Citation |

- Permanent Link:
- http://ufdc.ufl.edu/AA00024491/00001
## Material Information- Title:
- Direct-sequence code-division multiple-access overlay systems
- Creator:
- Rainbolt, Brad J, 1972-
- Publication Date:
- 1998
- Language:
- English
- Physical Description:
- ix, 86 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Bandwidth ( jstor )
Binary phase shift keying ( jstor ) Code division multiple access ( jstor ) Error rates ( jstor ) Narrowband ( jstor ) Random variables ( jstor ) Receivers ( jstor ) Signals ( jstor ) Simulations ( jstor ) Transmitters ( jstor ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1998.
- Bibliography:
- Includes bibliographical references (leaves 83-85).
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Brad J. Rainbolt.
## Record Information- Source Institution:
- University of Florida
- Rights Management:
- The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. This item may be protected by copyright but is made available here under a claim of fair use (17 U.S.C. Â§107) for non-profit research and educational purposes. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact the RDS coordinator (ufdissertations@uflib.ufl.edu) with any additional information they can provide.
- Resource Identifier:
- 41091692 ( OCLC )
ocm41091692
## UFDC Membership |

Downloads |

## This item has the following downloads: |

Full Text |

DIRECT-SEQUENCE CODE-DIVISION .MULTIPLE-ACCESS OVERLAY SYSTEMS BY BRAD .J. RAINBOLT 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 1998 I dedicate this work to my parents, Patricia A. and Ronald J. Rainbolt. ACKI\KNOWLEDGMENTS I would like to thank Professor Leoni \. Couch II, Professor Ulrich H. Kurzweg, Professor Haniph A. Latcluna, and Prof'essor .Jian Li for serving as members of my committee. I extend special thanks to imy adviser, Professor Scott L. Miller, not only for his time, but also for his expert guidance throughout my studies, as related both to research issues and to professional issues. I thank my family, my parents Patricia A. and Ronald J. Rainbolt in particular, for their support and encouragement throughout my studies. I also wish to acknowledge all of my friends at the University of Florida and elsewhere, especially my colleagues Ron Smith and Ali Almutairi. Finally, I acknowledge with gratitude financial support from the Robert C. Pittman Fellowship and from the National Science Foundation. 111 TABLE OF CONTENTS page ACKNOWLEDGMENTS ......... .................. iii LIST OF ABBREVIATIONS ............................. vi ABSTRACT ................. ................... viii CHAPTERS 1 INTRODUCTION ............................ 1 1.1 Code-Division Multil)h-Access ...... .............. 1 1.2 CDMA Overlay ............................ 5 1.3 Overview of the Dissertation . .................. 6 2 NARROWBAND SYSTEM PERFORMANCE IN THE AWGN CHANNEL ............................. 10 2.1 Effects of Overlay on a Narrowband User .......... 10 2.2 CDMA Transmitter Filtering .................. 15 2.3 Filtering Performance Criteria ................... 16 2.3.1 Gain of the BPSIK System ..................16 2.3.2 Effect on the PSD of the CDMA Signal ........ 17 2.3.3 Effect on the CDMA Code Sequence .......... 17 2.4 Filtering Methods and Results ................ 18 2.4.1 Eigenvector Filtering ................. 18 2.4.2 Null Filtering ....... ............... 20 2.4.3 Butterworth Filtering ................. 20 2.4.4 DFT-Based Filtering .................. 22 2.5 Summary ............. .................. 24 3 MMSE DETECTION OF FILTERED CDMA SIGNALS ...... 25 3.1 The MMSE Reciver ....... . .. ....... .. 25 3.2 MMSE Detection of Filtered CDMA Signals ........ 26 3.3 Simulation Results .... . . ................ 29 3.4 Summnary.......... ............................. 36 4 CDMA OVERLAY IN A CELLULAR. SYSTEM .......... 37 4.1 Characterization of the Cellular Environment ......... 38 iv 4.2 Performance of the Narrowband System ........... 45 4.3 Effects of CDMA Transmitter Notching ........... 47 4.3.1 Same-Link Assigninent . ................ 50 4.3.2 Staggereld-Link Assignment .............. 52 4.4 Simulations and Results . . .. .......... .. 53 4.5 Summary ...... ..... ................ 59 5 CELLULAR OVERLAY IN A FADING CHANNEL ......... 60 5.1 Limits on CDMA Capacity .................. 60 5.2 Limits on Narrowband Campacity. ... ....... ...... 65 5.3 Performance of the CDMA System .............. 71 5.4 Summary....................... ............... 77 6 CONCLUSIONS AND FUTURE WORK< .............. 78 6.1 Conclusions ......... ................. 78 6.2 Future Work ..... ...... ........... 81 REFERENCES ....................... .......... 83 BIOGRAPHICAL SKETCH .. ...... . ..... ............ 86 LIST OF ABBREVIATIONS AWGN: additive white Gaussian noise BPSK: binary phase-shift keying CDF: cumulative distribution function CDMA: code-division multiple-access DFT: discrete Fourier transform DS: direct-sequence FDMA: frequency-division multiple-access FH: frequency-hopped FIR: finite impulse response IEEE: The Institute of Electrical and Electronics Engineers, Inc. IIR: infinite impulse response ISI: inter-symbol interference LMS: least mean-squared LPD: low probability of detection LPI: low probability of intercept MAI: multi-access interference MC-CDMA: multi-carrier code-division nimultiple-access MMSE: minimum mean-squared error MSE: mean-squared error NBI: narrowband interference PDF: probability density function PSD: power spectral density RLS: recursive least-squares \'i SNR: signal-to-noise ratio SS: spread-spectrum vii1 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 DIRECT-SEQUENCE CODE-DIVISION MULTIPLE-ACCESS OVERLAY SYSTEMS Brad .J. Rainbolt December 1998 Chairman: Dr. Scott L. Miller Major Department: Electrical and Computer Engineering In this dissertation, the possibility of code-division multiple-access (CDMA) overlay, that is the simultaneous sharing of a frequency band by a narrowband system and a CDMA system, is investigated. In contrast to the majority of existing studies on overlay, which investigate almost exclusively the degradation experienced by CDMA users as a result of the narrowband users. the results presented here focus specifically on the effects of the overlay on the narrowband system. It is shown that even for CDMA systems operating with a small number of users in comparison to the system capacity in the absence of overlay, the narrow)and system will experience a substantial loss in performance. The use of notch-filtering in the CDMA users' transmitters in order to avoid active narrowband users can alleviate the degradation experienced by the narrowband system. In turn, the effects of the notching on the CDMA system itself are seen to be quite modest. v111 Overlay has great potential in a situation in which a frequency band which currently provides narrowband cellular service is designated to provide CDMA cellular service in the future. This transition can he made gradually with the implementation of overlay. Such a scenario is investigated in this research and promising results are presented. The use of multi-carrier CDNIA (MC-CDMA) is investigated for this purpose, and is seen to perform quite well. In a fading channel, diversity such as that offered by the use of multiple carriers improves the performance of a system operating at a given power level. In the overlay scenario in particular, there is the additional benefit to overall system performance. in terms of both CDMA and narrowband, that the CDMA users can lower their power in a imulti-carrier scenario and thus can reduce the amount of interference on the narrowband system. The feasibility of CDMA overlay is bolstered by the results presented in this dissertation, and a strong motivation for its use as a method of transition from narrowband service to CDMA service is argued as well. ix CHAPTER 1 INTRODUCTION In this chapter, the concept of code-division multiple-access (CDMA) will be presented. The benefits that CDMA offers as well as some of the main works of research in this field will be summarized. This chapter will also introduce CDMA overlay, the focus of this dissertation. which is the sharing of frequency spectrum by a CDMA system and a sparsely-populated narrowband system. An overview of the dissertation will conclude this introductory chapter. 1.1 Code-Division Multiple-Access CDMA is an emerging technology which employs spread-spectrum (SS) signaling, that is the intentional spreading of a digital signal over a bandwidth that is much greater than its information bandwidth. typically on the order of a 100-fold increase or more. Although spread-spectrumn systems are generally divided into two categories, direct-sequence (DS) and frequency-hopped (FH), depending on the method used to spread the user's bandwidth, this research will only be concerned with direct-sequence spread-spectrum, which is illustrated in Figure 1.1. In the direct sequence method, a digital signal is multiplied in the transmitter by a periodic waveform of narrow pulses called the spreading code. In the example shown, an unspread binary phase-shift keying (BPSK) signal is shown at baseband as a series of unit-amplitude square pulses of width Ti,, the bit time, and its power spectral density (PSD) has a null bandwidth of 1/Tb. The resulting waveform after spreading also consists of a series of pulses, but the width is reduced to T, = Tb/N, where T, and N are referred to as the c'lilp time and the processing gain, respectively, and N = 7 in this case. The PSD of the spread signal has the same form as that 1 2 -11 (a) (b) 0 Tb 2T1, 3TI, 4T 0 Tb 7/Tb Time Frequency + c(t)I c(t) + c(t)I + c(t) 0 Tb 2T,, 3T, 4T, o VTb 7/ b Time Frequency Figure 1.1: Illustration of DS-SS wavefornls and PSDs, 7 chips/bit. (a) Unspread signal waveform; (b) PSD of unspread signal: (c) Spread signal waveform; (d) PSD of spread signal. of the unspread signal, but has a null bandwidth of 1/T, = N/Tb, which is N times larger than the null bandwidth of the unslpread signal. Also note that since the total power is the same in the two signals, the height of the PSD is reduced. For many years, spread spectrum has been used in military applications, due mainly to three of its features. First, it has a resistance to jamming, a process in which an adversary transmits an interferei'nc signal, which is usually narrowband, in an attempt to destroy communication. but not to intercept and make sense of it. Spread spectrum also has low probability of intercept (LPI) capability, meaning that it is difficult for an adversary to receive and demodulate the signal without knowing the spreading code. Finally, because the power is spread over such a large bandwidth, the spectral height of the spread-spectrum signal is reduced significantly, possibly to the point where an adversary would not be able to distinguish it from the channel 3 noise and hence would not even know that a communication is taking place. This is referred to as low probability of detection (LPD) capability. In the past two decades, the research in this area has shifted from military to commercial applications, particularly in the study of CDMA systems, in which many direct-sequence spread-spectrum signals are transmitted in the same bandwidth and the code sequences are used as a means of providing separation between them. When the code sequences have some degree of orthogonality to each other, the job of the receiver is made easier, although exact orthogonality is not necessary, nor is it achievable in asynchronous communication systems. The problem of choosing good code sequences has been studied, and a good sunmmnary of many of the major results is found in Sarwate and Pursley [1]. CDMA is a very promising technology for several reasons. Most importantly, in a cellular scenario, there is the potential for a iany-fold increase in user capacity over traditional frequency-division imultile-access (FDMIA) systems. The cellular scenario will be looked at in detail in Chapters 4 and 3. Another advantage of using CDMA over FDMA is the inherent ability of a wideband signal, such as a CDMA signal, to realize diversity due to the frequency selectivity of the fading channel. An additional attribute of CDMA is the potential for privacy through the unique spreading codes of the different users. A survey of the receiver structures that have been proposed in the literature begins with the conventional matched filter receiver, which was analyzed by Pursley [2] and by Yao [3] for an additive white Gaussian noise (AWGN) channel. One of the principal shortcomings of this receiver is its susceptibility to the near-far problem, a situation in which one or more users are physically located much closer to the CDMA receiver than is the desired user. and the desired user's signal is thus overwhelmed by this multi-access interference (NIAI). In such an environment, the system must use power control, a process in which the base station, a centralized control center, 4 sends information back to the mobile users, telling them to either raise or lower their transmit powers. With perfect power control. all of the signals are received at the same power level at the base station. In response to the near-far problem, researchers have developed a large number of receiver structures with varying degrees of complexity and performance. A key feature of these near-far resistant receivers is that they utilize information about the other users' signals, either explicitly or implicitly in their processing. They take advantage of the known form of' the MAI and hence they also are referred to as multi-user detectors. The matched-filter receiver, on the other hand, treats the MAI as white noise which simply increases the therinal noise floor. The optimum multiple-user detector [4,5] consists of a bank of matched filters and uses a Viterbi algorithm to demodulate the data streams of all of the CDMA users. While it does minimize the probability of bit crror, its complexity increases exponentially with the number of users. rendering it infeasible in practice. The optimum detector is very important, however. because at the time of its inception, common sentiment was that the near-far problem was an inherent shortcoming of CDMA that could not be overcome. The existence of this near-far resistant detector served as motivation for the development of implementable receivers that could outperform the conventional matched-filter receiver. Because the optimum detector has such a high complexity, and requires a large amount of side information about all of the CDMIA users, several sub-optimum nearfar resistant receivers have been developed which have more manageable complexities. A few noteworthy receivers include the decorrelator [6.,7] and the multistage detector [8]. The minimum mean-squared error (MMSE) receiver [9-15] has received a great deal of attention during the past few years as it also offers near-far resistance and is very simple to implement. The MMSE also has an inherent resistance to narrowband interference (NBI), a feature that is very beneficial in an overlay scenario, which will 5 be discussed in the next section. It will also be seen that the MMSE can reject inter-symbol interference (ISI). A significant portion of the research presented in this dissertation focuses on the MMSE. 1.2 CDMA Overlay In communication systems, a major design factor involves the limited amount of frequency spectrum that is available. With this as motivation, the overlay of a CDMA system on a frequency buadi that is also populated by narrowband users from another system has been examined b1 Milstein et al. [16]. An actual CDMA overlay system was simulated in this field study, and gave preliminary indications that it is an attainable goal, as both the CDNIA users and the narrowband users were able to communicate reliably. It would be very beneficial if some of the frequency bands which are currently occupied by sparsely-plopulated narrowband systems could increase their overall capacity by adding CDMA technology. Of particular interest is the possibility of implementing overlay in a cellular scenario as a means of gradually phasing-out an existing narrowband cellular system in a frequency band which ultimately will support exclusively a CDMA cellular system. This will be looked at in detail in Chapters 4 and 5. In theory, the two systems, CDMA and narrowband, can coexist in the same frequency band as a result of their intrinsic properties. Consider first the effect of interference from a CDMA system on a narrowlband system. Using the same idea as that used in low probability of detection systems, CDMA users that are spread will have only a fraction of their power affecting the narrowband system which has a relatively small bandwidth. The effect of the CDMA system on the narrowband system may be tolerable if the processing gain is large enough and if the ratio of CDMA users to processing gain is small enough. Conversely, when narrowband interference appears at a CDMA receiver, its effect can be lessened to some extent by using various interference rejection techniques 6 that have been thoroughly studied [17-20]. These techniques in general exploit the predictability of the narrowband interference aiid tile unpredictability of the wideband CDMA signals to form an estiniate of the narrowband interference. This estimate is then subtracted off of the received signal. The resulting signal will have a weaker contribution from the narrowband interference, but will also have some distortion on the wideband signals. The net effect of the interference rejection on the CDMA system should be favorable, however. The design of an overlay system has many important trade-offs. In terms of the CDMA system, the ratio of users to processing gain should be maximized. But increasing this ratio too high might make the narrowband system inoperable. On the other hand, if the bandwidth of the narrowband users is increased too much, the interference rejection techniques employed in the CDMA system become less effective. And if either system operates at an excessive power level, it will render the other system inoperable. These considerations will be examined in detail in subsequent chapters. 