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Linear prediction of continuous-time bandlimited processes, with applications to fading mobile radio
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Lyman, Raphael J
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x, 94 leaves : ill. ; 29 cm.

## Subjects

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Error rates ( jstor )
Fourier transformations ( jstor )
Linear prediction ( 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 )
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non-fiction ( marcgt )

## Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 2000.
Bibliography:
Includes bibliographical references (leaves 88-93).
General Note:
Printout.
General Note:
Vita.
Statement of Responsibility:
by Raphael J. Lyman.

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University of Florida
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Full Text

LINEAR PREDICTION OF CONTINUOUS-TIME,
BANDLIMITED PROCESSES, WITH APPLICATIONS

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

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 Land-Mobile 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 frequency-selective 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 two-year-old daughter, Lena, who has had to put up with so much absence on my part.

Raphael J. Lyman

v

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 fiat-fading 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 fiat-fading 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 fading-envelope spectra . . . . . . . . . . . . . . . . . 77

4.10 A piecewise approximation of the fading-envelope spectrum . . . . . 77 4.11 Application of the fiat spectral density to non-adaptive 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 CONTINUOUS-TIME,

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 continuous-time 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, energy-constrained predictor when the process spectral density is flat within the band limits. As basis

ix

functions, we use time-shifted 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 decision-directed 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 decision-directed 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 minimum-phase and maximum-phase finite-energy 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 continuous-time 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 constrained-energy 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 time-shifted 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 decision-directed 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 continuous-time, 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.

8. An approach for applying the techniques of items 5-7 above to a problem

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 1-4. The techniques in items 5-7 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 mobile-radio 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 decision-directed 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 non-stochastic 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 uniform-sampling case, though it seems unclear whether the derivation is good for finite-energy or finite-power 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 uniformly-spaced 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 finite-power 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 time-shifted 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 signal-to-noise 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 finite-energy 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 continuous-time 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, lesser-known 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 well-written 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].

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 small-scale 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 mobile-radio 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 frequency-selective 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 non-coherent 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 decision-feedback equalization, which uses the output of the nonlinear detector to compensate for the dispersion in a manner that reduces the noise amplification. Maximum-likelihood 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 much-cited 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 maximum-likelihood sequence estimation, the channel impulse response is needed. Often, the channel is modeled as a linear finite-impulse response filter, and the adaptive algorithm is used to find the filter coefficients in a system-identification mode [54, Sec. 11-3].

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 higher-order 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 decision-directed 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 decision-feedback equalizers [30].

Overcoming the difficulties associated with decision-directed 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 decision-aided carrier tracking are included in the books by Proakis [54, Sec. 6-2-4] 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 well-ordered 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 filter-bank 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 continuous-time 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 frequency-domain 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)e--tdt,
1 /_oFwe tw
f(t) = F(w)ee
27r -oo

SThis chapter is based on "The Predictability of Continuous-Time, 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. 311-316, February 2000.

16

17

Now consider a real, continuous-time, wide-sense stationary, zero-mean
random process x(t), which is known on the T-length 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 T-length 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 frequency-domain 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 zero-phase all-pass 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)e--tdt *S(w)G(w)dw f L h(t) [f S(w)G(w)e-tdw] 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 Cauchy-Schwarz, 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 7-2(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 Cauchy-Schwarz, IH(w)12 = f + h(t)e-Iwtdt 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)e-e'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, wide-sense 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 J-m =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 time-shifted 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 time-shifted 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 O-bandlimited, finite-energy 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 finite-energy functions on [-, ]. This is

true even if such a function is not a segment of a bandlimited function. T6. Each function on(t) is Q-bandlimited. 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 finite-energy functions on [-Q, Q]. This

is true because a finite-energy function with the property F(w) = 0 for

IwI > Q has an inverse Fourier transform f(t) which is Q-bandlimited. 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 Time-Shifted Basis Functions
The basis functions we shall use to solve the linear prediction problem in Section 3.2 are time-shifted 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 O-bandlimited, finite-energy functions. To
see this, suppose f(t) is such a function. Then the time-shifted function
f (t + + I) is also Q-bandlimited. We may thus write
O0
(t +r + 7:) = Zan0n(t).
n=o
Substituting t' = t + r + T we have
00 00
(') = E-an, (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 finite-energy functions on [-r, -r + T]. This
is the interval of property T5 shifted according to (3.2).

