Englewood
Cliffs, NJ: PrenticeHall, 1988.
[29] S. M. Kay. Fundamentals of Statistical Signal Processing: Estimation Theory.
Englewood Cliffs, NJ: PrenticeHall, 1993.
[30] R. A. Kennedy, B. D. O. Anderson, and R. R. Bitmead. Blind adaptation
of decisionfeedback equalizers: Gross convergence properties. International Journal of Adaptive Control and Signal Processing, 7(6):497523, NovemberDecember 1993.
[31] J. J. Knab. Interpolation of bandlimited functions using the approximate
prolate series. IEEE Transactions on Information Theory, 25(6):717720,
November 1979.
[32] J. J. Knab. Noncentral interpolation of bandlimited signals. IEEE Transactions on Aerospace and Electronic Systems, 17(4):586591, July 1981.
[33] P. Koosis. Introduction to Hp Spaces, with an Appendix on Wolff's Proof of the
Corona Theorem. Cambridge: Cambridge University Press, 1980.
[34] M. B. Kozin, V. V. Volkov, and D. I. Svergun. A compact algorithm for
evaluating linear prolate functions. IEEE Transactions on Signal Processing,
45(4):10751078, April 1997.
[35] E. Kreyszig. Introductory Functional Analysis with Applications. New York:
Wiley, 1978.
[36] N. Levinson. Gap and Density Theorems. New York: American Mathematical
Society, 1940.
[37] M. Li, A. Bateman, and J. P. McGeehan. Analysis of decisionaided DPSK in
the presence of multipath fading. In 6th International Conference on Mobile
Radio and Personal Communications, pages 157162, Stevenage, England:
IEE, Michael Faraday House, 1991.
[38] D. W. Lozier and F. W. J. Olver. Numerical evaluation of special functions. In
W. Gautschi, editor, Mathematics of Computation 19431993: A HalfCentury
of Computational Mathematics, volume 48 of Proceedings of Symposia in
Applied Mathematics, pages 79125, Vancouver, BC, August 1993. American
Mathematical Society.
[39] D. G. Luenberger. Linear and Nonlinear Programming, 2nd ed. Reading, MA:
AddisonWesley, 1984.

Full Text 
66
Solving for c(n) we have
c(n) c(n) + + ..
u(n) u(n)
(4.46)
We use this value instead of c(n) in the fading compensator immediately preceding
the detector in Figure 4.5. Suppose that this results in the detector making the
correct decision (n). We switch back to adaptation and, using our adaptive
algorithm (4.14), we have
c(n + 1) = c(n) + = c(n) + (4.47)
u[n) u(n)
We see that, as long as we restart adaptation with a correct decision (n), the
effect of the big error e(n) disappears after one symbol, and since a(n) is above
the fade threshold a/, adaptation may proceed as in Section 4.2.3, and we consider
correct channel tracking to be recovered. Thus, a sufficient condition for recovery of
correct channel tracking is the correct detection of the last received symbol before
switching back from prediction to adaptation.
To quantify this condition we define
e(n) = c(n) c(n), (4.48)
which is the fadingenvelope prediction error at the instant the receiver emerges
from the fade. Now, when c(n) is used in the fading compensator, then the input
to the detector may be written as
(n)
c(n)
c(n)
u(n) +
v(n)
c(n)
u(ri)
Â£(n)
c(n)
u(n) +
v(n)
c(n)
because c(n) = c(n) + e(n). We define
(4.49)
S(n) = (n) u(n) +
c(n)
v(n)
c(n)'
(4.50)
26
But since J{hn) > 0, we have
a2w\\hn\\2
Thus,
y/k + e
IIM < n> m.
Since {hn} is norm bounded, we may choose a subsequence {hnk} which converges
to some weak limit h0. Then there is a sequence of finite sums {^} with
k+Mk
9k
l=k
(2.21)
where the coefficients c^k satisfy
fc+Mjfc
Ci,k >0 and Y = 1,
l=k
(2.22)
such that {pfc} converges strongly to h0. A function gk(t) satisfying (2.21) and
(2.22) is called a convex combination of {/i,}. Since a convex combination is a
particular type of linear combination, we have gk G C2T V k. Now because of
(2.20), for every e > 0, there is a p such that, for k > p we have
< J(9k)
1 rn
l rn
2n
1
k+Mk
1 Y cl,kHn,H
l=k
k+Mk
Y c/,41 Hni(uj)]
l=k
k+Mk
E cnjam(ui)
l=k
I a roo
j Sxx(u)\lGk{u))\2du) + r^L \Gk{aj)\2duj
Ztt Jn Ztt Joo
rU k+Mk 2 r
/ S(0 1 E /
Jn =ib 2tt Jc
, 2
l
= 2LnSxx{u)
k+Mk C 1 n 1 fc+M* f 1 /o 1
< Q,fc in,(o;)2ckj + Y Ci^\ySu \HnÂ¡^2dw\
Q2 roo
**+tL
k+Mk
k+Mk
y ^i,kHni{yj)
l=k
du>
2
k+Mk
Ys Cl,kJ{hni)
l=k
< k + e,
(2.23)
48
phenomenon known as multipath fading. A communication channel characterized
by multipath fading is called a fading channel.
In describing the fading channel, we shall use complex signal representation.
We shall assume that some form of quadrature modulation is being used; e.g.,
QPSK or QAM. Each signal is represented by a complex time function whose
real part represents the inphase part, and whose imaginary part represents the
quadrature part. Using this approach the fading channel can be modeled as a
timevarying linear filter. The received signal is given by
/OO
c(A; t)u(t X)dX + v(t), (4.1)
OO
where u(t) is the complex representation of the transmitted signal, c(A; t) is the
timevarying impulse response of the fading channel, and v(t) is Gaussian noise.
For simplicity, we will restrict our discussion to the case in which, for any t,
c(A; t) is nonzero over only a small range of A. In digital signaling, this occurs when
the nonzero range is small compared to the symbol interval, a condition otherwise
known as flat fading. In this case, the expression for the received signal becomes
r(t) = c(t)u(t) + v(t), (4.2)
where c(t) is simply a complex time function,
c(t) = x{t) + iy(t). (4.3)
The function c(t) is called the complex fading envelope, and x(t) and y(t) are called
the fading parameters. The complex envelope may also be represented in phasor
form,
c(t) = a{t)eie^\ (4.4)
where
a(t) = \Jx2(t) + y7(tj,
(4.5)
REFERENCES
[1] K. AbedMeraim, W. Qiu, and Y. Hua. Blind system identification. Proceed
ings of the IEEE, 85(8):13101322, August 1997.
[2] A. H. Aghvami. Digital modulation techniques for mobile and personal
communication systems. Electronics and Communication Engineering Journal,
5(3):125132, June 1993.
[3] P. A. Bello and B. D. Nelin. The effect of frequency selective fading on the
binary error probabilities of incoherent and differentially coherent matched
filter receivers. IEEE Transactions on Communication Systems, 11:170186,
June 1963.
[4] P. A. Bello and B. D. Nelin. Corrections to The effect of frequency selective
fading on the binary error probabilities of incoherent and differentially
coherent matched filter receivers. IEEE Transactions on Communication
Technology, 12:230231, December 1964.
[5] F. G. Beutler. Errorfree recovery of signals from irregularly spaced samples.
SIAM Review, 8(3):328335, July 1966.
[6] C. J. Bouwkamp. On spheroidal wave functions of order zero. Journal of
Mathematics and Physics, 26:7992, 1947.
[7] A. Broman. An Introduction to Partial Differential Equations, from Fourier
Series to Boundaryvalue Problems. New York: Dover, 1989.
[8] J. L. Brown. Uniform linear prediction of bandlimited processes from past
samples. IEEE Transactions on Information Theory, 18(5):662664, September
1972.
[9] J. L. Brown. On the prediction of a bandlimited signal from past samples.
Proceedings of the IEEE, 74(11):15961598, November 1986.
[10] J. A. Cadzow. An extrapolation procedure for bandlimited signals. IEEE
Transactions on Acoustics, Speech, and Signal Processing, 27(1):412, February
1979.
[11] R. H. Clarke. A statistical theory of mobile radio reception. Bell System
Technical Journal, 47(6):9571000, JulyAugust 1968.
88
where c is the timebandwidth product and r, given by (4.35), is already deter
mined. Also, using (4.44), we may define
70
(4.63)
We see from this that if one of the three parameters, 2, T or r, is known, the other
two may be determined from (4.62) or (4.63). Thus, only one of the three may be
considered to be a free variable. We choose 2, and use the notation 7min(D, E) to
explicitly indicate the dependence of the minimum mean squared prediction error
on this parameter.
We see from (4.57) that correct channel tracking will be recovered if
(4.64)
for some E > 0. Note that this condition depends only on 2, which is proportional
to the maximum Doppler frequency fm.
4.2.7 Computational Issues
It is important to note that the relationship between D, T and r indicated
by (4.62) and (4.63) offers a computational advantage. To see this, we consider the
7T
optimization problem (3.5). We scale the objective function by for normalization
u L
U)
(cf. Eq. 3.35), then substitute uj' = to obtain
u L
(4.65)
T T
Now let c = and d = be fixed numbers. Then, letting t' = Qt, the time
2 !/1
constraint may be written
h =0 t' [2cd, c(2d + 1)].
(4.66)
76
Though we are optimistic about solving this problem, we also would like
to consider how the solution obtained in Chapter 3 may itself be appropriately
applied. In order to do this, let us look again at the objective function of (2.10),
which we rewrite for convenience:
J = ^ I* S=(w)l HMfdw. (4.77)
Z7T JCl
We notice that the power spectral density Sxx(u>) serves only as a weighting
function. Regardless of the form of Sxx(uj), we still wish to bring H{u>) as close
as possible to 1 on w G [Cl, )]. The function Sxx(uj) specifies only the penalty
for deviation from 1 at each uj. The graph of Figure 4.9 compares a function of
form (4.74), in dotted line, with a function that is constant within the band limit,
represented by a solid line. For a fair comparison, each function is normalized with
respects to the process power. We see that the difference in the weighting is not
very large until u approaches the band limit Q, so it is possible that a predictor
optimized for a flat spectral density will also perform well for a process whose
density is of the form (4.74). For such a predictor, one of our objectives for future
research is to find a useful bound on the true minimum mean squared error, given
our computation based on the flat spectral density model.
If we wish to obtain a spectral shape that more closely resembles that of
Figure 4.3, we may consider the approach of Figure 4.10. In this case we combine
a flat, bandlimited spectrum with two sinusoidal components at uÂ¡ = ii. The
parameters of the sinusoidal components may be estimated using a correlation on
the interval of adaptive esitmates, and the result may be subtracted out before
applying the bandlimited predictor. Many variations on this approach are possible.
For instance, the flat segment need not occupy the the entire Q bandwidth, and
the sinusoidal components need not be placed precisely at the band limits. Also,
35
problem may be formulated by substituting Sxx(uj) = 1 in (2.10) and adding the
energy constraint as follows:
1 rn
minimize J = / 1 H(u>)\2du> w.r.t. hit)
27r Jq
subject to 1. h(t) =0 t [t,t + T] (3.5)
/OO
h2{t)dt = E.
OO
Note that we have written constraint 2 as an equality. We shall see later that
the solution to this problem is the same as if it were written with the inequality.
Our approach shall be to transform the integrals in this problem into algebraic
expressions by expanding in terms of the basis functions {(f>n(t)}. The algebraic
minimization may then be carried out by using a Lagrange multiplier.
3.2.1 Problem Transformation
In the expression for J in (3.5), we wish to expand the terms inside the
absolute value signs. In order to do this, we define
F(u) =
i m
0 elsewhere.
(3.6)
By properties $7 and $8, the basis function Fourier transforms form a
complete orthonormal set in finiteenergy functions on [2, ], so we may write
where
(3.7)
(3.8)
63
The overall bit error rate is given approximately by [65, p. 223]
Pb
.0007,
476
where the effective average SNR was employed. This yields
76 ~ 357
7t, 714.
(4.29)
(4.30)
Decisiondirected adaptation may continue as long as the symbol error rate is less
than Ps = .01. The value of a(n) that attains this error rate is called the fade
threshold, which we may write as ctÂ¡. Representing the corresponding SNR by 76,/,
we have
. a}Eb oi2fEb 76)/
76i/ 2Ebo\ + ol ~ 2ol ~ 2
Since we are interested in the symbol error rate only at the instant when a(n)
crosses the threshold a/, we do not use averaging. If we assume that 76,/ is high
enough that the probability of a simultaneous error in both the real and imaginary
parts of (n) is negligibly small, we have [65, p. 223]
Ps 2 Pr (singlebit error)
= 20(^2^;)
.01. (4.32)
This yields
76,/ ~ 3.32 => 76,/ ~ 6.64. (4.33)
4.2.4 Statistics of the Fading Envelope
Decisiondirected adaptation becomes unreliable when a(n), the magnitude of
the fading envelope, falls below the fading threshold a/. When this happens, as we
have discussed, we switch from adaptation to prediction in Figure 4.5. We wish to
resume adaptation as soon as a(n) rises above Oij again. If our predicted value c(n)
59
solving the bandlimited linear prediction problem for a more general case than that
of Chapter 3, remain important research objectives for us.
Even bypassing the theoretical objections, our work on this problem is
incomplete. We nevertheless present a simple case to illustrate how the techniques
of Chapter 3 may be applied in the analysis of a practical problem.
4.2.2 Problem Statement
To use the results of Chapter 3, we must know something of the error
statistics of the adaptive estimate c(nTs). This requires an explicit choice of an
adaptive algorithm in Figure 4.5. Also, our choice of the parameter r depends upon
the duration of the fade; that is, the length of time during which the probability of
error in (n) is high. In order to calculate the probability of error, we must decide
upon a modulation scheme.
These steps are carried out in the following sections. In each case the
choices are made with a view toward simplifying the analysis, in the hope that the
resulting presentation will provide clearer insight into the fundamental issues. For
notational convenience in what follows, we drop the explicit dependence on Ts and
refer, e.g., to u(n) and c(n). We now offer the following problem statement:
Consider the adaptive/predictive channel estimation method of Figures 4.5
and 4.6. Assume that the input sequence u(n) is uncorrelated QPSK, whose
symbols are chosen with equal probability from the set
u(n) 6 + iy/Ebi y/Eb iy/Eb, ~y/Eb ~ iy/Eb, ~y/E~b + iy/Eb} ,
where Eb is the transmitted energy per bit. Thus, \u(n)\ = \/2Eb for all n. The
fading envelope c(n) is a zeromean complex Gaussian process whose real and
imaginary parts are independent and identically distributed, and the signaltonoise
ratio at the receiver input, time averaged to include the effects of fading, is such
that the overall bit error rate is Pb = .0007.
REFERENCES 88
BIOGRAPHICAL SKETCH 94
Vll
67
The prediction c(n) is to be based on adaptive estimates c(n), a process
which is approximately zeromean Gaussian, as was discussed in Section 4.2.3.
Thus, the minimum mean squared error predictor is linear. Since c(n) is a linear
transformation of a Gaussian process, c(n) is also Gaussian, and since the real and
imaginary parts of c(n) are independent and identically distributed with zero mean,
the same holds for c(n). Therefore, the phase of c(n) is uniformly distributed on
[ 7T, 7r].
The Gaussian white noise v(n) is uncorrelated with, and hence independent
of c(n), which is a function of past values of c(n). The prediction error e(n) =
c(n) c(n) is a sum of Gaussian random variables; thus e(n) is Gaussian. From
the theory of mean square estimation, c(n) is the mean of c(n) conditioned on
the adaptive estimates c(n), so e(n) is zero mean. Also, e(n) must be orthogonal
to c(n) [25, p. 200]. Since e(n) is zeromean Gaussian, orthogonality implies
independence.
At the instant the receiver emerges from the fade, we have c(n) = af, but
the phase of c(n) is still randomized. Thus, and are independent, zero
mean, complex Gaussian random variables with variances ^ and p respectively.
From (4.50) we therefore have, recalling u(n) = \/2Eb,
at the instant the receiver emerges from the fade. The SNR per bit is
. Eb __ a}Eb 2 ^f l
76 a] 2Ebaj + al ^ 2% 2 Eb
To attain Pc = .9 we must have [54, p. 272]
(4.51)
(4.52)
Pc
1 Q V276
1 2
= .9.
(4.53)
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
William W. Edmonson, Chairman
Assistant Professor of Electrical
and Computer Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
John M. M. Anderson
Associate Professor of Electrical
and Computer Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Haniph A. Latchman
Associate Professor of Electrical
and Computer Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Tan F. Wong
Assistant Professor of Electrical
and Computer Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Scott McCullough
Associate Professor of Mathematics
5
4. If the prediction is based on noisy data, the mean squared prediction er
ror has a minimum, greater than zero, even without an energy constraint.
In the case of a bandlimited process whose power spectral density is flat within the
band limits, our contributions include:
5. A method for constructing the optimal linear predictor in the energy
constrained case.
6. A procedure for computing the minimum mean squared prediction error
under the energy constraint.
7. Assuming the process to be corrupted by white noise, a method for
determining whether a specified mean squared prediction error can be
attained.
Regarding the issue of mobile radio fading, our contributions include:
8. An approach for applying the techniques of items 57 above to a problem
in adaptive channel estimation.
9. The finding that, for the problem in item 8, success of the proposed
approach depends primarily on the maximum Doppler frequency, which is
proportional to the speed of the mobile receiver.
10.A method of computing the greatest value of maximum Doppler frequen
cy for which the proposed approach will work.
Chapter 2 contains proofs of items 14. The techniques in items 57 are
developed in Chapter 3. The problem in adaptive channel estimation is analyzed in
Section 4.2. Finally, some topics for future research are listed in Section 5.2.
25
Note that this is equivalent to (2.5) with the added term w(t) h(t). Using a line
of reasoning similar to that leading to (2.9), and noting that x(t) and w(t) are
uncorrelated, the mean squared error is given by
1 r 1 r
J = / Sxx(w)\lH(u)\2du + / Sww(u)\H(u)\2duj.
Z7T J oo 7T Joo
= J + l\\h\\2 (2.18)
where h2 = Since J and J both depend on h(t) we write
J = J(/.)+4h2. (2.19)
We know from Theorem 2 that J(h) has no minimum. We shall show, however,
that J(h) does have a minimum and that this minimum is greater than zero for any
nontrivial process.
Theorem 4 Let C2t T be as defined at the beginning of Section 2.2, and let
k = inf J(h).
^e^T,r
Then
1. There is an h0 G C?tT such that k = J(h0) .
* tfJSi Sxx(lj) du> > 0, we also have k > 0.
Proof: To prove (1.), choose a sequence {/n}, with hn G Â£2>T V n, such that
J(hn) > k. Then
Jim J(/in) +a2/in2 = k.
This means that, for every e > 0, there is an m such that
J(hn) + al,\\hn\\2 < k + e V n > m.
(2.20)
65
Recall from Section 4.2.2 that we wish to base our prediction on reliable
adaptive estimates of c(n) since the receiver emerged from the last fade. We are
thus interested in the mean time above the fade threshold, which we shall call tj.
To compute tj, we note that [27]
Nr =
1
0 2 _r2
'Â¡re
7T00
tf + tj
is the average number of times per second that a(n) crosses olj in the positive
direction. We thus have
1 
(4.41)
Making use of (4.34) and (4.39) we obtain
bo r
(4.42)
tt ~
Mo l
&2 r V 47r/m2 r'
Note that, taking the ratio of (4.34) and (4.43) we have
(4.43)
^ = er2 1. (4.44)
tf
This may be interpreted as the ratio of the mean time in the fade to the mean time
out of the fade. From (4.35) we see that this ratio depends on the SNR at the fade
threshold 7bj and the timeaverage SNR 75. It does not depend on the maximum
Doppler frequency fm or the shape of the power spectral density Sxx (u>). These
facts will ease our computations later (see Section 4.2.7).
4.2.5 Recovery of Channel Tracking
We now turn our attention to the instant at which the receiver emerges from
the fade and ask under what conditions can we ensure recovery of correct channel
tracking with a probability PR > .9. Let us consider the last predicted sample c(n)
before switching back to adaptation in Figure 4.6. For that sample we define
e(n) = r(n) y{n) = [u(n)c(n) + v(n)] u(n)c(n).
(4.45)
21
Now let r(u;) = 5(u>)G(u>), then
0
/oo r roo
h(t) / r(v)ewtdj
oo Uoo
/oo
h{t)^{t)dt.
oo
Because 7(i) is a bounded function which integrates against each h(t) Â£ Â£2r n Ll
to give zero, we conclude that 7(t) = 0 for t Â£ [r, r + T]. Further, since 7(i) is
bandlimited, it is an entire function, which implies that if 7(i) = 0 on any interval
of positive measure, it must be zero everywhere. So this must be true of T(o;) also.
Thus,
r(w) = S(u)G(u) = 0 V u.
This shows that G(cv) = 0, 5(A) > 0}. Accordingly, G(u;)2 = 0 for
u> in this set. Therefore, 5(w)G(u;)2du; = 0, and the theorem is proved.
The expression for the mean squared error in (2.9) is obtained simply by
letting F(uj) = 1 and scaling appropriately. Thus, we have shown that the infimum
of the mean squared prediction error is zero for bandlimited processes. Note that
this result holds for all r and positive T, i.e., it holds regardless of the length of the
known interval or how far in the future we wish to make the prediction. The same
argument holds even if we allow h(t) to be nonzero on the interval [r, 00).
Next, we shall show that under certain restrictions imposed on 5(w), no linear
predictor attains this infimum.
Theorem 2 Let t, T and 2 be fixed real numbers with r > 0 and T,fl> 0, and let
S(u>) Â£ Â£}, with S(u)) > 0 V u>, and 5(uj) > 0 for u Â£ M, where M C [A, tt] is
a set of positive measure. Then there is no h(t) Â£ Â£2 r such that
5(w)l H(uj)\2duj = 0.
(2.12)
62
We define
.. ... . e(n) . v(n)
6(n) u(n) u(n) rs u(n) + rT. (4.21)
c(n) c{n)
Then, holding a(n) = c(n) constant,
5{(n)2}
~ Â£ {]^)]2 [~e(n)u(n) + u(n)] [e*(n)ti*(n) + v*(n)]
= 7{e(n)tt(n)2 e(n)u(n)v*(n) e*(n)u*(n)u(n) + \v(n)2} .(4.22)
lc(n)r 1 J
Now,
5 {e(n)u(n)2} = u(n)2Â£ {e(n)2} = 2Eba2e. (4.23)
Further, since e(n) is a function of past values of c(n), u(n) and v(n), it is uncorre
lated with u(n) and v(n). Thus,
Â£{e(n)u(n)u*(n)} = Â£{e*(n)u*(n)u(n)} = 0. (4.24)
Therefore,
Â£ {(n)2} ft' 2E^ + ^
a2(n)
This is the effective noise power at the input of the detector, taking into account
the estimation errors in c(n). We define the received SNR per bit as
a2(n)Eb
(4.25)
7b(n) =
o
(4.26)
Then, accounting again for the errors in c(n), the effective SNR per bit at the input
of the detector is
ir,\
(4.27)
. a2(n)Eb a(n)Et 71(11)
7i(n)=k 5*=
If we let a2 be the timeaverage value of a2(n), then the timeaverage SNRs are
oi2Eb ot2Eb 7Â¡,
76 ~ 2Eba2 + a2 ~ ~ ~2
(4.28)
11
spaces, our main source is Koosis [33]. These books assume a background in
analysis and measure theory. The very popular book by Rudin provides a starting
point in gaining this background [57]. An interesting, lesserknown alternative is
Sprecher [63]. The distinguishing feature of this book is the manner in which the
author constructs the real numbers from Cauchy sequences of rational numbers. It
is a difficult approach, but provides useful insight for those interested in engineering
applications, since this theory is the basis of most iterative optimization algorithms
(see, e.g., Luenberger [39, Ch. 7]). Those interested in getting some background
in functional analysis without extensive prerequisites are referred to the very
wellwritten text by Kreyszig [35].
Functional analysis is very closely related to the subject of integral equations.
We see from Section 3.1, for example, that our basis functions may be defined as
eigenfunctions of a linear integral operator. Some knowledge of integral equations is
therefore of interest. An excellent text is the one by Hochstadt [26]. It is written at
an elementary level and covers many fundamental concepts of functional analysis.
Also, the treatment is confined to Hilbert spaces, which simplifies the presentation
and is perfectly adequate for our purposes.
Our basic problem formulation involves the minimization of an integral
(see Section 2.1). At the early stages of our research, an attempt was made at
performing this minimization using variational calculus. Although this was not the
approach that we ultimately adopted, it provided useful insight. Our references
were texts by Fox and Weinstock [18, 73].
1.3.3 Fading in Mobile Radio
Our research of bandlimited prediction was motivated by the problem of
fading in mobile radio. This is because the fading envelope may be modeled as
a bandlimited random process, as was mentioned in Section 1.1. A very good
treatment of fading on digital channels is offered by Proakis [54, Ch. 14]. The
31
are coefficients {(*} such that f(t) may be written
OO
f{t) = (t).
n0
4/4. The set {Â¡/Vi(i)} is also orthogonal on
?.?] with
0
m = n
m ^ n.
4>5. The set {ipn(t)} is complete in finiteenergy functions on [j, This is
true even if such a function is not a segment of a bandlimited function.
4>6. Each function ipn(t) is Qbandlimited. This means that each Fourier
transform has the property 4/n(a>) = 0 for u; > 2.
4/7. The set is orthonormal on [), ]. This may be seen
by applying Parsevals identity to property 4/2 and then applying
property 4/6.
4r8. The set {4/n(u>)} is complete in finiteenergy functions on [O, O]. This
is true because a finiteenergy function with the property F(uj) = 0 for
M > 2 has an inverse Fourier transform f(t) which is iibandlimited. By
property $3 we may write
OO
f(t) =
n=0
and taking the Fourier transform we have
OO
F(u) = 5>ntfn(u;).
n=0
4/9. Let us define
M)
VfcjtrunclO <
0 elsewhere.
82
white Gaussian noise added to the received signal. In electronic communication
systems, such noise arises from thermal effects in the receiver front end. Thus,
we may consider trying to reduce our prediction error by improving the input
signaltonoise ratio, either by increasing the transmit power or by redesigning the
receiver using more expensive lownoise devices. Would such expenditure be worth
it?
We hesitate to take such a bruteforce approach because an overall signalto
noise ratio is usually targeted to achieve a desired bit error rate. Raising the SNR
just to improve the performance of decisiondirected adaptation may result in a
BER that is much lower than necessary. Decisiondirected techniques are widely
employed, and are successful in situations where the channel conditions vary slowly,
and where deep fades are not encountered very frequently. In such cases, decision
directed adaptation may be thought of as incurring little cost since, e.g., it does
not require side information or additional bandwidth. In the case of mobileradio
fading, where such assumptions are not justified, our predictability analysis shows
how the formerly negligible costs can become significant.
If the costs are too high, then the adaptive/predictive approach of Section 4.2
will hold little advantage. Regardless of this, however, the implicit prediction
problem remains, since it is difficult to see how a decisiondirected approach can
otherwise be made to work in a fading environment without increasing power,
resorting to diversity, or applying some other resourceconsuming solution. Thus,
we see that the techniques of linear prediction developed here may be applied not
only to develop new approaches to fading compensation, but also to analyze the
feasibility of previously proposed methods which involve such an implicit prediction
problem.
We shall continue to pursue these matters, as we are sure others will continue
to pursue the general problem of fading. Whether these efforts will eventually
23
Proof: Define J: Be > 3ft by
J{h) = J P S{u)\l H(u>)\2du>.
27T J2
Since Be is weakly compact, we may prove that J{h) attains its infimum by
showing that J{h) is weakly continuous. This can be shown by first observing that
if h G Be, then, by CauchySchwarz,
l#MI2 =
<
<
Now suppose {hn} is a sequence from Be which converges to ha weakly. Then
{Hn} converges to H0 pointwise, since
[t+T .
Hn(u) = J hn{t)e~wtdt.
Thus, S(a))l Hn(u))\2 converges pointwise to 5'(o;)l H0(uj)2. From (2.14), we
also have
S(u)1 Hn{u)\2 < S(u){ 1 + TE)2.
We can thus apply the Lebesgue Dominated Convergence Theorem to conclude
that
J{K) > J(h0).
Thus, J is weakly continuous and the theorem is proved.
Theorem 3 implies that there is a function Jm\n(E). This function is the
minimum mean squared error obtainable when h(t) is constrained to have an
energy no greater than E. Note that Jm\n{E) is a nonincreasing function of E.
49
n()
r(t)
CM v(t)
WGN
Figure 4.1: Model of a flatfading mobile radio channel.
and
6(t) = arctan
(4.6)
The flatfading channel model is shown in Figure 4.1. A flatfading channel exhibits
high amplitude correlation across the entire bandwidth of the timevarying channel
frequency response. A channel for which the frequency response is likely to vary
significantly in amplitude across its bandwidth is called frequency selective. Such
channels cause a time dispersion or smearing of the received symbols, leading to
the undesirable phenomenon known as intersymbol interference. Although we shall
not be focusing on frequencyselective fading, the theory of the flatfading channel
may be used in the analysis of the more complicated frequencyselective case [54].
A popular model for a flatfading mobile radio channel was developed by
R. H. Clarke [11]. He viewed the received signal as a superposition of a large
number of vertically polarized electromagnetic waves arriving at random angles
with random phase shifts. Clarke analyzed the spatial correlation of the fields and
determined that, if the receiver is moving with a constant velocity, these fields
may be modeled as a stationary Gaussian random process. He also determined
the autocorrelation function for this process. Using Clarkes approach, we may
view x{t) and y{t) as independent Gaussian random processes with a normalized
autocorrelation function given by
(4.7)
41
2
7T
(3.30)
where
/t+T
I = 7(2, T, r) = J sinc2f2s ds.
Combining (3.30) and (3.27) with (3.23) and (3.24) yields
(3.31)
i n2 r
E
/l 7T
2 Q2
Jmin(E^)) Jmin(0) 27'
fl 7TZ
Solving (3.32) for // and substituting into (3.33) we get
.2
(3.32)
(3.33)
Jmin(^) *7min(0) 2y/l^E.
7T
(3.34)
We see that, for small E, JmÂ¡n is approximately linear in the square root of E.
Further, we see that the slope of the graph may be calculated without knowledge of
any (0) or Xn. If we normalize with respects to JmÂ¡n(0), which is also the power
of the process x(t), we get (cf. Equation 3.27)
Jmin(E) = Jmin(E) ^ 1 2VlVE = 1 GVe,
(3.35)
where G = 2\fl is the absolute value of the initial slope of plotted against
y/E.
When either Q, or T is fixed, G may not be made arbitrarily large. Indeed,
looking at (3.31), we see that, for fixed 2, I will be bounded by the case where
t > 0 and T > oo,
roo
7max(f^) = / sine 2Clsds. (3.36)
Jo
7T
We note again that the Fourier transform of sine fit is F(u) of (3.6). Using
d L
Parsevals relation we have
/o 1 rco 772
sine 2Qsds = / F2(u>)duj
oo 2WoofF v '
14
In some applications it is desirable to perform the equalization without
employing a training sequence. Approaches for such blind equalizers fall into three
broad categories: Bussgang techniques, of which the Godard algorithm is a widely
used example [20], methods based on higherorder statistics [24], and methods that
exploit the cyclostationarity properties of the transmitted signal [1].
Once the equalizer has converged, there is a high probability that the output
of the detector will be correct. Thus, the detected symbols may be used in place
of the training sequence so that the adaptive algorithm may track slow variations
in the channel impulse response while data are being transmitted. This is referred
to as decision direction. If a receiver operating in this mode enters a deep fade,
however, the detected symbols will no longer be reliable and the adaptive algorithm
may lose track of the channel. The conditions under which a decisiondirected
equalizer will converge is therefore of interest. Mazo provides an analysis of a
simplified but still interesting case [46]. Macchi and Eweda establish some sufficient
conditions for convergence in the case of linear equalizers [43], and Kennedy et al.
discuss decision direction for decisionfeedback equalizers [30].
Overcoming the difficulties associated with decisiondirected equalization in
a fading channel was the primary motivation for pursuing the prediction problem.
But first we wish to address the similar, but in some ways simpler problem of
carrier tracking. Interesting treatments of decisionaided carrier tracking are
included in the books by Proakis [54, Sec. 624] and by Macchi [42, Ch. 12]. Our
form of the problem is described in Section 4.2.
1.3.5 General References
We now discuss some references which have been of use throughout the
project. The broad field that encompasses our prediction problem is estimation
theory. Kay offers a strong theoretical and wellordered treatment of the subjec
t [29]. A strength of the text is the manner in which it carries a few examples
92
[56] A. A. G. Requicha. The zeros of entire functions: Theory and engineering
applications. Proceedings of the IEEE, 68(3):308328, March 1980.
[57] W. Rudin. Principles of Mathematical Analysis, 3rd ed. New York: McGraw
Hill, 1976.
[58] D. Slepian. On bandwidth. Proceedings of the IEEE, 64(3):292300, March
1976.
[59] D. Slepian. Prolate spheroidal wave functions, Fourier analysis and
uncertaintyV: The discrete case. Bell System Technical Journal, 57(5):1371
1430, MayJune 1978.
[60] D. Slepian, H. J. Landau, and H. 0. Poliak. Prolate spheroidal wave functions,
Fourier analysis and uncertaintyI & II. Bell System Technical Journal,
40(l):4384, January 1961.
[61] H. SoltanianZadeh and A. E. Yagle. Fast algorithm for extrapolation of
discretetime periodic bandlimited signals. Signal Processing, 33(2):183196,
August 1993.
[62] W. Splettstosser. On the prediction of bandlimited signals from past samples.
Information Sciences, 28(2):115130, November 1982.
[63] D. Sprecher. Elements of Real Analysis. New York: Academic Press, 1970.
[64] P. Stoica and R. L. Moses. Introduction to Spectral Analysis. Upper Saddle
River, NJ: PrenticeHall, 1997.
[65] G. L. Stuber. Principles of Mobile Communication. Boston: Kluwer, 1996.
[66] D. J. Thomson. Spectrum estimation and harmonic analysis. Proceedings of
the IEEE, 70(9): 10551096, September 1982.
[67] C. Trabelsi. Linear adaptive prediction using LMS algorithm over Rician
fading channels. European Transactions on Telecommunications, 7(2):193199,
MarchApril 1996.
[68] P. P. Vaidyanathan. On predicting a bandlimited signal based on past sample
values. Proceedings of the IEEE, 75(8):11251127, August 1987.
[69] A. L. Van Burn. A Fortran computer program for calculating the linear
prolate functions. Technical Report NRL 7994. Washington, DC: Naval
Research Laboratory, 1976. Available through NTIS, AN: ADA0252106XAB.
[70] A. L. Van Burn, B. J. King, and R. V. Baier. Tables of angular spheroidal
wave functions, volume 1: Prolate, m = 0. Technical report. Washing
ton, DC: Naval Research Laboratory, 1975. Available through NTIS, AN:
ADA0166694XAB.
CHAPTER 2
PROBLEM FORMULATION AND ANALYSIS
In this chapter we study the continuoustime bandlimited prediction problem
in its general form1 Our main objective is to justify the introduction of an energy
constraint, since the unconstrained problem fails to have an optimal solution. We
argue that the energy constraint is an intuitively satisfying one for the case in
which the prediction is to be based on past values that have been corrupted by
estimation errors.
In Section 2.1, we obtain a frequencydomain expression for the mean
squared prediction error and then formulate the problem as the minimization of an
integral. In Section 2.2, we prove our main results concerning the predictablility
of bandlimited processes. In Section 2.3, we discuss the case of corrupted past
values. Finally in Section 2.4, we summarize our results and motivate the problem
addressed in the following chapter.
2.1 Problem Formulation
In this dissertation, Fourier transforms will be denoted by capital letters, as
in the following pair:
/OO
/ (t)e~wtdt,
OO
m
1 This chapter is based on The Predictability of ContinuousTime, Bandlimited
Processes, by R. J. Lyman, W. W. Edmonson, S. McCullough, and M. Rao.
2000 IEEE. Reprinted, with permission, from IEEE Transactions on Signal Pro
cessing, vol. 48, no. 2, pp. 311316, February 2000.
16
44
again may be restricted to h(t) = 0, t Â£ [r, r + T]. As in Section 2.3 we may write
1 r 1 r
J(h) = Sxx(u)\lH(u;)\2du + SwwH\H(u)\2dc.
Z7T J oo Z7T J oo
/y2 /OO
= W+2?/_00l^(")l2dw
= J(/i) + o2wE. (3.43)
where E = h2(t)dt.
Note that, for fixed E, J(h) can be minimized by choosing h(t) such that
J{h) = Jm\n{E). Thus, we may write
= Jm\n{E) + (J^E. (3.44)
Suppose that, for given values of Cl, T and r, we wish to attain a mean squared
prediction error Jm\n(E)
+ a2wE < a2p. (3.45)
Rearranging this we have
JminiE)
As an example, consider the values of , T and r used in Figure 3.1, and suppose
cr = .002 and a2 = 0.2. The resulting function 1(E) is represented by the dashdot
curve at the bottom of Figure 3.1. For = 10, we see that condition (3.46) is met
for E in the interval 2.6 < \fE < 9.9 approximately. For Q = 20, the curve 1(E)
is nearly tangent to Jmin(E) at y/E 7. For significantly greater than 20, o2
cannot be attained for the given values of T, r and o2w.
Though Jm\n(E) may be driven arbitrarily close to zero, there is no finite E
that minimizes Jm\n(E), as we expect from the analysis of Chapter 2. On the other
hand, looking again at (3.44), we recall that Jm\n(E) > 0 is strictly decreasing in
E > 0, and we note that o2wE is strictly increasing in E since cr^ > 0. Thus, there
43
Figure 3.1: The minimum mean squared prediction error that is possible using lin
ear prediction on a bandlimited process whose power spectral density is fiat within
the band limits.
7T
process x(t), which is like that of Section 3.2 except that Sxx(u) = for o; < Cl.
u L
We use a linear predictor of the form (2.1), which yields a mean squared prediction
error J. The calligraphic J is used because the process x(t) is unity power.