1.3 Overview of the Dissertation The focus of this dissertation will be those issues arising when a CDMA system is overlaid on an existing narrowband system. The material can be divided broadly into three parts. First, overlay in an AWGN channel will be investigated in Chapters 2 and 3. This material will be extended to a cellular system in Chapter 4. Finally, in Chapter 5, the results for the cellular system will be extended to the fading channel, which is a much more realistic model for a \wireless environment. Conclusions and related topics of future research will be looked at in Chapter 6. Previous research in this area. has focused almost exclusively on the performance improvement of a CDMA system when techniques are applied to reject narrowband interference in the same frequency band. While this problem is certainly applicable and important, an equally and perhaps more important task is to quantify the effect 7 that CDMA overlay has on a typical narrowband system. Because the CDMA is being added to a frequency band that is already established, it seems logical that the operation of the existing narrowalnd system should be given top priority. A careful evaluation of the degradation due to overlay which is experienced by a BPSK user in an AWGN channel is given in Section 2.1. It is shown that the BPSK user can only tolerate the overlay if the loading of the CDMA system is extremely light. In this case, it is questionable as to whether or not it is cost-effective to even design and implement a new CDMA system when the added benefits seem to be very limited. A technique to alleviate the severe degradation seen by the narrowband system is presented in Section 2.2. By employing a notch filter in the CDMA transmitter, the frequency bands occupied by narrowband users can be avoided, and the effects of the CDMA overlay on the narrowband users will be reduced dramatically. The rationale behind this is that the wideband CDMA signals should still maintain their desirable properties to a large degree if the notch is not too wide. Some criteria to measure the effectiveness of different filtering methods are given in Section 2.3, and are applied to four filtering methods presented in Section 2.4. The narrowband system is found to benefit tremendously from this filtering, even if the CDMA users are received at large near-far ratios. It is found tha.t the narrowband system can tolerate the overlay of a CDMA system that is loaded up to levels far beyond that which the CDMA system itself can realistically handle. A significant conclusion of this dissertation is that the proposed filtering operations arc vital for the realization of CDMA overlay. Given that CDMA transmitter filtering is necessary in order for the narrowband system to work, the next step is to quantify the effects that the filtering will have on the performance of the CDMA system. This is done in Chapter 3 for the AWGN channel. With filtering added, the problem amounts to more than just receiving a CDMA signal corrupted by MAI and llrrowband noise, which has been studied extensively. The MMSE receiver, first mentioned in Section 1.1, is described in detail 8 in Section 3.1. Performance equations for the NIMSE receiver are presented in Section 3.2. Simulation results are then presented in Section 3.3, both for a receiver which is able to track the ideal WViener solution and for one which uses an adaptive algorithm. The MMSE receiver was compared to a conventional matched filter receiver, and was clearly superior, showing a capacity gain of 8-11 times when using the Wiener solution and 6-8 times when using an adaptive algorithm. It is also important to note that the MMSE receiver using an adaptive algorithnl was able to perform well although it did not even know that the code sequence was filtered, a very attractive feature in this scenario. The results of these simulations, combined with the results on the performance of the narrowband system presented in Chapter 2, present a strong case for the feasibility of CDMA overlay. The previous results simply demonstrated that overlay can be done if the CDMA users employ transmitter notching. In Chapter 4, a. compelling motivation for overlay is presented, as these results are extended to the cellular scenario, where overlay seems to have the most potential from a commercial point of view. If a frequency bandwidth which supports a narrowband cellular system is designated for conversion to a CDMA cellular system, overlay is an ideal way to make this transition gradually. Over the long term, the number of subscribers still using the narrowband products would shrink while the number of subscribers using the new CDMA products would increase. Conclusions similar to those reache(l in Chapters 2 and 3 will be realized in the cellular case. In Section 4.1. the cellular environment will be characterized. The effects of overlay on the narrowband system will be examined in Section 4.2, and it will be seen again that the narrowband users undergo a large amount of performance degradation as a result of the overlay. In Section 4.3, the idea of notch-filtering the CDMA signals will be applied, and will again provide tremendous benefits to the narrowband system. Simulations and results will be presented in Section 4.4. 9 In Chapter 5, the results of the cellular overlay system presented in Chapter 4, for which an AWGN channel model was used. will be extended to the fading channel, which is a more realistic model for \wireless communication. In a flat-fading channel, the number of CDMA and narrowband users which can be supported simultaneously is very low, as will be shown in Sections 5.1 and 5.2. The effect of the CDMA overlay on the narrowband system is substantial. and can be solved by CDMA transmitter notching once again. However, in order to support a reasonable amount of narrowband users, the CDMA users must do a large amount of notching. A solution to this is to use multi-carrier CDNIA (IC-CDMIA), which has recently become quite popular. The transmission of the CDNIA signal on several carriers, which are spaced in frequency so that the fading on each will be independent, allows for frequency diversity. Thus the CDMA users may transmit at lower powers, and will therefore not require as much notching as in the single-carrier case with a flat-fading channel. It will be seen that overlay is realizable in cellular scenarios, and can provide a very efficient means of transition from narrowband cellular to CDMA cellular. In Chapter 6, the overall conclusions of this research will be summarized by two major points. First, it is not practical in terms of maximizing user capacity to implement overlay without the use( of transmitter notching in the CDMA signals. Their effect otherwise is too harmful to the narrowband system. Second, there exists great potential for CD.MA overlay as a ineans for the cellular service provided in a given frequency bandwidth to be transitioned gradually from narrowband to CDMA. CHAPTER 2 NARROWBAND SYSTEM PERFORMANCE IN THE AWGN CHANNEL In an overlay scenario, it is especially important to make sure that the CDMA system does not degrade the narrowband system's performance to the point of inoperability. One possible situation occurs when a CDMA system is overlaid on a sparsely-populated narrowband systern that is already in operation. The designers of the CDMA system must work around the narrowband system and should not expect the designers of the narrowband system to make significant modifications to their design in order to accommodate the CDMA system. Another situation is one in which the two systems are to be designed jointly. It is the goal of this chapter to quantify the effect of the CDMA system on the performancene of a narrowband user. 2.1 Effects of' Overlay oil a Narrowband User As mentioned before, little attention has been given to quantifying the effects of CDMA interference on a narrowband user in an overlay scenario. The subject was treated by Davis [21] and it was reported that the CDMA overlay can cause problems for the narrowband users. Results were given by Pickholtz et al. [22] for the case of a single spread-spectrum user and actual measurements were reported in the CDMA overlay field study performed by Milstein et al. [16]. We will demonstrate in this paper exactly how much degradation is caused to a typical narrowband system in such an environment. Toward this end, consider the performance of a BPSK user operating in an overlay scenario in an AWGN channel. In order to hand-limit the BPSK user's signal, a rootraised cosine pulse shape will be used in tihe, transmitter and in the matched-filter receiver. Assuming equal carrier frequencies for the narrowband and CDMA systems, 10 11 the received signal at the input to the BPSIK matched filter after down-conversion is r(t) = V2 Eb d,,h(t iT,) + E 2P d,,ick(t iT- 7k) cos(Ok)+ n(t) i=- 00= 1 i=-co (2.1) where Eb is the average energy-per-bit of the BPSI( system, Tb is the bit time for both systems, and db,i and dk,i are the ith data bits of the BPSK user's and the kth CDMA user's data stream, and h(t) is the BPSIK user's root-raised cosine pulse shape, which is normalized so that the integral of h2(t) on (-oo, oo), its energy, is unity. Also, K is the number of' CDMA users. PA. is the average power of the kth CDMA user, ck(t) is the kth user's spreading waveform consisting of unit-amplitude positive and negative pulses of duration T,, the chip time, and N is the processing gain, or the number of chips/bit. The kth CDMA user's delay and phase, Tk and Ok, are taken as constant throughout transmission, as the channel is assumed to be non-fading and stationary. Also. ,(t) is a white Gaussian noise process with spectral height No/2 The jth sample of the output of the matched filter is Zj = 2d,j + V PAT cos(0k)Ik(j) + Nj (2.2) k=l where Nj is a zero-mean Gaussian random variable with variance No and the random process Ik(j) associated with the contribution from the kth CDMA user in the jth bit interval is given by I(j) = dk(j+[!LJ) k..,od(,,,N)h(nmTc Tk) (2.3) m=-LN and [xJ is the floor function. It is assumed in this expression that h(t) is negligible for Itl > LTb. Thus the interference contribution from each CDMA user is the sum of samples of the root-raised cosine pulse weighted by both a data bit and a spreading chip. It appears that the central limit th'or(emi may be invoked to approximate Ik as a Gaussian process since it is the sun of many random variables. Although this 12 (a) (b) (c) (d) _Illlli ll dII II IIIIII I Figure 2.1: Histograms for CDMA interference caused to a BPSK user. (a) Single CDMA user, BPSK user employing root-raised cosine pulses; (b) Single CDMA user, BPSK user employing square pulses; (c) 3 CDMA users, BPSIK user employing rootraised cosine pulses; (d) 3 CDNMA users, BPSIK user employing square pulses. approximation does not hold for each individual Ik the total CDMA interference is the sum of K of these terms, and for relatively small values of K, the distribution of the sum of the Ik terms does approach Gaussian. An example is shown in Figure 2.1. The histograms for a single CDMA user's interference contribution are shown for the case when the BPSIK signal uses root-raised cosine pulses, and also for the case when square pulses are used. In both cases, the interference is clearly nonGaussian. However, when the root-raised cosine filter is used, the interference does take on somewhat of a continuum of values concentrated in three areas, in contrast to the discrete 4-valued variable resulting in tihe square pulse case. As a result of this continuum, the sum of only 3 such variables is seen to be well-approximated by a Gaussian distribution, while in the square pulse case, the sum of 3 interference terms is still not close to Gaussian. Using the Gaussian approximation, and the fact that Ik(j) is of zero mean, the decision statistic from equation (2.2) can be rewritten as Zj = 2bb,j + N 0. 