46. Each function On(t) is O-bandlimited. 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 finite-energy 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 energy-constrained 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 finite-energy 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 finite-energy 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 E-0 ) (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 = JIir-F2 (w)d sinc20s ds = 2--F2 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, zero-mean, widesense stationary, white-noise 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 dash-dot 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 time-shifted 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 signal-processing problem. This is the topic of the following chapter.

CHAPTER 4

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 in-phase part, and whose imaginary part represents the quadrature part. Using this approach the fading channel can be modeled as a time-varying 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 time-varying 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

0(t) = arctan y(t) (4.6) a x(t)) (

The flat-fading channel model is shown in Figure 4.1. A flat-fading channel exhibits high amplitude correlation across the entire bandwidth of the time-varying 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 frequency-selective fading, the theory of the flat-fading channel may be used in the analysis of the more complicated frequency-selective case [54].

A popular model for a flat-fading 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 zeroth-order 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 16-QAM, 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 flat-fading 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 (single-weight) 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 decision-directed mode. This works well as long as the error rate is

54

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 flat-fading 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 decision-directed equalization, with which this problem bears some similarity).

The scheme of Figure 4.5 is a form of decision-aided carrier recovery. Other decision-aided approaches are offered by Proakis [54, Sec. 6-2-4] 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 decision-directed 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 decision-directed 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 r-predictor 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 *- r-Predictor (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 Decision-Directed I I

I

Prediction I I I I ] I

T t+

Figure 4.7: While in a deep fade, decision-directed 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 U-shaped 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 U-shaped 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 continuous-time analysis of Chapters 2 and 3 assumed that a sample function of x(t) was known over the entire continuum of a positive-length 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 discrete-time 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 continuous-time analysis was motivated by a desire to find a performance bound that was independent of T,. We assume that an optimal continuous-time predictor of the form (2.1) will not be outperformed by a discrete-time predictor. Although we have not proved this, our heuristic reasoning is that a continuous-time predictor makes use of all the information on the known interval, whereas the discrete-time predictor uses only a subset of this information. Extending our analysis to dicrete-time prediction, and establishing a rigorous connection between the discrete-time and continuous-time 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 zero-mean complex Gaussian process whose real and imaginary parts are independent and identically distributed, and the signal-to-noise 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 decision-directed 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 zero-mean 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 first-order term is retained because the zeroth-order 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 time-average value of a2(n), then the time-average 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) Decision-directed 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 r-2Es + 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 (single-bit 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

Decision-directed 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 re-r (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 time-average 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 fading-envelope 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 zero-mean 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 zero-mean 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 time-bandwidth 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(-2d-1))
- 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 decision-directed
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 decision-directed 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 = 1fl--w2 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 fading-envelope 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 fading-envelope 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 non-adaptive 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 non-adaptive 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 fading-envelope 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 real-world 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

Mobile-radio 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 fading-related 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 decision-directed 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 flat-fading channel model discussed in Section 4.1, including the characteristic U-shaped 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 non-bandlimited 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 signal-to-noise ratio, either by increasing the transmit power or by redesigning the receiver using more expensive low-noise devices. Would such expenditure be worth it?

We hesitate to take such a brute-force approach because an overall signal-tonoise ratio is usually targeted to achieve a desired bit error rate. Raising the SNR just to improve the performance of decision-directed adaptation may result in a BER that is much lower than necessary. Decision-directed 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 mobile-radio 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 decision-directed approach can otherwise be made to work in a fading environment without increasing power, resorting to diversity, or applying some other resource-consuming 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 finite-length 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 continuous-time 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 discrete-time case is of interest. We particularly want to know if the discrete-time solution converges to the continuous-time 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, decision-directed 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. Abed-Meraim, W. Qiu, and Y. Hua. Blind system identification. Proceedings of the IEEE, 85(8):1310-1322, August 1997.
[2] A. H. Aghvami. Digital modulation techniques for mobile and personal
communication systems. Electronics and Communication Engineering Journal,
5(3):125-132, 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, 11:170-186,
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, 12:230-231, December 1964.