Now suppose that, instead of x{t) itself, we have an estimate,
(7) = Â£(7) + 10(7), 7 e [t t T, t r], (3.41)
where the estimation error w(t) is considered to be a real, zeromean, wide
sense stationary, whitenoise process, uncorrelated with x(t), and having an
autocorrelation function given by
Rww(t) = alS(t), al > 0. (3.42)
We again use a predictor of form (2.1), resulting in a mean squared prediction error
J. Clearly, both J and J depend on the predictor impulse response h(t), which
CHAPTER 1
PREDICTION OF BANDLIMITED PROCESSES
Random processes are often classified as being either predictable or regular
[52, p. 420]. The future values of a predictable process may be estimated with
a mean squared error of zero if the past values of the process are known. It can
be shown that the spectrum of such a process can consist only of lines. Regular
processes, which contain no spectral lines, are not predictable in this sense.
Processes which contain no spectral lines, but whose spectrum vanishes
outside of a certain band limit, stand between these two extremes. The future of
these bandlimited processes cannot be predicted with zero error, but under certain
conditions, if one has sufficient knowledge of past values, the prediction error can
be made arbitrarily small [52, p. 380].
The aim of this research is to extend these findings and to discover how
they may be applied in the analysis of specific signal processing problems. The
motivating application was decisiondirected channel estimation in a fading, mobile
radio environment. A simple version of the problem is described and analyzed in
Section 4.2. Though we wish to maintain our focus on the issue of fading, it is
hoped that our results will show a broader potential for bandlimited modeling than
has been previously realized.
In the next section we summarize the fundamental ideas that have directed
our research, and discuss the significance of our key findings. Section 1.2 contains
a list of our research contributions. Finally, Section 1.3 is a detailed survey of the
relevant literature.
1
37
Applying (3.9), (3.12) and (3.13) yields
1 rSl
1 ru
2n Jn
EM0)9.(u)EfinXn^(a)
duj.
(3.15)
n=0 n=0
When we expand the integrand of (3.15), the cross terms vanish under the integral
sign because of orthogonality property $7. We are left with .
1 rn
oo
= Â£W) Aa,)2,
(3.16)
n=0
where the last step is made possible, again, by property $7.
The energy constraint on h(t), which is constraint 2 of (3.5), may also be
rewritten using (3.10),
E
f h2(t)dt
JOO
roo [ 00
/ Pn&kjt runc(0
J~ Ln=0
oo rr+T
EpI n(t)dt,
n Jr
dt
where the cross terms again vanish under the integral sign because of the orthogo
nality property $4. Using this same property again we have
OO
Â£An pl = E. (3.17)
n=0
Note that any set {pn} satisfying (3.17) will also satisfy (3.11), thus guaranteeing
the convergence of (3.10).
Using (3.16) and (3.17), we may transform (3.5) from an integral minimiza
tion to an algebraic one:
OO
minimize J = ^[^(O) Ap]2 w.r.t. {pn}
n=0
oo
Subject tO Xrxpl = Ei
n=0
(3.18)
4
consider the process to be corrupted by white noise, and show how to determine if
a specified mean squared prediction error can be attained.
Finally, we apply these prediction techniques to a problem in decisiondirected
adaptive channel estimation. We show how an adaptive estimator may lose track
of the channel when the mobile receiver enters a deep fade. We consider the
use of a predicted value of the fading envelope to restart the adaptation after
the fade. Our analysis determines the conditions under which we may expect to
recover correct channel tracking using the predicted value. The primary factor
is the maximum Doppler frequency, which is proportional to the speed of the
receiver. Our conclusion is that the tools we have developed for linear prediction of
bandlimited processes can be useful in the analysis of many approaches to fading
compensation.
1.2 Research Contributions
We now offer a summary of our contributions. Repeating for emphasis, we
consider a continuoustime, bandlimited process, a sample function of which is
known on an interval of positive length. We wish to estimate some future value
using a predictor which is linear with respects to the known interval. Our findings
include the following:
1. The mean squared prediction error of a bandlimited process may be made
arbitrarily small.
2. No linear predictor can make the error zero.
3. If an energy constraint is placed on the impulse response of the predictor,
then the mean squared prediction error has a minimum which is greater
than zero.
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Murali Rao
Professor of Mathematics
This dissertation was submitted to the Graduate Faculty of the College of
Engineering and to the Graduate School and was accepted as partial fulfillment of
the requirements for the degree of Doctor of Philosophy.
May 2000
M. Jack Ohanian
Dean, College of Engineering
Winfred M. Phillips
Dean, Graduate School
78
Figure 4.11: Application of the flat spectral density to nonadaptive prediction.
the magnitude of each component may be fixed by the model, or left as a random
variable to be estimated.
Finally, we note that optimal estimators are often realized using adaptive
filtering techniques. Suppose, however, that we prefer a nonadaptive predictor.
*
Then we argue that it makes sense to choose a flat spectral density in carrying out
the optimization procedure. The reason may be seen in Figure 4.11. We recall that
the fadingenvelope spectral density (4.9) depends upon the maximum Doppler
frequency fm, which is proportional to the speed of the receiver. As the receiver
speeds up and slows down, the bandwidth of the spectrum will correspondingly
increase and decrease. The characteristic U shape of the spectrum may therefore be
wider or narrower, as is shown by the dotted curves in the figure. Thus, a predictor
optimized to one of these spectra may perform poorly when the receiver changes
speed. A flat spectral density of sufficient bandwidth represents a reasonable
compromise, since it distributes an even weighting in the mean squared error
penalty across the entire band that is likely to be occupied by the fading envelope.
74
Figure 4.8: Predictability analysis for recovery of correct channel tracking.
perform the prediction based on a finite interval of estimated values, an interval
which was chosen somewhat arbitrarily for this example. We would like to extend
our approach to the case of prediction based on the infinite past. This would give
us a clearer indication of the viability of this approach.
Speaking more broadly, an analysis such as the one we have presented
can provide a starting point for investigating the viability of decisiondirected
fading compensation in general, since to date much of the research literature on
this subject has assumed that the detected symbols are error free (see, e.g., Li
et al. [37]).
4.3 Addressing the Model Mismatch
As was mentioned in Section 4.2.1, the spectral shape of Figure 4.3, which
is the most commonly used in modeling the mobile radio fading envelope, does
not match the flat spectrum assumed in Chapter 3. In this section, we consider
alternatives for addressing the model mismatch.
89
[12] J. B. Conway. A Course in Functional Analysis, 2nd ed. New York: Springer
Verlag, 1990.
[13] W. B. Davenport and W. L. Root. An Introduction to the Theory of Random
Signals and Noise. New York: McGrawHill, 1958.
[14] J. L. Doob. Stochastic Processes. New York: Wiley, 1953.
[15] T. T. Fjallbrandt. Interpolation and extrapolation in nonuniform sampling
sequences with average sampling rates below the Nyquist rate. Electronics
Letters, ll(12):264266, June 12, 1975.
[16] C. Flammer. Spheroidal Wave Functions. Stanford, CA: Stanford University
Press, 1957.
[17] G. D. Forney. Maximumlikelihood sequence estimation of digital sequences in
the presence of intersymbol interference. IEEE Transactions on Information
Theory, 18(3):363378, May 1972.
[18] C. Fox. An Introduction to the Calculus of Variations. London: Oxford
University Press, 1950.
[19] B. R. Frieden. Evaluation, design and extrapolation methods for optical
signals, based on use of the prolate functions. In Progress in Optics 9, edited
by E. Wolf, pages 311407. Amsterdam: NorthHolland, 1971.
[20] D. N. Godard. Selfrecovering equalization and carrier tracking in two
dimensional data communication systems. IEEE Transactions on Communica
tions, 28(11):18671875, November 1980.
[21] A. J. Goldsmith and S. G. Chua. Variablerate variablepower MQAM for
fading channels. IEEE Transactions on Communications, 45(10):12181230,
October 1997.
[22] G. H. Golub and C. F. Van Loan. Matrix Computations, 3rd ed. Baltimore,
MD: Johns Hopkins University Press, 1996.
[23] S. Hanish, R. V. Baier, A. L. Van Burn, and B. J. King. Tables of radial
spheroidal wave functions, volume 1: Prolate, m = 0. Technical Report NRL
7088. Washington, DC: Naval Research Laboratory, 1970. Available through
NTIS, AN: AD723836XAB.
[24] S. Haykin, editor. Blind Deconvolution. Englewood Cliffs, NJ: PrenticeHall,
1994.
[25] S. Haykin. Adaptive Filter Theory, 3rd ed. Upper Saddle River, NJ: Prentice
Hall, 1996.
[26]H. Hochstadt. Integral Equations. New York: Wiley, 1973.
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
LINEAR PREDICTION OF CONTINUOUSTIME,
BANDLIMITED PROCESSES, WITH APPLICATIONS
TO FADING IN MOBILE RADIO
By
Raphael J. Lyman
May 2000
Chairman: William W. Edmonson
Major Department: Electrical and Computer Engineering
In digital mobile radio, many techniques aimed at compensating for the
distorting effects of multipath fading could benefit from a prediction of the fad
ing envelope, a complex time function often modeled as a bandlimited random
process. We consider a continuoustime linear predictor applied to a bandlimited
process. We show that if the past values of the process are known over an inter
val of arbitrary positive length, then the mean squared prediction error may be
made arbitrarily small, regardless of how far in the future we wish to make the
prediction. We also show that this is no longer true when an energy constraint is
applied to the predictor, and we discuss what this means for the case in which the
prediction is based on past values that are corrupted by estimation errors.
We then go on to solve explicitly for the optimal, energyconstrained pre
dictor when the process spectral density is flat within the band limits. As basis
IX
22
Proof: Suppose such an h(t) exists. Then there is some H(u>) such that (2.12)
holds. Let s = [ Z, and let G(z) = H(s). Since H(u) G ?{2(UHP), we have
i z
G{uj) G ?2(IJD) [33, p. 158]. Due to (2.12), H{u) and thus G(cv) is equal to 1 on a
set of positive measure. But since G(cu) G %2(UD), this implies that G(u>) and thus
H{u) is identically 1. But this contradicts H(u>) G Â£2. Therefore no such H(lj) or
h(t) exists.
Since no allowable h(t) attains the infimum of J, by definition (2.10) has no
minimum. This theorem applies even when r = 0, in which case the value x(t) is
included on the known interval and may be obtained without error by sampling at
that instant. But in this case, h(t) must be interpreted as a delta function, which is
not in Â£2. For similar reasons, this theorem does not apply when a random process
consists of a finite sum of complex exponentials. It is known that such a process
may be predicted with a mean squared error of zero using a finite sum of past
values [52, p. 497]. But in this case, S(w) must be interpreted as a finite sum of
delta functions, which are not in C1.
Our next question is what happens to h(t) as J approaches zero. The next
theorem shows that, when h(t) is constrained in energy, the infimum mean squared
error of the resulting space of functions is attained.
Theorem 3 Let r, T, Cl and E be fixed real numbers with T,Cl,E > 0, and let
S(uj) G C1, with S(u) >0 V u. Now, consider the space Be of functions
h = h(t) G C2tT such that
rr+T
J h2(t)dt < E. (2.13)
Let
* = ,inf / S(u))\l H(uj)\2dw.
heBE Z7T JQ
Then there is an h G Be such that
i [ S(uj)\1 H(uj)\2duj = k.
27T Jn
PAGE 1
/,1($5 35(',&7,21 2) &217,182867,0( %$1'/,0,7(' 352&(66(6 :,7+ $33/,&$7,216 72 )$',1* ,1 02%,/( 5$',2 %\ 5$3+$(/ /<0$1 $ ',66(57$7,21 35(6(17(' 72 7+( *5$'8$7( 6&+22/ 2) 7+( 81,9(56,7< 2) )/25,'$ ,1 3$57,$/ )8/),//0(17 2) 7+( 5(48,5(0(176 )25 7+( '(*5(( 2) '2&725 2) 3+,/2623+< 81,9(56,7< 2) )/25,'$
PAGE 2
&RS\ULJKW E\ 5DSKDHO /\PDQ
PAGE 3
7KLV ZRUN LV GHGLFDWHG WR 0LVV /HQD 0DUJDUHW /\PDQ KDG D OLWWOH QXW WUHH QRWKLQJ ZRXOG LW EHDU %XW D VLOYHU QXWPHJ DQG D JROGHQ SHDU 7KH .LQJ RI 6SDLQfV GDXJKWHU FDPH WR YLVLW PH $QG DOO IRU WKH VDNH RI P\ OLWWOH QXW WUHH VNLSSHG RYHU ZDWHU GDQFHG RYHU VHD $QG DOO WKH ELUGV LQ WKH DLU FRXOGQfW FDWFK PH 0RWKHU *RRVH 5K\PH
PAGE 4
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n SOH $ IHZ RI WKHP VLPSO\ PXVW EH PHQWLRQHG 'U 6FRWW 0LOOHU DQG 'U 7DQ :RQJ KDYH EHHQ LQYDOXDEOH VRXUFHV RI LQIRUPDWLRQ UHJDUGLQJ FRPPXQLFDWLRQ WKHRU\ LQ JHQHUDO DQG ZLUHOHVV WHFKQRORJ\ LQ SDUWLFXODU %RWK KDYH RIIHUHG PDQ\ KHOSIXO FRPPHQWV DQG VXJJHVWLRQV &RQYHUVDWLRQV WKDW KDYH KDG ZLWK 'U %HUW 1HOLQ KDYH \LHOGHG LQWHUHVWLQJ LQVLJKWV LQWR WKH SKHQRPHQRQ RI IUHTXHQF\VHOHFWLYH IDGLQJ DOn VR WKDQN 'U 0 9 5DPDQD ZKR H[SUHVVHG HQWKXVLDVP IRU WKH SUREOHP DW DQ HDUO\ VWDJH DQG SDUWLFLSDWHG ZLWK PH LQ D PHPRUDEOH EUDLQVWRUPLQJ VHVVLRQ ,Q D PRUH JHQHUDO ZD\ 'U /HRQ : &RXFK ,, KDV DLGHG PH JUHDWO\ E\ PDNLQJ WKH EHQHILW RI KLV H[SHULHQFH DYDLODEOH ZKHQHYHU KDYH QHHGHG LW DQG DP PXFK REOLJHG $ ZRUG LV LQ RUGHU KHUH UHJDUGLQJ P\ UHODWLRQVKLS ZLWK P\ DGYLVRU (DUO\ RQ 'U (GPRQVRQ WRRN D FRQVLGHUDEOH ULVN E\ SURYLGLQJ PH ZLWK QRW RQO\ WKH ,9
PAGE 5
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n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
PAGE 6
7$%/( 2) &217(176 eDJH $&.12:/('*0(176 LY /,67 2) ),*85(6 YLLL $%675$&7 L[ &+$37(56 35(',&7,21 2) %$1'/,0,7(' 352&(66(6 3URMHFW 7KHVLV 5HVHDUFK &RQWULEXWLRQV /LWHUDWXUH 6XUYH\ 352%/(0 )2508/$7,21 $1' $1$/<6,6 3UREOHP )RUPXODWLRQ $QDO\VLV 'HDOLQJ ZLWK (VWLPDWLRQ (UURUV &RQFOXVLRQV 352&(66(6 :,7+ )/$7 63(&75$/ '(16,7,(6 %DVLV )XQFWLRQV 6ROXWLRQ RI WKH /LQHDU 3UHGLFWRU 0LQLPXP 0HDQ 6TXDUHG 3UHGLFWLRQ (UURU $ %DQGOLPLWHG 3URFHVV LQ :KLWH 1RLVH &RQFOXVLRQV $33/,&$7,216 72 )$',1* ,1 02%,/( 5$',2 0XOWLSDWK )DGLQJ LQ 0RELOH 5DGLR $GDSWLYH &KDQQHO (VWLPDWLRQ $GGUHVVLQJ WKH 0RGHO 0LVPDWFK &21&/86,216 $1' )8785( :25. &RQFOXVLRQV )XWXUH :RUN $33(1',; (9$/8$7,21 2) %$6,6 )81&7,216 YL
PAGE 7
5()(5(1&(6 %,2*5$3+,&$/ 6.(7&+ 9OO
PAGE 8
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
PAGE 9
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n LQJ HQYHORSH D FRPSOH[ WLPH IXQFWLRQ RIWHQ PRGHOHG DV D EDQGOLPLWHG UDQGRP SURFHVV :H FRQVLGHU D FRQWLQXRXVWLPH OLQHDU SUHGLFWRU DSSOLHG WR D EDQGOLPLWHG SURFHVV :H VKRZ WKDW LI WKH SDVW YDOXHV RI WKH SURFHVV DUH NQRZQ RYHU DQ LQWHUn YDO RI DUELWUDU\ SRVLWLYH OHQJWK WKHQ WKH PHDQ VTXDUHG SUHGLFWLRQ HUURU PD\ EH PDGH DUELWUDULO\ VPDOO UHJDUGOHVV RI KRZ IDU LQ WKH IXWXUH ZH ZLVK WR PDNH WKH SUHGLFWLRQ :H DOVR VKRZ WKDW WKLV LV QR ORQJHU WUXH ZKHQ DQ HQHUJ\ FRQVWUDLQW LV DSSOLHG WR WKH SUHGLFWRU DQG ZH GLVFXVV ZKDW WKLV PHDQV IRU WKH FDVH LQ ZKLFK WKH SUHGLFWLRQ LV EDVHG RQ SDVW YDOXHV WKDW DUH FRUUXSWHG E\ HVWLPDWLRQ HUURUV :H WKHQ JR RQ WR VROYH H[SOLFLWO\ IRU WKH RSWLPDO HQHUJ\FRQVWUDLQHG SUHn GLFWRU ZKHQ WKH SURFHVV VSHFWUDO GHQVLW\ LV IODW ZLWKLQ WKH EDQG OLPLWV $V EDVLV ,;
PAGE 10
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
PAGE 11
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
PAGE 12
3URMHFW 7KHVLV ,Q FRPPXQLFDWLRQV DQG VLJQDO SURFHVVLQJ RQH VRPHWLPHV HQFRXQWHUV SUREOHPV ZKLFK DUH FRQYHQLHQWO\ IRUPXODWHG LQ WHUPV RI EDQGOLPLWHG UDQGRP SURFHVVHV )RU H[DPSOH LQ ZLUHOHVV FRPPXQLFDWLRQV PXOWLSDWK IDGLQJ PD\ EH YLHZHG DV D PRGXODWLRQ RI WKH WUDQVPLWWHG VLJQDO E\ D FRPSOH[ WLPH IXQFWLRQ FDOOHG WKH IDGLQJ HQYHORSH > &K @ )RU PRELOH UDGLR WKH IDGLQJ HQYHORSH PD\ EH PRGHOHG DV D EDQGOLPLWHG UDQGRP SURFHVV ZLWK WKH EDQG OLPLWV GHSHQGLQJ XSRQ WKH VSHHG RI WKH PRELOH UHFHLYHU >@ 6RPH WHFKQLTXHV GHVLJQHG WR FRPSHQVDWH IRU WKH HIIHFWV RI IDGLQJ UHTXLUH DQ HVWLPDWH RI WKH FXUUHQW YDOXH RI WKH IDGLQJ HQYHORSH $Q H[DPSOH LV DGDSWLYH WUDQVPLVVLRQ ZKLFK XVHV WKLV HVWLPDWH WR PDNH DSSURSULDWH DGMXVWPHQWV WR WKH SRZHU RU UDWH RI WUDQVPLVVLRQ 8QIRUWXQDWHO\ DGDSWLYH WUDQVPLVVLRQ LV YHU\ VHQVLWLYH WR GHOD\V ZKLFK LQHYLWDEO\ RFFXU LQ IHHGLQJ EDFN HVWLPDWHV RI WKH IDGLQJ HQYHORSH IURP UHFHLYHU WR WUDQVPLWWHU >@ ,Q VXFK D FDVH ZH ZRXOG QDWXUDOO\ FRQVLGHU ZKHWKHU WKH FXUUHQW YDOXH FRXOG EH HVWLPDWHG EDVHG XSRQ RXU NQRZOHGJH RI WKH SDVW 6R DGDSWLYH WUDQVPLVVLRQ LV RQH H[DPSOH RI DQ DSSOLFDWLRQ WKDW FRXOG EHQHILW IURP WKH SUHGLFWLRQ RI D EDQGOLPLWHG SURFHVV 3UHGLFWLRQ SUREOHPV DUH RIWHQ DGGUHVVHG XVLQJ WKH WHFKQLTXHV RI :LHQHU ILOWHULQJ EXW WKH VROXWLRQ RI WKH :LHQHU SUHGLFWLRQ SUREOHP UHTXLUHV WKDW WKH SRZHU VSHFWUDO GHQVLW\ EH IDFWRUDEOH LQWR PLQLPXPSKDVH DQG PD[LPXPSKDVH ILQLWHHQHUJ\ SDUWV 8QIRUWXQDWHO\ WKLV IDFWRUL]DWLRQ LV LPSRVVLEOH ZKHQ WKH VSHFWUDO GHQVLW\ LV ]HUR RYHU D VHW RI SRVLWLYH PHDVXUH VR WKH :LHQHU DSSURDFK FDQQRW EH XVHG WR VROYH WKH SUHGLFWLRQ SUREOHP IRU EDQGOLPLWHG SURFHVVHV VHH HJ 3DSRXOLV > SS @f 3UHYLRXV DSSURDFKHV WR WKH SUREOHP KDYH IRFXVHG RQ WKH SUHGLFWLRQ RI IXWXUH YDOXHV E\ WDNLQJ OLQHDU FRPELQDWLRQV RI SDVW VDPSOH YDOXHV > @ 7KH TXHVWLRQV WKDW KDYH EHHQ GHDOW ZLWK LQFOXGH WKH FRQGLWLRQV XQGHU
PAGE 13
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
PAGE 14
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
PAGE 15
,I WKH SUHGLFWLRQ LV EDVHG RQ QRLV\ GDWD WKH PHDQ VTXDUHG SUHGLFWLRQ HUn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n F\ IRU ZKLFK WKH SURSRVHG DSSURDFK ZLOO ZRUN &KDSWHU FRQWDLQV SURRIV RI LWHPV 7KH WHFKQLTXHV LQ LWHPV DUH GHYHORSHG LQ &KDSWHU 7KH SUREOHP LQ DGDSWLYH FKDQQHO HVWLPDWLRQ LV DQDO\]HG LQ 6HFWLRQ )LQDOO\ VRPH WRSLFV IRU IXWXUH UHVHDUFK DUH OLVWHG LQ 6HFWLRQ
PAGE 16
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
PAGE 17
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fV UHIHUHQFHV %HXWOHUfV SURRI KRZHYHU GRHV QRW JLYH D SURFHGXUH IRU FDOFXODWLQJ WKH SUHGLFWRU FRHIILFLHQWV ,Q D FRUUHVSRQGHQFH %URZQ UHFRJQL]HG WKH ZRUNV RI :DLQVWHLQ DQG =XEDNRY DQG RI %HXWOHU DQG WKHQ RIIHUHG D SURFHGXUH IRU FDOFXODWLQJ WKH FRn HIILFLHQWV LQ WKH FDVH RI XQLIRUP VDPSOLQJ DW RQO\ WZLFH WKH 1\TXLVW UDWH >@ )MDOOEUDQGW FRQWULEXWHG D OHWWHU LQ GHDOLQJ ZLWK FHUWDLQ LQWHUSRODWLRQ DQG H[WUDSRODWLRQ SUREOHPV >@ $V DQ H[WUHPH FDVH KH ILQGV FRHIILFLHQWV IRU H[WUDSRODn WLRQ ZLWK XQLIRUP VDPSOLQJ DW DQ\ UDWH KLJKHU WKDQ WKH 1\TXLVW UDWH +H RIIHUV WKLV DV D FRQVWUXFWLYH SURRI RI %HXWOHUf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
PAGE 18
7KH QH[W HQWU\ LV D FRUUHVSRQGHQFH E\ .QDE ZKR DJDLQ PDNHV QR UHIHUHQFH WR DQ\ RI WKH DERYH ZRUN >@ +LV LQWHUHVW LV LQ WKH LQWHUSRODWLRQ RI D ILQLWHSRZHU EDQGOLPLWHG VLJQDO ZKHQ WKH VDPSOH WLPHV DUH QRW V\PPHWULF ZLWK UHVSHFWV WR WKH SRLQW WR EH HVWLPDWHG .QDEf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fV ILQGLQJV KDG EHHQ SURYHG E\ RWKHUV SUHYLn RXVO\ >@ +H WKHQ RIIHUHG KLV RZQ EULHI SURRI EDVHG RQ D WKHRUHWLFDO IUDPHZRUN GHVFULEHG LQ DQ DUWLFOH E\ 5HTXLFKD >@ ,Q WKH VDPH LVVXH %URZQ UHLWHUDWHG KLV HDUOLHU SURRI >@ DQG UHPLQGHG UHDGHUV RI %HXWOHUfV ZRUN >@ )LQDOO\ 9DLG\DQDWKDQ GUDZV DWWHQWLRQ WR VRPH SDVW OLWHUDWXUH DQG LQ DQ H[SOLFLWO\ VWRFKDVWLF IUDPHZRUN VKRZV KRZ WR FDOFXODWH SUHGLFWRU FRHIILFLHQWV WKDW ZRUN IRU DQ\ VDPSOLQJ UDWH DERYH WKH 1\TXLVW UDWH >@ 7KLV HQWU\ E\ 9DLG\DQDWKDQ SXEOLVKHG LQ LV WKH ODVW WKDW ZH KDYH IRXQG WR GHDO GLUHFWO\ ZLWK WKLV SUREOHP
PAGE 19
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f DUELWUDULO\ IDU DKHDG +RZHYHU WR VXEVWDQWLDOO\ LQFUHDVH WKH WLPH LQWHUYDO IRU ZKLFK WKH SUHGLFWLRQ LV SRVVLEOH ZH PXVW HQRUPRXVO\ LQFUHDVH WKH VLJQDOWRQRLVH UDWLR S f $OVR VHH 6OHSLDQ >@ IRU D PRUH JHQHUDO GLVFXVVLRQ RI EDQGOLPLWHG PRGHOLQJ ,Q RXU UHVHDUFK ZH DGGUHVV WKH LVVXH RI QRLVH VHQVLWLYLW\ E\ UHJXODUL]LQJ WKH SUREOHP ZLWK DQ HQHUJ\ FRQVWUDLQW VHH 6HFWLRQ f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n LQJ WKH H[WUDSRODWLRQ RQ D FRQWLQXRXVWLPH VLJQDO >@ +H VKRZHG WKHRUHWLFDOO\ WKDW WKH DOJRULWKP FRQYHUJHV EXW QXPHULFDO GLIILFXOWLHV OHG &DG]RZ WR GHYHORS D QHZ DOJRULWKP ZKLFK LV VWLOO ZLGHO\ XVHG >@ 6OHSLDQ HW DO DOVR KDG DGGUHVVHG WKH SUREOHP LQ DV DQ DSSOLFDWLRQ RI WKH SURODWH VSKHURLGDO ZDYH IXQFWLRQV > S @
PAGE 20
%DQGOLPLWHG H[WUDSRODWLRQ FRQWLQXHV WR EH DQ DFWLYH UHVHDUFK ILHOG DQG KDV DGPLWWHG RI D ZLGH YDULHW\ RI SUREOHP IRUPXODWLRQV ,Q SDUWLFXODU WKH FDVH RI SHULRGLF EDQGOLPLWHG IXQFWLRQV ZDV RI VRPH LQWHUHVW WR XV VHH HJ 6ROWDQLDQ =DGHK >@f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f ,Q &KDSWHU ZH XVH WKHVH IXQFWLRQV WR REWDLQ DQ H[SDQVLRQ WKDW WUDQVIRUPV D IXQFWLRQDO RSWLPL]DWLRQ LQWR DQ DOJHEUDLF RQH ZKLFK PD\ WKHQ EH VROYHG XVLQJ D /DJUDQJH PXOWLSOLHU 7KH DSSURDFK LV KLQWHG DW LQ D UHODWHG SUREOHP GHVFULEHG E\ 6OHSLDQ HW DO > S @ 7KH GHWDLOV PD\ EH IRXQG LQ D QXPHULFDO OLQHDU DOJHEUD WH[W E\ *ROXE DQG 9DQ /RDQ > S @ 7KH PRUH JHQHUDO WUHDWPHQW RI &KDSWHU UHOLHV KHDYLO\ RQ IXQFWLRQDO DQDO\VLV $ JRRG WH[W LV SURYLGHG E\ &RQZD\ >@ )RU VSHFLILF UHVXOWV FRQFHUQLQJ +S
PAGE 21
VSDFHV RXU PDLQ VRXUFH LV .RRVLV >@ 7KHVH ERRNV DVVXPH D EDFNJURXQG LQ DQDO\VLV DQG PHDVXUH WKHRU\ 7KH YHU\ SRSXODU ERRN E\ 5XGLQ SURYLGHV D VWDUWLQJ SRLQW LQ JDLQLQJ WKLV EDFNJURXQG >@ $Q LQWHUHVWLQJ OHVVHUNQRZQ DOWHUQDWLYH LV 6SUHFKHU >@ 7KH GLVWLQJXLVKLQJ IHDWXUH RI WKLV ERRN LV WKH PDQQHU LQ ZKLFK WKH DXWKRU FRQVWUXFWV WKH UHDO QXPEHUV IURP &DXFK\ VHTXHQFHV RI UDWLRQDO QXPEHUV ,W LV D GLIILFXOW DSSURDFK EXW SURYLGHV XVHIXO LQVLJKW IRU WKRVH LQWHUHVWHG LQ HQJLQHHULQJ DSSOLFDWLRQV VLQFH WKLV WKHRU\ LV WKH EDVLV RI PRVW LWHUDWLYH RSWLPL]DWLRQ DOJRULWKPV VHH HJ /XHQEHUJHU > &K @f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f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
PAGE 22
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fÂ§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f
PAGE 23
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fV GHWHFWRU LQSXW DQG WKH RULJLQDO NQRZQ VHTXHQFH DQG WKH HTXDOL]HU LV DGMXVWHG DFFRUGLQJO\ )RU OLQHDU DQG GHFLVLRQ IHHGEDFN HTXDOL]HUV WKH DGDSWLYH DOJRULWKPV PD\ DGMXVW WKH HTXDOL]HU WDS JDLQV GLUHFWO\ ,Q WKH FDVH RI PD[LPXPOLNHOLKRRG VHTXHQFH HVWLPDWLRQ WKH FKDQQHO LPn SXOVH UHVSRQVH LV QHHGHG 2IWHQ WKH FKDQQHO LV PRGHOHG DV D OLQHDU ILQLWHLPSXOVH UHVSRQVH ILOWHU DQG WKH DGDSWLYH DOJRULWKP LV XVHG WR ILQG WKH ILOWHU FRHIILFLHQWV LQ D V\VWHPLGHQWLILFDWLRQ PRGH > 6HF @
PAGE 24
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
PAGE 25
WKURXJK WKH HQWLUH GHYHORSPHQW ZKLFK KHOSV WR ILUP RQHV JUDVS RQ WKH UHODWLRQVKLSV DPRQJ WKH YDULRXV HVWLPDWLRQ WHFKQLTXHV ,W DVVXPHV D FRQVLGHUDEOH GHJUHH RI FRPn PLWPHQW RQ WKH SDUW RI WKH VWXGHQW KRZHYHU VLQFH WKH WKHRUHWLFDO DSSURDFK PHDQV WKDW VRPH RI WKH PRUH SRSXODU HVWLPDWLRQ PHWKRGV DUH GHDOW ZLWK LQ ODWHU FKDSWHUV $V ZDV PHQWLRQHG LQ 6HFWLRQ :LHQHU ILOWHU WKHRU\ SOD\V D FHQWUDO UROH LQ PDQ\ GLVFXVVLRQV DERXW SUHGLFWLRQ 7KLV VXEMHFW LV FRYHUHG LQ WH[WV RQ DGDSWLYH ILOWHU WKHRU\ DV ZHOO DV VSHFWUDO HVWLPDWLRQ > @ ,Q SDUWLFXODU .D\f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f 3DSRXOLV LV RXU VRXUFH IRU DOPRVW DOO PDWWHUV FRQFHUQLQJ VWRFKDVWLF SURFHVVn HV >@ 7KH IDPRXV ERRN E\ 'RRE LV PRUH FRPSOHWH EXW DJDLQ UHTXLUHV VXEVWDQWLDO EDFNJURXQG LQ KLJKHU PDWKHPDWLFV >@ $OVR 3DSRXOLV DQG 6OHSLDQ DUH WZR DXWKRUV WKDW KDYH VKRZQ VXVWDLQHG LQWHUHVW LQ EDQGOLPLWHG PRGHOLQJ > @ )RU LVVXHV UHODWHG WR GLJLWDO FRPPXQLFDWLRQV RXU SULPDU\ VRXUFH LV WKH ZHOO NQRZQ ERRN E\ 3URDNLV >@ ,Q LWV FRYHUDJH GHSWK DQG FODULW\ LW LV KDUG WR EHDW 7KH ERRN E\ 6WXEHU >@ LV DOVR H[FHOOHQW DQG LV JHDUHG PRUH WRZDUG PRELOH UDGLR
PAGE 26