2P,.T, cos' (0k)Var(Ik) + No (2.4) 13 where the notation N(m, a2) is used to denote a Gaussian random variable with mean m and variance a2. The probability of error can then be approximated as Pe 2 Q 2E, (2.5) S E 2PA.T cos-(O.)Var(Ik) + N0 where the Q-function is defined as 1 u Q(x) =e -) cdu (2.6) Equation (2.5) then simplifies to eA + N k Cos 2(Ok)l/ar(Ik) (2.7) where Pb = Eb/Tb is the BPSK user's average power. The variance of Ik, with the averaging done only with respect to the data bits, is then found as I. N l/Var( k) = (h(mT. )) LN + hm [(itT, T)h(ni2Tc Tk) (2.8) ml=-L A' X Ck,mod(m i, ) C('k.niod(i...V) Because the variance is dependent on the particular code sequence, it may take on a range of values. If the expectation is further takeii over all code sequences, the second term of equation (2.8) drops out, and the variance is equal to the sum of samples of h2(t) spaced at intervals of T_. Because h(t) was normalized such that the integral of h2(t) on (-oo, o) is unity, then this variance can be approximated by invoking 14 the definition of a Rieman sum, which states that b I f (x) dx = li a (a + (ba) (2.9) Thus for sufficiently large N. the variance is approximately 1/T,. Combining this with equation (2.7) gives Pe1 + cOS2 (0k)1/2 (2.10) Elk= 2E11 The same expression for the probability of error was obtained by Davis [21], and was also derived by Pickholtz et al. [22] for the single-user case, when square pulses are used in the transmitter, and the corresponding matched filter is an integrate-anddump. An approximation for the case of square pulses was also given as [21] P, mQ +2Ek)] -1/2 (2.11) This expression is much easier to evaluate than is equation (2.10), which requires a K-dimensional numerical integration over the IK CDMA users' random phases, a very time-consuming process even for a small nnullber of users. The numerical integration was performed, and a plot of the probability of error vs. Eb/No for the BPSK system is shown in Figure 2.2 for the case of 3 CDMA users of equal power ratio, (P/Pb), and a processing gain of 31 chips/bit. The different curves represent different values of (Pl/Pb), and the case of no overlay is shown for reference. The approximation of equation (2.11) is also shown and matches the numerical results pretty well. Although each CDMA signal may have only a small fraction of its energy in the same bandwidth as the BPSK user as a result of the processing gain, that energy obviously has a non-negligible effect. The collective effect of the 3 CDMA users clearly degrades the performance significantly even when they are received at 0 dB, a situation which would require strict power control working cooperatively with both the CDMA system 15 10- 2 . (Pc/Pb) 2 dB -2 dB 10 6 -6 dB No overlay -10 dB 10-8 -14 dB 7 8 9 10 11 12 Eb/No (dB) Figure 2.2: Probability of error, P, of BPSI< user vs. Eb/NO with CDMA overlay of 3 users with 31 chips/bit. Solid curve is for numerical integration, dashed curve is for approximation of equation (2.11). and the narrowband system. These results cast considerable doubt on the feasibility of CDMA overlay. It may not even be worthwhile to implement a CDMA system with a loading of K/N = 3/31, and these results indicate that this very lightly-loaded system still causes severe degradation to the narrowband system. Obviously, something must be done to lessen the effect of the CDMA interference on the narrowband system, and one such method will be investigated in the next section. 2.2 CDMA Transmitter Filtering In an effort to improve the previous results, which seem to preclude the chance of CDMA overlay, an idea which was suggested both by Milstein et al. [16] and by Davis [21] will be investigated. The energy from a CDMA signal that does appear in the same bandwidth of a narrowband user apparently requires more attenuation than that which results solely from the irocessiing gain. The attractive properties of a CDMA signal, such as its inherent separation from other CDMA signals as a result of the spreading codes, its inherent separation from narrowband signals as a result of the processing gain, and its robustness to multipath, may not be sacrificed too much if only a small notch is placed in its spectrum. And if the frequency ranges occupied 16 by narrowband signals are notched out. the CDMA signal will effectively avoid the narrowband system. To perform the notch filtering, the code sequence is altered to a new code sequence, which consists of pulses, most likely square, whose amplitudes are no longer constrained to the values 1. The filtered code sequence is then modulated with the data waveform and converted up to bandpass and transmitted, just as the unfiltered code sequence would be. Four methods of filtering the code sequence will be examined and will be compared based on three criteria which will be elaborated on in the next section: the gain realized in the BPSK system, the effect on the PSD of the CDMA signal, and the effect on the CDMA code sequence. 2.3 Filtering Performance Criteria In this section, three criteria will be presented which will measure the effectiveness of a given method of notching at the CDMA transmitter. The gain of the BPSK system, the effect on the PSD of the CDMA signal, and the effect on the CDMA code sequence will now be explained in detail. 2.3.1 Gain of the BPSK System The probability of error of the BPS system given in equation (2.7) depends on the variance of Ik, given in equation (2.8), which in turn depends on the users' code sequences. As will be seen later, two of the four filtering methods used in this work result in ISI, and hence the expression for Ilk must be modified in those cases to include the appropriate contributions which result from the spillover. The evaluation of the variance of Ik requires simulation. It is then used in equation (2.7) to get the probability of error. Because the variance is multiplied by the CDMA-to-BPSK power ratio (PA./PI,), if the filtering reduces the variance by a certain multiplicative factor, the power ratio can be increased by the same factor and the 17 performance will be the same as in the unfiltered case. Thus the ratio of the variance without filtering to the variance with filtering should be found, and it will indicate how much additional near-far effect the BPSK system can tolerate. The amount of gain required depends strongly upon environmental factors such as the cell geometry and path-loss exponent [16]. It is conceivable that gains on the order of 50 dB may be required. This could be the case when many CDMA users are present and/or a severe near-far problem results from one or more of these users transmitting from a location much closer to the narrowband receiver than that of the narrowband transmitter. If the CDMA users and the BPSIK user are not power-controlled by the same mechanism, then large near-far ratios are very possible. 2.3.2 Effect on the PSD of' the CDMA Signal To see how certain filters perform in notching the CDMA signal, consider the PSD of a CDMA user's signal which is given as IQq ~ A' 2 -2 SI ((f ) c,,e-1 f T ,=o (2.12) I (. C(f)l where Q(f) is the Fourier transform of the chip pulse shape, c, is the nth chip, and the number of chips in the spreading sequence is N.i, which may be greater than the processing gain N. From this expression, it is clear that the effects of filtering will appear only in the rightmost magnitude-squared term, IC(f)12 which can be plotted as a function of frequency to show these effects. 2.3.3 Effect on the CDMA Code Sequence In two of the four filtering methods examined in this dissertation, the output code sequences are of length greater than N, which means that they span more than a single bit. While this spillover greatly alleviates the effect of CDMA interference on 18 the BPSK system by making deep notches in the PSD, it may cause obvious problems in the CDMA receiver. Even if the receiver can handle this ISI, it is obviously desirable to minimize it. 2.4 Filtering Methods and Results In this section, four filtering methods will be presented and compared based on the three previously described criteria. Simulations were performed in which a BPSK user employed root-raised cosine pulse shapes with a rolloff of a = 0.35. The BPSK user and an interfering CDMA user were assumed to have the same data rate, 1/Tb, with the CDMA user employing a random code sequence of length N = 32 chips/bit, or a filtered version of this sequence, and a delay chosen from a uniform distribution on (0,Tc). Then 10000 trials were run. each of which generated a sample of Ik when filtering was employed and a sample when no filtering was employed. The ratio of sample variances of Ik for the filtered and unfiltered cases was then computed. This gain and a plot of |C(f)12 in equation (2.12) will be presented for each filtering method. A plot of the filtered code sequence will be presented for the two methods in which ISI results. 2.4.1 Eigenvector Filtering This method was proposed by Davis [21] and will be summarized briefly. The code sequence is clocked through a finite impulse response (FIR) filter with 2M + 1 taps, and the filtered code sequence will be taken from the delayed output. The weights are chosen so as to minimize the effects of the CDMA interference on the BPSK system. For the case when the BPSIK system uses square pulses, not root raised cosine pulses, the variance of the CDMA system's contribution to the BPSK system's matched filter was derived with and without transmitter filtering [21]. The two quantities were found to differ by a multiplicative factor of aBaT, where a is a 19 Table 2.1: Gains in dB for BPSIK system for eigenvector and null filters. Taps 6 7 8 9 10 Eigenvector 32.8 36.8 35.5 24.0 20.0 Null 9.7 14.9 20.9 15.5 10.6 row vector of tap weights, B is a matrix whose (i, j)th element is given by B(i,j) = V (2.13) where N is the processing gain and w is the difference of the two systems' carrier frequencies. If the quantity aBaT is minimized, then the effect of the CDMA interference on the BPSK system will also be minimized. This minimization is performed by letting a be the eigenvector of B corresponding to its smallest eigenvalue. Then the quantity aBaT is equal to this eigenvalue. It will be the convention throughout this dissertation to denote vector and matrix quantities in boldface type. In Table 2.1, the gains achieved by the eigenvector filter are shown when the number of taps, M, on each side of the center tap varies from 6 to 10, and the processing gain is 32 chips/bit. Although it seems that the gain should increase and eventually level off as the number of taps is increased, the maximum gain is 37 dB, occurring for AM = 7. An explanation for this behavior might be that the eigenvector filter was derived for the case when the BPSIK system uses square pulses, not root raised cosine pulses. In Figure 2.3, a plot of IC(f)12 for an cigenvector filter with M = 7 and Aw = 27r(0.125) shows that the filter severely distorts the CDMA signal. The desired response should have a deep inull of finite width around the desired frequency, and as little distortion as possible in the passband. This filter does not even have a parameter to control the width of the notch. Although it does provide a pretty sizable gain 20 of 37 dB for the BPSK system, the eigenvector filter does not seem to be a plausible option based on its detrimental effect on the CDMA signal. 2.4.2 Null Filtering In the second method proposed by Davis [21], the filter puts a spectral null at the BPSK system's carrier frequency. The tap weights a are given as 'o = 1 -2 cos(mAWT,.) m in = 1, 2,... M (2.14) 2M1 1 + a= 3/10 and the filtered code sequence is formed again from the delayed output of the FIR filter with tap weights given by a. The gains achieved by the null filter are shown in Table 2.1 for the same values of M as for the eigenvalue filter and also for a processing gain of 32 chips/bit. A maximum gain of 21 dB. significantly lower than that of the eigenvalue filter, occurs for I = 8. and again an explanation for this behavior could be that this filter was derived for the case of square pulses in the BPSK system. A plot of IC(f)12 for the null filter with 1 = 8 and Aw = 27r(0.125) is also shown in Figure 2.3. Again, the CDMA signal is somewhat distorted, and there is only a minor notch at the desired notched frequency, and no parameter to control this width explicitly. Based on this, the null filtering method does not seem like a plausible option, either. 2.4.3 Butterworth Filtering We next consider the use of a digital Butterworth notch filter, which is an infinite impulse response (IIR) filter. The response when one period of the code sequence appears at the input lasts forever in theory. although it is negligible beyond a few bits. Nonetheless, the filtered code sequence will spill over into other bit intervals, resulting in ISI. To form the output code sequence, the unfiltered code sequence, which is one bit long and has N chips, is padded with several bits of zeros and clocked into the IIR 21 S(a) S 10-5 0 -o) 10-5 0 (b) S iO- I 10-5 0! (d) 10-5 C'4i 100 S i i (e) 1 i I l l i 10-5 0 0.25 0.5 Digital frequency Figure 2.3: PSDs for notch-filtered signals. Processing gain is 32 chips/bit. (a) Eigenvector filter, MI = 7 taps on each side: (b) Null filter, M = 8 taps on each side; (c) Butterworth filter. 8th order. 3-dB BW 1/32; (d) DFT-based filtering, no zero-padding; (e) DFT-based filtering, zero-pLadded to 8 bits. Solid curves are for unfiltered, dashed curves are for filtered. 22 Table 2.2: Gains in (11dB for B3PSI systeni for Butterworth filter. BW 1/N 1.25/_N 1.5/AN Gain (dB) 50.5 67.9 80.5 > One bit -1 > One bit 0 Tb 2T, 3T, 3.5Tb 4.5Tb (a) (b) Figure 2.4: Real part of filtered code sequences. Processing gain is 32 chips/bit. (a) Butterworth filter, 8th order, 3-dB BW 1/32; (b) DFT-based filtering, zero-padded to 8 bits. filter. The gains realized by the But.terworth filter are shown in Table 2.2 for an 8th order filter with 3-dB bandwidths of 1/.N, 1.25/N. and 1.5/N. The plot of IC(f)12 in Figure 2.3 for a bandwidth of 1/N shows that the notch is very pronounced and that the signal in the passband is not distorted. As shown in Figure 2.4, also for a filter of bandwidth 1/N, the code sequence continues well beyond one bit time, but is negligible beyond about 2 bits. When a receiver is used that can handle sequences with spillover, the Butterworth filter seems like a very good option. It gives the BPSK system a large gain without distorting the CDMA signal's frequency response and without causing excessive spillover. 2.4.4 DFT-Based Filtering Another way to filter the code sequence is to use a Discrete Fourier Transform (DFT), which provides a representation of the code sequence in the frequency domain. Because the inverse DFT is the sum of discrete frequency components weighted by the corresponding value of the DFT coefficient, notch filtering of the code signal can 23 Table 2.3: Gains in dB for BPSI( system for DFT-based filter. Length (bits) 1 2 4 8 16 Gain (dB) 11.2 22.8 31.4 39.2 46.3 be performed by forcing to zero those values of the DFT that correspond to frequencies within the desired notching range and taking an inverse DFT. The frequencies represented are spaced by 1/N in the digital frequency domain, corresponding to a frequency spacing of 1/Tb in the analog domain. Thus if a notch of width 2/Tb is desired, there are three DFT coefficicnts. two surrounding the range and one in the middle of it, which should be set to zero. A plot of |C(f)' is shown in Figure 2.3, in the graph for no zero-padding. Notice that there are deep nulls, but there are also high peaks between the nulls. When an N-point DFT of the unfiltered code sequence is used, no more resolution is available for notching. However, if the code sequence is padded with zeros so that an N-point DFT with N > N is performed. there will be more resolution. As shown again in Figure 2.3 for the case when the sequence is padded to cover 8 bits, the resulting notch is much deeper. The DFT-based method was tested for an unpadded sequence and for sequences padded with zeros to cover 2, 4, 8. and 16 bits. The gains are shown in Table 2.3. To achieve a gain comparable to that of the Butterworth filtering method, a sequence padded with zeros up to 16 bits must be used. In Figure 2.4, the code sequence resulting from zero-padding up to 8 bits dies off pretty quickly, with much less spillover than was seen in the Butterworth case. The DFT-based filtering method also seems like a viable filtering method, perhaps in situations where the large gains given to the narrowband system by the Butterworth filtering are not needed. It will be seen later that the DFT-based filtering method is the better choice in terms of performance of the CDMA system. 24 2.5 Summary It has been shown in this chapter that CDMA overlay has quite a damaging effect on a typical narrowband user. A very good solution is to employ notch-filtering in the CDMA signals to avoid active iiarrowbanid users. For Butterworth notch-filtering and DFT-based notch-filtering, substantial benefits were seen by the narrowband users, while the CDMA signals experienced little distortion in their PSDs and a manageable amount of ISI. CHAPTER 3 MMSE DETECTION OF FILTERED CDMA SIGNALS In this chapter, we will focus on how the filtering described in Chapter 2 affects the performance of the CDMA system in an overlay scenario. It will be shown that the MMSE receiver performs very well in this environment. 