[5] F. G. Beutler. Error-free recovery of signals from irregularly spaced samples.
SIAM Review, 8(3):328-335, July 1966.

[6] C. J. Bouwkamp. On spheroidal wave functions of order zero. Journal of Mathematics and Physics, 26:79-92, 1947.
[7] A. Broman. An Introduction to Partial Differential Equations, from Fourier Series to Boundary-value 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):662-664, September
1972.

[91 J. L. Brown. On the prediction of a band-limited signal from past samples.
Proceedings of the IEEE, 74(11):1596-1598, November 1986.

[10] J. A. Cadzow. An extrapolation procedure for band-limited signals. IEEE
Transactions on Acoustics, Speech, and Signal Processing, 27(1):4-12, February
1979.

[11] R. H. Clarke. A statistical theory of mobile radio reception. Bell System
Technical Journal, 47(6):957-1000, July-August 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
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[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
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[16] C. Flammer. Spheroidal Wave Functions. Stanford, CA: Stanford University
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[17] G. D. Forney. Maximum-likelihood sequence estimation of digital sequences in
the presence of intersymbol interference. IEEE Transactions on Information
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[18] C. Fox. An Introduction to the Calculus of Variations. London: Oxford
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[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 311-407. Amsterdam: North-Holland, 1971.

[20] D. N. Godard. Self-recovering equalization and carrier tracking in twodimensional data communication systems. IEEE Transactions on Communications, 28(11):1867-1875, November 1980.

[21] A. J. Goldsmith and S. G. Chua. Variable-rate variable-power MQAM for
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[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

[24] S. Haykin, editor. Blind Deconvolution. Englewood Cliffs, NJ: Prentice-Hall,
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[29] S. M. Kay. Fundamentals of Statistical Signal Processing: Estimation Theory.
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[30] R. A. Kennedy, B. D. O. Anderson, and R. R. Bitmead. Blind adaptation
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[32] J. J. Knab. Noncentral interpolation of band-limited signals. IEEE Transactions on Aerospace and Electronic Systems, 17(4):586-591, 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
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[35] E. Kreyszig. Introductory Functional Analysis with Applications. New York:
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[37] M. Li, A. Bateman, and J. P. McGeehan. Analysis of decision-aided DPSK in
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[38] D. W. Lozier and F. W. J. Olver. Numerical evaluation of special functions. In
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[39] D. G. Luenberger. Linear and Nonlinear Programming, 2nd ed. Reading, MA:

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 fading-envelope 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^\yS-u \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
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 in-phase part, and whose imaginary part represents the
quadrature part. Using this approach the fading channel can be modeled as a
time-varying 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
time-varying 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. Abed-Meraim, 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):125-132, 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, 11:170-186,
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, 12:230-231, December 1964.
[5] F. G. Beutler. Error-free recovery of signals from irregularly spaced samples.
SIAM Review, 8(3):328-335, July 1966.
[6] C. J. Bouwkamp. On spheroidal wave functions of order zero. Journal of
Mathematics and Physics, 26:79-92, 1947.
[7] A. Broman. An Introduction to Partial Differential Equations, from Fourier
Series to Boundary-value 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):662-664, September
1972.
[9] J. L. Brown. On the prediction of a band-limited signal from past samples.
Proceedings of the IEEE, 74(11):1596-1598, November 1986.
[10] J. A. Cadzow. An extrapolation procedure for band-limited signals. IEEE
Transactions on Acoustics, Speech, and Signal Processing, 27(1):4-12, February
1979.
[11] R. H. Clarke. A statistical theory of mobile radio reception. Bell System
Technical Journal, 47(6):957-1000, July-August 1968.
88