&+$37(5 352%/(0 )2508/$7,21 $1' $1$/<6,6 ,Q WKLV FKDSWHU ZH VWXG\ WKH FRQWLQXRXVWLPH EDQGOLPLWHG SUHGLFWLRQ SUREOHP LQ LWV JHQHUDO IRUP 2XU PDLQ REMHFWLYH LV WR MXVWLI\ WKH LQWURGXFWLRQ RI DQ HQHUJ\ FRQVWUDLQW VLQFH WKH XQFRQVWUDLQHG SUREOHP IDLOV WR KDYH DQ RSWLPDO VROXWLRQ :H DUJXH WKDW WKH HQHUJ\ FRQVWUDLQW LV DQ LQWXLWLYHO\ VDWLVI\LQJ RQH IRU WKH FDVH LQ ZKLFK WKH SUHGLFWLRQ LV WR EH EDVHG RQ SDVW YDOXHV WKDW KDYH EHHQ FRUUXSWHG E\ HVWLPDWLRQ HUURUV ,Q 6HFWLRQ ZH REWDLQ D IUHTXHQF\GRPDLQ H[SUHVVLRQ IRU WKH PHDQ VTXDUHG SUHGLFWLRQ HUURU DQG WKHQ IRUPXODWH WKH SUREOHP DV WKH PLQLPL]DWLRQ RI DQ LQWHJUDO ,Q 6HFWLRQ ZH SURYH RXU PDLQ UHVXOWV FRQFHUQLQJ WKH SUHGLFWDEOLOLW\ RI EDQGOLPLWHG SURFHVVHV ,Q 6HFWLRQ ZH GLVFXVV WKH FDVH RI FRUUXSWHG SDVW YDOXHV )LQDOO\ LQ 6HFWLRQ ZH VXPPDUL]H RXU UHVXOWV DQG PRWLYDWH WKH SUREOHP DGGUHVVHG LQ WKH IROORZLQJ FKDSWHU 3UREOHP )RUPXODWLRQ ,Q WKLV GLVVHUWDWLRQ )RXULHU WUDQVIRUPV ZLOO EH GHQRWHG E\ FDSLWDO OHWWHUV DV LQ WKH IROORZLQJ SDLU 22 WfHaZWGW 22 P 7KLV FKDSWHU LV EDVHG RQ f7KH 3UHGLFWDELOLW\ RI &RQWLQXRXV7LPH %DQGOLPLWHG 3URFHVVHVf E\ 5 /\PDQ : : (GPRQVRQ 6 0F&XOORXJK DQG 0 5DR k ,((( 5HSULQWHG ZLWK SHUPLVVLRQ IURP ,((( 7UDQVDFWLRQV RQ 6LJQDO 3URn FHVVLQJ YRO QR SS )HEUXDU\
PAGE 27
1RZ FRQVLGHU D UHDO FRQWLQXRXVWLPH ZLGHVHQVH VWDWLRQDU\ ]HURPHDQ UDQGRP SURFHVV [Wf ZKLFK LV NQRZQ RQ WKH 7OHQJWK LQWHUYDO >W fÂ§ U fÂ§ 7W fÂ§ U@ ZLWK U 7 :H ZRXOG OLNH D SUHGLFWRU [Wf ZKLFK LV OLQHDU RQ WKH NQRZQ SRUWLRQ RI [Wf UW7 [Wf [;fKW fÂ§ ;fG? f WU7 ZKHUH KWf LV D UHDO DQG FRQWLQXRXV IXQFWLRQ RI W 1RWH WKDW ERWK [Wf DQG KWf PD\ DOVR GHSHQG RQ U DQG 7 :H ZLVK WR ILQG D IXQFWLRQ KWf ZKLFK PLQLPL]HV e ^>[Lf Âf@` f ZKHUH LV WKH PHDQ VTXDUHG SUHGLFWLRQ HUURU :H IRFXV RQ OLQHDU SUHGLFWRUV SDUWO\ EHFDXVH RI WKHLU VLPSOLFLW\ EXW DOVR ZKHQ WKH SURFHVV LV *DXVVLDQ QR SUHGLFWRU FDQ DFKLHYH D ORZHU PHDQ VTXDUHG HUURU WKDQ WKH RSWLPDO OLQHDU SUHGLFWRU LI LW H[LVWV > S @ 6XSSRVH ZH FRQVWUDLQ KWf WR EH ]HUR H[FHSW IRU WKH LQWHUYDO W f >WW 7@ 7KHQ ZH PD\ ZULWH f DV D FRQYROXWLRQ RR [;fKW ;fG; f RR [Wf r KWf ZKHUH KWf I >U W 7? f 7KXV DW HYHU\ W [Wf UHSUHVHQWV D SUHGLFWLRQ RI [Wf EDVHG XSRQ WKH NQRZQ LQWHUYDO >W fÂ§ 7 fÂ§ 7W fÂ§ Wf 1RWH WKDW LQ f KWf LV YLHZHG DV WKH LPSXOVH UHVSRQVH RI D OLQHDU WLPH LQYDULDQW ILOWHU 7KLV LV MXVWLILHG E\ WKH VWDWLRQDULW\ RI [Wf ,Q RXU GLVFXVVLRQ RI SUHGLFWLRQ KRZHYHU D SRVVLEOH SRLQW RI FRQIXVLRQ LV WKH UOHQJWK WLPH GHOD\ LQWURGXFHG E\ KWf LQ f $V ZH VKDOO VHH WKLV UHVXOWV LQ D VLPSOH IUHTXHQF\
PAGE 28
GRPDLQ IRUPXODWLRQ RI WKH SUREOHP DQG VHUYHV SHUIHFWO\ ZHOO WR DQVZHU RXU TXHVWLRQV DERXW WKH SUHGLFWRU GHILQHG LQ f :H QRZ SURFHHG WR ILQG D IUHTXHQF\GRPDLQ H[SUHVVLRQ IRU WKH PHDQ VTXDUHG SUHGLFWLRQ HUURU :H VWDUW E\ GHILQLQJ WKH HUURU HWf [Wf fÂ§ [Wf [Wf fÂ§ [Wf r KWf ,, FI r URA & f /HW XV IXUWKHU GHILQH WKH HUURU ILOWHU HWf 6Wf fÂ§ KWf f (Xf fÂ§ +XMf f ,W LV FOHDU WKDW WKH HUURU LV REWDLQHG E\ SDVVLQJ [Wf WKURXJK D ILOWHU ZLWK D IUHTXHQn F\ UHVSRQVH JLYHQ E\ f 1RZ VXSSRVH WKDW [Wf LV EDQGOLPLWHG WR _FR_ Â/ 7KHQ ZH PD\ REWDLQ WKH PHDQ VTXDUHG HUURU E\ ]WW Q 7 6[[Xf??+^Xf?GX ]WW Q f f ZKHUH 6[[X!f LV WKH SRZHU VSHFWUDO GHQVLW\ RI [Wf 7KXV WKH GHVLUHG SUHGLFWRU f PLQLPL]HV f ZLWK UHVSHFW WR KWf XQGHU WKH FRQVWUDLQW LPSRVHG E\ f $ FDUHIXO LQVSHFWLRQ RI f ZLOO VKRZ WKDW ZH DUH VHHNLQJ D WLPH IXQFWLRQ RI ILQLWH VXSSRUW KWf ZKRVH )RXULHU WUDQVIRUP DSSUR[LPDWHV WKH IUHTXHQF\ UHVSRQVH RI D ]HURSKDVH DOOSDVV ILOWHU LQ WKH IUHTXHQF\ UDQJH X >fÂ§6 IL@ 7KH WLPH GHOD\ LQFRUSRUDWHG LQ KWf PDNHV WKLV VLPSOH IRUP SRVVLEOH VLQFH RWKHUZLVH LW ZRXOG EH QHFHVVDU\ WR LQWURGXFH H[SRQHQWLDOV RI WKH IRUP HW:7 1RWH DOVR WKDW WKH IUHTXHQF\ UHVSRQVH IRU _FX_ 4 LV DUELWUDU\ EHFDXVH [Wf FRQWDLQV QR HQHUJ\ DW IUHTXHQFLHV
PAGE 29
RXWVLGH RI WKH 4 EDQG OLPLW 7KLV IDFW LV FULWLFDO WR WKH GLVFXVVLRQ LQ WKH QH[W VHFWLRQ $QDO\VLV ,Q WKLV VHFWLRQ ZH VKDOO UHIHU WR WKH IROORZLQJ VSDFHV RI IXQFWLRQV e` DQG e FRQWDLQ IXQFWLRQV ZKLFK DUH UHVSHFWLYHO\ LQWHJUDEOH DQG VTXDUH LQWHJUDEOH RQ WKH UHDO OLQH e>fÂ§U W@f FRQWDLQV IXQFWLRQV ZKLFK DUH VTXDUH LQWHJUDEOH RQ WKH LQWHUYDO >fÂ§U U@ e7 FRQWDLQV IXQFWLRQV If e VXFK WKDW Wf IRU W e >WW 7@ DQG &6f FRQWDLQV IXQFWLRQV )FYf VXFK WKDW 6X!f?)XMf?GX` RR 6XSSRVH WKDW )X!f e DQG IWf IRU W 7KHQ ZH VD\ WKDW )Xf LV LQ n+ RI WKH XSSHU KDOI SODQH ZULWWHQ )XMf "I8+3f 1RZ VXSSRVH *Xf e>fÂ§7 U@f DQG VXSSRVH WKDW WKH )RXULHU FRHIILFLHQWV ^f` RI *Zf DUH JLYHQ E\ Q U *^XMfHLXQGX fÂ§7 ,I f IRU Q ZH VD\ WKDW *Zf LV LQ + RI WKH XQLW GLVF ZULWWHQ *XMf +8'f )URP WKH SUHYLRXV VHFWLRQ ZH PD\ IRUPXODWH RXU SUHGLFWLRQ SUREOHP DV IROORZV UQ PLQLPL]H fÂ§ ,,Df_O fÂ§IRf_LLD ZUW KWf WW Q f VXEMHFW WR KWf W e >U U 7@ ZKHUH LV WKH PHDQ VTXDUHG SUHGLFWLRQ HUURU 2XU ILUVW TXHVWLRQ LV ZKHWKHU D PLQLPXP H[LVWV IRU f :H QRZ VKRZ WKDW WKHUH LV DQ DOORZDEOH KWf ZKLFK PDNHV DUELWUDULO\ VPDOO 7KHRUHP /HW W 7 DQG Â EH IL[HG UHDO QXPEHUV ZLWK 7 WW DQG OHW 6X!f & ZLWK 6Zf X! >fÂ§f IL@ DQG 6XMf X! A >fÂ§ Â@ 7KHQ IRU HYHU\
PAGE 30
)Xf e6f DQG HYHU\ H WKHUH LV DQ KWf eU VXFK WKDW 3 6Xf?)^Xf +^Xf?GX H Q 3URRI &RQVLGHU WKH VSDFH 7 RI IXQFWLRQV +^XMf IRU ZKLFK KWf e7 7R SURYH WKH WKHRUHP LW LV VXIILFLHQW WR VKRZ WKDW LI *X!f ef WKHQ IRU HYHU\ +Xf 7 LI 22 6Xf+rXf*XfGX 22 WKHQ 22 6Xf?*Xf?GX 22 1RZ VXSSRVH IAr26XMf+rLMMf*XMfGX 7KHQ RR U URR r KWfHaWXWGW 6^XMf*XMfGX RR 8fÂ§RR RR U URR K^Wf 6A*AHAGX GW RR 8fÂ§22 ,Q RUGHU WR MXVWLI\ WKH LQWHUFKDQJH RI LQWHJUDWLRQ ZH PXVW VKRZ WKDW 6Xf*LV DEVROXWHO\ LQWHJUDEOH 22 6nDf_*nDf_GD RR 22 7KDW WKLV LV VR IROORZV IURP &DXFK\6FKZDU] VLQFH 22 6^Xf?*^Xf?GX 22 9VPf YVKLJKLf GX B ^AM 6AGXAM 6Rf_*Df_GDA f %RWK RI WKH LQWHJUDOV RQ WKH ULJKW KDQG VLGH DUH ILQLWH E\ K\SRWKHVLV 7KXV f LV WUXH DQG WKH LQWHUFKDQJH LV MXVWLILHG
PAGE 31
1RZ OHW UXf X!f*X!f WKHQ RR U URR KWf UYfHZWGM RR 8fÂ§RR RR K^WfA^WfGW RR %HFDXVH Lf LV D ERXQGHG IXQFWLRQ ZKLFK LQWHJUDWHV DJDLQVW HDFK KWf e eU Q /O WR JLYH ]HUR ZH FRQFOXGH WKDW Wf IRU W e >U U 7@ )XUWKHU VLQFH Lf LV EDQGOLPLWHG LW LV DQ HQWLUH IXQFWLRQ ZKLFK LPSOLHV WKDW LI Lf RQ DQ\ LQWHUYDO RI SRVLWLYH PHDVXUH LW PXVW EH ]HUR HYHU\ZKHUH 6R WKLV PXVW EH WUXH RI 7Rf DOVR 7KXV UZf 6Xf*Xf 9 X 7KLV VKRZV WKDW *FYf $f ` $FFRUGLQJO\ _*Xf_ IRU X! LQ WKLV VHW 7KHUHIRUH Zf_*Xf_GX DQG WKH WKHRUHP LV SURYHGÂ’ 7KH H[SUHVVLRQ IRU WKH PHDQ VTXDUHG HUURU LQ f LV REWDLQHG VLPSO\ E\ OHWWLQJ )XMf DQG VFDOLQJ DSSURSULDWHO\ 7KXV ZH KDYH VKRZQ WKDW WKH LQILPXP RI WKH PHDQ VTXDUHG SUHGLFWLRQ HUURU LV ]HUR IRU EDQGOLPLWHG SURFHVVHV 1RWH WKDW WKLV UHVXOW KROGV IRU DOO U DQG SRVLWLYH 7 LH LW KROGV UHJDUGOHVV RI WKH OHQJWK RI WKH NQRZQ LQWHUYDO RU KRZ IDU LQ WKH IXWXUH ZH ZLVK WR PDNH WKH SUHGLFWLRQ 7KH VDPH DUJXPHQW KROGV HYHQ LI ZH DOORZ KWf WR EH QRQ]HUR RQ WKH LQWHUYDO >U f 1H[W ZH VKDOO VKRZ WKDW XQGHU FHUWDLQ UHVWULFWLRQV LPSRVHG RQ Zf QR OLQHDU SUHGLFWRU DWWDLQV WKLV LQILPXP 7KHRUHP /HW W 7 DQG Â EH IL[HG UHDO QXPEHUV ZLWK U DQG 7IO! DQG OHW 6X!f e e` ZLWK 6Xff 9 X! DQG XMf IRU X e 0 ZKHUH 0 & >fÂ§Â$ WW@ LV D VHW RI SRVLWLYH PHDVXUH 7KHQ WKHUH LV QR KWf e e U VXFK WKDW Zf_O fÂ§ +XMf?GXM f
PAGE 32
3URRI 6XSSRVH VXFK DQ KWf H[LVWV 7KHQ WKHUH LV VRPH +X!f VXFK WKDW f KROGV /HW V > = DQG OHW *]f +Vf 6LQFH +Xf "^8+3f ZH KDYH L fÂ§ ] *^XMf "Â,'f > S @ 'XH WR f +^Xf DQG WKXV *FYf LV HTXDO WR RQ D VHW RI SRVLWLYH PHDVXUH %XW VLQFH *FXf b8'f WKLV LPSOLHV WKDW *X!f DQG WKXV +^Xf LV LGHQWLFDOO\ %XW WKLV FRQWUDGLFWV +X!f e 7KHUHIRUH QR VXFK +OMf RU KWf H[LVWVÂ’ 6LQFH QR DOORZDEOH KWf DWWDLQV WKH LQILPXP RI E\ GHILQLWLRQ f KDV QR PLQLPXP 7KLV WKHRUHP DSSOLHV HYHQ ZKHQ U LQ ZKLFK FDVH WKH YDOXH [Wf LV LQFOXGHG RQ WKH NQRZQ LQWHUYDO DQG PD\ EH REWDLQHG ZLWKRXW HUURU E\ VDPSOLQJ DW WKDW LQVWDQW %XW LQ WKLV FDVH KWf PXVW EH LQWHUSUHWHG DV D GHOWD IXQFWLRQ ZKLFK LV QRW LQ e )RU VLPLODU UHDVRQV WKLV WKHRUHP GRHV QRW DSSO\ ZKHQ D UDQGRP SURFHVV FRQVLVWV RI D ILQLWH VXP RI FRPSOH[ H[SRQHQWLDOV ,W LV NQRZQ WKDW VXFK D SURFHVV PD\ EH SUHGLFWHG ZLWK D PHDQ VTXDUHG HUURU RI ]HUR XVLQJ D ILQLWH VXP RI SDVW YDOXHV > S @ %XW LQ WKLV FDVH 6Zf PXVW EH LQWHUSUHWHG DV D ILQLWH VXP RI GHOWD IXQFWLRQV ZKLFK DUH QRW LQ & 2XU QH[W TXHVWLRQ LV ZKDW KDSSHQV WR KWf DV DSSURDFKHV ]HUR 7KH QH[W WKHRUHP VKRZV WKDW ZKHQ KWf LV FRQVWUDLQHG LQ HQHUJ\ WKH LQILPXP PHDQ VTXDUHG HUURU RI WKH UHVXOWLQJ VSDFH RI IXQFWLRQV LV DWWDLQHG 7KHRUHP /HW U 7 &O DQG ( EH IL[HG UHDO QXPEHUV ZLWK 7&O( DQG OHW 6XMf & ZLWK 6Xf 9 X 1RZ FRQVLGHU WKH VSDFH %H RI IXQFWLRQV K KWf &W7 VXFK WKDW UU7 KWfGW ( f /HW r LQI Â 6Xff?O +XMf?GZ KH%( =7 4 7KHQ WKHUH LV DQ K %H VXFK WKDW L > 6XMf? fÂ§ +XMf?GXM N 7 Q
PAGE 33
3URRI 'HILQH %H fÂ§! IW E\ ^Kf 3 6^Xf?O +X!f?GX! 7 fÂ§Â 6LQFH %H LV ZHDNO\ FRPSDFW ZH PD\ SURYH WKDW ^Kf DWWDLQV LWV LQILPXP E\ VKRZLQJ WKDW ^Kf LV ZHDNO\ FRQWLQXRXV 7KLV FDQ EH VKRZQ E\ ILUVW REVHUYLQJ WKDW LI K %H WKHQ E\ &DXFK\6FKZDU] O0, 1RZ VXSSRVH ^KQ` LV D VHTXHQFH IURP %H ZKLFK FRQYHUJHV WR KD ZHDNO\ 7KHQ ^+Q` FRQYHUJHV WR + SRLQWZLVH VLQFH >Â‘W7 +QXf KQ^WfHaZWGW 7KXV 6Dff_O fÂ§ +QXff? FRQYHUJHV SRLQWZLVH WR nRf_O fÂ§ +XMf_ )URP f ZH DOVR KDYH 6Xf_ +Q^Xf? 6Xf^ 7(f :H FDQ WKXV DSSO\ WKH /HEHVJXH 'RPLQDWHG &RQYHUJHQFH 7KHRUHP WR FRQFOXGH WKDW ^.f Kf 7KXV LV ZHDNO\ FRQWLQXRXV DQG WKH WKHRUHP LV SURYHGÂ’ 7KHRUHP LPSOLHV WKDW WKHUH LV D IXQFWLRQ P?Q(f 7KLV IXQFWLRQ LV WKH PLQLPXP PHDQ VTXDUHG HUURU REWDLQDEOH ZKHQ KWf LV FRQVWUDLQHG WR KDYH DQ HQHUJ\ QR JUHDWHU WKDQ ( 1RWH WKDW P?Q^(f LV D QRQLQFUHDVLQJ IXQFWLRQ RI (
PAGE 34
'HDOLQJ ZLWK (VWLPDWLRQ (UURUV :H VKDOO GLVFXVV WKH ILQGLQJV RI WKH SUHYLRXV VHFWLRQ DV WKH\ UHODWH WR WKH SUHGLFWLRQ RI D EDQGOLPLWHG SURFHVV ZKRVH YDOXHV RQ WKH NQRZQ LQWHUYDO DUH FRUUXSWHG E\ HVWLPDWLRQ HUURUV &RQVLGHU WKH UDQGRP SURFHVV [Wf RI 6HFWLRQ DQG VXSSRVH WKDW ZH KDYH DQ HVWLPDWH RI WKDW SURFHVV L$f [$f LX$f $ f >W fÂ§ W fÂ§ 7 W fÂ§ W@ f ZKHUH ZWf UHSUHVHQWV WKH HVWLPDWLRQ HUURU :H FRQVLGHU ZWf WR EH D UHDO ]HUR PHDQ ZLGHVHQVH VWDWLRQDU\ SURFHVV XQFRUUHODWHG ZLWK [Wf DQG KDYLQJ DQ DXWRFRUUHODWLRQ IXQFWLRQ JLYHQ E\ 5ZZWf 2Z+Wf f 7KLV H[SUHVVLRQ LQGLFDWHV WKDW ZWf LV EHLQJ PRGHOHG DV ZKLWH QRLVH 2I FRXUVH WKH ZKLWH QRLVH DVVXPSWLRQ PD\ QRW EH YDOLG LQ VRPH DSSOLFDWLRQV 1HYHUWKHOHVV E\ FRQVLGHULQJ WKH FDVH RI ZKLWH QRLVH ZH FDQ VLPSOLI\ WKH PDWKHPDWLFDO WUHDWPHQW DQG VWLOO JDLQ LQVLJKW LQWR WKH HIIHFWV RI HVWLPDWLRQ HUURUV $OVR E\ XVLQJ WKLV DSSURDFK ZH FDQ VHH ZK\ ( DQ XSSHU ERXQG RQ HQHUJ\ LV D QDWXUDO FKRLFH IRU D FRQVWUDLQW RQ KWf /HW XV FRQVLGHU WKH OLQHDU SUHGLFWRU IURP f ZLWK LQSXW [Wf LQVWHDG RI [Wf 7KHQ WKH HUURU EHFRPHV HWf [Wf fÂ§ [Wf [Wf fÂ§ [Wf r KWf [Wf fÂ§ >[Wf ZWf@ r KWf [Wf fÂ§ [Wf r KWf fÂ§ ZWf r KWf [Wf r >If fÂ§ KWf@ fÂ§ ZWf r KWf f
PAGE 35
1RWH WKDW WKLV LV HTXLYDOHQW WR f ZLWK WKH DGGHG WHUP ZWf r KWf 8VLQJ D OLQH RI UHDVRQLQJ VLPLODU WR WKDW OHDGLQJ WR f DQG QRWLQJ WKDW [Wf DQG ZWf DUH XQFRUUHODWHG WKH PHDQ VTXDUHG HUURU LV JLYHQ E\ Urr Urr fÂ§ 6[[Zf?O+Xf?GX fÂ§ 6ZZXf?+Xf?GXM =7 fÂ§ RR Â7 fÂ§RR rO??K?? f ZKHUH __K__ 6LQFH DQG ERWK GHSHQG RQ KWf ZH ZULWH } f__K__ f :H NQRZ IURP 7KHRUHP WKDW Kf KDV QR PLQLPXP :H VKDOO VKRZ KRZHYHU WKDW Kf GRHV KDYH D PLQLPXP DQG WKDW WKLV PLQLPXP LV JUHDWHU WKDQ ]HUR IRU DQ\ QRQWULYLDO SURFHVV 7KHRUHP /HW &W 7 EH DV GHILQHG DW WKH EHJLQQLQJ RI 6HFWLRQ DQG OHW N LQI Kf AHA7U 7KHQ 7KHUH LV DQ K &"W7 VXFK WKDW N Kf rÂ‘ WI6L 6[[OMf GX! ZH DOVR KDYH N 3URRI 7R SURYH f FKRRVH D VHTXHQFH ^ÂQ` ZLWK KQ e!7 9 Q VXFK WKDW KQf fÂ§!Â‘ N 7KHQ LP LQf D__LQ__ N 7KLV PHDQV WKDW IRU HYHU\ H WKHUH LV DQ P VXFK WKDW KQf DO??KQ?? N H 9 Q P f
PAGE 36
%XW VLQFH ^KQf ZH KDYH DZ??KQ??N H 7KXV \N H ,,0 Q! P 6LQFH ^KQ` LV QRUP ERXQGHG ZH PD\ FKRRVH D VXEVHTXHQFH ^KQN` ZKLFK FRQYHUJHV WR VRPH ZHDN OLPLW K 7KHQ WKHUH LV D VHTXHQFH RI ILQLWH VXPV ^A` ZLWK N0N N fÂ§ O N f ZKHUH WKH FRHIILFLHQWV FAN VDWLVI\ IF0MIF &LN DQG < O N f VXFK WKDW ^SIF` FRQYHUJHV VWURQJO\ WR K $ IXQFWLRQ JNWf VDWLVI\LQJ f DQG f LV FDOOHG D FRQYH[ FRPELQDWLRQ RI ^Lf` 6LQFH D FRQYH[ FRPELQDWLRQ LV D SDUWLFXODU W\SH RI OLQHDU FRPELQDWLRQ ZH KDYH JN &7 9 N 1RZ EHFDXVH RI f IRU HYHU\ H WKHUH LV D S VXFK WKDW IRU N S ZH KDYH m Nf UQ O UQ Q N0N < FON+Q+ O N N0N < F +QLXMf@ O N N0N ( FQMDPXLf O N D URR fÂ§ fÂ§M 6[[Xf?OfÂ§*N^Xff?GXf UA/ ?*N^DMf?GXM =WW fÂ§Q =WW fÂ§RR U8 N0N U[ 6fm ( fÂ§Q Â LE WW fÂ§F O /Q6[[^Xf N0N & Q IF0r I R Â 4IF LQRf_FNM_ < &LA?\6X ?+QcAGZ? 4 URR rrW/ N0N N0N \ ALN+QL^\Mf O N GX! N0N fÂ§
PAGE 37
VLQFH WKH 4MWfV VXP WR 7KXV JNf fÂ§! N 1RZ ^JN` FRQYHUJHV VWURQJO\ WR K 6WURQJ FRQYHUJHQFH LPSOLHV FRQYHUJHQFH RI WKH QRUP VR __SMIF__ fÂ§!f ??K?? $QG VLQFH DQ\ VWURQJO\ FRQYHUJHQW VHTXHQFH DOVR VDWLVILHV WKH GHILQLWLRQ RI D ZHDNO\ FRQYHUJHQW VHTXHQFH ZH PD\ XVH WKH DUJXPHQW LQ WKH SURRI RI 7KHRUHP VXEVWLWXWLQJ ^JN` IRU ^Lf` WR VKRZ WKDW JNf fÂ§!Â‘ KDf :H WKXV KDYH Nf Nf A??N?? }f Kf A??Ke Kf $QG VLQFH ^JNf` FRQYHUJHV WR N ZH KDYH N fÂ§ KDf 7R SURYH f ZH QRWH IURP f WKDW LI N Kf WKHQ ??KR?? fÂ§ %XW WKHQ +TXf DOPRVW HYHU\ZKHUH 7KXV IURP f ZH KDYH f a > 6[[XfGX Â’ 1RWH WKDW WKH NH\ UHVXOW RI 7KHRUHP LV WKDW KDV D PLQLPXP HYHQ LI ZH GR QRW LPSRVH D FRQVWUDLQW RQ K^Wf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f ,I RQ WKH RWKHU
PAGE 38
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f ZDV FHQWUDO LQ WKH DQDO\VLV RI WKLV FDVH 7KXV LQ WKH FDVH ZKHUH YDOXHV RI [Wf RQ WKH NQRZQ LQWHUYDO DUH HUURU IUHH DQ XSSHU ERXQG RQ WKLV HQHUJ\ LV D QDWXUDO FKRLFH IRU D FRQVWUDLQW RQ KWf VLQFH ZRXOG QRW KDYH D PLQLPXP ZLWKRXW VRPH FRQVWUDLQW :H QRZ KDYH WZR REMHFWLYHV 2QH LV WR FRQVWUXFW D IXQFWLRQ KWf ZKLFK DWWDLQV PLQ^(f 7KH RWKHU LV WKH FRPSXWDWLRQ RI PLQ(f LWVHOI 7KLV ODWWHU JRDO ZRXOG EH XVHIXO LQ WKH FDVH ZKHUH D OLQHDU SUHGLFWRU LV GHVLUHG EXW UHDOL]LQJ WKH LPSXOVH UHVSRQVH KWf LV QRW SRVVLEOH RU SUDFWLFDO PDNLQJ LW QHFHVVDU\ WR UHVRUW WR VRPH VXERSWLPDO DSSURDFK .QRZOHGJH RI PP^(f ZRXOG EH XVHIXO LQ GHWHUPLQLQJ KRZ FORVHO\ WKLV VXERSWLPDO SUHGLFWRU DSSURDFKHV RSWLPDO SHUIRUPDQFH $V RI \HW QHLWKHU KWf QRU PcQ(f KDV EHHQ IRXQG IRU WKH JHQHUDO FDVH 6ROXWLRQV KDYH EHHQ IRXQG IRU D VSHFLDO FDVH KRZHYHU ZKLFK ZH GLVFXVV LQ WKH QH[W FKDSWHU
PAGE 39
&+$37(5 352&(66(6 :,7+ )/$7 63(&75$/ '(16,7,(6 ,Q WKLV FKDSWHU ZH VKDOO VROYH WKH OLQHDU SUHGLFWLRQ SUREOHP IRU EDQGOLPLWHG SURFHVVHV ZLWK IODW SRZHU VSHFWUDO GHQVLWLHV )RU VXFK SURFHVVHV ZH KDYH 6X!f IRU _FM_ 4 DQG 6XMf fÂ§ HOVHZKHUH ,Q VROYLQJ WKLV SUREOHP ZH VKDOO PDNH XVH RI D VHW RI EDVLV IXQFWLRQV ^Â!Nf` ZKLFK DUH WLPHVKLIWHG YHUVLRQV RI WKH SURODWH VSKHURLGDO ZDYH IXQFWLRQV ^LcfrLf`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f/LQHDU 3UHGLFWLRQ RI %DQGOLPLWHG 3URFHVVHV ZLWK )ODW 6SHFWUDO 'HQVLWLHVf E\ 5 /\PDQ DQG : : (GPRQVRQ k ,((( 8VHG E\ SHUPLVVLRQ 7KLV SDSHU LV FXUUHQWO\ XQGHU FRQVLGHUDWLRQ IRU SXEOLFDWLRQ LQ ,((( 7UDQVDFWLRQV RQ 6LJQDO 3URFHVVLQJ
PAGE 40
%DVLV )XQFWLRQV 'LVFXVVLRQV RI WKH SURODWH VSKHURLGDO ZDYH IXQFWLRQV DV ZHOO DV SURRIV RI PDQ\ RI WKHLU SURSHUWLHV DUH FRQWDLQHG LQ WUHDWPHQWV E\ 6OHSLDQ HW DO >@ 3DSRXOLV >@ DQG )ULHGHQ >@ :H VKDOO VXPPDUL]H VRPH NH\ SURSHUWLHV ZLWKRXW UHVWDWLQJ SURRIV IURP WKHVH UHIHUHQFHV 3URODWH 6SKHURLGDO :DYH )XQFWLRQV 7KH SURODWH VSKHURLGDO ZDYH IXQFWLRQV DUH VROXWLRQV RI WKH IROORZLQJ LQWHJUDO HLJHQYDOXH SUREOHP I? VfAG6 Lf 7KLV HTXDWLRQ KDV QRQWULYLDO VROXWLRQV IRU RQO\ D FRXQWDEOH VHW RI HLJHQYDOXHV ^$Q` (DFK $Q LV UHDO DQG SRVLWLYH VXFK WKDW WKH VHW ^$Q` PD\ EH RUGHUHG DV $ $L $ ! ,I VR RUGHUHG ZH DOVR KDYH OLP $Q QfÂ§ RR 7R HDFK $Q WKHUH FRUUHVSRQGV RQO\ RQH IXQFWLRQ LcfQWf ZLWKLQ D FRQVWDQW IDFWRU :LWK D SURSHU FKRLFH RI WKLV IDFWRU WKH IXQFWLRQV ^LfQWf` IRUP D VHW ZLWK WKH IROORZLQJ SURSHUWLHV (DFK IXQFWLRQ LSQWf LV UHDO YDOXHG A 7KH VHW ^LSQWf` LV RUWKRQRUPDO RQ >fÂ§@ WKDW LV 22 SP^WfOSQ^WfGW P Q P Q ?t 7KH VHW ^nLSQLWf` LV FRPSOHWH LQ Â7EDQGOLPLWHG ILQLWHHQHUJ\ IXQFWLRQV 7KLV PHDQV WKDW LI )Xf IRU _Z_ IW DQG IWfGW WKHQ WKHUH
PAGE 41
DUH FRHIILFLHQWV ^rf` VXFK WKDW IWf PD\ EH ZULWWHQ 22 I^Wf Wf QfÂ§ 7KH VHW ^Â‘c9LLf` LV DOVR RUWKRJRQDO RQ f""@} ZLWK P Q P A Q 7KH VHW ^LSQWf` LV FRPSOHWH LQ ILQLWHHQHUJ\ IXQFWLRQV RQ >fÂ§M 7KLV LV WUXH HYHQ LI VXFK D IXQFWLRQ LV QRW D VHJPHQW RI D EDQGOLPLWHG IXQFWLRQ (DFK IXQFWLRQ LSQWf LV 4EDQGOLPLWHG 7KLV PHDQV WKDW HDFK )RXULHU WUDQVIRUP KDV WKH SURSHUW\ QD!f IRU _X_ Â 7KH VHW LV RUWKRQRUPDO RQ >fÂ§f ÂÂ@ 7KLV PD\ EH VHHQ E\ DSSO\LQJ 3DUVHYDOfV LGHQWLW\ WR SURSHUW\ DQG WKHQ DSSO\LQJ SURSHUW\ U 7KH VHW ^QX!f` LV FRPSOHWH LQ ILQLWHHQHUJ\ IXQFWLRQV RQ >fÂ§2 2@ 7KLV LV WUXH EHFDXVH D ILQLWHHQHUJ\ IXQFWLRQ ZLWK WKH SURSHUW\ )XMf IRU 0 Â KDV DQ LQYHUVH )RXULHU WUDQVIRUP IWf ZKLFK LV LLEDQGOLPLWHG %\ SURSHUW\ ZH PD\ ZULWH 22 IWf Q DQG WDNLQJ WKH )RXULHU WUDQVIRUP ZH KDYH 22 )Xf !QWIQXf Q /HW XV GHILQH 0f 9fIFMWUXQFO2 fÂ§ HOVHZKHUH
PAGE 42
,I A$WUXQFZf LV WKH )RXULHU WUDQVIRUP RI 9IFWUXQFIf! WKHQ WUXQFf fÂ§ $Q?NQFL!f A A 1RWH WKDW n,nWUXQFf LV QRW ]HUR IRU _X!_ Â 7KLV SURSHUW\ VD\V QRWKLQJ DERXW WKH EHKDYLRU RI ?,IFWUXQFZf RXWVLGH RI WKH LQWHUYDO >fÂ§I ÂÂ@ ,W VKRXOG EH QRWHG WKDW LSQ^Wf GHSHQGV LPSOLFLWO\ RQ 4 DQG 7 DQG PD\ EH PD\ EH ZULWWHQ $QFf ,Q NHHSLQJ ZLWK WKH FRQYHQWLRQ RI 6OHSLDQ HW DO ZH VKDOO QRUPDOO\ VXSSUHVV WKLV DGGLWLRQDO QRWDWLRQ 7LPH6KLIWHG %DVLV )XQFWLRQV 7KH EDVLV IXQFWLRQV ZH VKDOO XVH WR VROYH WKH OLQHDU SUHGLFWLRQ SUREOHP LQ 6HFWLRQ DUH WLPHVKLIWHG YHUVLRQV RI ^9fQLf` :H GHILQH f )RU HDFK Q WKH IXQFWLRQ QLf VDWLVILHV WKH IROORZLQJ LQWHJUDO HTXDWLRQ f ZKHUH $f LV WKH QWK HLJHQYDOXH RI f 7R VHH WKLV ZH DSSO\ f WR f WR REWDLQ 1RZ ZLWK WKH VXEVWLWXWLRQV Vn V U A DQG Wn W U M ZH REWDLQ GVn $ QMfQWnf 6LQFH WKLV HTXDWLRQ PXVW KROG IRU DOO W DQG WKXV DOO Wn WKH SULPH QRWDWLRQ PD\ EH GURSSHG DQG f UHVXOWV 7KLV VKRZV WKDW f DQG f KDYH WKH VDPH VHW RI HLJHQYDOXHV ^$Q`
PAGE 43
7KH IXQFWLRQV ^!fef` KDYH WKH IROORZLQJ SURSHUWLHV PDQ\ RI ZKLFK IROORZ GLUHFWO\ IURP WKH SURSHUWLHV RI ^LSQWf`n (DFK IXQFWLRQ MfQWf LV UHDO YDOXHG 7KLV IROORZV GLUHFWO\ IURP SURSHUn W\ 9, 7KH VHW ^ILQWf` LV RUWKRQRUPDO RQ >fÂ§@ 6LQFH HDFK FMfQWf LV VKLIWHG E\ WKH VDPH DPRXQW WKH RUWKRJRQDOLW\ RI SURSHUW\ A LV SUHVHUYHG 7KH VHW ^c!Q^Wf` LV FRPSOHWH LQ 6OEDQGOLPLWHG ILQLWHHQHUJ\ IXQFWLRQV 7R VHH WKLV VXSSRVH IWf LV VXFK D IXQFWLRQ 7KHQ WKH WLPHVKLIWHG IXQFWLRQ I WU LV DOVR IEDQGOLPLWHG :H PD\ WKXV ZULWH 22 I^W 7 ,f 6WWQ9fQ Wf Q 6XEVWLWXWLQJ Wn W U A ZH KDYH 22 22 rnf e mQ9fQ I a 7 IDQfQWnf Q Q 6LQFH WKLV HTXDWLRQ PXVW EH WUXH IRU DOO Wn WKH SULPH QRWDWLRQ PD\ EH GURSSHG 7KH VHW ^!QLf` LV RUWKRJRQDO RQ >U U 7@ ZLWK L!PWfO!Q^WfGW $f P Q P Q 7KLV PD\ EH VHHQ E\ DSSO\LQJ WKH WLPH VKLIW RI f WR SURSHUW\ ?3 7KH VHW QLf` LV FRPSOHWH LQ ILQLWHHQHUJ\ IXQFWLRQV RQ >U U 7@ 7KLV LV WKH LQWHUYDO RI SURSHUW\ n VKLIWHG DFFRUGLQJ WR f (DFK IXQFWLRQ c!Q^Wf LV LLEDQGOLPLWHG 7KXV HDFK )RXULHU WUDQVIRUP KDV WKH SURSHUW\ QXf IRU _FX_ Â 7KLV IROORZV IURP SURSHUW\ A VLQFH D WLPH VKLIW FOHDUO\ GRHV QRW DIIHFW WKH EDQG OLPLWDWLRQ
PAGE 44
7KH VHW ^AAQ0` LV RUWKRQRUPDO RQ >Â Â@ 7KH UHDVRQLQJ LV WKH VLPLODU WR WKDW RI SURSHUW\ ?t 7KH VHW ^QXf` LV FRPSOHWH LQ ILQLWHHQHUJ\ IXQFWLRQV RQ >IL Â@ 6HH WKH DUJXPHQW IURP SURSHUW\ A /HW XV GHILQH tN WUXQF : fÂ§ WH>U7 7@ HOVHZKHUH ,I $WUXQFZf LV WKH )RXULHU WUDQVIRUP RI I!NW7XQF^Wf WKHQ AIFWUXQFAf fÂ§ $QQWMf _A_ A f 7R VHH WKLV ZH QRWH WUXQFWf 9fIFWUXQF W 7 Af f 7DNLQJ WKH )RXULHU WUDQVIRUPV ZH KDYH rtNWUXQF r A 7 WUXQF Af $f MR U AfXf` _X!_ 4 1RZ LX! fÂ§U fÂ§ ?NQRf LV WKH )RXULHU WUDQVIRUP RI LSQ W fÂ§ U fÂ§ MfQWf 7KXV LX W WIfXf fZf DQG f IROORZV 6ROXWLRQ RI WKH /LQHDU 3UHGLFWRU +DYLQJ GHILQHG RXU EDVLV IXQFWLRQV ^!fef` DQG GLVFXVVHG WKHLU NH\ SURSHUWLHV ZH QRZ UHWXUQ RXU DWWHQWLRQ WR WKH SUHGLFWLRQ SUREOHP IRU D EDQGOLPLWHG SURFHVV /HW XV FRQVLGHU WKH HQHUJ\FRQVWUDLQHG RSWLPDO OLQHDU SUHGLFWRU RI D EDQGOLPLWHG SURFHVV ZKRVH SRZHU VSHFWUDO GHQVLW\ LV IODW LQ WKH IUHTXHQF\ EDQG RI LQWHUHVW 7KH
PAGE 45
SUREOHP PD\ EH IRUPXODWHG E\ VXEVWLWXWLQJ 6[[XMf LQ f DQG DGGLQJ WKH HQHUJ\ FRQVWUDLQW DV IROORZV UQ PLQLPL]H fÂ§ fÂ§ +X!f?GX! ZUW KLWf U T VXEMHFW WR KWf W >WW 7@ f 22 K^WfGW ( 22 1RWH WKDW ZH KDYH ZULWWHQ FRQVWUDLQW DV DQ HTXDOLW\ :H VKDOO VHH ODWHU WKDW WKH VROXWLRQ WR WKLV SUREOHP LV WKH VDPH DV LI LW ZHUH ZULWWHQ ZLWK WKH LQHTXDOLW\ 2XU DSSURDFK VKDOO EH WR WUDQVIRUP WKH LQWHJUDOV LQ WKLV SUREOHP LQWR DOJHEUDLF H[SUHVVLRQV E\ H[SDQGLQJ LQ WHUPV RI WKH EDVLV IXQFWLRQV ^I!QWf` 7KH DOJHEUDLF PLQLPL]DWLRQ PD\ WKHQ EH FDUULHG RXW E\ XVLQJ D /DJUDQJH PXOWLSOLHU 3UREOHP 7UDQVIRUPDWLRQ ,Q WKH H[SUHVVLRQ IRU LQ f ZH ZLVK WR H[SDQG WKH WHUPV LQVLGH WKH DEVROXWH YDOXH VLJQV ,Q RUGHU WR GR WKLV ZH GHILQH )Xf L PVM HOVHZKHUH f %\ SURSHUWLHV DQG WKH EDVLV IXQFWLRQ )RXULHU WUDQVIRUPV IRUP D FRPSOHWH RUWKRQRUPDO VHW LQ ILQLWHHQHUJ\ IXQFWLRQV RQ >fÂ§Â ÂÂ@ VR ZH PD\ ZULWH ZKHUH f f
PAGE 46
7KXV )QX!f Q2fQDf f %\ SURSHUW\ WKH EDVLV IXQFWLRQV ^!QWf` IRUP D FRPSOHWH VHW LQ ILQLWHHQHUJ\ IXQFWLRQV RQ >U U 7@ VR ZH PD\ H[SDQG KWf DV ZHOO RR RR KWf ( KQWf \a 3QANWUXQFAff f Q Q ZKHUH !$WUXQFÂf LV WKH WUXQFDWHG EDVLV IXQFWLRQ DV GHILQHG LQ SURSHUW\ DQG WKH FRHIILFLHQWV ^"f` DUH WR EH GHWHUPLQHG :H FRQVLGHU QRZ WKH FRQGLWLRQV XQGHU ZKLFK f ZLOO FRQYHUJH 8VLQJ SURSHUW\ ZH KDYH RR UU7 Â$WUXQF 9 RR W 7KXV _A AIFWUXQFLf` LV RUWKRQRUPDO RQ >U U 7@ &RQYHUJHQFH RI f LV WKHUHIRUH JXDUDQWHHG LI 22 ( ;Q3O 22 f 1RZ OHWWLQJ WKH )RXULHU WUDQVIRUP RI WUXQFf EH GHQRWHG E\ !IFLWUXQFZf ZH PD\ ZULWH +QXMf WUXQF0 8 f 8VLQJ SURSHUW\ ZH KDYH AIFWUXQFAf fÂ§ _A_ fÂ§ :H PD\ QRZ UHZULWH WKH REMHFWLYH IXQFWLRQ RI f DV KV!A RR RR e )fXf e +f8f Q QfÂ§ UQ 7 GXf f f
PAGE 47
$SSO\LQJ f f DQG f \LHOGV U6O rr UX Q Q (0fXf(ILQ;QADf GXM f Q Q :KHQ ZH H[SDQG WKH LQWHJUDQG RI f WKH FURVV WHUPV YDQLVK XQGHU WKH LQWHJUDO VLJQ EHFDXVH RI RUWKRJRQDOLW\ SURSHUW\ :H DUH OHIW ZLWK UQ RR e:rf $fDf f Q ZKHUH WKH ODVW VWHS LV PDGH SRVVLEOH DJDLQ E\ SURSHUW\ 7KH HQHUJ\ FRQVWUDLQW RQ KWf ZKLFK LV FRQVWUDLQW RI f PD\ DOVR EH UHZULWWHQ XVLQJ f ( I KWfGW fÂ§22 URR > 3QtNMW UXQF arr /Q RR UU7 (S, W!QWfGW Q U GW ZKHUH WKH FURVV WHUPV DJDLQ YDQLVK XQGHU WKH LQWHJUDO VLJQ EHFDXVH RI WKH RUWKRJRn QDOLW\ SURSHUW\ 8VLQJ WKLV VDPH SURSHUW\ DJDLQ ZH KDYH 22 e$Q SO ( f Q 1RWH WKDW DQ\ VHW ^SQ` VDWLVI\LQJ f ZLOO DOVR VDWLVI\ f WKXV JXDUDQWHHLQJ WKH FRQYHUJHQFH RI f 8VLQJ f DQG f ZH PD\ WUDQVIRUP f IURP DQ LQWHJUDO PLQLPL]Dn WLRQ WR DQ DOJHEUDLF RQH 22 PLQLPL]H A>Am2f $fSf@ ZUW ^SQ` Q RR 6XEMHFW W2 ;U[SO (L Q f
PAGE 48
ZKHUH WKH WLPH OLPLWDWLRQ RQ KWf FRQVWUDLQW RI f LV LPSOLFLW LQ WKH H[SDQVLRQ f /DJUDQJH 0XOWLSOLHU 7KH WHFKQLTXH RI /DJUDQJH PXOWLSOLHUV ZLOO EH XVHG WR VROYH f 7KLV WHFKQLTXH FRPELQHV WKH REMHFWLYH IXQFWLRQ DQG WKH FRQVWUDLQW LQWR WKH IROORZLQJ XQFRQVWUDLQHG RSWLPL]DWLRQ SUREOHP ? PLQLPL]H SS Sf e>Aff a $Q3QI 3 AQf fÂ§ $fSf@ S$fSQ Q 23Q 7KH VROXWLRQ RI WKLV HTXDWLRQ LV S0 f L L ZKHUH WKH GHSHQGHQFH RQ S LV H[SOLFLWO\ QRWHG 7KH PXOWLSOLHU S LV WKHQ FKRVHQ VR WKDW WKH HQHUJ\ FRQVWUDLQW RI f LV PHW + ;Q>SQ^Sf` Q RR AQ2f, $Q 7 3B ( f 7KLV VHULHV FRQYHUJHV IRU DOO SRVLWLYH S DQG JLYHV D XQLTXH SRVLWLYH VROXWLRQ IRU HYHU\ SRVLWLYH ( 7KH QRQOLQHDU HTXDWLRQ f LV FDOOHG D VHFXODU HTXDWLRQ > S @ ,W PD\ EH VROYHG E\ D OLQH VHDUFK RQ WKH SRVLWLYH D[LV DQG WKHQ LWV VROXWLRQ PD\ EH VXEVWLWXWHG LQWR f WR REWDLQ YDOXHV IRU WKH SQfV 0LQLPXP 0HDQ 6TXDUHG 3UHGLFWLRQ (UURU ,Q WKH SUHYLRXV VHFWLRQ ZH VXFFHHGHG LQ VROYLQJ IRU WKH FRHIILFLHQWV ^SQ` VR WKH H[SDQVLRQ f IRU KWf LV XQLTXHO\ GHWHUPLQHG IRU DQ\ SRVLWLYH ( DQG WKH
PAGE 49
SUREOHP f LV VROYHG ,W ZLOO EH UHFDOOHG WKDW LQ D SUHGLFWRU RI IRUP f WKLV KWf DWWDLQV WKH PLQLPXP PHDQ VTXDUHG SUHGLFWLRQ HUURU XQGHU WKH FRQVWUDLQW WKDW +R K^WfGW ( $V ZDV VWDWHG LQ 6HFWLRQ WKHUH H[LVWV D IXQFWLRQ PLQ(f ZKLFK PDSV ( WR WKH PLQLPXP PHDQ VTXDUHG HUURU IRU JLYHQ YDOXHV RI Â 7 DQG U 6XEVWLWXWLQJ f LQWR f ZH VHH WKDW ZH FDQ FRPSXWH YDOXHV IRU PcQ(f XVLQJ WKH IROORZLQJ SURFHGXUH )RU ( f 1RWH WKDW LQ RUGHU WR FDUU\ RXW WKH SURFHGXUH f RQH QHHGV RQO\ WKH HLJHQYDOXH $Q DQG WKH VLQJOH IXQFWLRQ YDOXH Aff IRU HDFK Q :H FKRRVH WR IRFXV RQ PLQ(f EHFDXVH LW SURYLGHV D XVHIXO ORZHU ERXQG RQ WKH PHDQ VTXDUHG HUURU SHUIRUPDQFH RI OLQHDU SUHGLFWRUV )URP f ZH VHH WKDW P?D(f LV D VWULFWO\ GHFUHDVLQJ IXQFWLRQ RI ( 7KXV LQ f LI WKH YDOXH RI ( LV GHFUHDVHG WKH PLQLPXP YDOXH RI PXVW LQFUHDVH 7KLV MXVWLILHV WKH XVH RI WKH HTXDOLW\ LQ FRQVWUDLQW $OVR IURP f ZH QRWH WKDW PLQ^(f fÂ§! DV ( fÂ§! RR DV ZH H[SHFW IURP WKH DQDO\VLV LQ &KDSWHU 1RZ OHW XV FRQVLGHU WKH EHKDYLRU RI PcQ(f IRU VPDOO ( /RRNLQJ DW f ZH VHH WKDW LQ WKLV FDVH L ZLOO EH PXFK JUHDWHU WKDQ ZKLFK LV DQ XSSHU ERXQG IRU $f VHH 6HFWLRQ f )URP f ZH KDYH f DQG
PAGE 50
emIf RR R & RR n, a ( ÂQf ? ( ;QI!Of ? Q $ OQ f 1RZ ZH QRWH IURP f WKDW ZLWK ( SQ IRU DOO Q 6XEVWLWXWLQJ SQ LQWR WKH H[SUHVVLRQ IRU ZH VHH WKDW WKH VXP f LV VLPSO\ PcQf 7R HYDOXDWH WKH VXP H[SOLFLWO\ ZH XVH f WR FRPSXWH >Q ?)XMf?GX! Â Q :H WKHQ XVH f f DQG SURSHUW\ WR REWDLQ I ?)Xf?LGX! .