3.1 The IMMSE Receiver The MMSE receiver has received significant attention due to many of its features such as near-far resistance to multi-access interference [9-11]. For the case of a single spread-spectrum user in the presence of narrowband noise, it was observed by Pateros and Saulnier [13] that the frequency response of the MMSE filter will adapt such that there is a notch at the location of the narrowband noise, thus providing inherent resistance to this type of interference as well. It. was shown by Poor and Wang [23] that the MMSE outperforms conventional narrowband interference rejection techniques, and that it can simultaneously suppress MAI along with the narrowband interference. It will be seen that the MMSE receiver also has the ability to reject inter-symbol interference. Another feature of the MMISE that is useful in the overlay scenario is its ability to work without previous knowledge of the desired user's spreading code. In fact, it does not even need to know that the code has been filtered. A block diagram of the MMSE receiver is shown in Figure 3.1. The received signal is passed through a chip-lnatched filter, and the samples are stored until one bit time has passed. The contents of the equalizer are then correlated with the tap weights and compared to the desired output, which is the data bit of the desired user. For an initial training period, a known precanble is sent and thus the desired output is already known at the receiver. After trainingn. the taps should have had time to 25 26 t t = nTc Tc N-1 Wo t =nTb I C Decision Adaptive algorithm, update taps error signal Training Sequence Figure 3.1: Block diagram of the MMSE receiver. approach the ideal Wiener solution, but will fluctuate around it. Then the receiver uses decision feedback, in which the bit decision is used as the desired output. 3.2 MMSE Detection of Filtered CDMA Signals The performance of the MMSE receiver in an overlay scenario will now be examined. We will consider a K-user CDMA svsteni. which uses code sequences of length N chips/bit that are filtered to give sequences of length N, which will be greater than N. The kth user's filtered sequence will be denoted as ck = (ck,O, ck,1, ... Ck,l-)T, where ck,n is the nth filtered chip. Then cA. is partitioned into L = [N/N] sequences each of length N, giving ck = (Ck,,C1 .... /,Ck,L-1 (3.1) Ck.1 = (Ck,IN, CkN+1,. Ck,N+N-1 )T with I = 0,1,..., L 1. The last sequence, c..L-1, may be padded with zeros to make it of length N. It should be noted that in this application, it is desired to place a notch in the bandpass CDMA signal in order to avoid a frequency range that a narrowband user 27 occupies. If this notch is not to be placed symmetrically about the carrier frequency, as will generally be the case, then the baseband CDMA signal will have a PSD that is not symmetric about zero frequency. Thus the notched code sequence will be complex. For the Butterworth filter, instead of doing a bandstop filtering operation in the appropriate frequency range. the impulse response of a highpass filter with the same bandwidth can be found, and then multiplied by a complex exponential sequence to shift the notch to the appropriate frequency range. For DFT-based filtering, the DFT points represent both positive and negative frequency ranges, and thus only the appropriate one-sided range of coefficients should be set to zero. The received signal at baseband, denoted as r(t) in Figure 3.1, is given by r(t) = (E ,.c.(t iTb Tk) exp(jOk) (3.2) +2n(t) exp(-jc,ft) + 2.1(t) exp(-jwct) where Pk and Ok are the kth user's average power and phase, and dk(i) is the ith data bit of the kth user. Also in this equation, nl(t) is a white Gaussian noise process with spectral height No/2. The noise process resulting from a. narrowband user is given by j(t). Its PSD is modeled as a square pulse centered at an offset of Aw from the CDMA carrier frequency, and having width 2p. where p is a number between 0 and 1, most likely close to 0, that represents the fraction of the bandwidth (null-to-null) that the narrowband user occupies. The receiver forms a bank of samples by integrating r(t) over chip-length intervals, as shown in Figure 3.1. The vector of matched-filter samples is given by L-1 u(i) = T exp( 0I )d (i i)cl.I + n(i) + j(i) 1=0 -(3.3) + exp(j0s) [i.(i )fk,l + dk(i 1)gk,l] k=2 I=0 28 The cyclic shifts of the Ith partition of the kth CDMA user's code sequence are given by fk,l =(1 k)(0, 0,... 0, cA (ki+, . Ck,lN-pk-1)T + 6k(0, 0, . 0, c Ck,/i +,I .. Ck,lN-Pk_2)T (3.4) gk,t =(1 Sk)(Ck.N,-p.. CO.A-l,.+l1 ... Ck,lN-1, 0,0... O)T + 6k(Ck,N-Pl ,k1C.I p .-p. ,Ck,IN-1,0, 0, ... )T where the kth user's delay has been written as 7k = pkTc + 6k with pk an integer in the range (0, 1,... N 1) and rA. a. non-integer in the range (0, 1). Also in equation (3.3), n(i) is an N-length vector of independent complex Gaussian random variables with the real and the imaginary parts independent of each other and each having zero mean and variance o = Y/(2E,/No). The vector j(i) consists of complex narrowband interference samples with the real and imaginary parts independent of each other. The (i,j)th element of the matrix representing the correlation between the real parts of two samples is given by Rjj(i, j) = Sa(2p7r(i j)) cos((Aw)Te(i j)) (3.5) where Pj is the narrowband user's average power and the same result holds for the imaginary parts of the ith and jth samples. Also in equation 3.5, Sa(x) = sin(x)/x. Denoting the tap weights as w = (wo, wl, .. ,W -1)T, the output of the tapped delay line is given as Zi = w"u(i), which is generally a complex number. The receiver then compares Zi with the desired output dil(i). The tap weights wo which will minimize the mean-squared error between the two, J = E[IZ d1(i)12], are the solution to the Wiener-Hopf equation Rwo = p (3.6) where R = E[u(i)uH(i)] and p = E[dl(i)u(i)] = c1,o are the correlation matrix and steering vector, respectively. The bit decision is made as di(i) =sign(Re(Z})). 29 3.3 Simulation Results In order to get an idea of how filtering affects the performance of the MMSE receiver, an overlay system was simulated. The CDMA users operated with filtered random sequences and a processing gain of N = 32 chips/bit. The value of Eb/No was fixed at 10 dB. The powers of the interfering CDMA users were chosen from a log-normal distribution with a standard deviation of 1.5 dB, to simulate powercontrol error. The notch-filtering was done using a DFT-based filter with 8 bits of zero-padding. The narrowband users were generated by putting white Gaussian noise through a Butterworth bandpass filter with a, digital 3-dB bandwidth of 0.5/32. Their center frequencies were chosen to fall within the analog frequency range (-0.5/T, < fNB < 0.5/T,), which corresponds to the digital frequency range of (-0.5 < fNB,d < 0.5) as the sampling rate is 1/T,. Their powers (near-far ratios) were chosen from a distribution that results from a situation in which they are uniformly distributed spatially around the CDMA receiver, a path-loss exponent of n = 3 is used, and the maximum near-far ratio is set to 40 dB. For different numbers of narrowband users, the number of CDMA users was varied to find the maximum number that could be tolerated by the CDMA system. For each simulation, the code sequences, delays, and powers of the CDMA users as well as the frequency locations and powers of' the narrowband users were varied. The criterion chosen was that the blocking probability had to be less than 0.02, where a block was defined as a scenario in which the probability of bit error of the desired CDMA user, Pe,CDMA, was greater than 0.05. So the capacity of the CDMA system for a given number of narrowband users was the maxinmun number of CDMA users that could be present and still satisfy the performance criterion Pr(Pc,. A > 0.05) < 0.02 (3.7) 30 50 40 S 30 S(b) 2 (a) 0 20 10 (c) 0 2 4 6 8 10 Narrowband users Figure 3.2: Two-dimensional capacity plot for DFT-based filtering, zero-padded to 8 bits. Processing gain is 32 chips/bit. (a) MMSE receiver, minimizing J = E[IZi di(i)12]; (b) MMSE receiver, minimizing J = E[{Re(Zi dl(i))}2] ; (c) matched filter. In Figure 3.2, a two-dimensional capacity plot is shown for a system which employs DFT-based filtering, with zero-padding upl to 8 bits. The curve labeled (a) represents possible operating points for the system, in terms of the number of CDMA users and the number of narrowband users that can simultaneously use the frequency band and still satisfy the CDMA system's performance criterion given in equation (3.7). It should be noted that in this receiver, the blit decision is made by looking only at the real part of the filter output Zi = w" u(i). Hence better results would probably be obtained by choosing an algorithm to minimize J = E[{Re(Zi di(i))}2] as opposed to minimizing J = E[jZ di(i)12]. In this case, the imaginary part of the error can be ignored since it will not be used. With the same equalizer contents as in equation (3.3), the filter output can be written as Zi = (wo,.. + jwo.) (u.(i) + ju(i)) (3.8) 31 where wo,, wo,y, u.,(i), and u,(i) are all real vectors of length N. For the new definition of the error, we have J = E[{i (i) (wO,u,,,(i) + wTyuy(i))}2] (3.9) Now in choosing the taps to minimize J, it is clear that this is equivalent to a real Wiener filtering problem with 2N taps. The first N taps are filled with uz(i) while the last N taps are filled with u,(i). The \Vicner-Hopf equation given in (3.6) can then be applied, with the understanding that u(i) = [u7'(i), u'(i)]T and wo = -[w o, w ,]T and p = E[di(i)u(i)] are now real vectors of length 2N and R = E[u(i)uT(i)] becomes a 2N x 2N matrix. The results for this variation of the MMSE receiver are shown in Figure 3.2 as well, in the curve labeled (b). There is clearly an advantage realized over the case of minimizing the square of the absolute error. It is seen that between 1 and 2 additional narrowband users can be tolerated for a fixed number of CDMA users, for operating points at which there are 6 or less narrowband users. If the narrowband users employ spectrally efficient digital modulation at the same signaling rate as the CDMA system, then this range can fit N = 32 users without guard bands. So with 5 narrowband users, the band is about 135: populated, and the system can still support about 20 CDMA users. The results for the matched filter in which the receiver is matched to the filtered code sequence are also shown in Figure 3.2 for comparison, in the curve labeled (c). Note that there is a substantial performance improvement realized by the MMSE receiver over the matched filter. When there are 8 narrowband users present, the MMSE receiver performs as well as the inatched-filter performs in the case of no overlay. The performance of the DFT-based filtering method was also compared to the Butterworth filtering method in Figure 3.3. The DFT-based filtering method is clearly 32 40 ': 30 -' DFT-based Q 20 Butterworth 0 2 4 6 8 Narrowband users Figure 3.3: Two-dimensional capacity plot for MMSE receiver minimizing J = E[{Re(Zi di(i))}2]. Shown for DFT-based filtering, zero-padded to 8 bits, and Butterworth filter, 8th order, 3-dB BW 1/32. Processing gain is 32 chips/bit. Solid curves are for Wiener solutions, dashed curves are for adaptive solutions for LMS algorithm with step size 0.2/(total input power). superior, and this can be explained by examining the filtered code sequences in Figure 2.4. The code sequence resulting from the Butterworth filtering had more energy that spilled over into other bit intervals than did the code sequence that used the DFTbased filtering method, which obviously degrades the CDMA system's performance. Another important issue is the implecltation of these algorithms. The previous results assume that the receiver will be operating at the Wiener solution, and thus are best-case results. In practice, an adaptive algorithm, such as a least mean-squared (LMS) or recursive least-squares (RLS), would be used. Some simulations were performed to take this into account. In Figure 3.3, results are shown for both Butterworth filtering and DFT-bascd filtering, for the case when J = E[{Re(Zi d(i))}2] is minimized. As the LMS algorithm converges slowly, and the fast-converging RLS 33 (a) (b) 1 1 0.8 (3) 0.8 0.6 0.6 0 4 . 0.4 (2) 0.4 0.2 1 1 )Ji 0.2 mill i 0 100 200 0 100 200 Bits Bits Figure 3.4: Transient behavior of mean-squared error using the RLS algorithm and a training sequence. (1) is for minimizing J = E[{ Re(Zi-dj (i)) }2], (2) is for minimizing J = E[IZi di(i)12], (3) is resulting absolute squared-error under criterion of (1). (a) 20 CDMA users, 0 narrowband users and 0 notches; (b) 20 CDMA users, 2 narrowband users with near-far ratio of 20 dB. and 2 notches. algorithm may have stability problems. the training was avoided for now by initializing the tap weights to the Wiener solution and allowing an LMS algorithm to run with a known preamble for many bits. This allows the filter to reach a steady state solution which takes into account the excess mean-squared error which would result from an adaptive algorithm. For a step size of 0.1/tip, where tip is the total input power, almost no difference was seen in comparison to the Wiener solution. For a step size of 0.2/tip, some difference was seen as shown in Figure 3.3. Thus if the filter is trained well, results close to the ideal ones with the Wiener solution can be achieved for a small enough step size. 34 It would also be interesting to observe the transient behavior of the MMSE receiver in the overlay environment, in addition to its steady-state operation reported above. The performance of the RLS algorithm will be looked at for this purpose, as its speed of convergence does not depend on the inlut power as in the case of the LMS algorithm. For 200 trials, the RLS algorithms was run first with 20 CDMA users and without narrowband users and notches, and then with 2 narrowband users and notches added. The squared error was averaged over the 200 trials in Figure 3.4. Both of the previously described IMMSE criteria were used for comparison. In the first case with no narrowband users or notches, the mean-squared error is seen to converge to a steady-state value in about 150 bits. Notice that there is a significant improvement in the mean squared-error when the receiver minimizes J = E[{Re(Z; di(i))}2] as opposed to J = E[jZj di(i)12], but the required convergence time is about the same. Also notice that in the case when J = E[{Re(Z dl(i))}2] is minimized, the corresponding absolute value of the mean squared-error is very high. As stated before, the square of the real part of the error, which is used in making the bit decision, is minimized at the cost of allowing a large increase in the imaginary part of the error, which will not be used anyway. The same environment was simulated with 2 narrowband users received at a near-far ratio of 20 dB and with the corresponding notches added. The same amount of time, about 150 bits, was required for convergence to a steady-state value, which is seen to be somewhat higher than when no narrowband users are present. It is worth noting that the computational complexity of the two algorithms is the same, as every complex computation that is required for minimizing J = E[IZi di(i)12] requires two real computations in the other case. Thus minimizing J = E[{Re(Zi di(i))}2] offers a substantial improvement in performance with no added complexity or required training time. 35 (a) (b) (c) I i iI 0 0.05 0.1 0 0.05 0.1 0 0.05 0.1 Pe Pe Pe Figure 3.5: Histograms of CDMA probability of bit error, Pe, for overlay system with 42 CDMA users and 1 narrowband user. Processing gain is 32 chips/bit. MMSE receiver minimizes J = E[{Re(Z, dI(i))}2]. DFT-based filtering, zero-padded to 8 bits. Solid curve is for no narrowband signal present, dashed curve is for narrowband signal present with a given narrowband-to-CDMA near-far ratio. (a) Ratio is 0 dB; (b) Ratio is 20 dB; (c) Ratio is 40 dB. It is also interesting to look more closely at the ability of the MMSE receiver to inherently reject strong narrowband interference. It was determined previously that when J = E[{Re(Z -di (i))}2] is minimized, 42 CDMA users could share the channel with a single narrowband user. In order to examine the effect of the narrowband noise, a system was simulated with 42 CDMA users with a single notch and a narrowband user with varying levels of power. In each trial, a new set of code sequences, delays, and powers were chosen for the CDMA users and a new frequency location was chosen for the narrowband user, all using the same parameters as in the previous simulations. Then the probability of bit error was found for the CDMA system for narrowband-toCDMA near-far ratios of 0, 20, and 40 dB, and also for the case when the narrowband signal was not present but the notching was still performed. 