where c is the time-bandwidth 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 J-Cl
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 J-q
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 finite-energy 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)
Decision-directed 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 (single-bit error)
= 20(^2^;)
.01. (4.32)
This yields
76,/ ~ 3.32 => 76,/ ~ 6.64. (4.33)
4.2.4 Statistics of the Fading Envelope
Decision-directed 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 zero-mean complex Gaussian process whose real and
imaginary parts are independent and identically distributed, and the signal-to-noise
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 zero-mean 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 zero-mean 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.
8. An approach for applying the techniques of items 5-7 above to a problem
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 1-4. The techniques in items 5-7 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)\l-H(u)\2du + / Sww(u)\H(u)\2duj.
Z7T J oo 7T Joo
= J + l\\h\\2- (2.18)
where ||h||2 = Since J and J both depend on h(t) we write
J = J(/.)+4||h||2. (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||/in||2 = 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
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 time-average 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 time-average value of a2(n), then the time-average 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, lesser-known 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
well-written 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].
Our research of bandlimited prediction was motivated by the problem of
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 finite-energy 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 Q-bandlimited. 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 finite-energy functions on [O, O]. This
is true because a finite-energy function with the property F(uj) = 0 for
M > 2 has an inverse Fourier transform f(t) which is ii-bandlimited. 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
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
signal-to-noise ratio, either by increasing the transmit power or by redesigning the
receiver using more expensive low-noise devices. Would such expenditure be worth
it?
We hesitate to take such a brute-force approach because an overall signal-to-
noise ratio is usually targeted to achieve a desired bit error rate. Raising the SNR
just to improve the performance of decision-directed adaptation may result in a
BER that is much lower than necessary. Decision-directed 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 mobile-radio
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 decision-directed approach can
otherwise be made to work in a fading environment without increasing power,
resorting to diversity, or applying some other resource-consuming 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 Cauchy-Schwarz,
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
and
6(t) = arctan
(4.6)
The flat-fading channel model is shown in Figure 4.1. A flat-fading channel exhibits
high amplitude correlation across the entire bandwidth of the time-varying 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 frequency-selective fading, the theory of the flat-fading channel
may be used in the analysis of the more complicated frequency-selective case [54].
A popular model for a flat-fading 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) 2-y/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 77-2
sine 2Qsds = / F2(u>)duj
-oo 2W-oofF 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 higher-order 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 decision-directed
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 decision-feedback equalizers [30].
Overcoming the difficulties associated with decision-directed 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 decision-aided carrier tracking are
included in the books by Proakis [54, Sec. 6-2-4] 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 well-ordered 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):308-328, 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):292-300, March
1976.
[59] D. Slepian. Prolate spheroidal wave functions, Fourier analysis and
uncertainty-V: The discrete case. Bell System Technical Journal, 57(5):1371-
1430, May-June 1978.
[60] D. Slepian, H. J. Landau, and H. 0. Poliak. Prolate spheroidal wave functions,
Fourier analysis and uncertainty-I & II. Bell System Technical Journal,
40(l):43-84, January 1961.
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wave functions, volume 1: Prolate, m = 0. Technical report. Washing
ton, DC: Naval Research Laboratory, 1975. Available through NTIS, AN:

CHAPTER 2
PROBLEM FORMULATION AND ANALYSIS
In this chapter we study the continuous-time 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 frequency-domain 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
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 Continuous-Time, 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. 311-316, 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)\l-H(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 dash-dot
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, zero-mean, wide-
sense stationary, white-noise 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 decision-directed 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 J-n
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 decision-directed
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 continuous-time, 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

78
Figure 4.11: Application of the flat spectral density to non-adaptive 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 non-adaptive 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 fading-envelope 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 decision-directed
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]).
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
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[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 CONTINUOUS-TIME,
BANDLIMITED PROCESSES, WITH APPLICATIONS
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 continuous-time 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, energy-constrained 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 J-Q
Then there is an h G Be such that
-i- [ S(uj)\1 H(uj)\2duj = k.
27T J-n

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CHAPTER 4
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
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,
47