PAGE 51
f 7 f ZKHUH fW7 Â 7 Uf VLQFIV GV &RPELQLQJ f DQG f ZLWK f DQG f \LHOGV f f L Q U ( m OÂ 7Â 4 PLQ(Aff m PLQf n IO 7= 6ROYLQJ f IRU DQG VXEVWLWXWLQJ LQWR f ZH JHW Â f f PLQAf } rPLQf \OA( 7 f :H VHH WKDW IRU VPDOO ( PcQ LV DSSUR[LPDWHO\ OLQHDU LQ WKH VTXDUH URRW RI ( )XUWKHU ZH VHH WKDW WKH VORSH RI WKH JUDSK PD\ EH FDOFXODWHG ZLWKRXW NQRZOHGJH RI DQ\ !ff RU ;Q ,I ZH QRUPDOL]H ZLWK UHVSHFWV WR PcQf ZKLFK LV DOVR WKH SRZHU RI WKH SURFHVV [Wf ZH JHW FI (TXDWLRQ f PLQ(f PLQ(f f A m 9O9( *9H f ZKHUH ?IO LV WKH DEVROXWH YDOXH RI WKH LQLWLDO VORSH RI SORWWHG DJDLQVW \( :KHQ HLWKHU 4 RU 7 LV IL[HG PD\ QRW EH PDGH DUELWUDULO\ ODUJH ,QGHHG ORRNLQJ DW f ZH VHH WKDW IRU IL[HG Â ZLOO EH ERXQGHG E\ WKH FDVH ZKHUH W fÂ§! DQG 7 fÂ§! RR URR PD[IAf VLQH &OVGV f R 7 :H QRWH DJDLQ WKDW WKH )RXULHU WUDQVIRUP RI VLQH ILW LV fÂ§)Xf RI f 8VLQJ G / 3DUVHYDOfV UHODWLRQ ZH KDYH rR UFR VLQH 4VGV fÂ§ fÂ§)X!fGXM RR :RRI) Y n
PAGE 52
f 7KXV PDNLQJ XVH RI f DQG f ZH KDYH f 1RWH DOVR f 7KXV *A7f 9I f ,Q HYDOXDWLQJ $Q DQG AQf LSQfÂ§7 fÂ§ If ZH XVHG WKH H[WHQVLYH WDEOHV RI VSKHURLGDO ZDYH IXQFWLRQV FRPSLOHG LQ WKH fV DW WKH 1DYDO 5HVHDUFK /DERUDWRn U\ > @ 7KH SURFHGXUH LV GHVFULEHG LQ WKH DSSHQGL[ 8VLQJ $f DQG f LQ f DQG QRWLQJ WKDW 9fQ: rV V\PPHWULF LQ W IRU DOO Q ZH REWDLQ WKH FXUYHV IRU PLQ(f VKRZQ LQ )LJXUH 7KH VHULHV LQ f ZHUH WUXQFDWHG DW Q ZKLFK LV WKH KLJKHVW YDOXH RI Q WDEXODWHG LQ WKH 15/ WDEOHV ,Q WKH ILJXUH WKH VROLG FXUYHV UHSUHVHQW WKH IXQFWLRQ PLQ(f SORWWHG DJDLQVW WKH VTXDUH URRW RI ( IRU VDPSOH SDUDPHWHU YDOXHV 7 U DQG Âf 7KH GDVKHG VWUDLJKW OLQHV UHSUHVHQW WKH LQLWLDO VORSHV RI WKH FXUYHV FDOFXODWHG DFFRUGLQJ WR f DQG f 7KH VPDOO FLUFOHV LQGLFDWH WKH SRLQWV DW ZKLFK Q LQ f $V ( GHn FUHDVHV EHORZ WKLV SRLQW WKH FXUYHV TXLFNO\ DSSURDFK WKHLU OLQHDU DSSUR[LPDWLRQV DV H[SHFWHG $ %DQGOLPLWHG 3URFHVV LQ :KLWH 1RLVH $V VKRZQ LQ WKLV DQG WKH SUHYLRXV FKDSWHU WKH PHDQ VTXDUHG SUHGLFWLRQ HUURU RI D EDQGOLPLWHG SURFHVV PD\ EH PDGH DUELWUDULO\ VPDOO E\ DOORZLQJ WKH HQHUJ\ RI WKH SUHGLFWRU LPSXOVH UHVSRQVH WR LQFUHDVH EXW WKLV LV WUXH RQO\ LI D VDPSOH IXQFWLRQ RI WKH SURFHVV LV NQRZQ ZLWKRXW HUURU RQ DQ LQWHUYDO RI SRVLWLYH OHQJWK &RQVLGHU D
PAGE 53
)LJXUH 7KH PLQLPXP PHDQ VTXDUHG SUHGLFWLRQ HUURU WKDW LV SRVVLEOH XVLQJ OLQn HDU SUHGLFWLRQ RQ D EDQGOLPLWHG SURFHVV ZKRVH SRZHU VSHFWUDO GHQVLW\ LV ILDW ZLWKLQ WKH EDQG OLPLWV 7 SURFHVV [Wf ZKLFK LV OLNH WKDW RI 6HFWLRQ H[FHSW WKDW 6[[Xf fÂ§ IRU _R_ &O X / :H XVH D OLQHDU SUHGLFWRU RI WKH IRUP f ZKLFK \LHOGV D PHDQ VTXDUHG SUHGLFWLRQ HUURU 7KH FDOOLJUDSKLF LV XVHG EHFDXVH WKH SURFHVV [Wf LV XQLW\ SRZHU 1RZ VXSSRVH WKDW LQVWHDG RI [^Wf LWVHOI ZH KDYH DQ HVWLPDWH Âf ef f H >W W 7 W U@ f ZKHUH WKH HVWLPDWLRQ HUURU ZWf LV FRQVLGHUHG WR EH D UHDO ]HURPHDQ ZLGH VHQVH VWDWLRQDU\ ZKLWHQRLVH SURFHVV XQFRUUHODWHG ZLWK [Wf DQG KDYLQJ DQ DXWRFRUUHODWLRQ IXQFWLRQ JLYHQ E\ 5ZZWf DO6Wf DO f :H DJDLQ XVH D SUHGLFWRU RI IRUP f UHVXOWLQJ LQ D PHDQ VTXDUHG SUHGLFWLRQ HUURU &OHDUO\ ERWK DQG GHSHQG RQ WKH SUHGLFWRU LPSXOVH UHVSRQVH KWf ZKLFK
PAGE 54
DJDLQ PD\ EH UHVWULFWHG WR KWf W e >U U 7@ $V LQ 6HFWLRQ ZH PD\ ZULWH fÂ§ Urr m Urr Kf fÂ§ 6[[Xf?O+Xf?GX fÂ§ 6ZZ+?+Xf?GF =7 fÂ§RR =7 fÂ§RR \ 22 :"BOAfOGZ Lf RZ( f ZKHUH ( KWfGW 1RWH WKDW IRU IL[HG ( Kf FDQ EH PLQLPL]HG E\ FKRRVLQJ KWf VXFK WKDW ^Kf P?Q^(f 7KXV ZH PD\ ZULWH P?Q^(f A( f 6XSSRVH WKDW IRU JLYHQ YDOXHV RI &O 7 DQG U ZH ZLVK WR DWWDLQ D PHDQ VTXDUHG SUHGLFWLRQ HUURU P?Q(f R D :H WKHQ KDYH DZ( DS f 5HDUUDQJLQJ WKLV ZH KDYH PLQL(f DS DZ( (f f $V DQ H[DPSOH FRQVLGHU WKH YDOXHV RI 7 DQG U XVHG LQ )LJXUH DQG VXSSRVH FUÂ DQG D 7KH UHVXOWLQJ IXQFWLRQ (f LV UHSUHVHQWHG E\ WKH GDVKGRW FXUYH DW WKH ERWWRP RI )LJXUH )RU ZH VHH WKDW FRQGLWLRQ f LV PHW IRU ( LQ WKH LQWHUYDO ?I( DSSUR[LPDWHO\ )RU 4 WKH FXUYH (f LV QHDUO\ WDQJHQW WR PLQ(f DW \( m )RU VLJQLILFDQWO\ JUHDWHU WKDQ R FDQQRW EH DWWDLQHG IRU WKH JLYHQ YDOXHV RI 7 U DQG RZ 7KRXJK P?Q(f PD\ EH GULYHQ DUELWUDULO\ FORVH WR ]HUR WKHUH LV QR ILQLWH ( WKDW PLQLPL]HV P?Q(f DV ZH H[SHFW IURP WKH DQDO\VLV RI &KDSWHU 2Q WKH RWKHU KDQG ORRNLQJ DJDLQ DW f ZH UHFDOO WKDW P?Q(f LV VWULFWO\ GHFUHDVLQJ LQ ( DQG ZH QRWH WKDW RZ( LV VWULFWO\ LQFUHDVLQJ LQ ( VLQFH FUA 7KXV WKHUH
PAGE 55
PXVW EH VRPH YDOXH RI ( WKDW PLQLPL]HV PP(f 7KLV LV HTXLYDOHQW WR VD\LQJ WKDW IRU JLYHQ IO 7 U DQG DKf KDV D PLQLPXP HYHQ WKRXJK Kf GRHV QRW 7KLV LV DOVR LQ DFFRUGDQFH ZLWK WKH ILQGLQJV RI &KDSWHU &RQFOXVLRQV :H KDYH SUHVHQWHG WKH VROXWLRQ RI WKH OLQHDU SUHGLFWLRQ SUREOHP IRU D EDQG OLPLWHG SURFHVV ZKRVH VSHFWUDO GHQVLW\ LV IODW ZLWKLQ WKH EDQG OLPLWV 7KLV VSHFLDO FDVH SURYLGHV D FRQFUHWH H[DPSOH WKDW LOOXVWUDWHV PDQ\ RI WKH SURSHUWLHV VKRZQ WR KROG IRU WKH PRUH JHQHUDO EDQGOLPLWHG SUHGLFWLRQ SUREOHP GLVFXVVHG LQ &KDSWHU ,Q DGGLWLRQ ZH KDYH VKRZQ KRZ WR XVH WKH VROXWLRQ WR FRPSXWH YDOXHV RI WKH IXQFn WLRQ PLQ^(f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n PHWU\ H[WUHPDO DQG FRPSOHWHQHVV SURSHUWLHV RI WKHVH IXQFWLRQV DOORZHG IRU D VLPSOH VROXWLRQ XVLQJ D /DJUDQJH PXOWLSOLHU 7KHVH VDPH SURSHUWLHV DLGHG LQ GHWHUPLQn LQJ WKH EHKDYLRU RI WKH IXQFWLRQ P?Q(f IRU VPDOO ( &RPSXWDWLRQ RI WKH ZDYH IXQFWLRQ YDOXHV WKHPVHOYHV LV D FRPSOH[ QXPHULFDO SUREOHP > @ )RUWXQDWHO\ ZH ZHUH DEOH WR UHO\ XSRQ SXEOLVKHG WDEXODWHG YDOXHV > @ WKRXJK RQH PLJKW ZLVK WKDW TXDOLW\ VRIWZDUH IRU FRPSXWLQJ WKHVH IXQFWLRQV ZHUH PRUH ZLGHO\ DYDLODEOH VHH GLVFXVVLRQ LQ WKH DSSHQGL[f 7KLV FKDSWHU DQG WKH SUHYLRXV RQH RIIHU VRPH LQVLJKW LQWR WKH SUHGLFWDELOLW\ RI EDQGOLPLWHG SURFHVVHV 0XFK ZRUN UHPDLQV WR EH GRQH RQ WKLV VXEMHFW EXW DW WKLV SRLQW LW ZLOO EH KHOSIXO WR GLVFXVV KRZ RXU ILQGLQJV WR GDWH FRXOG EH XVHG LQ WKH
PAGE 56
DQDO\VLV RI D SUDFWLFDO VLJQDOSURFHVVLQJ SUREOHP 7KLV LV WKH WRSLF RI WKH IROORZLQJ FKDSWHU
PAGE 57
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
PAGE 58
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fXW fÂ§ ;fG; YWf f 22 ZKHUH XWf LV WKH FRPSOH[ UHSUHVHQWDWLRQ RI WKH WUDQVPLWWHG VLJQDO F$ Wf LV WKH WLPHYDU\LQJ LPSXOVH UHVSRQVH RI WKH IDGLQJ FKDQQHO DQG YWf LV *DXVVLDQ QRLVH )RU VLPSOLFLW\ ZH ZLOO UHVWULFW RXU GLVFXVVLRQ WR WKH FDVH LQ ZKLFK IRU DQ\ W F$ Wf LV QRQ]HUR RYHU RQO\ D VPDOO UDQJH RI $ ,Q GLJLWDO VLJQDOLQJ WKLV RFFXUV ZKHQ WKH QRQ]HUR UDQJH LV VPDOO FRPSDUHG WR WKH V\PERO LQWHUYDO D FRQGLWLRQ RWKHUZLVH NQRZQ DV IODW IDGLQJ ,Q WKLV FDVH WKH H[SUHVVLRQ IRU WKH UHFHLYHG VLJQDO EHFRPHV UWf FWfXWf YWf f ZKHUH FWf LV VLPSO\ D FRPSOH[ WLPH IXQFWLRQ FWf [^Wf L\Wf f 7KH IXQFWLRQ FWf LV FDOOHG WKH FRPSOH[ IDGLQJ HQYHORSH DQG [Wf DQG \Wf DUH FDOOHG WKH IDGLQJ SDUDPHWHUV 7KH FRPSOH[ HQYHORSH PD\ DOVR EH UHSUHVHQWHG LQ SKDVRU IRUP FWf D^WfHLHA? f ZKHUH DWf ?[Wf \WM f
PAGE 59
QÂf UWf &0 YWf :*1 )LJXUH 0RGHO RI D IODWIDGLQJ PRELOH UDGLR FKDQQHO DQG Wf DUFWDQ f 7KH IODWIDGLQJ FKDQQHO PRGHO LV VKRZQ LQ )LJXUH $ IODWIDGLQJ FKDQQHO H[KLELWV KLJK DPSOLWXGH FRUUHODWLRQ DFURVV WKH HQWLUH EDQGZLGWK RI WKH WLPHYDU\LQJ FKDQQHO IUHTXHQF\ UHVSRQVH $ FKDQQHO IRU ZKLFK WKH IUHTXHQF\ UHVSRQVH LV OLNHO\ WR YDU\ VLJQLILFDQWO\ LQ DPSOLWXGH DFURVV LWV EDQGZLGWK LV FDOOHG IUHTXHQF\ VHOHFWLYH 6XFK FKDQQHOV FDXVH D WLPH GLVSHUVLRQ RU fVPHDULQJf RI WKH UHFHLYHG V\PEROV OHDGLQJ WR WKH XQGHVLUDEOH SKHQRPHQRQ NQRZQ DV LQWHUV\PERO LQWHUIHUHQFH $OWKRXJK ZH VKDOO QRW EH IRFXVLQJ RQ IUHTXHQF\VHOHFWLYH IDGLQJ WKH WKHRU\ RI WKH IODWIDGLQJ FKDQQHO PD\ EH XVHG LQ WKH DQDO\VLV RI WKH PRUH FRPSOLFDWHG IUHTXHQF\VHOHFWLYH FDVH >@ $ SRSXODU PRGHO IRU D IODWIDGLQJ PRELOH UDGLR FKDQQHO ZDV GHYHORSHG E\ 5 + &ODUNH >@ +H YLHZHG WKH UHFHLYHG VLJQDO DV D VXSHUSRVLWLRQ RI D ODUJH QXPEHU RI YHUWLFDOO\ SRODUL]HG HOHFWURPDJQHWLF ZDYHV DUULYLQJ DW UDQGRP DQJOHV ZLWK UDQGRP SKDVH VKLIWV &ODUNH DQDO\]HG WKH VSDWLDO FRUUHODWLRQ RI WKH ILHOGV DQG GHWHUPLQHG WKDW LI WKH UHFHLYHU LV PRYLQJ ZLWK D FRQVWDQW YHORFLW\ WKHVH ILHOGV PD\ EH PRGHOHG DV D VWDWLRQDU\ *DXVVLDQ UDQGRP SURFHVV +H DOVR GHWHUPLQHG WKH DXWRFRUUHODWLRQ IXQFWLRQ IRU WKLV SURFHVV 8VLQJ &ODUNHfV DSSURDFK ZH PD\ YLHZ [^Wf DQG \^Wf DV LQGHSHQGHQW *DXVVLDQ UDQGRP SURFHVVHV ZLWK D QRUPDOL]HG DXWRFRUUHODWLRQ IXQFWLRQ JLYHQ E\ f
PAGE 60
,Q WKLV H[SUHVVLRQ LV WKH ]HURWKRUGHU %HVVHO IXQFWLRQ RI WKH ILUVW NLQG DQG IP LV WKH PD[LPXP 'RSSOHU IUHTXHQF\ ZKLFK LV JLYHQ E\ f ZKHUH IF LV WKH FDUULHU IUHTXHQF\ 9 LV WKH VSHHG RI WKH UHFHLYHU DQG F LV WKH VSHHG RI OLJKW ,Q WKH UHPDLQGHU RI WKH GLVFXVVLRQ ZH VKDOO SULPDULO\ GHDO ZLWK WKH UHDO FRPSRQHQW [Wf 7KH VDPH DQDO\VLV KRZHYHU DSSOLHV HTXDOO\ ZHOO WR \Wf $ JUDSK RI 5[[Wf LV VKRZQ LQ )LJXUH 7KH SRZHU VSHFWUDO GHQVLW\ RI [Wf LV UHSUHVHQWHG E\ WKH )RXULHU WUDQVIRUP RI f \LHOGLQJ 6[[^If fÂ§ r UPAO ,, f RWKHUZLVH f $ JUDSK RI 6[[If LV VKRZQ LQ )LJXUH 1RWH WKDW [Wf LV EDQGOLPLWHG WR WKH IUHTXHQFLHV __ IP $V DQ H[DPSOH FRQVLGHU D FDUULHU IUHTXHQF\ RI IF 0+] DQG D UHFHLYHU VSHHG RI 9 PSK 7KH PD[LPXP 'RSSOHU IUHTXHQF\ ZRXOG EH IP +] )LJXUH VKRZV D PVHF VHJPHQW RI D VLPXODWHG IDGLQJ HQYHORSH JHQHUDWHG ZLWK WKHVH YDOXHV 1RWLFH IURP WKH SORW RI WKH PDJQLWXGH DWf WKDW GXULQJ WKLV LQWHUYDO VHYHUDO IDGHV DUH HQFRXQWHUHG LQFOXGLQJ WZR WKDW IDOO EHORZ G% FRPSDUHG ZLWK WKH DYHUDJH VLJQDO SRZHU 'XULQJ D GHHS IDGH DOPRVW QR VLJQDO HQHUJ\ UHDFKHV WKH UHFHLYHU $V D UXOH RI WKXPE VRPH NLQG RI IDGH ZLOO EH HQFRXQWHUHG DERXW RQFH HYHU\ KDOI ZDYHOHQJWK 6XSSRVH WKDW LQ WKH DERYH H[DPSOH WKH V\PERO UDWH IV LV VXFK WKDW IV P RU HTXLYDOHQWO\ WKH IDGLQJ SDUDPHWHUV FKDQJH OLWWOH RYHU D VLQJOH V\PERO LQWHUYDO 6XFK D FRQGLWLRQ LV FDOOHG VORZ IDGLQJ
PAGE 61
)LJXUH $XWRFRUUHODWLRQ IXQFWLRQ RI D PRELOH UDGLR IDGLQJ SDUDPHWHU IP )LJXUH 3RZHU VSHFWUDO GHQVLW\ RI D IDGLQJ SDUDPHWHU
PAGE 62
)LJXUH &RPSOH[ IDGLQJ HQYHORSH IP fÂ§ +] ,Q WKLV ILJXUH [Wf LV WKH UHDO SDUW \Wf LV WKH LPDJLQDU\ SDUW DWf LV WKH PDJQLWXGH DQG Wf LV WKH SKDVH
PAGE 63
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f E\ LWV GLVFUHWH YHUVLRQ XQ7Vf ZKHUH 7V LV WKH V\PERO LQWHUYDO DQG SDVV LW WKURXJK WKH IODWIDGLQJ FKDQQHO RI )LJXUH ,W LV PXOWLSOLHG E\ FQ7Vf [Q7Vf L\Q7f DQG WKHQ ZKLWH *DXVVLDQ QRLVH LV DGGHG 7KH ORZHU SRUWLRQ RI )LJXUH VKRZV D DGDSWLYH VFKHPH IRU HVWLPDWLQJ FQ7Vf $W WKH EHJLQQLQJ RI WKH WUDQVPLVVLRQ XQ7Vf LV VHQW WKURXJK ERWK WKH FKDQQHO DQG D VLQJOHZHLJKWf DGDSWLYH ILOWHU DW WKH UHFHLYHU 7KLV PD\ EH DFFRPSOLVKHG E\ XVH RI D SUHDUUDQJHG WUDLQLQJ VHTXHQFH 7KH RXWSXWV UQf DQG \Qf DUH WKHQ FRPSDUHG DQG WKH HVWLPDWH F^Q7Vf LV XSGDWHG EDVHG XSRQ WKH GLIIHUHQFH HQf :H VKDOO FDOO F^Q7Vf WKH FKDQQHO HVWLPDWH $Q\ FRPPRQ DGDSWLYH DOJRULWKP VXFK DV /06 RU 5/6 PD\ EH HPSOR\HG 7KH UHFHLYHG VLJQDO UQf LV WKHQ PXOWLSOLHG E\ WKH UHFLSURFRO RI FQ7Vf WR UHPRYH PDJQLWXGH DQG SKDVH GLVWRUWLRQ )LQDOO\ WKH GHWHFWRU FKRRVHV WKH V\PERO Qf LQ WKH VLJQDO DOSKDEHW WKDW LV FORVHVW WR LWQf 2QFH WKH FKDQQHO KDV EHHQ DFTXLUHG WKH V\PEROV LWQf ZLOO EH FRUUHFW ZLWK KLJK SUREDELOLW\ DQG PD\ EH XVHG LQ SODFH RI XQ7Vf WR WUDFN VORZ FKDQQHO YDULDWLRQV LQ GHFLVLRQGLUHFWHG PRGH 7KLV ZRUNV ZHOO DV ORQJ DV WKH HUURU UDWH LV
PAGE 64
)DGLQJ &KDQQHO )LJXUH $GDSWLYH FKDQQHO HVWLPDWLRQ IRU D IODWIDGLQJ FKDQQHO FXWV $ % DQG & DUH IRU UHIHUHQFH LQ )LJXUH f ORZ ,W LV QRW FOHDU KRZ KLJK WKH HUURU UDWH FDQ EH EHIRUH WKH DGDSWLYH DOJRULWKP ORVHV WUDFN EXW HVWLPDWHV LQ WKH OLWHUDWXUH UDQJH EHWZHHQ 3D DQG 3V fÂ§ VHH +D\NLQ > S @ 0DFFKL DQG (ZHGD >@ DQG 7UDEHOVL >@ IRU GLVFXVVLRQV LQ WKH FDVH RI GHFLVLRQGLUHFWHG HTXDOL]DWLRQ ZLWK ZKLFK WKLV SUREOHP EHDUV VRPH VLPLODULW\f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f 6LQFH >Qf GRHV QRW HTXDO XQ7Vf FQ7Vf LV
PAGE 65
DGMXVWHG LQFRUUHFWO\ %\ WKH WLPH WKH UHFHLYHU FRPHV RXW RI WKH IDGH FQ7ff PD\ KDYH ZDQGHUHG IDU HQRXJK IURP LWV RSWLPDO YDOXH WKDW WKH GHFLVLRQGLUHFWHG FKDQQHO HVWLPDWRU IDLOV WR FRQYHUJH :H SURSRVH D SUHGLFWLYH PHWKRG IRU RYHUFRPLQJ WKLV SUREOHP DV VKRZQ LQ )LJXUH :LWK WKH VZLWFKHV LQ WKH SRVLWLRQ VKRZQ ZKLOH DQ7Vf ?FQ7Vf? LV VXIILFLHQWO\ ODUJH ZH VHH WKH DGDSWLYH VFKHPH RSHUDWLQJ QRUPDOO\ LQ GHFLVLRQ GLUHFWHG PRGH H[FHSW WKDW SDVW HVWLPDWHV FQ7Vf DUH FORFNHG LQWR D WDSSHG GHOD\ OLQH :KHQ DQ7Vf ?FQ7Vf? IDOOV EHORZ VRPH WKUHVKROG YDOXH WKH VZLWFK SRVLWLRQV FKDQJH WXUQLQJ RII WKH DGDSWDWLRQ ZKLFK LV QR ORQJHU UHOLDEOH $ UHJUHVVRU YHFWRU RI UHOLDEOH SDVW HVWLPDWHV RI FQ7Vf LV ODWFKHG DQG WKH DGDSWLYH HVWLPDWH LV UHSODFHG ZLWK D SUHGLFWLRQ FQ7Vf [Q7Vf L\Q7Vf RI WKH FXUUHQW YDOXH RI FQ7Vf EDVHG XSRQ WKH YHFWRU RI FQ7Vf 1RWH WKDW LQ WKH ILJXUH U UHSUHVHQWV KRZ PXFK WLPH KDV SDVVHG VLQFH WKH DGDSWDWLRQ ZDV WXUQHG RII :KHQ FQ7Vf ULVHV DERYH WKH IDGLQJ WKUHVKROG LQ DEVROXWH YDOXH ZH UHWXUQ WR GHFLVLRQGLUHFWHG WUDFNLQJ XVLQJ WKH ODVW SUHGLFWHG YDOXH DV WKH VWDUWLQJ SRLQW IRU WKH DGDSWDWLRQ ,I WKH SUHGLFWLRQ LV FORVH HQRXJK WR WKH WUXH YDOXH RI FQ7Vf ZKHQ WKH UHFHLYHU HPHUJHV IURP WKH IDGH WKHQ LW PD\ EH SRVVLEOH WR FRQWLQXH WUDFNLQJ WKH FKDQQHO ZLWKRXW WUDQVPLWWLQJ D QHZ WUDLQLQJ VHTXHQFH 7KLV LGHD LV IXUWKHU LOOXVWUDWHG LQ )LJXUH &OHDUO\ WKH SHUIRUPDQFH RI WKH USUHGLFWRU LQ )LJXUH LV NH\ )RU PRELOH UDGLR ZH QRWH WKDW EHFDXVH WKH UHDO DQG LPDJLQDU\ SDUWV RI FWf [Wf L\Wf DUH YLHZHG DV LQGHSHQGHQW SURFHVVHV WKH SUREOHP UHGXFHV WR WKH WZR HTXLYDOHQW SUREOHPV RI SUHGLFWLQJ [Wf DQG \Wf :H ZLVK WR XVH WKH DQDO\VLV WHFKQLTXHV RI &KDSWHU WR GHWHUPLQH WKH FRQGLWLRQV XQGHU ZKLFK WKH PHWKRG RI )LJXUH FDQ ZRUN 7ZR WKHRUHWLFDO GLIILFXOWLHV SUHVHQW WKHPVHOYHV )LUVW VLQFH ZH ZDQW D SUHGLFWLRQ RI [Wf WKH SRZHU VSHFWDO GHQVLW\ 6[[Xf LV LPSRUWDQW DV ZDV PDGH FOHDU LQ WKH DQDO\VLV RI &KDSWHU
PAGE 66
)LJXUH 3UHGLFWLYH PHWKRG IRU PDLQWDLQLQJ FRUUHFW FKDQQHO WUDFNLQJ WKURXJK GHHS IDGH FXWV $ % DQG & UHIHU WR )LJXUH f
PAGE 67
7 f U _fÂ§ W 7KUHVK )LJXUH :KLOH LQ D GHHS IDGH GHFLVLRQGLUHFWHG FKDQQHO WUDFNLQJ GRHV QRW ZRUN ZHOO 8VLQJ SUHGLFWLRQ LQ WKLV UHJLRQ PD\ DOORZ WKH WUDFNHU WR FRQYHUJH FRUUHFWO\ RQFH WKH VLJQDO FRPHV RXW RI WKH IDGH
PAGE 68
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f ZDV NQRZQ RYHU WKH HQWLUH FRQWLQXXP RI D SRVLWLYHOHQJWK LQWHUYDO 8VLQJ WKH DGDSWLYH WHFKQLTXH RI )LJXUH KRZHYHU WKH VDPSOH IXQFWLRQ ZLOO EH NQRZQ RQO\ DW GLVFUHWH WLPHV VLQFH WKH DGDSWLYH HVWLPDWH RI FIf [Wf L\Wf LV XSGDWHG RQFH HYHU\ V\PERO LQWHUYDO $ OLQHDU SUHGLFWRU ZRXOG WKHUHIRUH EH D VXP RI WKH IRUP 7VA[Q7VfKN7V fÂ§ Q7Vf ZKHUH Q7V WDNHV YDOXHV RI W RQ WKH NQRZQ LQWHUYDO RI [ Wf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f ZLOO QRW EH RXWSHUIRUPHG E\ D GLVFUHWHWLPH SUHGLFWRU $OWKRXJK ZH KDYH QRW SURYHG WKLV RXU KHXULVWLF UHDVRQLQJ LV WKDW D FRQWLQXRXVWLPH SUHGLFWRU PDNHV XVH RI DOO WKH LQIRUPDWLRQ RQ WKH NQRZQ LQWHUYDO ZKHUHDV WKH GLVFUHWHWLPH SUHGLFWRU XVHV RQO\ D VXEVHW RI WKLV LQIRUPDWLRQ ([WHQGLQJ RXU DQDO\VLV WR GLFUHWHWLPH SUHGLFWLRQ DQG HVWDEOLVKLQJ D ULJRURXV FRQQHFWLRQ EHWZHHQ WKH GLVFUHWHWLPH DQG FRQWLQXRXVWLPH FDVHV DV ZHOO DV
PAGE 69
VROYLQJ WKH EDQGOLPLWHG OLQHDU SUHGLFWLRQ SUREOHP IRU D PRUH JHQHUDO FDVH WKDQ WKDW RI &KDSWHU UHPDLQ LPSRUWDQW UHVHDUFK REMHFWLYHV IRU XV (YHQ E\SDVVLQJ WKH WKHRUHWLFDO REMHFWLRQV RXU ZRUN RQ WKLV SUREOHP LV LQFRPSOHWH :H QHYHUWKHOHVV SUHVHQW D VLPSOH FDVH WR LOOXVWUDWH KRZ WKH WHFKQLTXHV RI &KDSWHU PD\ EH DSSOLHG LQ WKH DQDO\VLV RI D SUDFWLFDO SUREOHP 3UREOHP 6WDWHPHQW 7R XVH WKH UHVXOWV RI &KDSWHU ZH PXVW NQRZ VRPHWKLQJ RI WKH HUURU VWDWLVWLFV RI WKH DGDSWLYH HVWLPDWH FQ7Vf 7KLV UHTXLUHV DQ H[SOLFLW FKRLFH RI DQ DGDSWLYH DOJRULWKP LQ )LJXUH $OVR RXU FKRLFH RI WKH SDUDPHWHU U GHSHQGV XSRQ WKH GXUDWLRQ RI WKH IDGH WKDW LV WKH OHQJWK RI WLPH GXULQJ ZKLFK WKH SUREDELOLW\ RI HUURU LQ Qf LV KLJK ,Q RUGHU WR FDOFXODWH WKH SUREDELOLW\ RI HUURU ZH PXVW GHFLGH XSRQ D PRGXODWLRQ VFKHPH 7KHVH VWHSV DUH FDUULHG RXW LQ WKH IROORZLQJ VHFWLRQV ,Q HDFK FDVH WKH FKRLFHV DUH PDGH ZLWK D YLHZ WRZDUG VLPSOLI\LQJ WKH DQDO\VLV LQ WKH KRSH WKDW WKH UHVXOWLQJ SUHVHQWDWLRQ ZLOO SURYLGH FOHDUHU LQVLJKW LQWR WKH IXQGDPHQWDO LVVXHV )RU QRWDWLRQDO FRQYHQLHQFH LQ ZKDW IROORZV ZH GURS WKH H[SOLFLW GHSHQGHQFH RQ 7V DQG UHIHU HJ WR XQf DQG FQf :H QRZ RIIHU WKH IROORZLQJ SUREOHP VWDWHPHQW &RQVLGHU WKH DGDSWLYHSUHGLFWLYH FKDQQHO HVWLPDWLRQ PHWKRG RI )LJXUHV DQG $VVXPH WKDW WKH LQSXW VHTXHQFH XQf LV XQFRUUHODWHG 436. ZKRVH V\PEROV DUH FKRVHQ ZLWK HTXDO SUREDELOLW\ IURP WKH VHW XQf L\(EL \(E fÂ§ L\(E a\(E a L\(E a\(aE L\(E` ZKHUH (E LV WKH WUDQVPLWWHG HQHUJ\ SHU ELW 7KXV ?XQf? ?(E IRU DOO Q 7KH IDGLQJ HQYHORSH FQf LV D ]HURPHDQ FRPSOH[ *DXVVLDQ SURFHVV ZKRVH UHDO DQG LPDJLQDU\ SDUWV DUH LQGHSHQGHQW DQG LGHQWLFDOO\ GLVWULEXWHG DQG WKH VLJQDOWRQRLVH UDWLR DW WKH UHFHLYHU LQSXW WLPH DYHUDJHG WR LQFOXGH WKH HIIHFWV RI IDGLQJ LV VXFK WKDW WKH RYHUDOO ELW HUURU UDWH LV 3E
PAGE 70
:H DVVXPH WKDW GHFLVLRQGLUHFWHG WUDFNLQJ PD\ SURFHHG DV ORQJ DV WKH LQVWDQWDQHRXV V\PERO HUURU UDWH GRHV QRW ULVH DERYH 3V :KHQ WKLV WKUHVKROG LV FURVVHG ZH FKDQJH SRVLWLRQ LQ )LJXUH IURP DGDSWDWLRQ WR SUHGLFWLRQ :H ZLVK WR UHVXPH DGDSWDWLRQ DV VRRQ DV WKH UHFHLYHU HPHUJHV IURP WKH IDGH VR IRU WKH SDUDPHWHU U ZH FKRRVH WKH PHDQ IDGH GXUDWLRQ WR EH FDOFXODWHG LQ 6HFWLRQ :H ZLVK WR XVH D SUHGLFWRU WKDW PLQLPL]HV WKH PHDQ VTXDUHG HUURU EDVHG RQ UHOLDEOH HVWLPDWHV RI FQf VLQFH WKH WLPH WKH UHFHLYHU HPHUJHG IURP WKH ODVW IDGH VR IRU WKH SDUDPHWHU 7 ZH FKRRVH WKH PHDQ WLPH DERYH WKH IDGH WKUHVKROG DOVR WR EH FDOFXODWHG LQ 6HFWLRQ :H ZLVK WR NQRZ XQGHU ZKDW FRQGLWLRQV WKLV DSSURDFK ZLOO DOORZ XV WR UHFRYHU FRUUHFW FKDQQHO WUDFNLQJ ZLWK D SUREDELOLW\ RI 3U (UURU 6WDWLVWLFV RI D 6LPSOH $GDSWLYH $OJRULWKP $V D ILUVW VWHS LQ DGGUHVVLQJ WKLV SUREOHP ZH FKRRVH D VLPSOH DGDSWLYH DOJRULWKP 7KHQ ZH ZLOO GHWHUPLQH WKH VWDWLVWLFDO EHKDYLRU RI FQf WKH DGDSWLYH HVWLPDWH RI FQf [Qf L\>Qf DV ZHOO DV WKH HIIHFW RI HVWLPDWLRQ HUURUV RQ Qf WKH LQSXW WR WKH GHWHFWRU 6LQFH ZH RQO\ XVH DGDSWDWLRQ ZKHQ WKH HUURU UDWH LV ORZ 3V f ZH PD\ DVVXPH LQ ZKDW IROORZV WKDW Qf XQf /RRNLQJ DW )LJXUH ZH VHH WKDW UQf XQfFQf YQf f DQG 1RZ XQf FQfU Qf FQf FQf XQf YQf FQfn HQf UQf fÂ§ \Qf >XQfFQf XQf@ fÂ§ XQfFQf 7KXV VROYLQJ IRU FQf ZH KDYH FQf FQf YQf XQf HUFf X^Qfn f f f
PAGE 71