36 The resulting histograms of the probability of bit error are shown in Figure 3.5 for the three different near-far ratios and each is compared to the case in which no narrowband signal is present. It is seen that when the narrowband signal is received at 0 dB, there is little difference, but the performance does degrade somewhat for the 20 dB case, and degrades significantly for the 40 dB case. Hence it appears that the MMSE receiver is relatively robust to the presence of the narrowband signal, but for large near-far ratios its performance will be degraded. This is consistent with the conclusions reached by Poor and WVang [23], that while the MMSE receiver does outperform conventional narrowband interference rejection schemes, the output signal-to-interference ratio of the hlMMSE receiver will degrade for high-powered narrowband interference. Fortunately, the probability that a narrowband user will have such a large near-far ratio is low, as this would occur only in a small geographic region around the CDMA receiver. 3.4 Summary In this chapter, we have shown that with the notch-filtering which was determined necessary in Chapter 2, the MMSE receiver can function quite well in an overlay environment. In the simulations performed here, the CDMA system could be loaded up to about 20 users for a processing gain of 32 chips/bit when 5 narrowband users were present, or about 15% of the band was jammed. It offers a substantial performance improvement over a conventional matched-filter receiver. These results, along with those of Chapter 2, are quite encouraging for the prospects of CDMA overlay. CHAPTER 4 CDMA OVERLAY IN A CELLULAR SYSTEM We will now extend the ideas of the previous two chapters to the cellular environment, an application for which CDMA overlay has perhaps the most potential for increasing user capacity. Because overlay is feasible only when the existing narrowband system is sparsely-populated, the cellular concept seems quite conducive to overlay, as each cell in a. narrowband system only utilizes a fraction of the system bandwidth, even when fully loaded. A cellular overlay system would be quite beneficial in a situation in which a narrowband cellular system is to be phased out in favor of a CDMA system [24]. With overlay, the transition could be gradual, as a new CDMA product could be introduced while the system still provides support to the existing narrowband products. In this chapter, it will be shown that overlay can be realized in the cellular scenario. The total system bandwidth of the narrowband system is divided into several frequency groups, each consisting of a number of narrowband channels separated in frequency so as to minimize adjacent-chanuel interference. The groups are assigned to the cells in an intelligent manner which minimizes the co-channel interference, that is interference caused by users in different cells using the same channel. Thus even when fully-loaded, each cell only utilizes a small fraction of the total system bandwidth, as would be necessary for overlay. A CDMA system could be implemented using the same cellular layout, with the CDMA users in each cell spread in frequency over the whole system bandwidth. 37 38 4.1 Characterization of the Cellular Environment In this section, some aspects of the cellular environment, such as the assignment of users to cells, the power control that is employed, and the interference that users cause to base stations other than their own will be described. We will look at a system with a frequency reuse of 1/7, which means that the total system bandwidth will be divided into seven groups which are then assigned to the cells in a manner which minimizes the co-channel interference. Each group will consist of a number of narrowband channels separated in frequency so as to minimize adjacent-channel interference. As shown in Figure 4.1, with cell 1 as the cell of interest, effects from users in those cells beyond the two outer layers of cells shown will be considered negligible. The individual cells, numbered 1-19, are also given a letter from A-G which indicates which of the seven channel groups it will use. The coverage areas are shown as circles for simplicity, but in practice and in this research, a user will be assigned to the cell for which the path between the mobile and the base station is least attenuated. This will depend not only on the distance of propagation, but also on log-normal shadowing, which may be severe. So a user often will be assigned to a base station that is not the closest one geographically. We begin by looking at the received power at a distance d, from a transmitter which may be either CDMA or narrowband. The power is typically modeled by first finding the received power at a close-in reference distance do, denoted Po. Then with n as the path-loss exponent and o, as the standard deviation of the log-normal shadowing, the received power at d,. is given in dB units as [25] P= P 10 log( +N(0,) (4.1) = [Po + 10, loglo(do)] + [N(0, ~) 10n loglo(d,)] 9- V~g-Ill~OL~l 39 12(F) I1(G) 10(E) 13(B) 4(D) 3(C) 9(F) d 14(G) 5(E) 1(A) 2(B) 8(D) 15(C) 6(F) 7(G) 19(E) 16(B) 17(D) 18(C) Cell # 1 (a) (b) Figure 4.1: Illustration of a narrowband cellular system. (a) Cellular layout for threelayer cluster. Cells numbered 1-19. Letter is the cell's channel-group assignment, chosen from A-G; (b) Magnified view of Cell 1. When comparing the paths to different base stations, it is only necessary to compare the rightmost term in the second line of equation (4.1), that is Gi = [N(0, a,) 10n loglo(di)] (4.2) where di is the distance from the mobile to the ith base station, normalized such that the cells are of unit radius. Also note that the quantity Gi in equation (4.2) will be referred to as a channel gain, as it is the sum of the path loss and shadowing, but this is not intended to imply that there is an amplification of the signal through the channel. This quantity will be used in comparing different paths on a relative basis only. Without loss of generality, it will be assumed that the user of interest is physically located in cell 1, and at a distance di and angle 01 as shown in Figure 4.1. The rest of the Gi quantities may be found in terms of d and 01 by using the law of cosines. We define the following quantities. also in dB units: I'V = max(G1, G2,..., CG9) (4.3) YI = max(GI, G,,... G19) #/_i 40 The meaning of the random variable 1T' should be clear. And the random variable Y3, for instance, is Y3 = max(G1, GC2, G4 .... G19) with G3 excluded from the argument of the max function. The use of the Yi allows the assignment of a user to the jth cell to be characterized simply by two exhaustive events, namely that the user is in fact assigned to the jth cell, (G- > ")), or is not, (G. < Yj). Certain pieces of information, such as the exact cell to which the user is assigned and the exact number of cells for which the corresponding Gi are greater than Gj, are not contained in the quantity Yj. In the results obtained in this work, this information is not necessary, and the corresponding simplification makes the analysis much more tractable. These random variables in equation (4.3). which depend implicitly on dl and 01, can be described by using the fact that for fixed values of dl and 01, the rest of the di will also be fixed and the Gi are then independent Gaussian random variables with means of mg,i = -10nloglo(d;) and each with variance 2. It is important to keep in mind that the independence holds only when the di are treated as fixed quantities and not as random variables themselves. The probability density function (PDF) of W can be shown to be i19 19 exp ( 2 The expressions for the PDFs of the ; are almost identical, with the only change being that the indices of the product and summnnation are adjusted so as not to include the ith term. As shown in Figure 4.2. for a user located at a distance of d = 0.75 and at an angle 01 = 0, and with aj = 8 dB and n = 3, the PDFs of W and Yi can be well-approximated as Gaussian. We can then find, for example, the probability that a user is assigned to the ith cell as F,i 9 P1 (G i > + 0,92 41 0.1 0.05 0 -15 25 dB Figure 4.2: Exact PDFs (solid) and Gaussian approximations (dashed) for Y and W, defined in equation (4.3), for a user at d1 = 0.75 and 01 = 0. where my,i and o ,i are the mean and variance of Y;, which again depend on the user's position, and must be found numerically. The variables in equations (4.2) and (4.3) can also be used to describe the effect that power control will have on the power levels at which users will be received at a base station. In practice, the transmitted powers of the mobiles assigned to a cell are adjusted so that the received powers of all of those users are the same at the base station. For both narrowband a.nd CDMA systems, the users employ power control as a means of conserving battery life. It is wasteful for a mobile that has a strong path to the base station to transmit as much power as does a user with a severely attenuated path. In CDMA systems only, there is the additional motivation of alleviating the near-far problem amongst a system's own users, so that one strong user does not disrupt communication for all the rest. Consider again a user located in cell 1 at a position of dl and 01 from Figure 4.1. As mentioned before, the user will be assigned to the ith cell, i = 1,2,... ,19, for which Gi in equation (4.2) is maximum. If the minimum acceptable received power 42 in dB at the base station is 7, the user's transmitted power will be adjusted to 7- W, where W is defined in equation (4.3), so that it will arrive at a power y at its own base station. At a base station to which the user is not assigned, in cell k, the received power is y + (Gk -Yk). Note that it is necessary to know Yk under this condition, that is (Yk > Gk), as it determines the user's transmitted power. But it is not necessary to know which of the base stations other than the kth that the user is assigned to. We must then find the PDF of HA., which is the interference power level at the kth base station relative to those users assigned to the kth base station. Under the condition that the user is not assigned to the kth base station, Hk will always be less than 0 dB. The PDF of Hk will have a discrete part, which is an impulse at 0 dB, with a weight equal to Pr(G- > Y.), the probability that the user is assigned to the kth cell. It will also have a continuous part. resulting from the event that it is not assigned to the kth cell, and appears as outer-cell interference. To find this part of the PDF, we first find the joint cumulative distribution function (CDF) Pr(Hk < h, H, < 0) =Q ,k - li(h, 0) (4.6) V ,k + og The derivative of equation (4.6) with respect to h gives the continuous part of the PDF of Hk, and the complete PDF is fHk (h),k h) 1 (h (mg,k y,k))2 u(-h) (4.7) + e+ xp 2 +2) (-h) where u(h) is the unit-step function. The continuous part of this PDF is just a Gaussian PDF that is truncated at h = 0. The expression in equation (4.7) is dependent on the exact position of the user. This will be useful in some cases, but it would also be helpful to have a PDF for 43 Table 4.1: Composite ,., mY,. and E[Y2] averaged with position. dB quanttiies m my a + m Cell 1 6.51 3.07 34.93 Layer 2 -9.03 9.28 151.56 Layer 3 -18.06 9.52 153.86 Over all 3 layers -13.92 9.11 146.88 which the position is somehow averaged out. For instance, it may be necessary to know the PDF of the interference level caused to the cell of interest for a user that is located uniformly in a second-layer cell, without having to know its position and then look up the corresponding values of mnj,; and a, in a table. A true averaging of the PDF in equation (4.7) can be done numerically, with the help of a table of my,i and a2,. values. However, it was observed that a very good approximation may be obtained by first averaging m,,i, m,,;. and the second moment ayi + mi with respect to position over the cell of interest. over a second-layer cell, and over a third-layer cell, and then forming for each an expression similar to equation (4.7), with the averaged parameters substituted appropriately. This approximation to the true averaging was done, and is shown to match pretty well with simulated results, which are shown in Figure 4.3. The average values are shown in Table 4.1. Then because there are 1, 6, and 12 cells in the 1st, 2nd, and 3rd layers, respectively, the averaging can be further extended to give averages which can be used to characterize the interference caused to cell 1 by a user located uniformly throughout the 3 layers of cells. This also matches the simulation results very well, as shown in Figure 4.3, and the averages with respect to position are also shown in Table 4.1. The approximate PDF is written below for use later: 1 1 (h + 23)2 fHk(h) -_ (h.) + 18exp 2(128) u(-h) (4.8) 19 V27(128) 2(128) It should be noted that while this function and the others in Figure 4.3 are good approximations, they are not truly valid PDFs as they do not integrate exactly to 44 0.05 2nd layer Cell 1 Po=0.05 Po =0.64 3rd layer (a) Po =0.007 / 0 -60 -40 -20 0 dB 0.05 Averaged over all 3 layers PO = 1/19 // '%, (b) 0 -60 -40 -20 0 dB Figure 4.3: Histograms and Gaussian approximations for interference caused to base station 1. (a) Interference from a user in cell 1, in a 2nd layer cell, and in a 3rd layer cell. For each, there is also an impulse at h = 0 with a weight of po; (b) Interference from a user located uniformly throughout the 3 layers. There is an impulse at h = 0, with weight po = 1/19. 45 unity, but close to it. The function in equation (4.8), for example, integrates to 1.032. It is simple to generate random variables with PDFs very close to these ones, however, and there is a substantial savings in time in comparison to the task of generating a user's position and shadowing processes to each base station. The approximate PDF of equation (4.8) will also be useful in the next section in investigating the effects of overlay on the existing narrowband system. 