61
As an algorithm for updating our estimate of c(n) we choose
-/ is \ e(n) / v(n)
c(n + l) = c(n) Hj-r = 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 zero-mean 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)'
(4-17)
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 first-order term is retained because the zeroth-order 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 U-shaped 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
U-shaped 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 continuous-time analysis of
Chapters 2 and 3 assumed that a sample function of x(t) was known over the
entire continuum of a positive-length 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 discrete-time 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 continuous-time analysis was motivated by
a desire to find a performance bound that was independent of Ts. We assume that
an optimal continuous-time predictor of the form (2.1) will not be outperformed
by a discrete-time predictor. Although we have not proved this, our heuristic
reasoning is that a continuous-time predictor makes use of all the information on
the known interval, whereas the discrete-time predictor uses only a subset of this
information. Extending our analysis to dicrete-time prediction, and establishing a
rigorous connection between the discrete-time and continuous-time 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 finite-energy
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 7-0
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 frequency-domain 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
c-f-
*
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 J-n
= T-/ Sxx(u)\\-H{u)\2du,
ztt J-n
(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 zero-phase all-pass 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
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 minimum-phase and maximum-phase
finite-energy 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 decision-directed
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£4-27+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.
J-n
We then use (3.7), (3.9) and property \$7 to obtain
f \F(u)\idu> = 2KY'4i{0).
J-a 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 non-stochastic 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 uniform-sampling case, though
it seems unclear whether the derivation is good for finite-energy or finite-power
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 uniformly-spaced 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 time-shifted 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 decision-directed
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
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 Sl-bandlimited, finite-energy functions. To
see this, suppose f(t) is such a function. Then the time-shifted function
f (t-1-r + is also f2-bandlimited. 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 finite-energy 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 ii-bandlimited. 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 signal-processing 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):260-269, April 1967.
[72] L. A. Wainstein and V. D. Zubakov. Extraction of Signals from Noise.
Englewood Cliffs, NJ: Prentice-Hall, 1962.
[73] R. Weinstock. Calculus of Variations, with Applications to Physics and
Engineering. New York: McGraw-Hill, 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
finite-power 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 time-shifted 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
(4-6i)
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 finite-energy 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 energy-constrained 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 decision-directed 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
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 finite-length 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 continuous-time 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 discrete-time case is of inter
est. We particularly want to know if the discrete-time solution converges to the
continuous-time 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]
*>=)<4-34>
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 decision-feedback equalization,
which uses the output of the nonlinear detector to compensate for the dispersion
in a manner that reduces the noise amplification. Maximum-likelihood 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 much-cited
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-
directly. In the case of maximum-likelihood sequence estimation, the channel im
pulse response is needed. Often, the channel is modeled as a linear finite-impulse
response filter, and the adaptive algorithm is used to find the filter coefficients in a
system-identification mode [54, Sec. 11-3].

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 = i3-1)
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, finite-energy functions.
This means that if F(u) = 0 for |w| > ft and f2(t)dt < 00, then there