$V DQ DOJRULWKP IRU XSGDWLQJ RXU HVWLPDWH RI FQf ZH FKRRVH LV ? HQf YQf FQ Of FQf +fÂ§MU FQf XQf aY n XQf 7KLV FKRLFH VLPSOLILHV WKH FKDUDFWHUL]DWLRQ RI HVWLPDWLRQ HUURUV 7R VHH WKLV QRWH WKDW LI WKH YDULDWLRQ LQ FQf LV VORZ HQRXJK WKHQ WKH HUURU LV GRPLQDWHG E\ WKH Y^Qf f X^Qf WHUP 7KXV FQf FQf HQf FQf YQ fÂ§ Of ZKHUH HQf FQf fÂ§ FQf XQ fÂ§ Of f YQ fÂ§ Of f f XQ fÂ§ Off 1RZ VLQFH XQf LV DQ XQFRUUHODWHG 436. VHTXHQFH ZLWK _XQf_ ?I(L LW LV FOHDU IURP f WKDW HQf LV DSSUR[LPDWHO\ D ]HURPHDQ ZKLWH *DXVVLDQ QRLVH VHTXHQFH ZLWK RI A ZKHUH RI L^_FQf_` DQG DO e^_XQf_` (W 6XEVWLWXWLQJ f LQ OOf ZH KDYH f ? FQf ? YQf XQf fÂ§cfÂ§U AXQf FQf HQf FQf HQfn f 6LQFH ZH DUH FRQVLGHULQJ WKH FDVH LQ ZKLFK WKH HVWLPDWLRQ HUURUV HQf DQG DGGLWLYH QRLVH YQf DUH QRW ODUJH HQRXJK WR FDXVH D KLJK V\PERO HUURU UDWH ZH DVVXPH _FQf_ } _HQf_ DQG _FQf_ ?YQf? 7KXV FAf B L B L0 FQf HQf FQf f f ZKHUH WKH ILUVWRUGHU WHUP LV UHWDLQHG EHFDXVH WKH ]HURWKRUGHU WHUP ZLOO EH VXEWUDFWHG RXW VKRUWO\ $OVR FQf HQf FQfn 6XEVWLWXWLQJ f DQG f LQWR f ZH REWDLQ XQf HQf FQfB XQf YQf FQfn f f
PAGE 72
:H GHILQH HQf YQf Qf fÂ§ XQf XQf UV fÂ§XQf U7 f FQf F^Qf 7KHQ KROGLQJ DQf _FQf_ FRQVWDQW ^_ÂQf_` a e ^@Af@ >aHQfXQf XQf@ >HrQfWLrQf YrQf@_ ^_HQfWWQf_ HQfXQfYrQf HrQfXrQfXQf ?YQf_` f OFQfU 1RZ ^_HQfXQf_` _XQf_e ^_HQf_` (EDH f )XUWKHU VLQFH HQf LV D IXQFWLRQ RI SDVW YDOXHV RI FQf XQf DQG YQf LW LV XQFRUUHn ODWHG ZLWK XQf DQG YQf 7KXV e^HQfXQfXrQf` e^HrQfXrQfXQf` f 7KHUHIRUH e ^_ÂQf_` IWn (A A DQf 7KLV LV WKH HIIHFWLYH QRLVH SRZHU DW WKH LQSXW RI WKH GHWHFWRU WDNLQJ LQWR DFFRXQW WKH HVWLPDWLRQ HUURUV LQ FQf :H GHILQH WKH UHFHLYHG 615 SHU ELW DV DQf(E f EQf R f 7KHQ DFFRXQWLQJ DJDLQ IRU WKH HUURUV LQ FQf WKH HIIHFWLYH 615 SHU ELW DW WKH LQSXW RI WKH GHWHFWRU LV LU? f DQf(E DfQf(W f LQf N r fÂ§Â‘ ,I ZH OHW D EH WKH WLPHDYHUDJH YDOXH RI DQf WKHQ WKH WLPHDYHUDJH 615V DUH fÂ§ RL(E B RW(E B c a (ED D a a a f
PAGE 73
7KH RYHUDOO ELW HUURU UDWH LV JLYHQ DSSUR[LPDWHO\ E\ > S @ 3E Â m ZKHUH WKH HIIHFWLYH DYHUDJH 615 ZDV HPSOR\HG 7KLV \LHOGV a W m f f 'HFLVLRQGLUHFWHG DGDSWDWLRQ PD\ FRQWLQXH DV ORQJ DV WKH V\PERO HUURU UDWH LV OHVV WKDQ 3V 7KH YDOXH RI DQf WKDW DWWDLQV WKLV HUURU UDWH LV FDOOHG WKH IDGH WKUHVKROG ZKLFK ZH PD\ ZULWH DV FWc 5HSUHVHQWLQJ WKH FRUUHVSRQGLQJ 615 E\ ZH KDYH B D`(E B RLI(E f L B (ER? RO a RO a f 6LQFH ZH DUH LQWHUHVWHG LQ WKH V\PERO HUURU UDWH RQO\ DW WKH LQVWDQW ZKHQ DQf FURVVHV WKH WKUHVKROG D ZH GR QRW XVH DYHUDJLQJ ,I ZH DVVXPH WKDW LV KLJK HQRXJK WKDW WKH SUREDELOLW\ RI D VLPXOWDQHRXV HUURU LQ ERWK WKH UHDO DQG LPDJLQDU\ SDUWV RI Qf LV QHJOLJLEO\ VPDOO ZH KDYH > S @ 3V m 3U VLQJOHELW HUURUf AAf m f 7KLV \LHOGV a a f 6WDWLVWLFV RI WKH )DGLQJ (QYHORSH 'HFLVLRQGLUHFWHG DGDSWDWLRQ EHFRPHV XQUHOLDEOH ZKHQ DQf WKH PDJQLWXGH RI WKH IDGLQJ HQYHORSH IDOOV EHORZ WKH IDGLQJ WKUHVKROG D :KHQ WKLV KDSSHQV DV ZH KDYH GLVFXVVHG ZH VZLWFK IURP DGDSWDWLRQ WR SUHGLFWLRQ LQ )LJXUH :H ZLVK WR UHVXPH DGDSWDWLRQ DV VRRQ DV DQf ULVHV DERYH 2LM DJDLQ ,I RXU SUHGLFWHG YDOXH FQf
PAGE 74
LV FORVH HQRXJK WR FQf DW WKDW LQVWDQW ZH PD\ EH DEOH WR UHFRYHU FRUUHFW FKDQQHO WUDFNLQJ 5HFDOO WKDW WKH SDUDPHWHU U UHSUHVHQWV KRZ IDU LQ WKH IXWXUH ZH ZLVK WR SUHGLFW FLf ZKHUH W Q7V ZLWK UHVSHFWV WR WKH LQWHUYDO RI DGDSWLYHO\ HVWLPDWHG YDOXHV ,I WKLV LQWHUYDO LV FRQVLGHUHG WR HQG DW WKH LQVWDQW WKDW DQf FURVVHV EHORZ FWI WKHQ U VKRXOG EH FKRVHQ WR HTXDO WKH H[SHFWHG IDGH GXUDWLRQ WKDW LV WKH H[SHFWHG WLPH LQWHUYDO GXULQJ ZKLFK DQf UHPDLQV EHORZ Dc :H FDOO WKH H[SHFWHG IDGH GXUDWLRQ WI ,W PD\ EH FRPSXWHG E\ > S @ r! f! ZKHUH f DQG UQIP fÂ§ 6[[XfXQGXM f UP :H FRQVLGHU [Wf WR EH D EDQGOLPLWHG SURFHVV ZLWK D IODW VSHFWUDO GHQVLW\ 6LQFH ZH DUH RQO\ LQWHUHVWHG LQ WKH UDWLR RI E WR LQ f WKH VFDOLQJ RI f LV DUELWUDU\ 7KXV ZH OHW ÂAXf XM >fÂ§UP UP@ :H KDYH f DQG 7KXV DQG If WW fUIP .XGX fÂ§ rIP 7 / 8 rIP 7 2 77 QIP R 0R WW .IPf ? VaI.IP [IU f f f
PAGE 75
5HFDOO IURP 6HFWLRQ WKDW ZH ZLVK WR EDVH RXU SUHGLFWLRQ RQ UHOLDEOH DGDSWLYH HVWLPDWHV RI FQf VLQFH WKH UHFHLYHU HPHUJHG IURP WKH ODVW IDGH :H DUH WKXV LQWHUHVWHG LQ WKH PHDQ WLPH DERYH WKH IDGH WKUHVKROG ZKLFK ZH VKDOO FDOO WM 7R FRPSXWH WM ZH QRWH WKDW >@ 1U BU nfÂ§cUH 7 WI WM LV WKH DYHUDJH QXPEHU RI WLPHV SHU VHFRQG WKDW DQf FURVVHV ROM LQ WKH SRVLWLYH GLUHFWLRQ :H WKXV KDYH f 0DNLQJ XVH RI f DQG f ZH REWDLQ ER U f WW a 0R O t U 9 UP Un 1RWH WKDW WDNLQJ WKH UDWLR RI f DQG f ZH KDYH f A HU f WI 7KLV PD\ EH LQWHUSUHWHG DV WKH UDWLR RI WKH PHDQ WLPH LQ WKH IDGH WR WKH PHDQ WLPH RXW RI WKH IDGH )URP f ZH VHH WKDW WKLV UDWLR GHSHQGV RQ WKH 615 DW WKH IDGH WKUHVKROG EM DQG WKH WLPHDYHUDJH 615 ,W GRHV QRW GHSHQG RQ WKH PD[LPXP 'RSSOHU IUHTXHQF\ IP RU WKH VKDSH RI WKH SRZHU VSHFWUDO GHQVLW\ 6[[ X!f 7KHVH IDFWV ZLOO HDVH RXU FRPSXWDWLRQV ODWHU VHH 6HFWLRQ f 5HFRYHU\ RI &KDQQHO 7UDFNLQJ :H QRZ WXUQ RXU DWWHQWLRQ WR WKH LQVWDQW DW ZKLFK WKH UHFHLYHU HPHUJHV IURP WKH IDGH DQG DVN XQGHU ZKDW FRQGLWLRQV FDQ ZH HQVXUH UHFRYHU\ RI FRUUHFW FKDQQHO WUDFNLQJ ZLWK D SUREDELOLW\ 35 /HW XV FRQVLGHU WKH ODVW SUHGLFWHG VDPSOH FQf EHIRUH VZLWFKLQJ EDFN WR DGDSWDWLRQ LQ )LJXUH )RU WKDW VDPSOH ZH GHILQH HQf UQf fÂ§ \^Qf >XQfFQf YQf@ fÂ§ XQfFQf f
PAGE 76
6ROYLQJ IRU FQf ZH KDYH FQf fÂ§ FQf XQf XQf f :H XVH WKLV YDOXH LQVWHDG RI FQf LQ WKH IDGLQJ FRPSHQVDWRU LPPHGLDWHO\ SUHFHGLQJ WKH GHWHFWRU LQ )LJXUH 6XSSRVH WKDW WKLV UHVXOWV LQ WKH GHWHFWRU PDNLQJ WKH FRUUHFW GHFLVLRQ Qf :H VZLWFK EDFN WR DGDSWDWLRQ DQG XVLQJ RXU DGDSWLYH DOJRULWKP f ZH KDYH FQ f FQf FQf f X>Qf XQf :H VHH WKDW DV ORQJ DV ZH UHVWDUW DGDSWDWLRQ ZLWK D FRUUHFW GHFLVLRQ Qf WKH HIIHFW RI WKH fELJf HUURU HQf GLVDSSHDUV DIWHU RQH V\PERO DQG VLQFH DQf LV DERYH WKH IDGH WKUHVKROG D DGDSWDWLRQ PD\ SURFHHG DV LQ 6HFWLRQ DQG ZH FRQVLGHU FRUUHFW FKDQQHO WUDFNLQJ WR EH UHFRYHUHG 7KXV D VXIILFLHQW FRQGLWLRQ IRU UHFRYHU\ RI FRUUHFW FKDQQHO WUDFNLQJ LV WKH FRUUHFW GHWHFWLRQ RI WKH ODVW UHFHLYHG V\PERO EHIRUH VZLWFKLQJ EDFN IURP SUHGLFWLRQ WR DGDSWDWLRQ 7R TXDQWLI\ WKLV FRQGLWLRQ ZH GHILQH HQf FQf fÂ§ FQf f ZKLFK LV WKH IDGLQJHQYHORSH SUHGLFWLRQ HUURU DW WKH LQVWDQW WKH UHFHLYHU HPHUJHV IURP WKH IDGH 1RZ ZKHQ FQf LV XVHG LQ WKH IDGLQJ FRPSHQVDWRU WKHQ WKH LQSXW WR WKH GHWHFWRU PD\ EH ZULWWHQ DV Qf FQf FQf XQf YQf FQf XULf eQf FQf XQf YQf FQff EHFDXVH FQf FQf HQf :H GHILQH f 6Qf Qf fÂ§ XQf fÂ§ FQf YQf FQfn f
PAGE 77
7KH SUHGLFWLRQ FQf LV WR EH EDVHG RQ DGDSWLYH HVWLPDWHV FQf D SURFHVV ZKLFK LV DSSUR[LPDWHO\ ]HURPHDQ *DXVVLDQ DV ZDV GLVFXVVHG LQ 6HFWLRQ 7KXV WKH PLQLPXP PHDQ VTXDUHG HUURU SUHGLFWRU LV OLQHDU 6LQFH FQf LV D OLQHDU WUDQVIRUPDWLRQ RI D *DXVVLDQ SURFHVV FQf LV DOVR *DXVVLDQ DQG VLQFH WKH UHDO DQG LPDJLQDU\ SDUWV RI FQf DUH LQGHSHQGHQW DQG LGHQWLFDOO\ GLVWULEXWHG ZLWK ]HUR PHDQ WKH VDPH KROGV IRU FQf 7KHUHIRUH WKH SKDVH RI FQf LV XQLIRUPO\ GLVWULEXWHG RQ > 7 U@ 7KH *DXVVLDQ ZKLWH QRLVH YQf LV XQFRUUHODWHG ZLWK DQG KHQFH LQGHSHQGHQW RI FQf ZKLFK LV D IXQFWLRQ RI SDVW YDOXHV RI FQf 7KH SUHGLFWLRQ HUURU HQf FQf fÂ§ FQf LV D VXP RI *DXVVLDQ UDQGRP YDULDEOHV WKXV HQf LV *DXVVLDQ )URP WKH WKHRU\ RI PHDQ VTXDUH HVWLPDWLRQ FQf LV WKH PHDQ RI FQf FRQGLWLRQHG RQ WKH DGDSWLYH HVWLPDWHV FQf VR HQf LV ]HUR PHDQ $OVR HQf PXVW EH RUWKRJRQDO WR FQf > S @ 6LQFH HQf LV ]HURPHDQ *DXVVLDQ RUWKRJRQDOLW\ LPSOLHV LQGHSHQGHQFH $W WKH LQVWDQW WKH UHFHLYHU HPHUJHV IURP WKH IDGH ZH KDYH _FQf_ DI EXW WKH SKDVH RI FQf LV VWLOO UDQGRPL]HG 7KXV DQG DUH LQGHSHQGHQW ]HUR PHDQ FRPSOH[ *DXVVLDQ UDQGRP YDULDEOHV ZLWK YDULDQFHV A DQG S UHVSHFWLYHO\ )URP f ZH WKHUHIRUH KDYH UHFDOOLQJ _XQf_ ?(E DW WKH LQVWDQW WKH UHFHLYHU HPHUJHV IURP WKH IDGH 7KH 615 SHU ELW LV B (E BB D`(E rAI rO D@ (EDM DO A b (E 7R DWWDLQ 3F ZH PXVW KDYH > S @ f f 3F fÂ§ 4 9 f
PAGE 78
7KLV \LHOGV f ,Q WKH QH[W VHFWLRQ ZH VKDOO VHH WKDW LW LV XQQHFHVVDU\ WR FRPSXWH WKH YDOXH RI RI H[SOLFLWO\ 3UHGLFWDELOLW\ $QDO\VLV ,Q WKLV VHFWLRQ ZH VKDOO XVH WKH SUHGLFWDELOLW\ DQDO\VLV WHFKQLTXHV GHYHORSHG LQ &KDSWHU WR GHWHUPLQH XQGHU ZKDW FRQGLWLRQV WKH PHDQ VTXDUHG SUHGLFWLRQ HUURU FDQ EH PDGH OHVV WKDW RI DV JLYHQ E\ f 1RZ OHW PLQ^(f EH WKH PLQLPXP PHDQ VTXDUHG SUHGLFWLRQ HUURU RI [Wf WKH UHDO SDUW RI FWf :H QRWH WKDW WKH DGDSWLYH HVWLPDWLRQ HUURUV HQf DQG WKH SUHGLFWLYH HVWLPDWLRQ HUURU ÂQf DUH FRPSOH[ SURFHVVHV ZLWK WKHLU SRZHUV VSOLW HYHQO\ EHWZHHQ WKH UHDO DQG LPDJLQDU\ SDUWV 8VLQJ WKH DSSURDFK RI 6HFWLRQ ZH ZDQW FI (T f f 1RZ D e ^_FLf_` LV WZR WLPHV WKH DYHUDJH SRZHU RI [Wf 7KXV LI PP^(f LV WKH PLQLPXP PHDQ VTXDUHG SUHGLFWLRQ HUURU RI D QRUPDOL]HG SURFHVV ZH KDYH f RU f 1RZ IURP f ZH KDYH f
PAGE 79
DQG VLQFH IURP 6HFWLRQ ZH NQRZ WKDW (EMÂ LV DSSUR[LPDWHO\ HTXDO WR WKH QRLVH YDULDQFH ZH XVH f WR REWDLQ $OVR IURP f ZH KDYH (E D f DM D E (E D EI fÂ‘ t 9 f f ZKHUH XVH ZDV PDGH RI f f DQG f 6XEVWLWXWLQJ f DQG f LQWR f ZH KDYH Lf 1RZ E E DQG b ZHUH GHWHUPLQHG LQ SUHYLRXV VHFWLRQV :H ZLVK WR NQRZ XQGHU ZKDW FRQGLWLRQV ZLOO EH OHVV WKDQ RU HTXDO WR (f DV JLYHQ E\ f IRU DW OHDVW VRPH YDOXH RI ( ,W ZLOO EH UHFDOOHG WKDW PP(f GHSHQGV LPSOLFLWO\ RQ 2 7 DQG U ZKHUH LV WKH EDQG OLPLW RI WKH SURFHVV [Wf 7 LV WKH OHQJWK RI WKH LQWHUYDO RI NQRZQ YDOXHV RQ ZKLFK WKH SUHGLFWLRQ LV WR EH EDVHG DQG U LV KRZ IDU LQ WKH IXWXUH ZH ZLVK WR PDNH WKH SUHGLFWLRQ ,Q DFFRUGDQFH ZLWK WKH GLVFXVVLRQ LQ 6HFWLRQ WKH IDGLQJ HQYHORSH FWf [Wf L\Wf LV EDQGOLPLWHG WR WKH PD[LPXP 'RSSOHU IUHTXHQF\ IP :H WKXV FKRRVH 4 fÂ§ UP 7KH SUDPHWHUV U DQG 7 DUH VHW HTXDO WR WKH PHDQ IDGH GXUDWLRQ W DQG WKH PHDQ WLPH DERYH WKH IDGH WKUHVKROG WM UHVSHFWLYHO\ DV GLVFXVVHG LQ 6HFWLRQ 1RZ QRWH IURP f WKDW Oa 47 f 9 OUUQ U 9 U f
PAGE 80
ZKHUH F LV WKH WLPHEDQGZLGWK SURGXFW DQG U JLYHQ E\ f LV DOUHDG\ GHWHUn PLQHG $OVR XVLQJ f ZH PD\ GHILQH f :H VHH IURP WKLV WKDW LI RQH RI WKH WKUHH SDUDPHWHUV Â 7 RU U LV NQRZQ WKH RWKHU WZR PD\ EH GHWHUPLQHG IURP f RU f 7KXV RQO\ RQH RI WKH WKUHH PD\ EH FRQVLGHUHG WR EH D IUHH YDULDEOH :H FKRRVH Â DQG XVH WKH QRWDWLRQ mPLQ' (f WR H[SOLFLWO\ LQGLFDWH WKH GHSHQGHQFH RI WKH PLQLPXP PHDQ VTXDUHG SUHGLFWLRQ HUURU RQ WKLV SDUDPHWHU :H VHH IURP f WKDW FRUUHFW FKDQQHO WUDFNLQJ ZLOO EH UHFRYHUHG LI f IRU VRPH ( 1RWH WKDW WKLV FRQGLWLRQ GHSHQGV RQO\ RQ Â ZKLFK LV SURSRUWLRQDO WR WKH PD[LPXP 'RSSOHU IUHTXHQF\ IP &RPSXWDWLRQDO ,VVXHV ,W LV LPSRUWDQW WR QRWH WKDW WKH UHODWLRQVKLS EHWZHHQ 7 DQG U LQGLFDWHG E\ f DQG f RIIHUV D FRPSXWDWLRQDO DGYDQWDJH 7R VHH WKLV ZH FRQVLGHU WKH 7 RSWLPL]DWLRQ SUREOHP f :H VFDOH WKH REMHFWLYH IXQFWLRQ E\ fÂ§ IRU QRUPDOL]DWLRQ X / 8f FI (T f WKHQ VXEVWLWXWH XMn fÂ§ WR REWDLQ X / f 7 7 1RZ OHW F DQG G fÂ§ EH IL[HG QXPEHUV 7KHQ OHWWLQJ Wn 4W WKH WLPH FRQVWUDLQW PD\ EH ZULWWHQ K Wn >FG FG f@ f
PAGE 81
$OVR ZLWK WKH VXEVWLWXWLRQ Wn 4W WKH HQHUJ\ FRQVWUDLQW EHFRPHV \\^4GWn m! 1RZ OHW *XMf fÂ§ +4XMf J^Wf AKAf 2XU RSWLPL]DWLRQ SUREOHP EHFRPHV UO PLQLPL]H fÂ§ *LYf?FNX VXEMHFW WR JWf W e >FG FG f@ f 22 S P D :H VHH IURP WKLV WKDW f 7KXV ZH QHHG XVH WKH SURFHGXUH f RQO\ RQFH VFDOLQJ DSSURSULDWHO\ WR REWDLQ LPLQO (f WKHQ XVH f WR FRPSXWH (f IRU DUELWUDU\ Âf :H UHFDOO IURP 6HFWLRQ VHH SDJH f WKDW [MQWf GHSHQGV LPSOLFLWO\ RQ 4 DQG 7 :H PD\ WKHUHIRUH ZULWH f DV b!f 7 L W If f 1RZ ZH OHW DQG QRWH IURP f WKDW 7 F $OVR IURP f ZH KDYH fÂ§U fÂ§ I FfÂ§G fÂ§ f f 7KXV PDNLQJ XVH RI $f IURP WKH DSSHQGL[ ZH KDYH f OFFfÂ§GOff AÂ>6aFf@.fFO cf@ f ZKHUH 6QFU@f DQG 5AQF ef DUH WKH DQJXODU DQG UDGLDO IXQFWLRQV GLVFXVVHG LQ WKH DSSHQGL[ 1RWH WKDW LQ f ZH XVH WKH IDFW WKDW LS7Wf LV HYHQ LQ W 2XU
PAGE 82
FRPSXWDWLRQDO SURFHGXUH WKHQ EHFRPHV 67(3 6ROYH A AF FG Off$QFf M ? ? ( IRU Q 9$QFf9 67(3 7KHQ A(f Q If A F FG ff IO f Q 9 $QFf Wrf 7 ZKHUH WKH OHDGLQJ IDFWRU U fÂ§ LQ VWHS LV QHFHVVDU\ EHFDXVH P?Q^O(f LV G / QRUPDOL]HG )RU WKLV SUREOHP ZH XVH f f DQG f WR FRPSXWH U 7KHQ XVLQJ f DQG f ZH FRPSXWH F DQG G )RU RXU FRPSXWDWLRQV ZH URXQG F WR DQG G WR 6XPPDU\ RI $QDO\VLV 3URFHGXUH DQG 'LVFXVVLRQ $ VXPPDU\ RI RXU DQDO\VLV SURFHGXUH LV JLYHQ LQ 7DEOH 8VLQJ YDOXHV RI t DQG FRPSXWHG LQ f f DQG f DQG DOVR XVLQJ URXQGHG YDOXHV RI F DQG G ZH REWDLQ WKH JUDSK RI )LJXUH :H VHH WKDW ZH PD\ UHFRYHU FRUUHFW FKDQQHO WUDFNLQJ ZLWK SUREDELOLW\ 3U IRU YDOXHV RI 2 OHVV WKDW RU HTXDO WR 7KLV LV D PD[LPXP 'RSSOHU IUHTXHQF\ IP fÂ§ RI OHVV WKDQ Q RU HTXDO WR +] 7KLV GRHV QRW LQGLFDWH JRRG SHUIRUPDQFH EHFDXVH LQ PRELOH UDGLR WKH YDOXH RI IP FDQ EH PXFK KLJKHU RQ WKH RUGHU RI WHQV RI KHUW] $V VWDWHG SUHYLRXVO\ WKRXJK RXU ZRUN RQ WKLV SUREOHP LV LQFRPSOHWH 7KH FDVH SUHVHQWHG KHUH LV LQWHQGHG PDLQO\ DV DQ LOOXVWUDWLRQ RI KRZ WKH WHFKQLTXHV GHYHORSHG LQ SUHYLRXV FKDSWHUV FDQ EH DSSOLHG WR WKH DQDO\VLV RI D SUDFWLFDO SUREOHP 7KHUH DUH PDQ\ DSSURDFKHV ZH PLJKW WDNH WR LPSURYH WKH SHUIRUPDQFH RI RXU DGDSWLYHSUHGLFWLYH FKDQQHO WUDFNHU )RU H[DPSOH LQVWHDG RI UHVWDUWLQJ WKH DGDSWDWLRQ DV VRRQ DV RXU SUHGLFWLRQ FQf FURVVHV WKH IDGLQJ WKUHVKROG ZH PLJKW ZDLW D IHZ V\PERO LQWHUYDOV IRU WKH VLJQDO WR VWUHQJWKHQ ,I WKH TXDOLW\ RI RXU SUHGLFWLRQ KDV QRW GHJUDGHG WRR PXFK ZH PLJKW KDYH D EHWWHU FKDQFH RI FRUUHFW UHFRYHU\ DW WKH KLJKHU 615 :H DOVR QRWH WKDW RXU WHFKQLTXHV RQO\ DOORZ XV WR
PAGE 83
7DEOH 6XPPDU\ RI 3UHGLFWDELOLW\ $QDO\VLV 3URFHGXUH %DVHG RQ WKH GHVLUHG RYHUDOO ELW HUURU UDWH XVH f DQG f WR GHWHUPLQH r %DVHG RQ WKH PD[LPXP ELW HUURU UDWH WR PDLQWDLQ FRUUHFW GHFLVLRQGLUHFWHG FKDQQHO WUDFNLQJ XVH f DQG f WR GHWHUPLQH bM %DVHG RQ WKH GHVLUHG SUREDELOLW\ RI UHFRYHULQJ FRUUHFW FKDQQHO WUDFNLQJ DIWHU D GHHS IDGH 3U XVH f WR GHWHUPLQH b 8VH f WR GHWHUPLQH U 7KHQ XVH f DQG f WR GHWHUPLQH F DQG G 8VLQJ f IRU ( SORW DOO SRVLWLYH YDOXHV RI (f RQ D JUDSK 8VLQJ f DQG f RQH WLPH DQG WKHQ f ILQG D YDOXH 4 VXFK WKDW LPLQAf (f SORWWHG DJDLQVW ( LV WDQJHQW WR WKH FXUYH IRU (f ,I WKH PD[LPXP 'RSSOHU IUHTXHQF\ IP LV OHVV WKDQ RU HTXDO WR ILU FRUUHFW FKDQQHO WUDFNLQJ PD\ EH UHFRYHUHG ZLWK SUREDELOLW\ 3U
PAGE 84
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f $GGUHVVLQJ WKH 0RGHO 0LVPDWFK $V ZDV PHQWLRQHG LQ 6HFWLRQ WKH VSHFWUDO VKDSH RI )LJXUH ZKLFK LV WKH PRVW FRPPRQO\ XVHG LQ PRGHOLQJ WKH PRELOH UDGLR IDGLQJ HQYHORSH GRHV QRW PDWFK WKH IODW VSHFWUXP DVVXPHG LQ &KDSWHU ,Q WKLV VHFWLRQ ZH FRQVLGHU DOWHUQDWLYHV IRU DGGUHVVLQJ WKH PRGHO PLVPDWFK
PAGE 85
)LUVW OHW XV FRQVLGHU WKH SURVSHFWV RI H[WHQGLQJ WKH VROXWLRQ RI 6HFWLRQ WR SURFHVVHV ZLWK VSHFWUDO GHQVLWLHV RI WKH IRUP f )RU VLPSOLFLW\ OHW 6[[ f fÂ§ r ?O fÂ§ 8f 0 L RWKHUZLVH f 6XEVWLWXWLQJ LQWR WKH REMHFWLYH IXQFWLRQ RI f DQG QRWLQJ WKDW Â ZH REWDLQ UEAnA f ,Q 6HFWLRQ ZH H[SDQGHG +RMf LQ WHUPV RI WKH SURODWH VSKHURLGDO ZDYH IXQFn WLRQV DOORZLQJ XV WR FRQYHUW WKH REMHFWLYH IXQFWLRQ LQWR DQ DOJHEUDLF H[SUHVVLRQ 8QIRUWXQDWHO\ WKH SURODWH IXQFWLRQV DUH QRW RUWKRJRQDO ZLWK UHVSHFWV WR WKH ZHLJKWLQJ IXQFWLRQ 6[[Xf DV JLYHQ E\ f :H PD\ FRQVLGHU H[SDQGLQJ +Xf LQ WHUPV RI IXQFWLRQV VXFK DV WKH 7FKHE\ FKHII SRO\QRPLDOV ZKLFK KDYH WKH DSSURSULDWH RUWKRJRQDOLW\ SURSHUWLHV > S @ $OWHUQDWLYHO\ ZH PD\ VXEVWLWXWH FRV ?O fÂ§ X DQG VLQ FX LQ f WR REWDLQ Aa Â +^VLQf?G f 7 r 7KH IXQFWLRQ *^f +^VLQf PD\ QRZ EH H[SDQGHG LQ WHUPV RI DQ\ FRPSOHWH RUWKRJRQDO VHW RQ >fÂ§ii@f 7KHVH PD\ EH WKH SURODWH IXQFWLRQV /HJHQGUH IXQFWLRQV RU VLPSO\ VLQXVRLGV 7KH IXQFWLRQ KWf ZLOO DOVR EH H[SDQGHG LQ WHUPV RI DQ RUWKRJRQDO VHW RQ >U U 7@ VR HYDOXDWLQJ WKH H[SDQVLRQ FRHIILFLHQWV IRU *f ZRXOG UHTXLUH D FURVV SURGXFW EHWZHHQ WKH WZR VHWV RI EDVLV IXQFWLRQV 7KLV LV SRWHQWLDOO\ WKH PRVW WHGLRXV VWHS 2QFH WKLV LV GRQH WKH /DJUDQJH PXOWLSOLHU FDQ EH DSSOLHG DQG D V\VWHP RI OLQHDU HTXDWLRQV VXFK DV WKH RQH SUHFHGLQJ f ZLOO UHVXOW 8QOLNH WKDW FDVH KRZHYHU ZH H[SHFW PRUH WKDQ RQH XQNQRZQ LQ DGGLWLRQ WR WKH /DJUDQJH PXOWLSOLHU WR DSSHDU LQ HDFK HTXDWLRQ
PAGE 86
7KRXJK ZH DUH RSWLPLVWLF DERXW VROYLQJ WKLV SUREOHP ZH DOVR ZRXOG OLNH WR FRQVLGHU KRZ WKH VROXWLRQ REWDLQHG LQ &KDSWHU PD\ LWVHOI EH DSSURSULDWHO\ DSSOLHG ,Q RUGHU WR GR WKLV OHW XV ORRN DJDLQ DW WKH REMHFWLYH IXQFWLRQ RI f ZKLFK ZH UHZULWH IRU FRQYHQLHQFH A ,r 6 Zf_O +0IGZ f =7 &O :H QRWLFH WKDW WKH SRZHU VSHFWUDO GHQVLW\ 6[[X!f VHUYHV RQO\ DV D ZHLJKWLQJ IXQFWLRQ 5HJDUGOHVV RI WKH IRUP RI 6[[XMf ZH VWLOO ZLVK WR EULQJ +^X!f DV FORVH DV SRVVLEOH WR RQ Z >fÂ§&O Âf@ 7KH IXQFWLRQ 6[[XMf VSHFLILHV RQO\ WKH SHQDOW\ IRU GHYLDWLRQ IURP DW HDFK XM 7KH JUDSK RI )LJXUH FRPSDUHV D IXQFWLRQ RI IRUP f LQ GRWWHG OLQH ZLWK D IXQFWLRQ WKDW LV FRQVWDQW ZLWKLQ WKH EDQG OLPLW UHSUHVHQWHG E\ D VROLG OLQH )RU D IDLU FRPSDULVRQ HDFK IXQFWLRQ LV QRUPDOL]HG ZLWK UHVSHFWV WR WKH SURFHVV SRZHU :H VHH WKDW WKH GLIIHUHQFH LQ WKH ZHLJKWLQJ LV QRW YHU\ ODUJH XQWLO X DSSURDFKHV WKH EDQG OLPLW 4 VR LW LV SRVVLEOH WKDW D SUHGLFWRU RSWLPL]HG IRU D IODW VSHFWUDO GHQVLW\ ZLOO DOVR SHUIRUP ZHOO IRU D SURFHVV ZKRVH GHQVLW\ LV RI WKH IRUP f )RU VXFK D SUHGLFWRU RQH RI RXU REMHFWLYHV IRU IXWXUH UHVHDUFK LV WR ILQG D XVHIXO ERXQG RQ WKH WUXH PLQLPXP PHDQ VTXDUHG HUURU JLYHQ RXU FRPSXWDWLRQ EDVHG RQ WKH IODW VSHFWUDO GHQVLW\ PRGHO ,I ZH ZLVK WR REWDLQ D VSHFWUDO VKDSH WKDW PRUH FORVHO\ UHVHPEOHV WKDW RI )LJXUH ZH PD\ FRQVLGHU WKH DSSURDFK RI )LJXUH ,Q WKLV FDVH ZH FRPELQH D IODW EDQGOLPLWHG VSHFWUXP ZLWK WZR VLQXVRLGDO FRPSRQHQWV DW Xc sLL 7KH SDUDPHWHUV RI WKH VLQXVRLGDO FRPSRQHQWV PD\ EH HVWLPDWHG XVLQJ D FRUUHODWLRQ RQ WKH LQWHUYDO RI DGDSWLYH HVLWPDWHV DQG WKH UHVXOW PD\ EH VXEWUDFWHG RXW EHIRUH DSSO\LQJ WKH EDQGOLPLWHG SUHGLFWRU 0DQ\ YDULDWLRQV RQ WKLV DSSURDFK DUH SRVVLEOH )RU LQVWDQFH WKH IODW VHJPHQW QHHG QRW RFFXS\ WKH WKH HQWLUH 4 EDQGZLGWK DQG WKH VLQXVRLGDO FRPSRQHQWV QHHG QRW EH SODFHG SUHFLVHO\ DW WKH EDQG OLPLWV $OVR
PAGE 87
& V &2 O I ? ? ? ? ? ? 9 f f f ? fÂ§fÂ§ 6f RQ )LJXUH &RPSDULVRQ RI IDGLQJHQYHORSH VSHFWUD 7KH GRWWHG OLQH LQGLFDWHV WKH VSHFWUXP PRVW FRPPRQO\ DVVXPHG LQ PRELOH UDGLR 7KH VROLG OLQH LQGLFDWHV WKH IODW EDQGOLPLWHG VSHFWUXP RI &KDSWHU D FR L L L ? L L ? ? M ? c! V M S FR6 )LJXUH $ SLHFHZLVH DSSUR[LPDWLRQ RI WKH IDGLQJHQYHORSH VSHFWUXP XVLQJ D IODW VHJPHQW DQG WZR VLQXVRLGDO FRPSRQHQWV
PAGE 88
)LJXUH $SSOLFDWLRQ RI WKH IODW VSHFWUDO GHQVLW\ WR QRQDGDSWLYH SUHGLFWLRQ WKH PDJQLWXGH RI HDFK FRPSRQHQW PD\ EH IL[HG E\ WKH PRGHO RU OHIW DV D UDQGRP YDULDEOH WR EH HVWLPDWHG )LQDOO\ ZH QRWH WKDW RSWLPDO HVWLPDWRUV DUH RIWHQ UHDOL]HG XVLQJ DGDSWLYH ILOWHULQJ WHFKQLTXHV 6XSSRVH KRZHYHU WKDW ZH SUHIHU D QRQDGDSWLYH SUHGLFWRU r 7KHQ ZH DUJXH WKDW LW PDNHV VHQVH WR FKRRVH D IODW VSHFWUDO GHQVLW\ LQ FDUU\LQJ RXW WKH RSWLPL]DWLRQ SURFHGXUH 7KH UHDVRQ PD\ EH VHHQ LQ )LJXUH :H UHFDOO WKDW WKH IDGLQJHQYHORSH VSHFWUDO GHQVLW\ f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