4.2 Performance of the Narrowband System Given the characterization of the mobiles' received powers, the effect that the CDMA overlay has on the narrowband system can now be examined. It would be best if the overlay necessitated as little change as possible for the narrowband system, which is assumed to already be in existence. However, it will cause degradation to some extent, and it is thus necessary to define an acceptable level of degradation, and then to quantify the number of CDMA users for which this level is not exceeded. We begin by considering the effects of overlay, either in the single- or multi-cell case, on the performance of a single BPSIK user operating in the presence of additive white Gaussian noise of spectral height No/2. The probability of error of the BPSK matched-filter receiver was found in equation (2.7) which will be repeated here with a slight modification. Pe Q1 cos2() 1 2 (4.9) with Vk = TcVar(Ik), which is a random variable with respect to the CDMA users' codes and delays. As verified by simulation, the 1'. variables can be well-approximated as unit-mean exponential random variables. Now a criterion must be established to determine if the overlay causes too much degradation to the narrowband user. Such a criterion is described by Milstein et al. [16], which states that if the BPSIK user's probability of error is increased from 46 10-6 to greater than 10-5 as a. result only of the overlay, that is while its signal-tonoise-ratio (SNR) remains constant, then the overlay is excessive. This criterion was employed in conjunction with a convolutional code of rate 7- = 2/3 and a constraint length m = 9. The optimum code has a free distance dfree = 8, as was found through an exhaustive computer search [26]. A tight lower bound for the probability of bit error of the coded system is 1 E P, > Q 2rdfree (4.10) In 1 NO The values of 2Eb/No required for this bound to achieve bit error rates of 10-6 and 10-5, denoted SNR6 and SNR5 respectively, are 5.43 dB and 4.27 dB. In order to apply this to the overlay system, note that the square of the argument of the Q-function in equation (4.9) must be greater than SNR5, and also note that SNR6 = 2Eb/No, to give the necessary condition 1 P_ 1 1 1 cos2(0k)n < = 0.09 (4.11) k= l SR SNR for which the overlay is not excessive. The number of CDMA users that can satisfy this condition for all but a. specified percentage statistically, say 2%, will be the limit on the CDMA capacity from the narrowband system's perspective. This should not be confused with a different limit. which may or may not be greater than this one, which is the number of users for which the MAI eventually becomes too severe for the CDMA system itself. That limit will be looked at in Section 4.4. Before the capacity limit defined by equation (4.11) can be found, it is necessary to specify that, in this chapter, it will be assumed that the received powers of the CDMA and narrowband users at their respective assigned base stations are powercontrolled to the same level. The approximate PDF of the received power given in equation (4.8) was used to generate realizations of the CDMA-to-BPSK near far ratio Pk/Pb. Then 10000 different realizations of the sum on the left-hand side of equation 47 (4.11) were formed and compared with the value 0.09 for a fixed value of the number of users K, and for a processing gain N = 32. Then K was varied until the sum was greater than 0.09 for more than 2% of the 10000 realizations. It was found that K = 14 users was the maximum number which could meet the criterion. Note that there are 14 users spread over the three layers, or 19 cells, which is 0.75 users/cell, which is extremely low for a processing gain of 32. A similar conclusion, that the overlay is excessive even for a lightly-loaded CDMA system, was observed in Chapter 2 for the single-cell case. One way to improve this loading relative to a fixed processing gain is to increase substantially the power level at which the narrowband users arrive at their own base stations. Doing so would not lead to increased co-channel interference amongst the narrowband users, because all the narrowband users in the system would increase their powers equally. It would, however, decrease their battery life, which is inconsistent with the idea that the overlay should require that the narrowband system change as little as possible. Another solution is to employ notch-filtering in the CDMA transmitters to avoid certain narrowband users, as was done in Chapter 2 for the single-cell case. This will be explored in the next section. 4.3 Effects of CDMA Transmitter Notching In an effort to alleviate the severe degradation caused to the narrowband system as a result of the CDMA overlay, the idea of notching the CDMA signals at their transmitters in order to avoid narrowband users was investigated in Chapter 2 for the single-cell case. The DFT-based filtering method, described in Section 2.4.4, will be employed here in the cellular case. The use of notching is more involved than in the single-cell case, where the CDMA signals were notched to avoid all of the narrowband users present. In the cellular case, it is very possible that in addition to those narrowband users located near to it physically, a CDMA mobile may have to avoid a narrowband user that is assigned 48 to an adjacent cell or even to a cell that is two layers away. But it obviously cannot notch for all of the narrowband users in the system, as that would likely constitute the entire system bandwidth, nor does it need to. As a result of the shadowing described in Section 4.1, there will be in some cases narrowband users near to the CDMA mobile that are not significantly degraded by the CDMA signal, and hence a notch is not necessary for them. This also holds for a significant percentage of the users that are not near to the CDMA mobile. The total CDMA interference seen by a narrowband user depends on the CDMA signals' received powers, phases, and on the Vs7 variables according to equation (4.9). When deciding whether or not a notch is necessary, the power can be estimated and the information exchanged between base stations, but the phases and the Vk obviously will be unknown. Hence a specific criterion that will be used is that a CDMA signal must be notched if the power level at which it arrives at the corresponding narrowband receiver relative to the narrowband signal is higher than a given threshold. From Table 2.3, when the DFT-based filtering method is used with 8 bits of zeropadding, the interference contribution is reduced by about 40 dB. The experiment of Section 4.2 was repeated, that is the number of CDMA users for which the excessive overlay criterion of equation (4.11) could be met was found, with transmitter filtering employed in those users for which the narrowband-to-CDMA power ratio at the narrowband receiver was less than a threshold T, given in dB. The results are shown in Figure 4.4 for T = 7, 9. 11 dB. In contrast to the 0.75 CDMA users/cell that was found in the unfiltered case, it is possible that with T = 9 dB, 19 users/cell can be tolerated by the narrowband system when filtering is employed, a tremendous increase in capacity. As the threshold is raised to T = 11 dB, and hence a CDMA user is more likely to need a notch, the amount of CDMA loading that can be tolerated by the narrowband system increases substantially to 29 users/cell, a level at which the CDMA system itself probably cannot function. And if it is reduced to T = 7 dB, the 49 30 20 S10 10 No notching 0.75 users/cell 0 7 9 11 Notching threshold (dB) Figure 4.4: Maximum CDMA loading tolerable to narrowband system. Processing gain is 32 chips/bit. CDMA user notched if NB-to-CDMA power ratio is less than the notching threshold. possible loading drops to 13 users/cell. In the next section, the capacity limit will be looked at based on how much notching the CDMA system can handle. It should be pointed out that there is a good deal of information that must be exchanged between base stations, such as the received powers of both the CDMA and narrowband users. This combined with possible inaccuracies in estimating the received powers could lead to cases in which the CDMA users need to notch for certain narrowband users, but they do not. To measure this effect, we will declare that the CDMA users miss a necessary notch with probability p,,,. In Figure 4.5, the number of CDMA users/cell that can be tolerated is shown to degrade quite rapidly as Pm increases, which indicates that careful attention must be given to the estimation of the received powers. Next, the amount of notching that must be done by the CDMA signals will be determined. This brings up an important design issue involving whether or not the 50 30 T= lldB S20 T = 9 dB 10 T = 7 dB 0 0.01 0.02 Pl = Pr(missed notch) Figure 4.5: Effects of missing notches on CDMA loading that narrowband system can tolerate. Processing gain is 32 chips/bit. CDMA user notched if NB-to-CDMA power ratio is less than the notching threshold, T = 7, 9, 11 dB. two systems should use the same frequency band for the forward link, and correspondingly for the reverse link, or if they should be staggered such that one band covers the forward link of one system and the reverse link of the other. These two scenarios will be referred to as the same-link and staggered-link cases, and will be compared now based on the reverse link of the CDMA system. 4.3.1 Same-Link Assignment In this case, the CDMA mobiles and the narrowband mobiles are transmitting in the same band, and thus the narrowband base station is taken to be the receiver. To determine the required notching, consider again a CDMA user located at a distance dl from the first base station as shown in Figure 4.1. We will find the probability that a CDMA user located at this position must notch for a single narrowband user that is located uniformly within the hexagonal enclosure of the three layers of cells. This will require for i = 1,2, ... ,19, the values of my,i and o,2 which are numerically calculated, as well as the m.q,i, which are easily found using the law of cosines. 51 First we define I = (Received narrowband power) (Received CDMA power), where the received power refers to the power in dB at the base station to which the narrowband user is assigned. A notch will then be required if I < T, with T given in dB. Recall that at this base station, the narrowband user's received power will be a specified value, denoted y. It was stated earlier that the CDMA and narrowband users would be power-controlled to the same level if assigned to the same base station. In that case, I = 0 and there is certainly a notch required. For a CDMA user that is not assigned to the same cell as the narrowband user, denoted cell k, the conditional PDF of the CIDMA received power at the narrowband user's base station is exp (h-( m y,k))2 fHk (h(Gk < ) 2(a +,k ) u(-h) (4.12) 27r(a, + c(),k)(Pr(Gk < 1k)) where again nzg,k, ,.k, and T., depend on (11, and Pr(Gk < I) is the probability that the CDMA user is not assigned to the kth cell. The notching probability, p,, is then found as 19 .= [ Pr(I < T, NB to cell k, CDMA to cell k) k=1 (4.13) + Pr(I < T, NB to cell k, CDMA not to cell k)] which simplifies to 19 PT =T + Q -Q (4.14) The notching probability is plotted in Figure 4.6, as a function of dl, for T = 1,3,... ,15 dB. As expected, it is highest at the edge of the cell, as those users are more likely to be transmitting at a higher power than are those nearer to the base station, as a result of the power control. For a number of narrowband users located throughout the region, that is the density of users/cell multiplied by 19 for 52 T = 11 dE 0.3 S0.2 ST = 7 dB 0.1 T= l dB 0 0.2 0.4 0.6 0.8 1 d, Figure 4.6: Notching probability as a function of dl for same-link case. Notching thresholds are T = 1, 3,... ,15 dB. the number of cells, the number of notches required for the CDMA signal is then a binomial random variable with success probability p,,. 4.3.2 Staggered-Link Assignment In contrast to the previous case, the CDMA mobiles and the narrowband base station are now transmitting in the same band. It is not necessary to perform a detailed analysis in this case. The signal on the narrowband forward link will have a composite gain from the base station's transmitter antenna and the mobile's receiver antenna, while the interference from a CDMA mobile to the narrowband mobile will only be amplified by its own transmit antenna and the narrowband mobile's receiver antenna. Obviously, the base station antennas will be much larger in effective area than will a size-limited mobile antenna. So the narrowband forward-link signal is likely to arrive at the mobile at a much higher level than would the interference from the CDMA mobile. Hence it is possible that notching is not required of the CDMA mobiles in the staggered-link case. 53 The major drawback to this configuration, however, is the severe near-far problem that results at the CDMA base station as it must receive signals from its mobiles in the presence of the signals sent from the narrowband base station, located a short distance away on the same tower, to its own mobiles. It will be seen later that this disadvantage outweighs by far the advantage of needing few, if any, notches in the CDMA mobiles, and that the same-link assignment is the better of the two configurations. 4.4 Simulations and Results In order to investigate the perforinanice of the CDMA system, an environment similar to that used in the single-cell results of Chapter 3 was constructed, incorporating the cellular layout of Figure 4.1, and log-normal shadowing and power control as described earlier. The CDMA system had a processing gain of N = 32 chips/bit, an Eb/NO value of 10 dB, and used the DFT-based filtering method with 8 bits of zero-padding when filtering was necessary. For a 1/7 frequency reuse system, each cell would have at most 4-5 narrowband users. In a "brute-force" simulation of the cellular overlay system, the coverage area that must be considered would consist of 6 layers of cells surrounding the center cell of interest. It was observed that a CDMA user might have to notch for a narrowband user that is 2 layers away from its geographic location. And any user could be assigned to a cell at most usually 2 layers away from its geographic location. So a CDMA user in the second layer out might have to notch for a narrowband user in the fourth layer out whose base station could be in the sixth laver out. The total number of cells in the geographic coverage area multiplied by the density of users/cell gives the total number of users, either CDMA or narrowband, which would then be distributed uniformly about the coverage area. Each user would then be assigned to the cell at which its received power is maximum, and power control would be implemented. After it is determined how much of an increase or decrease 54 in transmitted power from the mobiles is required, the necessary notching for each CDMA mobile would be found by comparing its received power with that of each of the narrowband mobiles assigned to one of the cells within a three-layer cluster of the CDMA mobile's base station. This procedure would obviously require a great deal of simulation time, which can be decreased significantly with some simplifications. The first simplification results from the observation, which was verified by simulation, that if an interfering CDMA user is left unfiltered, there is no noticeable difference in its effect on the desired CDMA user. The notching probability for a CDMA user depends jointly on its position and its transmitted power. But since it does not need to be known in the simulations, it is sufficient only to know the interferer's received power relative to the desired user, and its position is unimportant. So we can further simplify the simulations by using equation (4.8), which gives the approximate PDF of the received power for a user that is located uniformly throughout a three-layer cluster around the cell of interest, taking into account both cell assignment and power control. This was used to generate the near-far ratios of both the interfering CDMA users and the narrowband users. For the desired user, it was assigned to the center cell, and based on its position, its notching probability was found from a numerically-evaluated table. The number of notches required was then a binomial random variable dependent on the density of narrowband users. Each CDMA user was given a random code sequence, delay, and phase, and each narrowband user was assigned to a random frequency location. The desired user was demodulated using an MMSE receiver, with the true Wiener solution. As explained in Section 3.3, the best performance is obtained by choosing the tap weights to minimize in expected value the square of the real part of the error as opposed to the square of its absolute value. The system capacity in CDMA users/cell was determined to be the maximum density for which the following blocking criterion 55 was satisfied: Pr((Pe,CDAA > 0.05) U (#notches > 10)) < 0.02 (4.15) In this criterion, a CDMA user will be blocked if its bit error rate is too high, or if it needs more than 10 notches. As the sampling rate of the system is 1/Tc, the range in which notching can be done is actually (-0.5/T, < f < 0.5/Te). Any frequency locations outside of this range must be notched in the appropriate location mirrored around 0.5/T,. Thus with a processing gain of 32, there are effectively only 16 notching locations. With more than 60% of the CDMA signal notched out, the user must be dropped. A two-dimensional capacity plot is shown in Figure 4.7 for the same-link case, with notching thresholds of T = 7, 9, 11 dB. The notching threshold may affect the capacity criterion of equation (4.15) in two ways. First, as more notching becomes necessary, it is more likely that the desired user will have more than 10 notches and will be dropped. Also, it should be more difficult to demodulate the desired user's signal as more notches are added, and hence the probability of error should be higher on average. Notice that when the narrowband system is lightly-loaded, there is little difference in the amount of CDMA loading possible for each threshold. It was observed that when the loading was at most 1.5 narrowband users/cell, no drops occurred as a result of excessive notching. The fact that the threshold had little effect at these loading levels suggests that the notching has a minimal effect on the bit error rate, an idea which later will be investigated further. As the narrowband loading increases beyond 1.5 users/cell, the system in which a notch is placed if the narrowband-to-CIDMA power ratio is less than T = 11 dB immediately shows the effects of having users dropped as there now are some instances in which the desired user requires more than 10 notches. For the other values T = 7, 9 dB, the effects of dropping users show up for higher values of narrowband loading. 