by
Raphael J. Lyman

54
Figure 4.5: Adaptive channel estimation for a flat-fading 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 decision-directed equalization, with which this problem bears some
similarity).
The scheme of Figure 4.5 is a form of decision-aided carrier recovery. Other
decision-aided approaches are offered by Proakis [54, Sec. 6-2-4] 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 fading-envelope 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 fading-envelope 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 time-shifted 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 continuous-time, bandlimited processes. IEEE Transactions on Signal
Processing, 48(2):311-316, 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 self-adaptive equalizers.
IEEE Transactions on Information Theory, 30(2):161176, March 1984.
[44] S. L. Marple Jr. Digital Spectral Analysis with Applications. Englewood Cliffs,
NJ: Prentice-Hall, 1987.
[45] F. Marvasti. Comments on A note on the predictability of band-limited
processes. Proceedings of the IEEE, 74(11):1596, November 1986.
[46] J. E. Mazo. Analysis of decision-directed equalizer convergence. Bell System
Technical Journal, 59(10): 1857-1876, December 1980.
[47] Y. Okumura, E. Ohmori, T. Kawano, and K. Fukuda. Field strength and its
variability in VHF and UHF land-mobile radio service. Review of the Electrical
Communication Laboratory, 16(9-10):825-873, September-October 1968.
[48] A. Papoulis. A new algorithm in spectral analysis and band-limited extrapola
tion. IEEE Transactions on Circuits and Systems, 22(9):735-742, September
1975.
[49] A. Papoulis. Signal Analysis. New York: McGraw-Hill, 1977.
[50] A. Papoulis. A note on the predictability of band-limited processes. Proceed
ings of the IEEE, 73(8):1332-1333, August 1985.
[51] A. Papoulis. Predictable processes and Wolds decomposition: A review.
IEEE Transactions on Acoustics, Speech, and Signal Processing, 33(4):933-938,
August 1985.
[52] A. Papoulis. Probability, Random Variables, and Stochastic Processes, 3rd ed.
New York: McGraw-Hill, 1991.
[53] P. Z. Peebles. Digital Communication Systems. Englewood Cliffs, NJ:
Prentice-Hall, 1987.
[54] J. G. Proakis. Digital Communications, 3rd ed. New York: McGraw-Hill, 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 Land-Mobile 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 frequency-selective 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 zeroth-order 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
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
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 16-QAM, 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 flat-fading 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
(single-weight) 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 decision-directed 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
view of the markets apparently insatiable appetite for mobile communication
services, it seems likely that the urgency for dealing squarely with fading-related
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 decision-directed 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 filter-bank methods, some of which employ the discrete prolate spheroidal
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 continuous-time 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 constrained-energy 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 time-shifted 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)e-iuJndu.
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 J-n (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
motivated our interest in bandlimited processes. Section 1.3.4 includes sources
on equalization and carrier recovery, including blind techniques and the issue of
decision-directed 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 4.3: 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, continuous-time, wide-sense stationary, zero-mean
random process x(t), which is known on the T-length 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),
rt-T
x(t) = / x(X)h(t X)d\, (2.1)
Jt-r-T
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 r-length 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 flat-fading channel model discussed in Section 4.1, including the charac
teristic U-shaped 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
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 non-bandlimited 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, decision-directed 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
4.2 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
4.6 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 fading-envelope spectra 77
4.10 A piecewise approximation of the fading-envelope spectrum 77
4.11 Application of the flat spectral density to non-adaptive 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 real-world 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
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(2d-l))
= ^[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, decision-directed 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.
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 time-shifted 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 time-shifted 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
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[28] S. M. Kay. Modem Spectral Estimation: Theory and Application. Englewood
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[30] R. A. Kennedy, B. D. O. Anderson, and R. R. Bitmead. Blind adaptation
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Journal of Adaptive Control and Signal Processing, 7(6):497-523, November-
December 1993.
[31] J. J. Knab. Interpolation of band-limited functions using the approximate
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[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):1075-1078, 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 decision-aided DPSK in
the presence of multipath fading. In 6th International Conference on Mobile
Radio and Personal Communications, pages 157-162, Stevenage, England:
[38] D. W. Lozier and F. W. J. Olver. Numerical evaluation of special functions. In
W. Gautschi, editor, Mathematics of Computation 1943-1993: A Half-Century
of Computational Mathematics, volume 48 of Proceedings of Symposia in
Applied Mathematics, pages 79-125, Vancouver, BC, August 1993. American
Mathematical Society.
[39] D. G. Luenberger. Linear and Nonlinear Programming, 2nd ed. Reading, MA:

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], (2-15)
where w(t) represents the estimation error. We consider w(t) to be a real, zero-
mean, wide-sense 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
3.1.2 Time-Shifted Basis Functions
The basis functions we shall use to solve the linear prediction problem in
Section 3.2 are time-shifted 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.
J-n
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 Cauchy-Schwarz, 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 signal-to-noise 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 finite-energy 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 continuous-time 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].

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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 small-scale 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 mobile-radio 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
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 non-coherent and differentially

LINEAR PREDICTION OF CONTINUOUS-TIME,
BANDLIMITED PROCESSES, WITH APPLICATIONS
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

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
4.3 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
have wandered far enough from its optimal value that the decision-directed 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.
decision-directed 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 r-predictor 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 two-year-old daughter, Lena, who
has had to put up with so much absence on my part.
Raphael J. Lyman
v