PAGE 89
,Q VXPPDU\ LW LV OLNHO\ WKDW WKH JDS EHWZHHQ WKH IODW VSHFWUDO GHQVLW\ PRGHO DQG WKDW RI )LJXUH ZLOO EH FORVHG VRRQ %XW LQ WKH LQWHULP WKH VROXWLRQ RI &KDSWHU ZLOO VWLOO EH D XVHIXO WRRO LQ DQDO\]LQJ UHDOZRUOG IDGLQJ SUREOHPV
PAGE 90
&+$37(5 &21&/86,216 $1' )8785( :25. ,Q WKLV GLVVHUWDWLRQ ZH KDYH GLVFRYHUHG VRPH NH\ IDFWV DERXW WKH SUHGLFWDELOn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f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
PAGE 91
,Q HYDOXDWLQJ WKH SHUIRUPDQFH RI D UHFHLYHU ZH PD\ ZRQGHU ZKLFK HIIHFW HVWLPDWLRQ HUURUV RU GHOD\V LV PRUH LPSRUWDQW EXW RXU ZRUN VKRZV WKDW LI WKH SDUDPHWHU WR EH HVWLPDWHG PD\ EH PRGHOHG DV D EDQGOLPLWHG SURFHVV WKHQ WKH WZR IDFWRUV DUH DFWXDOO\ UHODWHG EHFDXVH LI WKH HVWLPDWHV DUH HUURU IUHH WKHQ ZH FDQ RYHUFRPH WKH GHOD\ ZLWK D SUHGLFWLRQ ZKRVH HUURU PD\ EH PDGH DUELWUDULO\ VPDOO 7KH IODWIDGLQJ FKDQQHO PRGHO GLVFXVVHG LQ 6HFWLRQ LQFOXGLQJ WKH FKDUDFn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n WLRQ ZDV LQWHQGHG PDLQO\ DV DQ LOOXVWUDWLRQ ,Q WKDW FDVH WKH VLPSOLFLW\ RI WKH DGDSWLYH DOJRULWKP FKRVHQ DOORZHG XV WR WUDFH WKH PDLQ VRXUFH RI HUURUV LW LV WKH
PAGE 92
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
PAGE 93
VXFFHHG LQ RSHQLQJ XS WHUUHVWULDO PRELOH FKDQQHOV WR WKH NLQGV RI LQIRUPDWLRQ UDWHV WKDW DUH QRZ FRPPRQ LQ RWKHU W\SHV RI FRPPXQLFDWLRQ LV QRW \HW FHUWDLQ 6WLOO LI RXU FRQWULEXWLRQ FDQ EH XVHG WR FODULI\ LVVXHV DQG DYRLG VRPH ZDVWHG HIIRUW WKHQ RXU ZRUN ZLOO KDYH EHHQ ZRUWKZKLOH )XWXUH :RUN 2XU LPPHGLDWH UHVHDUFK REMHFWLYH LV WR H[WHQG WKH VROXWLRQ RI WKH EDQGOLPLWHG SUHGLFWLRQ SUREOHP WR SURFHVVHV ZLWK VSHFWUDO GHQVLWLHV RI WKH IRUP f DV ZDV GLVFXVVHG LQ 6HFWLRQ ,Q WKLV ZD\ RXU ILQGLQJV ZRXOG EH PRUH GLUHFWO\ DSSOLFDEOH WR PRELOH UDGLR IDGLQJ $V ZDV PHQWLRQHG LQ WKH VDPH VHFWLRQ ZH ZRXOG DOVR OLNH WR ILQG D XVHIXO ERXQG RQ WKH PLQLPXP PHDQ VTXDUHG SUHGLFWLRQ HUURU RI VXFK D SURFHVV JLYHQ D FRPSXWDWLRQ EDVHG RQ WKH IODW VSHFWUDO GHQVLW\ PRGHO $OVR WKH WHFKQLTXHV WKDW ZH KDYH GHYHORSHG DOORZ XV WR FDUU\ RXW D SUHGLFn WLRQ EDVHG RQ D ILQLWHOHQJWK LQWHUYDO RI NQRZQ YDOXHV :H PD\ ZRQGHU ZKHWKHU VXFK D SUHGLFWLRQ PLJKW EH LPSURYHG LI ZH FRQVLGHUHG DOO SDVW YDOXHV WR EH NQRZQ 7KXV ZH DOVR SODQ WR H[WHQG RXU VROXWLRQ WR WKLV FDVH DV ZHOO :H SXUVXHG D FRQWLQXRXVWLPH DQDO\VLV LQ RUGHU WR ILQG D SUHGLFWRU SHUIRUn PDQFH ERXQG WKDW LV LQGHSHQGHQW RI WKH V\PERO LQWHUYDO ,Q GLJLWDO PRGXODWLRQ KRZHYHU WKH XVH RI DGDSWLYH WHFKQLTXHV ZLOO \LHOG HVWLPDWHV RI WKH IDGLQJ HQYHn ORSH DW GLVFUHWH WLPHV DQG WKHVH ZLOO IRUP WKH EDVLV RI RXU SUHGLFWLRQ 7KLV ZLOO EHFRPH HVSHFLDOO\ LPSRUWDQW DV ZH H[SORUH WKH SUREOHP RI DGDSWLYH HTXDOL]DWLRQ )RU WKLV UHDVRQ DQ H[WHQVLRQ RI WKH VROXWLRQ WR WKH GLVFUHWHWLPH FDVH LV RI LQWHUn HVW :H SDUWLFXODUO\ ZDQW WR NQRZ LI WKH GLVFUHWHWLPH VROXWLRQ FRQYHUJHV WR WKH FRQWLQXRXVWLPH VROXWLRQ LQ VRPH VHQVH DV WKH V\PERO LQWHUYDO LV VKUXQN WR ]HUR 2XU DQDO\VLV DVVXPHV WKDW WKH SUHGLFWLRQ LV WR EH EDVHG RQ D VDPSOH IXQFWLRQ WKDW KDV EHHQ FRUUXSWHG E\ ZKLWH QRLVH 0DQ\ HVWLPDWLRQ WHFKQLTXHV UHVXOW LQ HUURUV WKDW DUH QRW ZHOO PRGHOHG DV ZKLWH QRLVH 7KXV RQH RI RXU UHVHDUFK JRDOV LV
PAGE 94
WR GHWHUPLQH KRZ WR DSSO\ WKH VROXWLRQ RI WKH EDQGOLPLWHG SUHGLFWLRQ SUREOHP ZKHQ WKH FRUUXSWLQJ QRLVH LV FRORUHG ,Q DGGLWLRQ WR FRQVLGHULQJ H[WHQVLRQV RI WKH VROXWLRQ LQ &KDSWHU ZH DOVR ZLVK WR DGGUHVV TXHVWLRQV UDLVHG LQ WKH DQDO\VLV RI 6HFWLRQ GHDOLQJ ZLWK DGDSn WLYH FKDQQHO HVWLPDWLRQ LQ D IDGLQJ HQYLURQPHQW $W WKH HQG RI WKH VHFWLRQ ZH OLVWn HG VRPH VWHSV ZH PD\ WDNH WR LPSURYH WKH SHUIRUPDQFH RI RXU DGDSWLYHSUHGLFWLYH DSSURDFK WR FDUULHU WUDFNLQJ $OVR LW ZLOO EH UHFDOOHG WKDW ZH DVVXPHG WKURXJKRXW WKDW D V\PERO HUURU SUREDELOLW\ RI OHVV WKDQ LV D VXIILFLHQW FRQGLWLRQ IRU PDLQn WDLQLQJ FRUUHFW GHFLVLRQGLUHFWHG WUDFNLQJ $OWKRXJK YDULRXV ILJXUHV DUH FLWHG LQ WKH OLWHUDWXUH VHH GLVFXVVLRQ RQ SDJH f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
PAGE 95
ZLGH RSHQ KHUH DQG ZH KDYH FRQVLGHUHG QRQOLQHDU DSSURDFKHV DV ZHOO DV DOWHUQDWLYH PRGHOV
PAGE 96
$33(1',; (9$/8$7,21 2) %$6,6 )81&7,216 7KLV DSSHQGL[ GHVFULEHV KRZ WR FRPSXWH YDOXHV RI $Q DQG LMMQWf DV GHVFULEHG E\ 6OHSLDQ HW DO >@ XVLQJ WKH WDEOHV RI VSKHURLGDO ZDYH IXQFWLRQV SXEOLVKHG E\ WKH 1DYDO 5HVHDUFK /DERUDWRU\ > @ 7KHVH LQFOXGH WKH DQJXODU IXQFWLRQV 6QF UMf >@ DQG WKH UDGLDO IXQFWLRQV 5QFf >@ :H DGKHUH WR WKH QRWDWLRQDO 47 FRQYHQWLRQV RI 6OHSLDQ HW DO ZLWK F ,Q WKH 15/ WDEOHV WKH V\PEROV O DQG ÂL K DUH XVHG LQ SODFH RI Q DQG F UHVSHFWLYHO\ 7KH HLJHQYDOXHV $QFf DUH DSSHQGHG WR WKH WDEOHV RI F f > S @ 7KH IXQFWLRQ LSQeO7Wf PD\ EH IRXQG XVLQJ > S [Y@ 0$f ^L >Q&!6f@UIV` ,Q WKH 15/ WDEOHV WKH DQJXODU IXQFWLRQV DUH QRUPDOL]HG VXFK WKDW > S [L@ D! VR WKH GHQRPLQDWRU RI $Of LV HDVLO\ KDQGOHG 8QIRUWXQDWHO\ IRU RXU SUHGLFWLRQ SUREOHP ZH DUH LQWHUHVWHG LQ YDOXHV RI LSQWf IRU _L_ a EXW 6IMfF UMf LV RQO\ WDEXODWHG IRU W A 7KLV PD\ EH KDQGOHG E\ XVLQJ WKH UHODWLRQ > S @ Â‘6RQF! ]f mQFf 56, & ]f $f ZKHUH WKH DQJXODU DQG UDGLDO IXQFWLRQV KDYH EHHQ H[WHQGHG WR HQWLUH IXQFWLRQV RI WKH FRPSOH[ YDULDEOH ] 7KH IXQFWLRQ QQFf LV FDOOHG D MRLQLQJ IDFWRU DQG PD\ EH
PAGE 97
HYDOXDWHG E\ $f :H DOVR PDNH XVH RI > S [Y@ $fFf f .f!0f) $f 6ROYLQJ $f IRU F f DQG XVLQJ $f $f ZLWK ] AW $f DQG $Of ZH REWDLQ $f 7KH IXQFWLRQ L"AF ef LV WDEXODWHG IRU I bcW ! W W ZKLFK LV WKH LQWHUYDO RI LQWHUHVW :RUNLQJ ZLWK WDEXODWHG YDOXHV FDQ EH WHGLRXV DQG LW VKRXOG EH QRWHG WKDW VRPH FRPSXWHU SURJUDPV IRU HYDOXDWLQJ WKH SURODWH VSKHURLGDO ZDYH IXQFWLRQV DUH DYDLODEOH > @ EXW PDQ\ RI WKHVH DUH ZULWWHQ LQ RXWGDWHG ODQJXDJHV DUH QRW SRUWDEOH RU KDYH QRW EHHQ WHVWHG WKRURXJKO\
PAGE 98
5()(5(1&(6 >@ $EHG0HUDLP : 4LX DQG < +XD %OLQG V\VWHP LGHQWLILFDWLRQ 3URFHHGn LQJV RI WKH ,((( ffÂ§ $XJXVW >@ $ + $JKYDPL 'LJLWDO PRGXODWLRQ WHFKQLTXHV IRU PRELOH DQG SHUVRQDO FRPPXQLFDWLRQ V\VWHPV (OHFWURQLFV DQG &RPPXQLFDWLRQ (QJLQHHULQJ RXUQDO f XQH >@ 3 $ %HOOR DQG % 1HOLQ 7KH HIIHFW RI IUHTXHQF\ VHOHFWLYH IDGLQJ RQ WKH ELQDU\ HUURU SUREDELOLWLHV RI LQFRKHUHQW DQG GLIIHUHQWLDOO\ FRKHUHQW PDWFKHG ILOWHU UHFHLYHUV ,((( 7UDQVDFWLRQV RQ &RPPXQLFDWLRQ 6\VWHPV XQH >@ 3 $ %HOOR DQG % 1HOLQ &RUUHFWLRQV WR f7KH HIIHFW RI IUHTXHQF\ VHOHFWLYH IDGLQJ RQ WKH ELQDU\ HUURU SUREDELOLWLHV RI LQFRKHUHQW DQG GLIIHUHQWLDOO\ FRKHUHQW PDWFKHG ILOWHU UHFHLYHUVf ,((( 7UDQVDFWLRQV RQ &RPPXQLFDWLRQ 7HFKQRORJ\ 'HFHPEHU >@ ) %HXWOHU (UURUIUHH UHFRYHU\ RI VLJQDOV IURP LUUHJXODUO\ VSDFHG VDPSOHV 6,$0 5HYLHZ f XO\ >@ & %RXZNDPS 2Q VSKHURLGDO ZDYH IXQFWLRQV RI RUGHU ]HUR RXUQDO RI 0DWKHPDWLFV DQG 3K\VLFV >@ $ %URPDQ $Q ,QWURGXFWLRQ WR 3DUWLDO 'LIIHUHQWLDO (TXDWLRQV IURP )RXULHU 6HULHV WR %RXQGDU\YDOXH 3UREOHPV 1HZ @ / %URZQ 8QLIRUP OLQHDU SUHGLFWLRQ RI EDQGOLPLWHG SURFHVVHV IURP SDVW VDPSOHV ,((( 7UDQVDFWLRQV RQ ,QIRUPDWLRQ 7KHRU\ f 6HSWHPEHU >@ / %URZQ 2Q WKH SUHGLFWLRQ RI D EDQGOLPLWHG VLJQDO IURP SDVW VDPSOHV 3URFHHGLQJV RI WKH ,((( f 1RYHPEHU >@ $ &DG]RZ $Q H[WUDSRODWLRQ SURFHGXUH IRU EDQGOLPLWHG VLJQDOV ,((( 7UDQVDFWLRQV RQ $FRXVWLFV 6SHHFK DQG 6LJQDO 3URFHVVLQJ f )HEUXDU\ >@ 5 + &ODUNH $ VWDWLVWLFDO WKHRU\ RI PRELOH UDGLR UHFHSWLRQ %HOO 6\VWHP 7HFKQLFDO RXUQDO f XO\$XJXVW
PAGE 99
>@ % &RQZD\ $ &RXUVH LQ )XQFWLRQDO $QDO\VLV QG HG 1HZ @ : % 'DYHQSRUW DQG : / 5RRW $Q ,QWURGXFWLRQ WR WKH 7KHRU\ RI 5DQGRP 6LJQDOV DQG 1RLVH 1HZ @ / 'RRE 6WRFKDVWLF 3URFHVVHV 1HZ @ 7 7 )MDOOEUDQGW ,QWHUSRODWLRQ DQG H[WUDSRODWLRQ LQ QRQXQLIRUP VDPSOLQJ VHTXHQFHV ZLWK DYHUDJH VDPSOLQJ UDWHV EHORZ WKH 1\TXLVW UDWH (OHFWURQLFV /HWWHUV OOf XQH >@ & )ODPPHU 6SKHURLGDO :DYH )XQFWLRQV 6WDQIRUG &$ 6WDQIRUG 8QLYHUVLW\ 3UHVV >@ )RUQH\ 0D[LPXPOLNHOLKRRG VHTXHQFH HVWLPDWLRQ RI GLJLWDO VHTXHQFHV LQ WKH SUHVHQFH RI LQWHUV\PERO LQWHUIHUHQFH ,((( 7UDQVDFWLRQV RQ ,QIRUPDWLRQ 7KHRU\ f 0D\ >@ & )R[ $Q ,QWURGXFWLRQ WR WKH &DOFXOXV RI 9DULDWLRQV /RQGRQ 2[IRUG 8QLYHUVLW\ 3UHVV >@ % 5 )ULHGHQ (YDOXDWLRQ GHVLJQ DQG H[WUDSRODWLRQ PHWKRGV IRU RSWLFDO VLJQDOV EDVHG RQ XVH RI WKH SURODWH IXQFWLRQV ,Q 3URJUHVV LQ 2SWLFV HGLWHG E\ ( :ROI SDJHV $PVWHUGDP 1RUWK+ROODQG >@ 1 *RGDUG 6HOIUHFRYHULQJ HTXDOL]DWLRQ DQG FDUULHU WUDFNLQJ LQ WZR GLPHQVLRQDO GDWD FRPPXQLFDWLRQ V\VWHPV ,((( 7UDQVDFWLRQV RQ &RPPXQLFDn WLRQV f 1RYHPEHU >@ $ *ROGVPLWK DQG 6 &KXD 9DULDEOHUDWH YDULDEOHSRZHU 04$0 IRU IDGLQJ FKDQQHOV ,((( 7UDQVDFWLRQV RQ &RPPXQLFDWLRQV f 2FWREHU >@ + *ROXE DQG & ) 9DQ /RDQ 0DWUL[ &RPSXWDWLRQV UG HG %DOWLPRUH 0' RKQV +RSNLQV 8QLYHUVLW\ 3UHVV >@ 6 +DQLVK 5 9 %DLHU $ / 9DQ %XUQ DQG % .LQJ 7DEOHV RI UDGLDO VSKHURLGDO ZDYH IXQFWLRQV YROXPH 3URODWH P 7HFKQLFDO 5HSRUW 15/ :DVKLQJWRQ '& 1DYDO 5HVHDUFK /DERUDWRU\ $YDLODEOH WKURXJK 17,6 $1 $';$% >@ 6 +D\NLQ HGLWRU %OLQG 'HFRQYROXWLRQ (QJOHZRRG &OLIIV 13UHQWLFH+DOO >@ 6 +D\NLQ $GDSWLYH )LOWHU 7KHRU\ UG HG 8SSHU 6DGGOH 5LYHU 13UHQWLFH +DOO >@+ +RFKVWDGW ,QWHJUDO (TXDWLRQV 1HZ
PAGE 100
>@ : & DNHV HGLWRU 0LFURZDYH 0RELOH &RPPXQLFDWLRQV 1HZ @ 6 0 .D\ 0RGHP 6SHFWUDO (VWLPDWLRQ 7KHRU\ DQG $SSOLFDWLRQ (QJOHZRRG &OLIIV 13UHQWLFH+DOO >@ 6 0 .D\ )XQGDPHQWDOV RI 6WDWLVWLFDO 6LJQDO 3URFHVVLQJ (VWLPDWLRQ 7KHRU\ (QJOHZRRG &OLIIV 13UHQWLFH+DOO >@ 5 $ .HQQHG\ % 2 $QGHUVRQ DQG 5 5 %LWPHDG %OLQG DGDSWDWLRQ RI GHFLVLRQIHHGEDFN HTXDOL]HUV *URVV FRQYHUJHQFH SURSHUWLHV ,QWHUQDWLRQDO RXUQDO RI $GDSWLYH &RQWURO DQG 6LJQDO 3URFHVVLQJ f 1RYHPEHU 'HFHPEHU >@ .QDE ,QWHUSRODWLRQ RI EDQGOLPLWHG IXQFWLRQV XVLQJ WKH DSSUR[LPDWH SURODWH VHULHV ,((( 7UDQVDFWLRQV RQ ,QIRUPDWLRQ 7KHRU\ f 1RYHPEHU >@ .QDE 1RQFHQWUDO LQWHUSRODWLRQ RI EDQGOLPLWHG VLJQDOV ,((( 7UDQVDFn WLRQV RQ $HURVSDFH DQG (OHFWURQLF 6\VWHPV f XO\ >@ 3 .RRVLV ,QWURGXFWLRQ WR +S 6SDFHV ZLWK DQ $SSHQGL[ RQ :ROIIfV 3URRI RI WKH &RURQD 7KHRUHP &DPEULGJH &DPEULGJH 8QLYHUVLW\ 3UHVV >@ 0 % .R]LQ 9 9 9RONRY DQG 6YHUJXQ $ FRPSDFW DOJRULWKP IRU HYDOXDWLQJ OLQHDU SURODWH IXQFWLRQV ,((( 7UDQVDFWLRQV RQ 6LJQDO 3URFHVVLQJ f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
PAGE 101
>@5 /\PDQ DQG : : (GPRQVRQ /LQHDU SUHGLFWLRQ RI EDQGOLPLWHG SURFHVVHV ZLWK IODW VSHFWUDO GHQVLWLHV 6XEPLWWHG WR ,((( 7UDQVDFWLRQV RQ 6LJQDO 3URFHVVLQJ >@ 5 /\PDQ : : (GPRQVRQ 0 5DR DQG 6 0F&XOORXJK 7KH SUHGLFWDELOn LW\ RI FRQWLQXRXVWLPH EDQGOLPLWHG SURFHVVHV ,((( 7UDQVDFWLRQV RQ 6LJQDO 3URFHVVLQJ f )HEUXDU\ >@ 2 0DFFKL $GDSWLYH 3URFHVVLQJ 7KH /HDVW 0HDQ 6TXDUHV $SSURDFK ZLWK $SSOLFDWLRQV LQ 7UDQVPLVVLRQ 1HZ @ 2 0DFFKL DQG ( (ZHGD &RQYHUJHQFH DQDO\VLV RI VHOIDGDSWLYH HTXDOL]HUV ,((( 7UDQVDFWLRQV RQ ,QIRUPDWLRQ 7KHRU\ ffÂ§ 0DUFK >@ 6 / 0DUSOH U 'LJLWDO 6SHFWUDO $QDO\VLV ZLWK $SSOLFDWLRQV (QJOHZRRG &OLIIV 13UHQWLFH+DOO >@ ) 0DUYDVWL &RPPHQWV RQ f$ QRWH RQ WKH SUHGLFWDELOLW\ RI EDQGOLPLWHG SURFHVVHVf 3URFHHGLQJV RI WKH ,((( f 1RYHPEHU >@ ( 0D]R $QDO\VLV RI GHFLVLRQGLUHFWHG HTXDOL]HU FRQYHUJHQFH %HOO 6\VWHP 7HFKQLFDO RXUQDO f 'HFHPEHU >@ < 2NXPXUD ( 2KPRUL 7 .DZDQR DQG )XNXGD )LHOG VWUHQJWK DQG LWV YDULDELOLW\ LQ 9+) DQG 8+) ODQGPRELOH UDGLR VHUYLFH 5HYLHZ RI WKH (OHFWULFDO &RPPXQLFDWLRQ /DERUDWRU\ f 6HSWHPEHU2FWREHU >@ $ 3DSRXOLV $ QHZ DOJRULWKP LQ VSHFWUDO DQDO\VLV DQG EDQGOLPLWHG H[WUDSRODn WLRQ ,((( 7UDQVDFWLRQV RQ &LUFXLWV DQG 6\VWHPV f 6HSWHPEHU >@ $ 3DSRXOLV 6LJQDO $QDO\VLV 1HZ @ $ 3DSRXOLV $ QRWH RQ WKH SUHGLFWDELOLW\ RI EDQGOLPLWHG SURFHVVHV 3URFHHGn LQJV RI WKH ,((( f $XJXVW >@ $ 3DSRXOLV 3UHGLFWDEOH SURFHVVHV DQG :ROGfV GHFRPSRVLWLRQ $ UHYLHZ ,((( 7UDQVDFWLRQV RQ $FRXVWLFV 6SHHFK DQG 6LJQDO 3URFHVVLQJ f $XJXVW >@ $ 3DSRXOLV 3UREDELOLW\ 5DQGRP 9DULDEOHV DQG 6WRFKDVWLF 3URFHVVHV UG HG 1HZ @ 3 = 3HHEOHV 'LJLWDO &RPPXQLFDWLRQ 6\VWHPV (QJOHZRRG &OLIIV 13UHQWLFH+DOO >@ 3URDNLV 'LJLWDO &RPPXQLFDWLRQV UG HG 1HZ @ 6 8 + 4XUHVKL $GDSWLYH HTXDOL]DWLRQ 3URFHHGLQJV RI WKH ,((( f 6HSWHPEHU
PAGE 102
>@ $ $ 5HTXLFKD 7KH ]HURV RI HQWLUH IXQFWLRQV 7KHRU\ DQG HQJLQHHULQJ DSSOLFDWLRQV 3URFHHGLQJV RI WKH ,((( f 0DUFK >@ : 5XGLQ 3ULQFLSOHV RI 0DWKHPDWLFDO $QDO\VLV UG HG 1HZ @ 6OHSLDQ 2Q EDQGZLGWK 3URFHHGLQJV RI WKH ,((( f 0DUFK >@ 6OHSLDQ 3URODWH VSKHURLGDO ZDYH IXQFWLRQV )RXULHU DQDO\VLV DQG XQFHUWDLQW\9 7KH GLVFUHWH FDVH %HOO 6\VWHP 7HFKQLFDO RXUQDO f 0D\XQH >@ 6OHSLDQ + /DQGDX DQG + 3ROLDN 3URODWH VSKHURLGDO ZDYH IXQFWLRQV )RXULHU DQDO\VLV DQG XQFHUWDLQW\, t ,, %HOO 6\VWHP 7HFKQLFDO RXUQDO Of DQXDU\ >@ + 6ROWDQLDQ=DGHK DQG $ ( @ : 6SOHWWVWRVVHU 2Q WKH SUHGLFWLRQ RI EDQGOLPLWHG VLJQDOV IURP SDVW VDPSOHV ,QIRUPDWLRQ 6FLHQFHV f 1RYHPEHU >@ 6SUHFKHU (OHPHQWV RI 5HDO $QDO\VLV 1HZ @ 3 6WRLFD DQG 5 / 0RVHV ,QWURGXFWLRQ WR 6SHFWUDO $QDO\VLV 8SSHU 6DGGOH 5LYHU 13UHQWLFH+DOO >@ / 6WXEHU 3ULQFLSOHV RI 0RELOH &RPPXQLFDWLRQ %RVWRQ .OXZHU >@ 7KRPVRQ 6SHFWUXP HVWLPDWLRQ DQG KDUPRQLF DQDO\VLV 3URFHHGLQJV RI WKH ,((( f 6HSWHPEHU >@ & 7UDEHOVL /LQHDU DGDSWLYH SUHGLFWLRQ XVLQJ /06 DOJRULWKP RYHU 5LFLDQ IDGLQJ FKDQQHOV (XURSHDQ 7UDQVDFWLRQV RQ 7HOHFRPPXQLFDWLRQV f 0DUFK$SULO >@ 3 3 9DLG\DQDWKDQ 2Q SUHGLFWLQJ D EDQGOLPLWHG VLJQDO EDVHG RQ SDVW VDPSOH YDOXHV 3URFHHGLQJV RI WKH ,((( f $XJXVW >@ $ / 9DQ %XUQ $ )RUWUDQ FRPSXWHU SURJUDP IRU FDOFXODWLQJ WKH OLQHDU SURODWH IXQFWLRQV 7HFKQLFDO 5HSRUW 15/ :DVKLQJWRQ '& 1DYDO 5HVHDUFK /DERUDWRU\ $YDLODEOH WKURXJK 17,6 $1 $'$;$% >@ $ / 9DQ %XUQ % .LQJ DQG 5 9 %DLHU 7DEOHV RI DQJXODU VSKHURLGDO ZDYH IXQFWLRQV YROXPH 3URODWH P 7HFKQLFDO UHSRUW :DVKLQJn WRQ '& 1DYDO 5HVHDUFK /DERUDWRU\ $YDLODEOH WKURXJK 17,6 $1 $'$;$%
PAGE 103
>@ $ 9LWHUEL (UURU ERXQGV IRU FRQYROXWLRQDO FRGHV DQG DQ DV\PSWRWLFDOO\ RSWLPXP GHFRGLQJ DOJRULWKP ,((( 7UDQVDFWLRQV RQ ,QIRUPDWLRQ 7KHRU\ f $SULO >@ / $ :DLQVWHLQ DQG 9 =XEDNRY ([WUDFWLRQ RI 6LJQDOV IURP 1RLVH (QJOHZRRG &OLIIV 13UHQWLFH+DOO >@ 5 :HLQVWRFN &DOFXOXV RI 9DULDWLRQV ZLWK $SSOLFDWLRQV WR 3K\VLFV DQG (QJLQHHULQJ 1HZ @ 6 =KDQJ DQG LQ &RPSXWDWLRQ RI 6SHFLDO )XQFWLRQV 1HZ
PAGE 104
%,2*5$3+,&$/ 6.(7&+ 5DSKDHO /\PDQ JUHZ XS LQ WKH 7DPSD %D\ DUHD RI )ORULGD +H UHFHLYHG WKH %6 GHJUHH LQ HOHFWULFDO HQJLQHHULQJ IURP WKH 8QLYHUVLW\ RI +RXVWRQ LQ DQG WKH 06 GHJUHH LQ HOHFWULFDO HQJLQHHULQJ IURP WKH 8QLYHUVLW\ RI 6RXWK )ORULGD LQ +H KDV WHQ \HDUV RI LQGXVWULDO H[SHULHQFH LQ WKH PDQXIDFWXUH RI HOHFWURQLF DVVHPEOLHV HVSHFLDOO\ LQ WKH ILHOGV RI TXDOLW\ DQG WHVW 6LQFH KH KDV EHHQ SXUn VXLQJ GRFWRUDO VWXGLHV LQ WKH 'HSDUWPHQW RI (OHFWULFDO DQG &RPSXWHU (QJLQHHULQJ DW WKH 8QLYHUVLW\ RI )ORULGD +LV UHVHDUFK LQWHUHVWV LQFOXGH PRELOH UDGLR VLJQDO SURFHVVLQJ DQG HVWLPDWLRQ WKHRU\ +H LV PDUULHG DQG KDV RQH GDXJKWHU
PAGE 105
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
PAGE 106
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
CHAPTER 4
APPLICATIONS TO FADING IN MOBILE RADIO
It is well known that mobile radio is plagued by the distorting effects of
multipath fading. As we shall see, this problem may be characterized by a complex
time function called the fading envelope, which is often modeled as a bandlimited
random process [11]. In Section 1.1, we mentioned adaptive transmission as
an application that could benefit from the prediction of the fading envelope.
The theoretical discussions in Chapters 2 and 3 provided us with some tools for
predicting bandlimited processes. This chapter is dedicated to a discussion of
fading and the possible application of bandlimited prediction to a problem in
adaptive channel estimation.
4.1 Multipath Fading in Mobile Radio
Our discussion of fading will be very brief. Additional details may be found
in Proakis [54, Ch. 14] and in Stuber [65, Ch. 2,5,6].
Ideally, in a wireless communication system, there would always be an
unobstructed line of sight between the transmitting and receiving antennas.
Unfortunately, this is rarely practical for mobile communication, in which the
mobile receiver is often embedded in a clutter of obstructing objects such as
buildings, trees, hills and vehicles. The direct signal path is often completely cut
off, such that nearly all of the received signal energy is reflected toward the receiver
from these surrounding objects. The different reflections may have comparable
amplitudes, but each may have undergone a random shift in phase, as well as a
time delay that is longer or shorter than other reflections. At the receiver location,
these various phases may add constructively or destructively, leading to the
47
61
As an algorithm for updating our estimate of c(n) we choose
/ is \ e(n) / v(n)
c(n + l) = c(n) Hjr = c(n) +
u(n) ~v"7 u(n)
This choice simplifies the characterization of estimation errors. To see this, note
that if the variation in c(n) is slow enough, then the error is dominated by the
v{n)
(4.14)
u{n)
term. Thus,
c(n) = c(n) + e(n) c(n) +
v(n l)
where
e(n) = c(n) c(n)
u(n l)
v(n l)
(4.15)
(4.16)
u(n l)
Now since u(n) is an uncorrelated QPSK sequence with u(n) = \f2Ei, it is clear
from (4.16) that e(n) is approximately a zeromean white Gaussian noise sequence
2
with of = 7^, where of = i{c(n)2} and al = Â£{u(n)2}.
2 Et,
Substituting (4.15) in (4.ll) we have
/ \ c(n) / \ v(n)
u(n) = Â¡r 7^u(n) +
c(n) + e(n)
c(n) + e(n)'
(417)
Since we are considering the case in which the estimation errors e(n) and additive
noise v(n) are not large enough to cause a high symbol error rate, we assume
c(n) e(n) and c(n) > \v(n)\. Thus,
c(^) i iM
c(n) + e(n) c(n)
(4.18)
where the firstorder term is retained because the zerothorder term will be
subtracted out shortly. Also,
c(n) + e(n) c(n)'
Substituting (4.18) and (4.19) into (4.17) we obtain
u(n)
1 
e(n)
c(n)_
u(n) +
v(n)
c(n)'
(4.19)
(4.20)
68
This yields
76 = 1.3321.
(4.54)
In the next section, we shall see that it is unnecessary to compute the value of of
explicitly.
4.2.6 Predictability Analysis
In this section we shall use the predictability analysis techniques developed in
Chapter 3 to determine under what conditions the mean squared prediction error
can be made less that of as given by (4.52). Now let Jmin{E) be the minimum
mean squared prediction error of x(t), the real part of c(t). We note that the
adaptive estimation errors e(n) and the predictive estimation error (n) are complex
processes with their powers split evenly between the real and imaginary parts.