56 20 T = 9 dB CT = 7 dB T= 11 dB U 2.3 NB i 2.8 NB :3.5 NE 0 1 2 3 4 NB users/cell Figure 4.7: Capacity plot for same-link case. Processing gain is 32 chips/bit. Eb/No of CDMA system is 10 dB. Notching thresholds are T = 7, 9, 11 dB. For 2.3, 2.8, and 3.5 NB users, respectively, the CDDMA system reaches capacity in terms of those drops which result from needing too many notches. Notice that the CDMA system is constrained by the fact that for a high enough narrowband loading, the probability that a CDMA user is dropped is greater than 2%, and the criterion of equation (4.15) cannot be satisfied for any amount of CDMA loading. From Figure 4.7, this occurs at 3.5, 2.8, and 2.3 narrowband users/cell respectively for T = 7, 9, 11 dB. The results of Figure 4.7 characterize the joint capacity limits of the system as dictated by the CDMA system. In conjunction with the limits resulting from considerations of the narrowband system as shown in Figures 4.4 and 4.5, it is clear that the notching threshold T = 9 dB is the best choice, as its limits according to each of the two systems are relatively close. From Figure 4.5, if the probability of missing a notch is 1%, then about 15 CDMA users/cell can be tolerated by the narrowband system. And if the narrowband system is about half-loaded with 2 users/cell, then the CDMA system itself can support about 15 users/cell according to Figure 4.7. 57 This is a significant capacity improvement which would be very useful as mentioned in the transition from a narrowband cellular system to a CDMA cellular system. It would also be of interest to investigate separately the effects of notching and the effects of narrowband noise on the CDMA system. This was done by comparing results for the staggered-link case, which naturally has a very high level of narrowband interference, and the same-link case explicitly in terms of the number of notches in the desired user and in the number of narrowband users which interfere with the CDMA signal. Recall that these will likely be different as the CDMA mobiles often must notch for narrowband users that are assigned to other cells. Also, although it was stated earlier that notching will not likely be required in the staggered-link case, it will be done here only in an effort to look specifically at the difference in the effects of notching and narrowband noise. In the capacity plot shown in Figure 4.8, results for the staggered-link case are shown as solid lines while the one dashed line applies to the same-link case. For the staggered-link results, the number of narrowband users assigned to the desired user's cell was held constant at 0, 1, 2, 3. or 4, while the number of notches in the desired user was varied, and the CDMA density was found according to only the bit error rate part of the criterion of equation (4.15). Recall that in this configuration, the signals on the narrowband system's forward link will appear on the CDMA reverse link as interference at a very high near-far ratio. Assuming that the problem can be alleviated to some extent by shielding the two antennas, the near-far ratio was fixed at 20 dB. In going downward from curve to curve, it is seen that the effect of an additional narrowband user is to decrease the CDMA density by about 4 users/cell, which is significant. For the same-link case, when there are no narrowband users assigned to the desired CDMA user's cell, the capacity curve is the same as in the staggered-link case. The narrowband users that are assigned to the cell will arrive at 0 dB with respect to the 58 0 NB users Same-link 20 4 NB users 1 2 10 ' 0 3 0 4 8 12 Notches in desired user Figure 4.8: CDMA users/cell vs. number of notches in the desired user. Solid line is staggered-link case, with narrowband-to-CDMA near far ratio at CDMA base station of 20 dB, and 0, 1, 2, 3, or 4 narrowband users assigned to cell of interest. Dashed line is same-link case, with 4 narrowband users with near-far ratio of 0 dB assigned to cell of interest. Eb/NO of CDMA system is 10 dB. CDMA user's power level, as mentioned before. The plot when there are 4 narrowband users assigned to the cell of interest is shown, and the plots for 1, 2, and 3 users will naturally fall between it and the plot for no narrowband users present. It is seen that 4 narrowband users received at a near-far ratio of 0 dB have the effect of decreasing the CDMA capacity by only about 4 users/cell, as compared to a degradation of about 4 CDMA users/cell for each narrowband user in the staggered-link case. These results are not surprising, but they do give useful insight as to how much effect the narrowband interference has. It is interesting that in all of the curves shown, the CDMA capacity degrades slowly as the number of notches is increased to around 7 or 8, and then degrades more rapidly. Clearly, the presence of narrowband interference has more effect on CDMA capacity than does the notching. 59 4.5 Summary In this chapter, we have looked at applying CDMA overlay to the cellular case, as narrowband cellular systems by design only use a fraction of the system bandwidth in each cell. The CDMA overlay, if not very lightly-loaded, was shown to have an adverse effect on the existing narrowband system. The use of notch-filtering at the CDMA transmitters to avoid active narrowband users can greatly increase the amount of loading that the narrowband system can tolerate. Simulations showed that while the CDMA signals may at times require a large amount of notching to avoid narrowband users located throughout the cellular coverage region, the performance degradation is not severe. It was observed that the CDMA system is much more sensitive to the presence of strong narrowband interference than it is to notching. It was also found that the CDMA and narrowband users should have their forward links supported by the same frequency band. The results indicate that CDMA overlay is very promising for cellular systems, particularly as a means of transition from a system which exclusively supports narrowband service to one which supports CDMA service but can still meet previous commitments to narrowband subscribers. CHAPTER 5 CELLULAR OVERLAY IN A FADING CHANNEL In this chapter, the use of CDMA overlay in a cellular scenario will be extended to the fading channel. As has been the case throughout this research, the overlay will again cause severe problems for the existing narrowband system. The narrowband users also will benefit once again from notch-filtering in the CDMA transmitters. But in a flat fading channel, there is a large amount of notching necessary in order that enough relief is provided to the narrowband system. This places a strict constraint on user capacity in terms of the number of CDMIA and narrowband users which can simultaneously share the spectrum. The use of multi-carrier CDMA, however, allows the CDMA users to realize frequency diversity, and thus they may transmit at lower power levels. This reduces the amount of notching necessary, and provides a substantial improvement in joint user capacity. 5.1 Limits on CDMA Capacity The idea of multi-carrier CDMA has recently received significant attention as an alternative to traditional single-carrier CDIMA [27]. The transmission of a CDMA signal on two or more disjoint carrier frequencies, with enough frequency spacing so that the fading on each signal is independent, allows for the possibility of frequency diversity, and therefore an increase in user capacity. In this section, a development of user capacity limits for overlay in the fading channel begins with a quantification of how many CDMA users can be tolerated by the narrowband system before its performance is degraded too severely. We will then look at the resulting improvement due to the CDMA users employing notch-filtering in their transmitters. 60 61 Throughout this chapter, a cellular system identical to the one in Chapter 4, with the addition of fading, will be considered. From Figure 4.1, interference from cells within two outer layers of the center cell of interest will be taken into account. The PDF of the near-far ratio of an interfering user, either CDMA or narrowband, located uniformly throughout the three-layer area, was approximated in equation (4.8) as 1 1 (h + 23)2 fHk(h) I 6(h) + 2(128) xp(- 2(128) )u(-h) (5.1) 19 27(12 2(128) We will now look at the performance of a BPSIK user received in the presence of overlay in a fading channel. A general multi-carrier CDMA format will be used, where there are Q carriers, and the total power of each CDMA user is divided equally amongst the Q carriers. The identical CDMA signal will be transmitted simultaneously on each carrier. The fading process on each carrier will be taken as frequencynonselective, and independent of the fading processes on other paths. The CDMA bandwidth in the single-carrier case must be smaller than, or on the order of, the coherence bandwidth of the channel in order that the fading be frequency-nonselective. Thus in the realization of multi-carrier CDMA, the carriers must have sufficient frequency spacing between themselves in order for the fading on each carrier to be independent. It will be assumed that only the particular CDMA signal which overlaps the BPSK signal at bandpass will pass through the BPSK receiver, and thus we will concentrate only on that signal in the equations which follow. The BPSK user employs square pulses of width T, the bit time, and a matched-filter receiver. The complex baseband received signal can be written as 00 r(t) = V 7Pby,(i)d,(i)I((t iTb)/Tb) + 2n(t) exp(-jwt) i=-OO S-0 K (5.2) + E1 2PA/Qkk(i)dk(i)ck(t iTb rk) i=-oo k=l 62 where II(t/Tb) is a unit-amplitude square pulse of width Tb, Pb is the BPSK user's average power, Pk is the kth CDMA user's composite average power from all of its carriers, yb(i) and 7k(i) are the BPSIK user's and kth CDMA user's fading process during the ith bit interval, with each a zero-mean complex Gaussian random variable. The kth user's spreading waveform, A.(t), has a period of Tb and consists of unitamplitude square pulses of width QT. where the single-carrier chip-time is T, = Tb/N and N is the composite processing gain. The output of the BPSK user's matched-filter in the ith bit interval is (i+1)Tb K Z(i)= 1 T r(t) dt = y(i)dP(i) + Pk (iTb iTb (5.3) where Ne(i) and N,(i) are zero-mean Gaussian random variables each with variance r2 = (2Eb/No)-1, Eb is the average energy-per-bit of the BPSIK system, and the term due to interference from the kth CDMA user during the ith bit interval, Ik(i), is T, Tk Ik(i) = dk (i 1) C (t) d dk () f Ck (t) dt (5.4) rA. 0 Denoting the nth chip of the kth user's spreading code as ck,,2, and assuming that the per-carrier processing gain N/Q is an integer, we have Ik(i) Q S= d(i 1) 6kCkN/Q-pk-1 + 1 Ckj + =b N/N Qj=NQ-pk (5.5) dk (i) Ck,j + (1 6k)Ck,N/QPk-1 j=0 where the delay has been written as h. = (p + 6k)QTc with pk an integer in the range (0, 1, . (N/Q) 1) and 5, a non-integer in the range (0, 1). The variance of this term, for fixed codes and delays, and hence with the expectation taken only with respect to the data bits, is itself a random variable. From Section 4.2, it can be 63 verified by simulation that with respect to the codes and delays, the variance can be well-approximated as an exponential random variable. Assuming coherent detection for the BPSK user, the decision on the ith BPSK data bit is db(i) = sign(Re[Z exp(-jZy,(i))]). Invoking a Gaussian approximation on the CDMA interference contribution, the probability of bit error, conditioned on the BPSK user's fading process ye(i), is Pr(error/yb(i)) = Q (- Ai 2 (5.6) k= 1 where (Eb/NO)b has been clarified to apply to the BPSK user, not to be confused with the CDMA system, for which (Eb/No)c will be used. Also, Vk is a unit-mean exponential random variable. Note that the probability of bit error for the BPSK user does not depend on the number of carriers used in the multi-carrier CDMA signaling, which checks with intuition. When the number of carriers is doubled, for example, the CDMA users on each carrier are only spread by half of the original processing gain, but in turn, they transmit only half of the power on each carrier. In order to determine how many CDMA users may be tolerated by the BPSK system before the overlay is excessive, consider the probability of error in the presence of no overlay, conditioned again on the BPSIK user's fading process, which is Pr(error/ hb(i)) Q ( 2 () b (i)2 (5.7) We denote SNR as the square of the argument of the Q-function of equation (5.7) and SNReff as the same for equation (5.6). Thus with the addition of the overlay, the BPSK user must increase its power by a factor of SNR r= 1 + 7.