Using the approach of Section 3.4 we want (cf. Eq. 3.45)
(4.55)
Now a2 = Â£ {c(i)2} is two times the average power of x(t). Thus, if Jmm{E) is
the minimum mean squared prediction error of a normalized process we have
2
(4.56)
or
(4.57)
Now, from (4.28) we have
(4.58)
58
For mobile radio, however, we normally assume the Ushaped spectral density of
Figure 4.3, whereas in Chapter 3 we assumed a flat spectral density.
We have not solved the bandlimited linear prediction problem for the
Ushaped spectrum. We nevertheless go forward with the analysis under the
assumption that the resulting inaccuracies in our results will not be too great.
We do this in order to illustrate a technique that may be applied to bandlimited
processes with spectral densities that are not flat once the linear prediction
problem is solved for those cases.
The second theoretical difficulty is that the continuoustime analysis of
Chapters 2 and 3 assumed that a sample function of x(t) was known over the
entire continuum of a positivelength interval. Using the adaptive technique of
Figure 4.5, however, the sample function will be known only at discrete times, since
the adaptive estimate of c(f) = x(t) + iy(t) is updated once every symbol interval.
A linear predictor would therefore be a sum of the form Ts^2x(nTs)h(kTs nTs),
where nTs takes values of t on the known interval of x (t) and kTs = r is how far in
the future, with respects to the known interval, we wish to make the prediction.
An analysis of prediction based on discretetime samples will depend upon
the symbol interval Ta. Thus, if a sufficient predictor performance cannot be
attained at a given symbol rate, we may wonder if better performance might result
from sampling at a different rate. Our continuoustime analysis was motivated by
a desire to find a performance bound that was independent of Ts. We assume that
an optimal continuoustime predictor of the form (2.1) will not be outperformed
by a discretetime predictor. Although we have not proved this, our heuristic
reasoning is that a continuoustime predictor makes use of all the information on
the known interval, whereas the discretetime predictor uses only a subset of this
information. Extending our analysis to dicretetime prediction, and establishing a
rigorous connection between the discretetime and continuoustime cases, as well as
39
problem (3.5) is solved. It will be recalled that, in a predictor of form (2.1), this
h(t) attains the minimum mean squared prediction error under the constraint that
Ho h2{t)dt = E. As was stated in Section 2.2, there exists a function Jmin(E)
which maps E to the minimum mean squared error for given values of 2, T and
r. Substituting (3.20) into (3.18), we see that we can compute values for JmÂ¡n(E)
using the following procedure:
For E > 0,
(3.22)
Note that, in order to carry out the procedure (3.22), one needs only the
eigenvalue An and the single function value <^(0) for each n. We choose to focus
on Jmin(E) because it provides a useful lower bound on the mean squared error
performance of linear predictors.
From (3.22), we see that Jm\a(E) is a strictly decreasing function of E. Thus,
in (3.5), if the value of E is decreased, the minimum value of J must increase.
This justifies the use of the equality in constraint 2. Also from (3.22), we note that
Jmin{E) > 0 as E > oo, as we expect from the analysis in Chapter 2.
Now let us consider the behavior of JmÂ¡n(E) for small E. Looking at (3.22) we
see that in this case, /i will be much greater than 1, which is an upper bound for A
(see Section 3.1.1). From (3.22) we have
(3.23)
and
36
Thus,
Fn(u>) = 0n(O)$n(a;).
(3.9)
By property $5, the basis functions {(/>n(t)} form a complete set in finiteenergy
functions on [r, r + T], so we may expand h(t) as well,
oo oo
h(t) = E hn(t) y~! Pn^k,trunc(^)) (3.10)
n=0 n=0
where A:,trunc() is the truncated basis function as defined in property $9 and the
coefficients {/?} are to be determined.
We consider now the conditions under which (3.10) will converge. Using
property $4 we have
/oo rr+T
A,trunc = / = V
oo Jt
Thus, ^=^fc,trunc(i)} is orthonormal on [r, r + T]. Convergence of (3.10) is
therefore guaranteed if
OO
E XnPl < OO. (3.11)
71=0
Now, letting the Fourier transform of trunc) be denoted by 4>fcitrunc(w), we may
write
Hn(uj) = truncM, U = 0, 1, 2, . (3.12)
Using property $9 we have
^fc.trunc(^) ^
We may now rewrite the objective function of (3.5) as
J = hs>^
oo oo
Â£ F(u) Â£ H(U)
n=0 n0
1 rn
27T 70
du).
(3.13)
(3.14)
18
domain formulation of the problem, and serves perfectly well to answer our
questions about the predictor defined in (2.1).
We now proceed to find a frequencydomain expression for the mean squared
prediction error. We start by defining the error,
e(t) = x(t) x(t)
= x(t) x(t) h(t)
II
cf
*
ro^
1
C+
(2.5)
Let us further define the error filter,
e(t) = S(t) h(t)
(2.6)
E(u) = 1 H(uj).
(2.7)
It is clear that the error is obtained by passing x(t) through a filter with a frequen
cy response given by (2.7). Now, suppose that x(t) is bandlimited to co < L Then
we may obtain the mean squared error by
ztt Jn
= T/ Sxx(u)\\H{u)\2du,
ztt Jn
(2.8)
(2.9)
where Sxx(u>) is the power spectral density of x(t). Thus, the desired predictor
(2.3) minimizes (2.9) with respect to h(t) under the constraint imposed by (2.4).
A careful inspection of (2.9) will show that we are seeking a time function of
finite support, h(t), whose Fourier transform approximates the frequency response
of a zerophase allpass filter in the frequency range u G [S7, fi]. The time delay
incorporated in h(t) makes this simple form possible, since otherwise it would be
necessary to introduce exponentials of the form etWT. Note also that the frequency
response for cu > Q is arbitrary, because x(t) contains no energy at frequencies
2
1.1 Project Thesis
In communications and signal processing, one sometimes encounters problems
which are conveniently formulated in terms of bandlimited random processes.
For example, in wireless communications, multipath fading may be viewed as a
modulation of the transmitted signal by a complex time function called the fading
envelope [54, Ch. 14]. For mobile radio, the fading envelope may be modeled as a
bandlimited random process, with the band limits depending upon the speed of the
mobile receiver [11].
Some techniques designed to compensate for the effects of fading require
an estimate of the current value of the fading envelope. An example is adaptive
transmission, which uses this estimate to make appropriate adjustments to the
power or rate of transmission. Unfortunately, adaptive transmission is very
sensitive to delays which inevitably occur in feeding back estimates of the fading
envelope from receiver to transmitter [21]. In such a case we would naturally
consider whether the current value could be estimated based upon our knowledge
of the past. So adaptive transmission is one example of an application that could
benefit from the prediction of a bandlimited process.
Prediction problems are often addressed using the techniques of Wiener
filtering, but the solution of the Wiener prediction problem requires that the
power spectral density be factorable into minimumphase and maximumphase
finiteenergy parts. Unfortunately, this factorization is impossible when the
spectral density is zero over a set of positive measure, so the Wiener approach
cannot be used to solve the prediction problem for bandlimited processes (see, e.g.,
Papoulis [52, pp. 402, 493]).
Previous approaches to the problem have focused on the prediction of future
values by taking linear combinations of past sample values [72, 5, 8, 9, 59, 62, 50,
45, 68]. The questions that have been dealt with include the conditions under
73
Table 4.1: Summary of Predictability Analysis Procedure
1.
Based on the desired overall bit error rate, use (4.29) and (4.28) to determine 7*,.
2.
Based on the maximum bit error rate to maintain correct decisiondirected
channel tracking, use (4.32) and (4.31) to determine 7%j.
3.
Based on the desired probability of recovering correct channel tracking after
a deep fade, Pr, use (4.53) to determine %.
4.
Use (4.35) to determine r. Then use (4.62) and (4.63) to determine c and d.
5.
Using (4.61), for E > 0, plot all positive values of 1(E) on a graph.
6.
Using (4.73) and (4.72) one time, and then (4.69), find a value Q0 such that
i7min(^0) E) plotted against E is tangent to the curve for 1(E).
7.
If the maximum Doppler frequency fm is less than or equal to fi0/27r, correct
channel tracking may be recovered with probability Pr.
40
= Â£427+f)
oo o C oo 'I
~ E n(0)  \ E Xnl(0) \ .
n=0 A4 ln=0 J
(3.24)
Now we note from (3.18) that, with E = 0, pn = 0 for all n. Substituting
pn = 0 into the expression for J, we see that the sum (/>2 (0) is simply JmÂ¡n(0).
To evaluate the sum explicitly, we use (3.6) to compute
[n \F(uj)\2du> = 22.
Jn
We then use (3.7), (3.9) and property $7 to obtain
f \F(u)\idu> = 2KY'4i{0).
Ja ri=0
Setting (3.25) equal to (3.26) we have
(3.25)
(3.26)
Cl
4,1.(0) = Â£ (0) =
n0
7r
(3.27)
Note that this is true regardless of the value of r or T, so by (3.2) we have
= for ah t. To evaluate the braced sum in (3.23) and (3.24) we
7T
use (3.3) to obtain
Â£A^(0) = tM0) rT
n=0 n0 Jt
fT+T ( ^2, 1 sin Qs nn.
= / 4 E (s) a (3.28)
^ U=o J 7rs
From (3.9), we know that 0(O)n(a>) = F(u;) as defined in (3.6). The braced
expression in (3.28) is therefore the inverse Fourier transform of F(u),
00 Cl
Y; tniWnit) = sine Clt.
n=0 *
(3.29)
Thus, substituting into (3.28) we have
Q2 rT+T
E^^n(0) = / sinc2f2s ds
7
coefficients may be calculated. The problem is not central to their development,
but the book is referred to in many of the later entries.
In 1966, Beutler [5] addressed the problem in the general context of the
recovery of bandlimited signals from irregularly spaced past samples. His treatment
includes both stochastic and nonstochastic signals, and uniform sampling may be
considered a special case. Using the gap and density theorems of Levinson [36],
he showed that it is theoretically possible to drive the prediction error to zero if
the past samples are taken at any rate higher than the Nyquist rate. This rate
improves on that of Wainstein and Zubakov by a factor of three, though the book
by those authors is not listed among Beutlers references. Beutlers proof, however,
does not give a procedure for calculating the predictor coefficients.
In a 1972 correspondence, Brown recognized the works of Wainstein and
Zubakov, and of Beutler, and then offered a procedure for calculating the co
efficients in the case of uniform sampling at only twice the Nyquist rate [8].
Fjallbrandt contributed a letter in 1975 dealing with certain interpolation and
extrapolation problems [15]. As an extreme case, he finds coefficients for extrapola
tion with uniform sampling at any rate higher than the Nyquist rate. He offers this
as a constructive proof of Beutlers results for the uniformsampling case, though
it seems unclear whether the derivation is good for finiteenergy or finitepower
signals, and the stochastic case is not referred to explicitly.
Slepian discusses the problem, without reference to the previous authors, in
a 1978 article on the discrete prolate spheroidal sequences [59]. In this case, he
considers the optimal predictor for N uniformlyspaced samples of a bandlimited
process whose spectral density is flat within the band limits. He shows that if the
sampling rate exceeds the Nyquist rate, the mean squared error of the prediction
approaches zero exponentially in N.
functions, we use timeshifted versions of the prolate spheroidal wave functions,
leading to a simple algebraic optimization problem which may be solved using a
Lagrange multiplier. We show how to use the solution to compute the minimum
mean squared prediction error under the energy constraint. Then we discuss the
case of a bandlimited process emebedded in white noise, showing how to determine
if a specified mean squared prediction error can be attained.
Finally, we apply these prediction techniques to a problem in decisiondirected
adaptive channel estimation. We show how an adaptive estimator may lose track of
the channel when the mobile receiver enters a deep fade. We consider the use of a
predicted value of the fading envelope to restart the adaptation after the fade. Our
analysis determines the conditions under which we may expect to recover correct
carrier tracking using the predicted valu. The primary factor is the maximum
Doppler frequency, which is proportional to the speed of the receiver.
x
85
wide open here, and we have considered nonlinear approaches, as well as alternative
models.
33
The functions {(Â£)} have the following properties, many of which follow
directly from the properties of {ipn(t)}'
$1. Each function (j)n(t) is real valued. This follows directly from proper
ty VI.
$2. The set {(fin(t)} is orthonormal on [00,00]. Since each cj)n(t) is shifted
by the same amount, the orthogonality of property ^2 is preserved.
$3. The set {(Â¡>n{t)} is complete in Slbandlimited, finiteenergy functions. To
see this, suppose f(t) is such a function. Then the timeshifted function
f (t1r + is also f2bandlimited. We may thus write
OO
f{t + T+ I) = SttnVn (t).
n=0
Substituting t' = t + r + ^ we have
OO OO
/(*') = Â£ nVn (f ~ T fan(/)n(t').
n=0 n=0
Since this equation must be true for all t', the prime notation may be
dropped.
$4. The set {(/>n(i)} is orthogonal on [r, r + T], with
m(t)n{t)dt
A m = n
<
0 m n.
This may be seen by applying the time shift of (3.2) to property \P4.
$5. The set (0n(i)} is complete in finiteenergy functions on [r, r I T], This
is the interval of property '1/5 shifted according to (3.2).
$6. Each function (Â¡>n{t) is iibandlimited. Thus, each Fourier transform has
the property $n(u;) = 0 for cu > 1 This follows from property ^6, since
a time shift clearly does not affect the band limitation.
46
analysis of a practical signalprocessing problem. This is the topic of the following
chapter.
93
[71] A. J. Viterbi. Error bounds for convolutional codes and an asymptotically
optimum decoding algorithm. IEEE Transactions on Information Theory,
13(2):260269, April 1967.
[72] L. A. Wainstein and V. D. Zubakov. Extraction of Signals from Noise.
Englewood Cliffs, NJ: PrenticeHall, 1962.
[73] R. Weinstock. Calculus of Variations, with Applications to Physics and
Engineering. New York: McGrawHill, 1952.
[74] S. Zhang and J. Jin. Computation of Special Functions. New York: Wiley,
1996.
8
The next entry is a 1981 correspondence by Knab who, again, makes no
reference to any of the above work [32]. His interest is in the interpolation of a
finitepower bandlimited signal when the sample times are not symmetric with
respects to the point to be estimated. Knabs approach is interesting because he
imposes no necessary stochastic interpretation on the problem, and for this reason,
his contribution is difficult to place. He deals with endpoint extrapolation as an
extreme case, in the manner of Fjallbrandt. Another interesting point is that he
uses a timeshifted approximation of the prolate spheroidal wave functions [31],
with an approach that bears some similarity to ours in Chapter 3.
In a 1982 paper, Spletstosser [62] reviews the work of Brown, as well as
Wainstein and Zubakov, and then shows how to calculate the coefficients for
sampling at 1.5 time the Nyquist rate. The problem was addressed again in
1985, this time in a letter by Papoulis [50]. He shows the possibility of prediction
with arbitrarily small error using an approach substantially different from that
of Beutler. His treatment is quite brief, though it is clearly laid out, but of the
authors we have discussed, he mentions only Wainstein and Zubakov in his
references.
This led to a number of followup letters in the same journal, each making
some attempt to set the record straight, and some making additional contributions.
Marvasti claimed that all of Papouliss findings had been proved by others previ
ously [45]. He then offered his own brief proof based on a theoretical framework
described in an article by Requicha [56]. In the same issue, Brown reiterated his
earlier proof [9], and reminded readers of Beutlers work [5]. Finally, Vaidyanathan
draws attention to some past literature and, in an explicitly stochastic framework,
shows how to calculate predictor coefficients that work for any sampling rate above
the Nyquist rate [68]. This entry by Vaidyanathan, published in 1987, is the last
that we have found to deal directly with this problem.
and since, from Section 4.2.3, we know that 2Eb
noise variance we use (4.28) to obtain
69
Also, from (4.52) we have
1
2Eb a2
276
aj 1
a2 27b
2 Eb a2
7b,f 1
1
76 276
276
1 (76,/
)
27& V 76
(4.59)
(4.60)
where use was made of (4.58), (4.28) and (4.31). Substituting (4.59) and (4.60)
into (4.57) we have
(46i)
Now 7b, 7b,/ and % were determined in previous sections. We wish to know under
what conditions will be less than or equal to 1(E) as given by (4.61) for at
least some value of E.
It will be recalled that Jmm(E) depends implicitly on O, T and r, where is
the band limit of the process x(t), T is the length of the interval of known values
on which the prediction is to be based, and r is how far in the future we wish to
make the prediction. In accordance with the discussion in Section 4.1, the fading
envelope c(t) = x(t) + iy(t) is bandlimited to the maximum Doppler frequency
fm. We thus choose Q, 27r/m. The prameters r and T are set equal to the mean
fade duration t/ and the mean time above the fade threshold tj respectively, as
discussed in Section 4.2.4.
Now note from (4.43) that
l~3 1 QT 1371
V 4lr/rn r 2 V 4 r
(4.62)
34
$7. The set {^^nM} is orthonormal on [2, 2]. The reasoning is the
similar to that of property \&7.
$8. The set {$n(u;)} is complete in finiteenergy functions on [fi, 2]. See
the argument from property ^8.
$9. Let us define
&k, trunc W
te[r,T + T]
0 elsewhere.
If $A:,trunc(w) is the Fourier transform of k,tTunc{t), then
^fc.trunc(^) An$n(tj), ^ ^ .
(3.4)
To see this, we note
trunc(t) = Vfc,trunc (t T 2^)
Taking the Fourier transforms we have
*&k,trunc 0*0 = ^ ( T trunc (^)
= A jo; (r ^(u;)} u> < Q.
Now iu> (r \kn(o;) is the Fourier transform of ipn (t r
(j)n(t). Thus,
iu ( t tf(u;) = $(w)
and (3.4) follows.
3.2 Solution of the Linear Predictor
Having defined our basis functions {(Â£)} and discussed their key properties,
we now return our attention to the prediction problem for a bandlimited process.
Let us consider the energyconstrained optimal linear predictor of a bandlimited
process whose power spectral density is flat in the frequency band of interest. The
42
(3.37)
Thus, making use of (3.36) and (3.37) we have
(3.38)
Note also,
(3.39)
Thus,
G^(T) = 2Vf.
(3.40)
In evaluating An and <^n(0) = ipn(T f), we used the extensive tables of
spheroidal wave functions compiled in the 1970s at the Naval Research Laborato
ry [70, 23]. The procedure is described in the appendix. Using (A.6) and (3.2) in
(3.22), and noting that VnW *s symmetric in t for all n, we obtain the curves for
is the highest value of n tabulated in the NRL tables. In the figure, the solid curves
represent the function Jmin(E) plotted against the square root of E for sample
parameter values T = 0.2, r = .04 and ) = 10,20,..., 50. The dashed straight
lines represent the initial slopes of the curves, calculated according to (3.35) and
(3.31). The small circles indicate the points at which n = 1 in (3.22). As E de
creases below this point, the curves quickly approach their linear approximations,
as expected.
3.4 A Bandlimited Process in White Noise
As shown in this and the previous chapter, the mean squared prediction error
of a bandlimited process may be made arbitrarily small by allowing the energy of
the predictor impulse response to increase, but this is true only if a sample function
of the process is known without error on an interval of positive length. Consider a
60
We assume that decisiondirected tracking may proceed as long as the
instantaneous symbol error rate does not rise above Ps = .01. When this threshold
is crossed, we change position in Figure 4.6 from adaptation to prediction. We
wish to resume adaptation as soon as the receiver emerges from the fade, so for the
parameter r we choose the mean fade duration, to be calculated in Section 4.2.4.
We wish to use a predictor that minimizes the mean squared error based on reliable
estimates of c(n) since the time the receiver emerged from the last fade, so for
the parameter T we choose the mean time above the fade threshold, also to be
calculated in Section 4.2.4. We wish to know under what conditions this approach
will allow us to recover correct channel tracking with a probability of Pr > 0.9.
4.2.3 Error Statistics of a Simple Adaptive Algorithm
As a first step in addressing this problem, we choose a simple adaptive
algorithm. Then we will determine the statistical behavior of c(n), the adaptive
estimate of c(n) = x(n) + iy[n), as well as the effect of estimation errors on (n),
the input to the detector. Since we only use adaptation when the error rate is
low (Ps < .01), we may assume in what follows that (n) = u(n). Looking at
Figure 4.5, we see that
r(n) = u(n)c(n) + v(n), (4.10)
and
Now,
u(n)
c(n)r
(n)
c(n)
c(n)
u(n) +
v(n)
c(n)'
e(n) = r(n) y(n) = [u(n)c(n) + u(n)] u(n)c(n).
Thus, solving for c(n), we have
c(n) = c(n) +
v(n)
u(n)
e(rc)
u{n)'
(4.11)
(4.12)
(4.13)
83
succeed in opening up terrestrial mobile channels to the kinds of information rates
that are now common in other types of communication is not yet certain. Still, if
our contribution can be used to clarify issues and avoid some wasted effort, then
our work will have been worthwhile.
5.2 Future Work
Our immediate research objective is to extend the solution of the bandlimited
prediction problem to processes with spectral densities of the form (4.9), as was
discussed in Section 4.3. In this way, our findings would be more directly applicable
to mobile radio fading. As was mentioned in the same section, we would also like
to find a useful bound on the minimum mean squared prediction error of such a
process, given a computation based on the flat spectral density model.
Also, the techniques that we have developed allow us to carry out a predic
tion based on a finitelength interval of known values. We may wonder whether
such a prediction might be improved if we considered all past values to be known.
Thus, we also plan to extend our solution to this case as well.
We pursued a continuoustime analysis in order to find a predictor perfor
mance bound that is independent of the symbol interval. In digital modulation,
however, the use of adaptive techniques will yield estimates of the fading enve
lope at discrete times, and these will form the basis of our prediction. This will
become especially important as we explore the problem of adaptive equalization.
For this reason, an extension of the solution to the discretetime case is of inter
est. We particularly want to know if the discretetime solution converges to the
continuoustime solution in some sense as the symbol interval is shrunk to zero.
Our analysis assumes that the prediction is to be based on a sample function
that has been corrupted by white noise. Many estimation techniques result in
errors that are not well modeled as white noise. Thus, one of our research goals is
64
is close enough to c(n) at that instant, we may be able to recover correct channel
tracking.
Recall that the parameter r represents how far in the future we wish to
predict c(i), where t = nTs, with respects to the interval of adaptively estimated
values. If this interval is considered to end at the instant that a(n) crosses below
ctf, then r should be chosen to equal the expected fade duration; that is, the
expected time interval during which a(n) remains below aÂ¡. We call the expected
fade duration tf. It may be computed by [27, p. 36]
*>=)<434>
where
(4.35)
and
1 r2nfm
= / Sxx(u)unduj. (4.36)
2/K J 27r/m
We consider x(t) to be a bandlimited process with a flat spectral density. Since
we are only interested in the ratio of b0 to 62 in (4.34), the scaling of (4.36) is
arbitrary. Thus, we let ^(u;) = K, uj G [27r/m, 27r/m]. We have
(4.37)
and
Thus,
and
= f
2tt J
2rfm K
Ku2du =
2*fm 27T
L3
U!
2*fm 87T2
O J TT
2nfm o
Mo
62
tt(2 Kfm)
\ s~fKfm3
47xfr
2 >
(4.38)
(4.39)
(4.40)
13
1.3.4 Equalization and Carrier Recovery
Although the term equalization derives from linear filter theory, it now applies
to any scheme aimed at compensating for the effects of a dispersive channel. For
general background on equalization, we again recommend Proakis [54, Ch. 10, 11],
Linear techniques have found widespread use in telecommunications, but they
often prove inadequate in wireless channels, where they can lead to inordinate
noise amplification [65, p. 264]. One alternative is decisionfeedback equalization,
which uses the output of the nonlinear detector to compensate for the dispersion
in a manner that reduces the noise amplification. Maximumlikelihood sequence
estimation is a technique which is in a sense optimal and may be carried out
using the Viterbi algorithm [17], which was originally developed for decoding
convolutional codes [71]. The Viterbi algorithm has the disadvantage of imposing a
heavy computational burden. It also requires an accurate estimate of the channel
impulse response, about which we shall say more shortly.
Both telecommunications and wireless channels are characterized by vari
/
ablility, so equalization is normally carried out adaptively. Qureshi is a muchcited
reference [55]. At this point a brief explanation is in order. The conventional
approach to adaptive equalization is to transmit a prearranged training sequence
of symbols, which the receiver attempts to equalize using its initial setting. The
error is then measured between the receivers detector input and the original
known sequence, and the equalizer is adjusted accordingly. For linear and decision
feedback equalizers, the adaptive algorithms may adjust the equalizer tap gains
directly. In the case of maximumlikelihood sequence estimation, the channel im
pulse response is needed. Often, the channel is modeled as a linear finiteimpulse
response filter, and the adaptive algorithm is used to find the filter coefficients in a
systemidentification mode [54, Sec. 113].
30
3.1 Basis Functions
Discussions of the prolate spheroidal wave functions, as well as proofs of many
of their properties, are contained in treatments by Slepian et al. [60], Papoulis [49],
and Frieden [19]. We shall summarize some key properties without restating proofs
from these references.
3.1.1 Prolate Spheroidal Wave Functions
The prolate spheroidal wave functions, are solutions of the following
integral eigenvalue problem:
f\ s)^dS = i31)
This equation has nontrivial solutions for only a countable set of eigenvalues {An}.
Each An is real and positive, such that the set {An} may be ordered as
1 > A0 > Ai > A2 > ... > 0.
If so ordered we also have
lim An = 0.
n oo
To each An there corresponds only one function iÂ¡)n(t) within a constant factor.
With a proper choice of this factor, the functions {i/)n(t)} form a set with the
following properties:
$1. Each function ipn(t) is real valued.
^2. The set {ipn(t)} is orthonormal on [00,00]; that is,
/OO
1pm{t)lpn{t)dt
00
1 m = n
<
0 m / n.
\&3. The set {'ipnit)} is complete in Tbandlimited, finiteenergy functions.
This means that if F(u) = 0 for w > ft and f2(t)dt < 00, then there
Copyright 2000
by
Raphael J. Lyman
54
Fading Channel
Figure 4.5: Adaptive channel estimation for a flatfading channel (cuts A, B and C
are for reference in Figure 4.6).
low. It is not clear how high the error rate can be before the adaptive algorithm
loses track, but estimates in the literature range between Pa = .1 and Ps .01
(see Haykin [25, p. 37], Macchi and Eweda [43] and Trabelsi [67] for discussions
in the case of decisiondirected equalization, with which this problem bears some
similarity).
The scheme of Figure 4.5 is a form of decisionaided carrier recovery. Other
decisionaided approaches are offered by Proakis [54, Sec. 624] and Macchi [42,
Ch. 12]. Often, the tasks of recovering carrier phase and compensating for channel
gain are handled separately. Our approach tracks both simultaneously. We do this
only to simplify the presentation, since other approaches introduce nonlinearities
that complicate the analysis.
4.2.1 Prediction of the Fading Envelope
Now, when the receiver enters a deep fade, the signal will be overcome by
noise, causing frequent errors in (n). Since [n) does not equal u(nTs), c(nTs) is
77
5
4.5
4
3.5
C! 3
"3 2.5
7s
CO
2
1.5
1
0.5
0
1
1
l
f
1
1
1
\
1
.\
\
1
1
\
1
1
\
\
/
/
V
\
/
S
1
0.5
0
(o/n
0.5
1.5
Figure 4.9: Comparison of fadingenvelope spectra. The dotted line indicates the
spectrum most commonly assumed in mobile radio. The solid line indicates the
flat, bandlimited spectrum of Chapter 3.
a
5
4.5
4
3.5
3
3,2.5
3
co
2
1.5
1
0.5
0
1
i
i
i
\
i
i
\
\
/
/
\
Â¡> "
/
s
p
1 0.5 0 0.5
coJS1
1.5
Figure 4.10: A piecewise approximation of the fadingenvelope spectrum, using a
flat segment and two sinusoidal components.
45
must be some value of E that minimizes Jmm(E). This is equivalent to saying that,
for given fl, T, r and aJ(h) has a minimum even though J(h) does not. This is
also in accordance with the findings of Chapter 2.
3.5 Conclusions
We have presented the solution of the linear prediction problem for a band
limited process whose spectral density is flat within the band limits. This special
case provides a concrete example that illustrates many of the properties shown to
hold for the more general bandlimited prediction problem discussed in Chapter 2.
In addition, we have shown how to use the solution to compute values of the func
tion Jmin{E), which for given values of Q, T and r as defined in Section 2.1, is the
minimum mean squared prediction error that may be attained when the predictor
impulse response is constrained to have energy E. We then used this function
to analyze the case of a bandlimited process which has been corrupted by white
noise, showing how to determine if a specified mean squared prediction error can be
attained.
In solving the problem, we used a timeshifted version of the prolate
spheroidal wave functions as basis functions. The special orthogonality, sym
metry, extremal and completeness properties of these functions allowed for a simple
solution, using a Lagrange multiplier. These same properties aided in determin
ing the behavior of the function Jm\n(E) for small E. Computation of the wave
function values themselves is a complex numerical problem [6, 16]. Fortunately, we
were able to rely upon published tabulated values [70, 23], though one might wish
that quality software for computing these functions were more widely available (see
discussion in the appendix).
This chapter, and the previous one, offer some insight into the predictability
of bandlimited processes. Much work remains to be done on this subject, but at
this point it will be helpful to discuss how our findings to date could be used in the
[40]R. J. Lyman and W. W. Edmonson. Linear prediction of bandlimited
processes with flat spectral densities. Submitted to IEEE Transactions on
Signal Processing.
91
[41] R. J. Lyman, W. W. Edmonson, M. Rao, and S. McCullough. The predictabil
ity of continuoustime, bandlimited processes. IEEE Transactions on Signal
Processing, 48(2):311316, February 2000.
[42] O. Macchi. Adaptive Processing: The Least Mean Squares Approach with
Applications in Transmission. New York: Wiley, 1995.
[43] O. Macchi and E. Eweda. Convergence analysis of selfadaptive equalizers.
IEEE Transactions on Information Theory, 30(2):161176, March 1984.
[44] S. L. Marple Jr. Digital Spectral Analysis with Applications. Englewood Cliffs,
NJ: PrenticeHall, 1987.
[45] F. Marvasti. Comments on A note on the predictability of bandlimited
processes. Proceedings of the IEEE, 74(11):1596, November 1986.
[46] J. E. Mazo. Analysis of decisiondirected equalizer convergence. Bell System
Technical Journal, 59(10): 18571876, December 1980.
[47] Y. Okumura, E. Ohmori, T. Kawano, and K. Fukuda. Field strength and its
variability in VHF and UHF landmobile radio service. Review of the Electrical
Communication Laboratory, 16(910):825873, SeptemberOctober 1968.
[48] A. Papoulis. A new algorithm in spectral analysis and bandlimited extrapola
tion. IEEE Transactions on Circuits and Systems, 22(9):735742, September
1975.
[49] A. Papoulis. Signal Analysis. New York: McGrawHill, 1977.
[50] A. Papoulis. A note on the predictability of bandlimited processes. Proceed
ings of the IEEE, 73(8):13321333, August 1985.
[51] A. Papoulis. Predictable processes and Wolds decomposition: A review.
IEEE Transactions on Acoustics, Speech, and Signal Processing, 33(4):933938,
August 1985.
[52] A. Papoulis. Probability, Random Variables, and Stochastic Processes, 3rd ed.
New York: McGrawHill, 1991.
[53] P. Z. Peebles. Digital Communication Systems. Englewood Cliffs, NJ:
PrenticeHall, 1987.
[54] J. G. Proakis. Digital Communications, 3rd ed. New York: McGrawHill, 1995.
[55] S. U. H. Qureshi. Adaptive equalization. Proceedings of the IEEE, 73(9):1349
1387, September 1985.
ACKNOWLEDGMENTS
I have rarely paid much attention to acknowledgments in the books and
articles that I have read, viewing them mostly as formalities, but now that I am
assuming the role of presenter, I would not dream of submitting this dissertation
without offering my deepest gratitude to those who have made this work possible.
The research described here is the result of a team effort. In addition to myself, the
team includes my advisor, Dr. William Edmonson, as well as two faculty members,
Dr. Scott McCullough and Dr. Murali Rao, from the Department of Mathematics.
I am also grateful to Dr. John M. M. Anderson, Dr. Tan F. Wong, and Dr. Haniph
A. Latchman, for graciously agreeing to serve on my supervisory committee.
Generous support for this project, and especially for my participation in it, was
provided by the Motorola LandMobile Products Sector.
At various times during my work I have received assistance from many peo
ple. A few of them simply must be mentioned. Dr. Scott Miller and Dr. Tan Wong
have been invaluable sources of information regarding communication theory in
general and wireless technology in particular. Both have offered many helpful
comments and suggestions. Conversations that I have had with Dr. Bert Nelin have
yielded interesting insights into the phenomenon of frequencyselective fading. I al
so thank Dr. M. V. Ramana, who expressed enthusiasm for the problem at an early
stage, and participated with me in a memorable brainstorming session. In a more
general way, Dr. Leon W. Couch II has aided me greatly by making the benefit of
his experience available whenever I have needed it, and I am much obliged.
A word is in order here regarding my relationship with my advisor. Early
on, Dr. Edmonson took a considerable risk by providing me with not only the
IV
28
hand, the bandlimited model is valid, then the estimation errors on the known
interval become the primary factor which limits predictability, as was discussed
in Section 2.3. In fact, the choice of an appropriate model may be influenced
by whether the estimation errors on the known interval are expected to be the
dominant source of prediction errors.
If the estimation errors can be modeled as white noise with known positive
variance, then J, the mean squared prediction error, has a minimum that is greater
than zero. We recall that the energy of h(t) was central in the analysis of this
case. Thus, in the case where values of x(t) on the known interval are error free,
an upper bound on this energy is a natural choice for a constraint on h(t), since J
would not have a minimum without some constraint.
We now have two objectives. One is to construct a function h(t) which
attains Jmin{E). The other is the computation of Jmin(E) itself. This latter goal
would be useful in the case where a linear predictor is desired, but realizing the
impulse response h(t) is not possible or practical, making it necessary to resort to
some suboptimal approach. Knowledge of Jmm{E) would be useful in determining
how closely this suboptimal predictor approaches optimal performance.
As of yet, neither h(t) nor JmÂ¡n(E) has been found for the general case.
Solutions have been found for a special case, however, which we discuss in the next
chapter.
This work is dedicated to Miss Lena Margaret Lyman.
I had a little nut tree, nothing would it bear
But a silver nutmeg and a golden pear.
The King of Spains daughter came to visit me,
And all for the sake of my little nut tree.
I skipped over water, I danced over sea,
And all the birds in the air couldnt catch me.
Mother Goose Rhyme
87
evaluated by
(A.4)
We also make use of [70, p. xv]
A(c) = (!) K>M)F
(A.5)
Solving (A.5) for (c, 1), and using (A.4), (A.3) with z = ^t, (A.2) and (A.l) we
obtain
(A.6)
The function i?^(c, Â£) is tabulated for f = %Â¡t > 1 => t >
t > which is the interval
of interest.
Working with tabulated values can be tedious, and it should be noted that
some computer programs for evaluating the prolate spheroidal wave functions are
available [38, 69, 74, 34], but many of these are written in outdated languages, are
not portable, or have not been tested thoroughly.
52
Figure 4.4: Complex fading envelope, fm 12 Hz. In this figure, x(t) is the real
part, y(t) is the imaginary part, a(t) is the magnitude and 6(t) is the phase.
50
In this expression, J0 is the zerothorder Bessel function of the first kind, and fm is
the maximum Doppler frequency, which is given by
(4.8)
where fc is the carrier frequency, V is the speed of the receiver, and c is the speed
of light. In the remainder of the discussion, we shall primarily deal with the real
component, x(t). The same analysis, however, applies equally well to y(t). A graph
of Rxx(t) is shown in Figure 4.2.
The power spectral density of x(t) is represented by the Fourier transform of
(4.7), yielding
Sxx{f) *
1
27r/m^l
0
I/I < /
otherwise.
(4.9)
A graph of Sxx(f) is shown in Figure 4.3. Note that x(t) is bandlimited to the
frequencies / < fm.
As an example, consider a carrier frequency of fc = 400 MHz and a receiver
speed of V = 20 m.p.h. The maximum Doppler frequency would be fm = 12 Hz.
Figure 4.4 shows a 400 msec segment of a simulated fading envelope generated with
these values. Notice from the plot of the magnitude a(t) that, during this interval,
several fades are encountered, including two that fall below 20 dB compared with
the average signal power. During a deep fade, almost no signal energy reaches the
receiver. As a rule of thumb, some kind of fade will be encountered about once
every half wavelength.
Suppose that, in the above example, the symbol rate fs is such that fs /m;
or equivalently, the fading parameters change little over a single symbol interval.
Such a condition is called slow fading.
53
4.2 Adaptive Channel Estimation
If a digital radio signal is to be detected coherently, some means must be
provided for estimating the carrier frequency and phase. This task is called carrier
recovery. For some signal constellations, such as 16QAM, we must also estimate
the attenuation of the channel, or more commonly its reciprocal, the channel gain.
Suppose you have a digital communication system in a fading channel. We
assume a perfect frequency lock, but we still have to estimate and compensate for
errors in phase and gain. If we further assume that the primary source of these
errors is the fading channel itself, then these quantities are just the phase and
amplitude of the complex fading envelope mentioned in the previous section.
Consider the simple channel acquisition and tracking scheme shown in
Figure 4.5. Here, we model u(t) by its discrete version u(nTs), where Ts is the
symbol interval, and pass it through the flatfading channel of Figure 4.1. It is
multiplied by c(nTs) = x(nTs) + iy(nT3), and then white Gaussian noise is added.
The lower portion of Figure 4.5 shows a adaptive scheme for estimating c(nTs). At
the beginning of the transmission, u(nTs) is sent through both the channel and a
(singleweight) adaptive filter at the receiver. This may be accomplished by use of
a prearranged training sequence. The outputs r(n) and y(n) are then compared and
the estimate c{nTs) is updated based upon the difference e(n). We shall call c{nTs)
the channel estimate.
Any common adaptive algorithm, such as LMS or RLS, may be employed.
The received signal r(n) is then multiplied by the reciprocol of c(nTs) to remove
magnitude and phase distortion. Finally, the detector chooses the symbol (n) in
the signal alphabet that is closest to it(n).
Once the channel has been acquired, the symbols it(n) will be correct
with high probability and may be used in place of u(nTs) to track slow channel
variations in decisiondirected mode. This works well as long as the error rate is
CHAPTER 5
CONCLUSIONS AND FUTURE WORK
In this dissertation, we have discovered some key facts about the predictabil
ity of bandlimited processes, we have solved the bandlimited prediction problem
for processes with flat spectral densities, and we have shown how these findings
may be applied in the analysis of a problem in multipath fading. We believe that
the approach outlined here has potentially a very wide applicability, not only to
fading compensation, but to other problems where the bandlimited model applies.
Clearly, though, there is much that remains to be done. In this chapter, we offer
some conclusions based on our work and then outline some of our plans for future
research.
5.1 Conclusions
Mobileradio fading is an extremely difficult problem to overcome. Still, in
view of the markets apparently insatiable appetite for mobile communication
services, it seems likely that the urgency for dealing squarely with fadingrelated
issues will only increase with time.
It is not surprising that many techniques for fading compensation have been
proposed in the literature. Often, as we have discussed, these techniques require an
estimate of the current value of the fading envelope. We have mentioned adaptive
transmission, and dealt at some length with decisiondirected adaptive channel
tracking. As we have seen, such approaches often invlove an implicit prediction
problem, since performace analyses assuming the availability of perfect channel
estimates fail to account for the effects of estimation errors, as well as delays in
obtaining the estimates.
80
15
through the entire development, which helps to firm ones grasp on the relationships
among the various estimation techniques. It assumes a considerable degree of com
mitment on the part of the student, however, since the theoretical approach means
that some of the more popular estimation methods are dealt with in later chapters.
As was mentioned in Section 1.1, Wiener filter theory plays a central role in
many discussions about prediction. This subject is covered in texts on adaptive
filter theory, as well as spectral estimation [25, 44, 28, 64]. In particular, Kays
book is again very strong in theory, especially in his emphasis on vector space
concepts [28, Sec. 6.3.3]. The organization, however, does not seem as good as his
previously mentioned text [29], leading, perhaps, to unnecessary repetition and too
great a reliance on illustrative simulations. The more recent book by Stoica [64]
offers the virtue of succintness as well as coverage of more recent approaches,
including filterbank methods, some of which employ the discrete prolate spheroidal
sequences (see also Thomson [66]).
Papoulis is our source for almost all matters concerning stochastic process
es [52]. The famous book by Doob is more complete but, again, requires substantial
background in higher mathematics [14]. Also, Papoulis and Slepian are two authors
that have shown sustained interest in bandlimited modeling [52, 51, 49, 58].
For issues related to digital communications our primary source is the well
known book by Proakis [54]. In its coverage, depth and clarity it is hard to beat.
The book by Stuber [65] is also excellent, and is geared more toward mobile radio.
75
First, let us consider the prospects of extending the solution of Section 3.2 to
processes with spectral densities of the form (4.9). For simplicity, let
Sxx ) *
1
\/l U)2
0
M < i
otherwise.
(4.74)
Substituting into the objective function of (2.10) and noting that 2 = 1 we obtain
rb11^'2^ (475)
In Section 3.2, we expanded H(oj) in terms of the prolate spheroidal wave func
tions, allowing us to convert the objective function into an algebraic expression.
Unfortunately, the prolate functions are not orthogonal with respects to the
weighting function Sxx(u) as given by (4.74).
We may consider expanding H(u) in terms of functions, such as the Tcheby
cheff polynomials, which have the appropriate orthogonality properties [7, p. 54],
Alternatively, we may substitute cos# = \/l u2 and sin9 = cu in (4.75) to obtain
J = ^~ 2 1 H{sin6)\2d6. (4.76)
27T J*
The function G{9) = H{sin6) may now be expanded in terms of any complete
orthogonal set on [Â§,Â§] These may be the prolate functions, Legendre functions,
or simply sinusoids. The function h(t) will also be expanded in terms of an
orthogonal set on [r, r + T], so evaluating the expansion coefficients for G(9) would
require a cross product between the two sets of basis functions. This is potentially
the most tedious step. Once this is done, the Lagrange multiplier can be applied
and a system of linear equations, such as the one preceding (3.20), will result.
Unlike that case, however, we expect more than one unknown, in addition to the
Lagrange multiplier, to appear in each equation.
3
which the prediction error may be made to approach zero, as well as procedures
for calculating the predictor coefficients. All of these works assume that samples of
known values may be taken arbitrarily far in the past. None addresses the question
of how good a prediction is possible if the known past values are confined to a
given interval.
In this work, we consider the predictability of a bandlimited random process
using a continuoustime linear predictor, rather than a linear combination of past
samples. We suppose that a sample function of the process is known over an
interval of arbitrary positive length, and we show that future values of that sample
function may be predicted with a mean squared error that is arbitrarily small,
regardless of how far in the future we wish to make the prediction.
We also show that this is no longer true when we apply an energy constraint
to the impulse response of the linear predictor. The constrainedenergy problem
is used to analyze the case in which linear prediction is to be based on past values
which have been corrupted by estimation errors. If such errors are modeled as
white noise, we can then show that they impose a fundamental limitation on the
predictablility of the process.
An important contribution of the work, as described in Chapter 3, is the
solution of the bandlimited linear prediction problem in the case where the power
spectral density is constant within the band limits. The parameters of the problem
are the band limits, the length of the interval of known past values, how far in
the future the prediction is to be made, and the energy constraint. With these
parameters given, we show how to obtain an expansion for the optimal predictor
in terms of a set of basis functions that are timeshifted versions of the prolate
spheroidal wave functions [60]. We include a procedure for computing the minimum
mean squared error as a function of the energy applied in the constraint. Then we
outside of the Q band limit. This fact is critical to the discussion in the next
section.
19
2.2 Analysis
In this section, we shall refer to the following spaces of functions: Â£} and Â£2
contain functions which are, respectively, integrable and square integrable on the
real line. Â£2([7r, 7t]) contains functions which are square integrable on the interval
[7r, 7r]. Â£2T contains functions /(f) G Â£2 such that /(t) = 0 for t Â£ [t,t + T], and
C2(S) contains functions F(cv) such that S(u>)\F(uj)\2du} < oo.
Suppose that F(u>) G Â£2 and f(t) = 0 for t < 0. Then we say that
F(u) is in 'H2 of the upper half plane, written F(uj) G ?f2(UHP). Now suppose
G(u) G Â£2([7T, 7r]), and suppose that the Fourier coefficients {7} of G(w) are
given by
7n = r G{uj)eiuJndu.
J7T
If 7 = 0 for n < 0, we say that G(w) is in H2 of the unit disc, written G(uj) G
H2(UD).
From the previous section, we may formulate our prediction problem as
follows:
1 rn
minimize J = / 5II(a;)l /f(o;)2iia; w.r.t. h(t)
2tt Jn (2.10)
subject to h(t) = 0, t Â£ [r, r + T],
where J is the mean squared prediction error. Our first question is whether a
minimum exists for (2.10). We now show that there is an allowable h(t) which
makes J arbitrarily small.
Theorem 1 Let t, T and 2 be fixed, real numbers with T, tt > 0, and let S(u>) G
C.1, with S(w) >0, u> G [), fi] and S(uj) = 0, u> ^ [, 2]. Then for every
6
1.3 Literature Survey
In this section we shall review the literature that has proved relevant to
our research. The survey is in five parts. Section 1.3.1 includes previous material
dealing directly with the prediction and extrapolation of bandlimited signals. In
Section 1.3.2 we discuss sources for the various mathematical techniques that we
have employed in seeking a solution to this problem. Then, in Section 1.3.3, we
consider treatments of various issues in mobileradio fading, the problem which
motivated our interest in bandlimited processes. Section 1.3.4 includes sources
on equalization and carrier recovery, including blind techniques and the issue of
decisiondirected convergence. Finally, in Section 1.3.5, we discuss some general
references.
1.3.1 Bandlimited Prediction and Extrapolation
Up until now, investigations of bandlimited prediction have focused on the
following question: Suppose you have a stationary random process whose spectral
density vanishes outside of some finite interval, and suppose you may obtain sample
values of this process arbitrarily far in the past. You wish to predict a future value
as a linear combination of the past samples. Under what conditions is it possible to
make the prediction error approach zero, and how can the predictor coefficients be
calculated, knowing only the band limits and not the exact spectral shape? Note
that such coefficients may not be optimal for a given spectral density and set of
sample values.
An early treatment of this problem can be found in a 1962 text by Wainstein
and Zubakov [72, p. 70]. They suppose that the past values of the process are
uniformly sampled at a rate higher than three time the Nyquist rate, which is six
times the spectral band limit. They show that the mean squared prediction error
may be made arbitrarily small by taking linear combinations of values that extend
further and further into the past. The proof is constructive, showing how the
Figure 4.2: Autocorrelation function of a mobile radio fading parameter, fm
Figure 4.3: Power spectral density of a fading parameter.
10
Bandlimited extrapolation continues to be an active research field, and has
admitted of a wide variety of problem formulations. In particular, the case of
periodic bandlimited functions was of some interest to us (see, e.g., Soltanian
Zadeh [61]) because we briefly considered this as an alternative to the stochastic
model described in Chapter 2.
1.3.2 Mathematical Techniques
In solving the prediction problem of Chapter 3, we made use of the prolate
spheroidal wave functions. Interest in these functions increased after Bouwkamp
discovered a numerical technique for evaluating them [6, 16]. Their original
application was the solution of wave equations with boundary conditions defined
on prolate spheroids. Later, Slepian et al. wrote a series of articles describing their
properties and pointing out several interesting applications to signal processing
problems [60]. Papoulis has summarized many of the important properties very
succinctly while avoiding any mathematics above the level of basic calculus [49,
p. 205]. Frieden also offers a thorough tutorial [19].
The most extensive and authoritative tabulation of the prolate spheroidal
wave functions is that compiled by the Naval Research Laboratory [70, 23]. We
have relied upon these even though some software for evaluating the functions
is available [38, 69, 74, 34], since we had difficulty adapting the software to our
application (see discussion in the appendix).
In Chapter 3 we use these functions to obtain an expansion that transforms
a functional optimization into an algebraic one, which may then be solved using a
Lagrange multiplier. The approach is hinted at in a related problem described by
Slepian et al. [60, p. 53]. The details may be found in a numerical linear algebra
text by Golub and Van Loan [22, p. 582].
The more general treatment of Chapter 2 relies heavily on functional analysis.
A good text is provided by Conway [12]. For specific results concerning Hp
17
Now consider a real, continuoustime, widesense stationary, zeromean
random process x(t), which is known on the Tlength interval [t r T,t r], with
r, T > 0. We would like a predictor x(t), which is linear on the known portion of
x(t),
rtT
x(t) = / x(X)h(t X)d\, (2.1)
JtrT
where h(t) is a real and continuous function of t. Note that both x(t) and h(t) may
also depend on r and T. We wish to find a function h(t) which minimizes
J = Â£ {[x(i) ()]2} (2.2)
where J is the mean squared prediction error. We focus on linear predictors partly
because of their simplicity, but also, when the process is Gaussian, no predictor
can achieve a lower mean squared error than the optimal linear predictor, if it
exists [13, p. 231].
Suppose we constrain h(t) to be zero except for the interval t [t,t + T],
Then we may write (2.1) as a convolution,
/oo
x(X)h(t X)dX (2.3)
oo
= x(t) h(t),
where
h(t)=0, f $ [r, t + T\. (2.4)
Thus, at every t, x(t) represents a prediction of x(t) based upon the known interval
[t T T,t t).
Note that in (2.3), h(t) is viewed as the impulse response of a linear, time
invariant filter. This is justified by the stationarity of x(t). In our discussion
of prediction, however, a possible point of confusion is the rlength time delay
introduced by h(t) in (2.4). As we shall see, this results in a simple frequency
81
In evaluating the performance of a receiver, we may wonder which effect,
estimation errors or delays, is more important, but our work shows that if the
parameter to be estimated may be modeled as a bandlimited process, then the two
factors are actually related, because if the estimates are error free, then we can
overcome the delay with a prediction whose error may be made arbitrarily small.
The flatfading channel model discussed in Section 4.1, including the charac
teristic Ushaped spectral density, has been widely applied since its introduction
in 1968 [11, 27]. Clearly, this is a bandlimited model, and it has been known since
at least 1962 that such bandlimited random processes are predictable in the mean
square sense [72], but this result does not seem to be well known among those
researching the fading problem.
The predictability of bandlimited processes, as discussed in Chapter 2, would
seem to offer hope that the implicit prediction problems mentioned above can be
solved, but as we have seen, these findings must be interpreted with care. After all,
many common phenomena, including human speech, can reasonably be modeled
as bandlimited. Does this mean that such phenomena can really be predicted with
arbitrarily small error?
The answer is no, if there is any error in the process values on which the
prediction is to be based, as was made clear in Chapter 2. And such error is always
present, whether caused by an estimation procedure, a measurement technique,
or simply the quantization error of storing a number in memory. In applying
Wiener analysis to nonbandlimited processes, the effects of noise are often ignored,
because the spectrum itself limits predictability. But in bandlimited processes the
noise, no matter how small, is always the dominant factor in prediction error.
The application of our techniques to adaptive channel estimation in Sec
tion 4.2 was intended mainly as an illustration. In that case, the simplicity of the
adaptive algorithm chosen allowed us to trace the main source of errors; it is the
57
T ) r  t
Thresh.
Figure 4.7: While in a deep fade, decisiondirected channel tracking does not work
well. Using prediction in this region may allow the tracker to converge correctly
once the signal comes out of the fade.
LIST OF FIGURES
Figure page
3.1 Minimum mean squared prediction error of a bandlimited process. . 43
4.1 Model of a flatfading mobile radio channel 49
4.2 Autocorrelation function of a fading parameter 51
4.3 Power spectral density of a fading parameter 51
4.4 Example of a complex fading envelope 52
4.5 Adaptive channel estimation for a flatfading channel 54
4.6 Predictive method for maintaining correct channel tracking 56
4.7 Linear prediction of a fading parameter 57
4.8 Predictability analysis for recovery of correct channel tracking 74
4.9 Comparison of fadingenvelope spectra 77
4.10 A piecewise approximation of the fadingenvelope spectrum 77
4.11 Application of the flat spectral density to nonadaptive prediction. . 78
Vlll
Figure 4.6: Predictive method for maintaining correct channel tracking through
deep fade (cuts A, B and C refer to Figure 4.5).
BIOGRAPHICAL SKETCH
Raphael J. Lyman grew up in the Tampa Bay area of Florida. He received
the B.S. degree in electrical engineering from the University of Houston in 1983,
and the M.S. degree in electrical engineering from the University of South Florida
in 1988. He has ten years of industrial experience in the manufacture of electronic
assemblies, especially in the fields of quality and test. Since 1995, he has been pur
suing doctoral studies in the Department of Electrical and Computer Engineering
at the University of Florida. His research interests include mobile radio, signal
processing, and estimation theory. He is married and has one daughter.
94
38
where the time limitation on h(t), constraint 1 of (3.5), is implicit in the expansion
(3.10).
3.2.2 Lagrange Multiplier
The technique of Lagrange multipliers will be used to solve (3.18). This
technique combines the objective function and the constraint into the following
unconstrained optimization problem:
00 / 00 \
minimize p(p, p) = Â£[^(0) ~ AnPnf + P ( Yh E ) (3.19)
n=0 \n=0 /
where p is the Lagrange multiplier, a scalar which is to be determined.
We solve first formally for the pns, setting the appropriate partial derivatives
to zero,
= 2A[^n(0) Ap] + 2pApn = 0, n = 0,1,2,...
OPn
The solution of this equation is
pM = (3.20)
i /i
where the dependence on p is explicitly noted. The multiplier p is then chosen so
that the energy constraint of (3.18) is met,
H Xn[pn{p)}2
n=0
oo
^n(O)I"
An T P_
= E.
(3.21)
This series converges for all positive p and gives a unique positive solution for every
positive E. The nonlinear equation (3.21) is called a secular equation [22, p. 582].
It may be solved by a line search on the positive axis, and then its solution may be
substituted into (3.20) to obtain values for the pns.
3.3 Minimum Mean Squared Prediction Error
In the previous section, we succeeded in solving for the coefficients {pn}, so
the expansion (3.10) for h(t) is uniquely determined for any positive E and the
27
since the Qjts sum to 1. Thus, J(gk) > k.
Now {gk} converges strongly to h0. Strong convergence implies convergence of
the norm so pjfc > \\h0\\. And since any strongly convergent sequence also satisfies
the definition of a weakly convergent sequence, we may use the argument in the
proof of Theorem 3, substituting {gk} for {/i} to show that J(gk) > J(ha). We
thus have
J(9k) = J(9k) + ^\\9k\\2  J(h0) + ^\\hÂ£ = J(h0).
And since {J(gk)} converges to k, we have
k J (ha).
To prove (2.), we note from (2.19) that, if k = J(h0) = 0, then \\ho\\ 0. But then
Hq(u) = 0 almost everywhere. Thus, from (2.9) we have
J(0) = ~ [ Sxx(u)du = 0.
Note that the key result of Theorem 4 is that J has a minimum even if we do not
impose a constraint on h{t).
2.4 Conclusions
The results presented in this chapter show that there is no theoretical
limit to the predictability of a bandlimited random process, even if knowledge
of the process is restricted to a short interval. The usefulness of these findings
depends on a number of factors. One of these is the validity of modeling a physical
phenomenon as bandlimited [58]. Also, as we have discussed, there is the problem
of estimation errors on the known interval.
If the bandlimited model is invalid, then Theorem 1 does not apply, and it
is possible that the predictability of the process is limited even if we have perfect
knowledge of the infinite past (see, e.g., Papoulis [52, Sec. 14.2]). If, on the other
79
In summary, it is likely that the gap between the flat spectral density model
and that of Figure 4.3 will be closed soon. But in the interim, the solution of
Chapter 3 will still be a useful tool in analyzing realworld fading problems.
72
computational procedure then becomes
STEP 1: Solve ^ ^(1,2c, c(2d l))An(c) j \ \ =E for//
n=0 VAn(c)+/V
STEP 2: Then J^E) = n f) ^(1, 2c, c(2d 1)) fl ) ,
n=0 V An(c) + t*J
(4.73)
7T
where the leading factor 7r = in step 2 is necessary because Jm\n{l,E) is
d L
normalized.
For this problem, we use (4.30), (4.33) and (4.35) to compute r = .0964.
Then, using (4.62) and (4.63) we compute c = 15.92 and d = .00934. For our
computations, we round c to 16 and d to .01.
4.2.8 Summary of Analysis Procedure and Discussion
A summary of our analysis procedure is given in Table 4.1. Using values
of 7&, 76,/ and computed in (4.30), (4.33) and (4.54), and also using rounded
values of c = 16.0 and d = .01, we obtain the graph of Figure 4.8. We see that we
may recover correct channel tracking with probability Pr > .9 for values of O less
that or equal to .52. This is a maximum Doppler frequency fm = of less than
2n
or equal to .08 Hz. This does not indicate good performance because, in mobile
radio, the value of fm can be much higher, on the order of tens of hertz. As stated
previously, though, our work on this problem is incomplete. The case presented
here is intended mainly as an illustration of how the techniques developed in
previous chapters can be applied to the analysis of a practical problem.
There are many approaches we might take to improve the performance of
our adaptive/predictive channel tracker. For example, instead of restarting the
adaptation as soon as our prediction c(n) crosses the fading threshold, we might
wait a few symbol intervals for the signal to strengthen. If the quality of our
prediction has not degraded too much, we might have a better chance of correct
recovery at the higher SNR. We also note that our techniques only allow us to
71
Also, with the substitution t' = Qt, the energy constraint becomes
yy{Qdt'=K <7>
Now let G(uj) H(Q.uj) ==> g{t) = ^h(^). Our optimization problem becomes
1 rl
minimize J = / 1 G(iv)\2cku
subject to 1. g(t) = 0 t Â£ [2cd, c(2d + 1)] (4.68)
/OO p.
.Jm=a
We see from this that
= (4.69)
Thus, we need use the procedure (3.22) only once, scaling appropriately to obtain
i7min(l, E), then use (4.69) to compute E) for arbitrary ). We recall from
Section 3.1.1 (see page 32) that xJjn(t) depends implicitly on Q, and T. We may
therefore write (3.2) as
= %/> (2, T, i t f) (4.70)
Now we let = 1 and note from (4.62) that T = 2c. Also, from (4.63) we have
r f = c(2d 1).
(4.71)
Thus, making use of (A.6) from the appendix, we have
4(0) = 4(l,2c,c(2dl))
= ^[S~(c.1)]2K)(c,l + 2<Â¡)]2. (4.72)
where S$n(c,r]) and R^n(c, Â£) are the angular and radial functions discussed in the
appendix. Note that in (4.72), we use the fact that ip2(,T,t) is even in t. Our
84
to determine how to apply the solution of the bandlimited prediction problem when
the corrupting noise is colored.
In addition to considering extensions of the solution in Chapter 3, we also
wish to address questions raised in the analysis of Section 4.2, dealing with adap
tive channel estimation in a fading environment. At the end of the section, we list
ed some steps we may take to improve the performance of our adaptive/predictive
approach to carrier tracking. Also, it will be recalled that we assumed throughout
that a symbol error probability of less than .01 is a sufficient condition for main
taining correct, decisiondirected tracking. Although various figures are cited in
the literature (see discussion on page 54), there seems to be little experimental or
theoretical justification for them. We wish to address this issue more rigorously,
since clearly the greater the symbol error rate that is acceptable, the easier our
prediction will be.
Adaptive channel tracking is not the only fading compensation approach that
could make use of a prediction of the fading envelope. Many techniques require
a current estimate of the channel impulse response, and performance analyses
of these techniques often assume that a perfect channel estimate is available.
In Section 1.1 we discussed the case of adaptive transmission. Unfortunately,
estimation necessarily involves some delay. As we discussed, we would like to
explore the application of our predictive techniques to the problem of overcoming
this delay, in adaptive transmission as well as other approaches requiring channel
information.
A final, broad, and very important question is how the prediction of a
bandlimited process is actually to be carried out. Although we do not wish to
pursue a detailed discussion of this subject here, it is not clear that a direct
realization of the optimal predictor impulse response is always possible or desirable.
Also, we would like to know how to make such a predictor adaptive. The door is
CHAPTER 3
PROCESSES WITH FLAT SPECTRAL DENSITIES
In this chapter we shall solve the linear prediction problem for bandlimited
processes with flat power spectral densities1 For such processes we have S(u>) = 1
for cj < Q and S(uj) 0 elsewhere. In solving this problem, we shall make use
of a set of basis functions, {<>k()}, which are timeshifted versions of the prolate
spheroidal wave functions {iÂ¡)*(i)}. As we shall see, these are eigenfunctions of
a particular linear integral operator. They have many orthogonality, symmetry
and extremal properties that make them very well suited to certain problems in
Fourier analysis, especially ones in which a specific time and frequency interval are
of interest simultaneously.
In Section 3.1, we discuss the key properties of the prolate spheroidal wave
functions and the timeshifted basis functions based upon them. In Section 3.2,
we solve the bandlimited linear prediction problem for the flat spectral density
case. In Section 3.3, we focus on the computation of the minimum mean squared
prediction error. A discussion of the prediction problem for a bandlimited process
embedded in white noise follows in Section 3.4. Finally, we offer some conclusions
in Section 3.5.
1 This chapter is based on Linear Prediction of Bandlimited Processes with
Flat Spectral Densities, by R. J. Lyman and W. W. Edmonson. 2000 IEEE.
Used by permission. This paper is currently under consideration for publication in
IEEE Transactions on Signal Processing.
29
90
[27] W. C. Jakes, editor. Microwave Mobile Communications. New York: Wiley,
1974.
[28] S. M. Kay. Modem Spectral Estimation: Theory and Application. Englewood
Cliffs, NJ: PrenticeHall, 1988.
[29] S. M. Kay. Fundamentals of Statistical Signal Processing: Estimation Theory.
Englewood Cliffs, NJ: PrenticeHall, 1993.
[30] R. A. Kennedy, B. D. O. Anderson, and R. R. Bitmead. Blind adaptation
of decisionfeedback equalizers: Gross convergence properties. International
Journal of Adaptive Control and Signal Processing, 7(6):497523, November
December 1993.
[31] J. J. Knab. Interpolation of bandlimited functions using the approximate
prolate series. IEEE Transactions on Information Theory, 25(6):717720,
November 1979.
[32] J. J. Knab. Noncentral interpolation of bandlimited signals. IEEE Transac
tions on Aerospace and Electronic Systems, 17(4):586591, July 1981.
[33] P. Koosis. Introduction to Hp Spaces, with an Appendix on Wolffs Proof of the
Corona Theorem. Cambridge: Cambridge University Press, 1980.
[34] M. B. Kozin, V. V. Volkov, and D. I. Svergun. A compact algorithm for
evaluating linear prolate functions. IEEE Transactions on Signal Processing,
45(4):10751078, April 1997.
[35] E. Kreyszig. Introductory Functional Analysis with Applications. New York:
Wiley, 1978.
[36] N. Levinson. Gap and Density Theorems. New York: American Mathematical
Society, 1940.
[37] M. Li, A. Bateman, and J. P. McGeehan. Analysis of decisionaided DPSK in
the presence of multipath fading. In 6th International Conference on Mobile
Radio and Personal Communications, pages 157162, Stevenage, England:
IEE, Michael Faraday House, 1991.
[38] D. W. Lozier and F. W. J. Olver. Numerical evaluation of special functions. In
W. Gautschi, editor, Mathematics of Computation 19431993: A HalfCentury
of Computational Mathematics, volume 48 of Proceedings of Symposia in
Applied Mathematics, pages 79125, Vancouver, BC, August 1993. American
Mathematical Society.
[39] D. G. Luenberger. Linear and Nonlinear Programming, 2nd ed. Reading, MA:
AddisonWesley, 1984.
24
2.3 Dealing with Estimation Errors
We shall discuss the findings of the previous section as they relate to the
prediction of a bandlimited process whose values on the known interval are
corrupted by estimation errors. Consider the random process x(t) of Section 2.1,
and suppose that we have an estimate of that process,
i(A) = x(A) + iu(A), A [t t T, t t], (215)
where w(t) represents the estimation error. We consider w(t) to be a real, zero
mean, widesense stationary process, uncorrelated with x(t), and having an
autocorrelation function given by
Rww(t) = OwHt), > 0. (2.16)
This expression indicates that w(t) is being modeled as white noise. Of course, the
white noise assumption may not be valid in some applications. Nevertheless, by
considering the case of white noise, we can simplify the mathematical treatment
and still gain insight into the effects of estimation errors. Also, by using this
approach, we can see why E, an upper bound on energy, is a natural choice for a
constraint on h(t).
Let us consider the linear predictor from (2.3) with input x(t) instead of x(t).
Then the error becomes
e(t) = x(t) x(t)
= x(t) x(t) h(t)
= x(t) [x(t) + w(t)] h(t)
= x(t) x(t) h(t) w(t) h(t)
= x(t) [5(f) h(t)] w(t) h(t).
(2.17)
32
If ^A:,trunc(w) is the Fourier transform of V,fc,trunc(f)> then
trunc) An\kn(ci>), 1^1 ^
Note that 'I'trunc) is not zero for u> > 1 This property says nothing
about the behavior of \I,fc,trunc(w) outside of the interval [f2, ].
It should be noted that ipn{t) depends implicitly on Q and T, and may be
may be written An(c). In keeping with the convention of Slepian et al., we shall
normally suppress this additional notation.
3.1.2 TimeShifted Basis Functions
The basis functions we shall use to solve the linear prediction problem in
Section 3.2 are timeshifted versions of {Vn(i)} We define
(3.2)
For each n, the function 0n(i) satisfies the following integral equation:
(3.3)
where A is the nth eigenvalue of (3.1). To see this, we apply (3.2) to (3.1) to
obtain
Now, with the substitutions s' = s + r + ^ and t' = t + r + j we obtain
ds' = A n(j)n(t').
Since this equation must hold for all t and thus all t', the prime notation may be
dropped and (3.3) results. This shows that (3.1) and (3.3) have the same set of
eigenvalues {An}.
APPENDIX
EVALUATION OF BASIS FUNCTIONS
This appendix describes how to compute values of An and ijjn(t), as described
by Slepian et al. [60], using the tables of spheroidal wave functions published by
the Naval Research Laboratory [70, 23]. These include the angular functions,
S$n(c, rj) [70], and the radial functions, R$n(c,() [23]. We adhere to the notational
QT
conventions of Slepian et al., with c = In the NRL tables, the symbols l and
i
h are used in place of n and c respectively. The eigenvalues An(c) are appended to
the tables of (c, 77) [70, p. 404].
The function ipn(Â£l,T,t) may be found using [70, p. xv]
= M(A1)
{/i [50n(C>S)]2rfs}2
In the NRL tables, the angular functions are normalized such that [70, p. xi]
so the denominator of (A.l) is easily handled. Unfortunately, for our prediction
problem we are interested in values of ipn(t) for i > ~, but Sfj(c, rj) is only
tabulated for 77 = < 1 t < ^. This may be handled by using the relation [16,
p. 32]
Son(c> z) = 0n(c) RSI (C, z), (A.3)
where the angular and radial functions have been extended to entire functions of
the complex variable z. The function n0n(c) is called a joining factor, and may be
86
20
F(u) G Â£2(S) and every e > 0, there is an h(t) G Â£2r such that
P S(u)\F{u) H{u)\2du < e.
Jn
Proof: Consider the space T of functions H{uj) for which h(t) G Â£2T. To prove
the theorem it is sufficient to show that if G(u>) G Â£2(5), then for every H(u) G T
if
/OO
S(u)H*(u)G(u)du = 0,
OO
then
/OO
S(u)\G(u)\2du = 0.
OO
Now, suppose f^0OS(uj)H*(ijj)G(uj)du = 0. Then
/oo r roo 1*
/ h(t)e~tutdt S{uj)G(uj)du
oo Uoo
/oo r roo
h{t) / S^G^e^du dt.
oo UOO
In order to justify the interchange of integration, we must show that
S(u)G(is absolutely integrable,
/OO
S'(a;)G'(a;)da; < oo.
OO
That this is so follows from CauchySchwarz, since
/OO
S{u)\G{u)\du
OO
= (Vsm) (v/shighi) du
1_
< {^j S^du^j S(o;)G(a;)2da;^
(2.11)
Both of the integrals on the right hand side are finite by hypothesis. Thus (2.11) is
true and the interchange is justified.
9
This survey shows that contributions dealing with bandlimited prediction
have been infrequent, often marked by brevity, and characterized by a lack of
discussion concerning specific engineering applications. Often, the topic has been
introduced as an adjunct to a more general discussion. In addition, there seems to
be some suspicion of bandlimited modeling in general. For example, Wainstein and
Zubakov [72] note that the addition of even a small amount of white noise to the
bandlimited signal destroys its predictability properties. They add,
Of course, as the intensity of the noise approaches zero, we obtain the
formal possibility of predicting m(t) arbitrarily far ahead. However,
to substantially increase the time interval for which the prediction is
possible, we must enormously increase the signaltonoise ratio, (p. 73)
Also, see Slepian [58] for a more general discussion of bandlimited modeling. In our
research, we address the issue of noise sensitivity by regularizing the problem with
an energy constraint (see Section 2.3).
In contrast to the work on bandlimited prediction, the related but distinct
problem of bandlimited extrapolation has been much more extensively studied.
This problem in its basic form may be stated as follows: Suppose you have a
segment of a finiteenergy signal whose Fourier transform vanishes outside of
some finite interval. How can you use that segment to calculate the values of the
function that are not on the known segment?
We mention only a few examples that have come to our attention in the
course of our work. In 1975, Papoulis described an iterative procedure for perform
ing the extrapolation on a continuoustime signal [48]. He showed theoretically
that the algorithm converges, but numerical difficulties led Cadzow to develop a
new algorithm, which is still widely used [10]. Slepian et al. also had addressed
the problem in 1961 as an application of the prolate spheroidal wave functions [60,
p. 46].
xml version 1.0 encoding UTF8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID EMADJBLYX_PHRLQ1 INGEST_TIME 20150205T21:28:53Z PACKAGE AA00024530_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
12
fading environment is particularly harsh in terrestrial mobile communications. In a
1968 paper, Okumura et al. covered many aspects of mobile radio propagation [47],
including smallscale fading caused by multipath.
Clarke dealt directly with the multipath problem, developing a mathematical
formulation that resulted in the bandlimited model mentioned above [11]. Jakes
incorporated the model in his 1974 book, showing how it could be used to simulate
a fading signal in the laboratory [27, Ch. 1]. For this reason, the model itself is
often attributed to Jakes.
The effects of fading on a given channel are intimately connected with
the modulation scheme, and for this reason a good knowledge of modulation is
absolutely essential for understanding the fading problem. The treatment by
Proakis is characteristically thorough [54, Ch. 4]. Alternatives include Peebles [53,
Ch. 5], Stuber [65, Ch. 4] and a review article by Aghvami [2]. The latter two are
7r
geared more specifically towards mobile radio, and include coverage of QPSK
and GMSK modulation, which are incorporated in current mobileradio TDMA
standards.
Adaptive transmission was mentioned in Section 1.1 as a possible means of
fading compensation. The paper by Goldsmith contains a literature review, as well
as an analysis of the effects of delayed channel estimates [21].
When multipath effects cause a significant time dispersion of the transmitted
signal, the received signal is said to undergo frequencyselective fading. This
problem was our initial research interest. Proakis includes the subject in his
chapter on fading [54], We also mention an interesting early research paper by
Bello and Nelin [3], which specifically addresses noncoherent and differentially
coherent modulation (see also the followup [4]).
LINEAR PREDICTION OF CONTINUOUSTIME,
BANDLIMITED PROCESSES, WITH APPLICATIONS
TO FADING IN MOBILE RADIO
By
RAPHAEL J. LYMAN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2000
TABLE OF CONTENTS
Â£age
ACKNOWLEDGMENTS iv
LIST OF FIGURES viii
ABSTRACT ix
CHAPTERS
1 PREDICTION OF BANDLIMITED PROCESSES 1
1.1 Project Thesis 2
1.2 Research Contributions 4
1.3 Literature Survey 6
2 PROBLEM FORMULATION AND ANALYSIS 16
2.1 Problem Formulation 16
2.2 Analysis 19
2.3 Dealing with Estimation Errors 24
2.4 Conclusions . 27
3 PROCESSES WITH FLAT SPECTRAL DENSITIES 29
3.1 Basis Functions 30
3.2 Solution of the Linear Predictor 34
3.3 Minimum Mean Squared Prediction Error 38
3.4 A Bandlimited Process in White Noise 42
3.5 Conclusions 45
4 APPLICATIONS TO FADING IN MOBILE RADIO 47
4.1 Multipath Fading in Mobile Radio 47
4.2 Adaptive Channel Estimation 53
4.3 Addressing the Model Mismatch 74
5 CONCLUSIONS AND FUTURE WORK 80
5.1 Conclusions 80
5.2 Future Work 83
APPENDIX EVALUATION OF BASIS FUNCTIONS 86
vi
55
adjusted incorrectly. By the time the receiver comes out of the fade, c(nT) may
have wandered far enough from its optimal value that the decisiondirected channel
estimator fails to converge.
We propose a predictive method for overcoming this problem, as shown in
Figure 4.6. With the switches in the position shown, while a(nTs) = \c(nTs)\
is sufficiently large we see the adaptive scheme operating normally in decision
directed mode, except that past estimates c(nTs) are clocked into a tapped delay
line. When a(nTs) = \c(nTs)\ falls below some threshold value, the switch positions
change, turning off the adaptation, which is no longer reliable. A regressor vector
of reliable past estimates of c(nTs) is latched, and the adaptive estimate is replaced
with a prediction c(nTs) = x(nTs) + iy(nTs) of the current value of c(nTs) based
upon the vector of c(nTs). Note that, in the figure, r represents how much time has
passed since the adaptation was turned off.
When c(nTs) rises above the fading threshold in absolute value, we return to
decisiondirected tracking, using the last predicted value as the starting point for
the adaptation. If the prediction is close enough to the true value of c(nTs) when
the receiver emerges from the fade, then it may be possible to continue tracking
the channel without transmitting a new training sequence. This idea is further
illustrated in Figure 4.7. Clearly, the performance of the rpredictor in Figure 4.6
is key. For mobile radio we note that, because the real and imaginary parts of
c(t) = x(t) + iy(t) are viewed as independent processes, the problem reduces to the
two equivalent problems of predicting x(t) and y(t).
We wish to use the analysis techniques of Chapter 3 to determine the
conditions under which the method of Figure 4.6 can work. Two theoretical
difficulties present themselves. First, since we want a prediction of x(t), the power
spectal density Sxx(u) is important, as was made clear in the analysis of Chapter 2.
utmost freedom in seeking a solution for our problem, but also considerable voice
in defining the problem itself. His approach throughout has been one of unbounded
enthusiasm and support, and I have always sought to make his risk pay off. It
would seem that in the academic world a dose of sobriety, when needed, is always
available. But in the moments of greatest doubt, when intuition is your only
guide, it is probably safe to say that encouragement is what is needed most, and
Dr. Edmonson provided it.
Of course, not all the challenges in producing a doctoral dissertation are
academic ones, but thanks to the efforts of Ron Smith in creating the DT^X class
ufthesis, the mechanics of manuscript preparation have been considerably eased
for me, and for many other graduate students at the University of Florida.
I am grateful to the Institute of Electrical and Electronic Engineers for grant
ing permission to reuse copyrighted material [41, 40]. Thanks to the administrative
staff of the Department of Electrical and Computer Engineering, especially Sharon
Bosarge, Janet Burke, Janet Holman, Linda Kahila, Chris Reardon, Alice Riegel,
Greta Sbrocco, and Wes Shamblin, who have made my life easier in an uncountable
infinity of ways. Thanks also to Frances Smith, who has been scrupulous in her
efforts to keep our working environment tidy, an underappreciated task, to be sure.
Finally, I owe a very special debt of gratitude to my wife, Chulalak, for her
patience and support, and beg the pardon of my twoyearold daughter, Lena, who
has had to put up with so much absence on my part.
Raphael J. Lyman
v