(i) cos2(ZC S (i)) Vk (5.8) SNRIf f ,vo b k= Pb 64 As mentioned before, it should be stipulated that the overlay cause only a minor amount of degradation to the existing narrowband system. For a fixed number of CDMA users, the quantity in equation (5.8) is a random variable depending on the CDMA users' fading processes and near-far ratios (with respect to the BPSK user) and also on the exponentially-distributed 1,. variables. As the criterion here, we will say that the number of CDMA users is excessive if the random variable of equation (5.8) is greater than 3 dB more than 2% of the time. That is, we find the maximum value of K such that Pr (: A(i) cos(Z b(i)) ) > 1 < 0.02 (5.9) The capacity limits dictated by equation (5.9) will be found for a system with a composite processing gain of N = 32 chips/bit and with (Eb/No)b = 14 dB for the BPSK system. This choice results from averaging the conditional probability of error of equation (5.7) over the Rayleigh fading process 7b(i), which gives the well-known result 0.25 P 0.5 (5.10) (Es,/No)b and choosing (Eb/No)b = 14 dB to get a bit error rate of 0.01 in the absence of overlay. The maximum number of users K will be found for a range of values of (Eb/No)c. Recall that the results will not depend on the number of carriers used. However, the added frequency diversity resulting from multiple carriers allows for a smaller value of (Eb/No)c in order to achieve the same probability of error with the same number of CDMA users. For a CDMA user which is assigned to one of the cells within three layers of the center cell of interest at random, its near-far ratio in dB, (Pk/Pb), will come from the approximate PDF of equation (5.1), with an adjustment to account for the difference in the values of E,/No for a CDMA and a BPSK user assigned to the same cell. 65 Without the CDMA notching that has been mentioned, the amount of CDMA loading that the narrowband system can tolerate is practically zero. To implement the notching, when a CDMA signal is received such that the BPSK user's received power is less than T dB above that CDMA signal, a. notch is placed in the CDMA signal, and the near-far ratio, with the PDF given by equation (5.1), is then reduced by about 40 dB. This reduction assumes that the code sequences will be filtered when necessary by using the DFT-based filtering from Section 2.4.4, with the filtered code sequences zero-padded to 8 bits. In Figure 5.1, for a range of values for the notching threshold T and for several values of the difference (E,1/No), (E,/No)c, the amount of CDMA loading tolerable to the BPSK user was found such that the criterion of equation (5.9) was satisfied. As expected, for larger values of (E1/No)b (E,/No), the PDF of (Pk/Pb) tends to shift toward lower values and thus more CDMIA users can be tolerated by the BPSK user. And as the notching threshold T increases, and the CDMA users are therefore more likely to place notches. the BPSIK user also can tolerate more CDMA users. In order to make use of these results we must next investigate how much notching the CDMA system can handle before its signals become too distorted to be received reliably. 5.2 Limits on Narrowband Capacity In this section, we will look at the effects of notching from the perspective of the CDMA system. As more notches are necessary, there will be a point at which an excessive amount of some of the CDMA signals' spectra must be notched out, and hence many of the CDMA users would have to be dropped. This would occur not only in cases for which the notching threshold T is large, but would also occur when the quantity (Eb/No)b (Ebi/No)c is decreased, or equivalently the value of (Eb/No)c is increased. 66 25 (Eb/NO)b(Eb /)C = 7 dB 5 dB 20 0-3 dB 4 15 10 5 0 "j 5 7 9 11 13 15 17 T (dB) Figure 5.1: CDMA users/cell tolerable to the BPSK system. CDMA user is notched if BPSK power is less than T dB above CDMA power. Curves are for (Eb/NO)b (Eb/No)c values of 7,5,... ,-3 dB. (E,/No)b and (Eb/No)c represent values after power control at the base station to which the BPSK or CDMA mobile is assigned. This problem was examined in Section 4.3.1, where a narrowband user was located at random such that it was equally likely to be assigned to any of the cells within two outer layers of a center cell of interest. Then a CDMA user located at a normalized distance 0 < d < 1 from the center cell of interest, must notch for that narrowband user with probability p,,, which is plotted in Figure 4.6. It must be noted however, that in Chapter 4, the narrowband and CDMA users, when assigned to the same base station, were power-controlled so as to arrive at that base station at the same level. In this chapter, that level will be allowed to vary, and hence the quantity T in Figure 4.6 will actually be replaced by T ((E,/No)b (Eb/No)). We will now look at several different multi-carrier CDMA schemes, and for each the number of narrowband users which can be present so that the CDMA users do not require an excessive amount of notching will be found. A different quantification of how much notching is excessive will be given for each case. 67 Later in this chapter, the performance of the CDMA system employing the MMSE receiver will be investigated. The sampling rate used in the MMSE is 1/QTc, or 1/T, in the single-carrier case. The CDMA signals, with spectrally efficient pulse shapes, will typically be contained within the frequency range (-1/Tc < f < 1/T,). However, the notching can really only be done uniquely within the range (-0.5/T, < f < 0.5/T) as a result of the sampling rate. Each notch within this range, therefore, gives rise to a second notch outside of this range, but still within the range (-1/T, < f < 1/T,). Thus, recalling that the BPSIK and CDMA systems have the same data rate, there are only 16 unique notching locations when the processing gain is 32 chips/bit. In the single-carrier case, a CDMA user will be dropped if more than half of its signal must be notched, or 8 unique notching locations. The density of narrowband users/cell for which this criterion can be satisfied was found for a range of values of T((Eb/NO)b- (Eb/NO)c), in dB, and the results appear in Figure 5.2. Notice as expected that as T gets larger, and hence more notching is necessary, fewer narrowband users can be present. Also, as ((Eb/No)b (Eb/No)c) gets larger, the CDMA users are less likely to need notches and more narrowband users can be present. In Figure 5.3, a combination of the results of Figures 5.1 and 5.2 is shown for the single-carrier case. Several two-dimensional capacity curves, each with a constant value of ((Eb/NO)b (Eb/NoT)). were formed by finding the tolerable densities of CDMA users/cell from Figure 5.1 and narrowband users/cell from Figure 5.2 for a given value of the notching threshold T. This was repeated for a range of values of T. As ((Eb/NO)b (Eb/NTO)) gets larger, the CDMA users are less likely to interfere with the BPSK user and are less likely to require notches, and thus more users of each type should be supportable. It is important to keep in mind that we have not yet considered how much selfinterference the CDMA system can handle, nor the effects of narrowband interference 68 5 Single-carrier 4. S4 barriers 3 2 carriers .0 2 8 carriers, n o notching SI III 4 carriers, no notching 0 2 4 6 8 10 12 14 T- ((Eb/No)b (EbNo),), (dB) Figure 5.2: Narrowband users/cell tolerable to CDMA system before too much notching is required. Shown are single-carrier and multi-carrier cases. The "no notching" curves represent cases in which a carrier is dropped if even one notch is necessary. 25 25 25 +7 +7: +7 +5 +5 20 20 20 -1 +3 +3 15 15 15--10 .-- :10 10 +1 5 --- 5 5 ---0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 NB users/cell NB users/cell NB users/cell (a) (b) (c) Figure 5.3: Two-dimensional capacity curves combining Figures 5.1 and 5.2. Labels on curves indicate the value of (Ei,/No)b (E,1/No) Processing gain is 32 chips/bit. (a) Single-carrier case; (b) 2-carrier case; (c) 4-carrier case. 69 and notching on CDMA performance. We have only considered two things, which nonetheless do impose some limitations on user capacity: the number of CDMA users for which the narrowband system's performance is severely degraded according to the criterion of equation (5.9) and the number of narrowband users for which the CDMA users would simply require too much notching. In the next section, we will look at the CDMA performance in more detail. We next consider using multi-carrier CDMA with 2 carriers, and hence a processing gain per path of 16 chips/bit. In deciding how much notching is excessive here, we must note that only 8 unique notching locations are available in each carrier, and hence if there are more than 4 notches required in either carrier, the CDMA user will be dropped. With only one of the two carriers in use, this is equivalent to the singlecarrier case with only half of the processing gain, but the same amount of loading. The results for this case are also shown in Figure 5.2. For a given value of T, the probability that at least one of the two carriers will have more than 4 notches out of a possible 8 is greater than the probability that there will be 8 notches out of a possible 16 in the single-cell case, which agrees with intuition. In Figure 5.3, two-dimensional capacity curves for the 2-carrier case indicate performance inferior to the single-carrier case for a given value of ((Eb/No)b (E,/No)c). However, there is no significant conclusion to be drawn from this, and these results are only presented for future use. It will be seen later that when CDMA receiver performance is taken into account, the multi-carrier systems are able to operate at much lower values of (Eb/No)c than can the single-carrier case, as a result of the added frequency diversity. Thus it would be more fair to compare a curve in the single-carrier case with one of those from the multi-carrier case with a significantly larger value of ((Eb/No)b (Eb/No)c), which would lean more favorably toward the nmulti-carrier case. Finally, we consider using 4 carriers, which means that the CDMA signals will have a processing gain per path of 8 chips/bit. Thus a carrier will be dropped if 70 more than 2 notches are necessary. In contrast to the 2-carrier case, if only one of the carriers must be dropped, 3 of the 4 carriers would still remain, and it might be possible that the CDMA receiver could still function reliably. This will be looked at in Section 5.3, but for now, the criterion that we will employ is that a CDMA user will be dropped if at least 2 of the 4 carriers have more than 2 notches. These results are shown in Figures 5.2 and 5.3 as well. In Figure 5.3, there is an improvement over the 2-carrier case, in which the CDMA user was dropped even if only one of the carriers needed significant notching. It has been suggested previously that with the use of multi-carrier CDMA, it is possible to avoid the narrowband users in an overlay scenario by simply not transmitting on those carriers which might interfere with a narrowband user [27]. This possibility was also examined here for comparison. We will first utilize the previous notching criteria, and declare that if the CDMA signal on a given carrier is received at a high power level in terms of the notching threshold T, the carrier will not be used, analogous to simply placing a notch in the previous scenarios. We will consider both 4 and 8 carriers in this type of system. It does not seem fruitful to raise the number of carriers beyond 8, with a composite processing gain of 32 chips/bit, for several reasons. First, even assuming that the CDMA signal on each carrier would experience independent fading, the incremental diversity advantage realized by using more carriers diminishes with such a high number of carriers. Second, it has been assumed that the CDMA system, when used with multiple carriers, can be split such that transmission takes place on disjoint frequency bands, so that independent fading can be realized. While this seems plausible with 2 or even 4 carriers, the possibility becomes less likely with a large number of carriers. Thus some of the carriers will undergo correlated fading, which will not give nearly as much diversity as will independent fading. Finally, the process of tracking the fading processes, which is 71 not considered in this paper, would become tremendously complicated with so many carriers. It was observed in later simulations that the CDMA system suffered a large performance loss, in terms of the number of supportable users, if half of its carriers must be dropped. This would be essentially equivalent to doubling the load on a system with all of its carriers operational. Thus we can obtain an upper bound on capacity limits in this section by finding how many narrowband users may be present so that at least half of the carriers are still operational. As it is shown in Figure 5.2, for 4 and 8 carriers, even these upper bounds fall far below the performance of the systems which use notching. This suggests that a hybrid of these ideas must be used. When one or more of the carriers simply requires too much notching, transmission on it can be avoided, but the remaining carriers may still have some notches. Again, this must be tested in terms of the effect on CDMA performance, which will be done in the next section. 5.3 Performance of the CDMA System In this section, the capacity limits of multi-carrier CDMA will be examined further. Results from Sections 5.1 and 5.2 will be extended to include the effects on CDMA performance of MAI, NBI, notching, and the possibility of operating on fewer carriers than the maximum. We consider the use of the MMSE receiver, which is well-suited to the overlay environment. Several of its desirable properties were mentioned in Section 3.1, such as its ability to reject MAI, NBI, and ISI as well as its ability to adapt to the desired user's signal without even knowing that its code has been filtered. It was shown by Miller et al. [28] that the MMSE can successfully realize diversity in a frequency-selective fading channel, and that there is a substantial performance loss when all of the paths of all of the interfering users are not tracked explicitly in forming the Wiener solution. The performance of the MMSE was evaluated in the multi-carrier case by Miller and Rainbolt [29], and it was reaffirmed that 72 all of the paths of all of the CDMA users must be tracked in order to avoid a sizable performance loss. The performance in the overlay environment is a relatively straightforward extension of the analyses given by Miller et al. [28] and by Miller and Rainbolt [29]. The desired CDMA user now may operate with a filtered code sequence and narrowband noise must be added when necessary. We consider a general multi-carrier system with Q carriers, and a processing gain per path of N/Q. It is also assumed that the desired user's code sequence in the qth carrier will be filtered (when necessary) using the DFT-based filtering method from Section 2.4.4, which results in a code sequence of length greater than the processing gain, N/Q in this case. It will be expressed as the cascade of L individual sequences of length N/Q, as C1,q = (Cq_,-L/2 ... CT,q,-1, CT,q,0, CT,q, C,q,L/ T, with Cl,q,O in the middle associated with the desired component and the other sequences corresponding to ISI. Note that on each carrier, the filtered code sequence will be different as the notching necessary on each carrier is generally not the same. Also, for simplicity, it was verified in Section 4.4 that there is no significant difference in the reception of the desired user if the interfering CDMA users are left unfiltered. For the qth carrier, the samples of a chip-matched-filter bank will be collected during the ith bit interval, resulting in a column-vector of N/Q samples given by L/2 rq(i) = 7Y,q(i)di (i)c1,q,o + E 71(cl(i 1m)ci,q,m + nq(i) + jq(i) + m=-L/2 m O (5.11) K P-k,qk (i) +d(i-l)gk] E V Y-1"(Z) [dk(i)fk + dlk(i-- 1 k=2 where yk,q(i) is the fading process on the qth carrier of the kth user during the ith bit interval, and the fading processes for the same user on each carrier are independent. Also, fk and gk are the even and odd cyclic shifts of the kth user's code sequence, which were defined in equation (3.4) and nq(i) is a vector of length N/Q of independent 73 complex Gaussian noise samples, with the real and imaginary parts each having variance of a2 = N/(2Eb/No). The vector jq(i) consists of the sum of samples of all of the narrowband noise processes present in the qth carrier, if any. Each process is complex, with the real and imaginary parts independent, and each with a correlation matrix given by (for the mth narrowband user) R,(i, Pb) = Sa(2(i- cos(QTAw(i j)) (5.12) for the (i,j)th element, where (PI/PI) is the narrowband-to-CDMA near-far ratio, and Aw is the frequency difference between the location of the narrowband user and the CDMA carrier frequency. It was shown by Miller and Rainbolt [29] that the receiver will work best if the Q different received vectors of equation (5.11) are cascaded into a single composite vector of length N, given by r(i) = (r'(i), r'(i),... ,rQ(i)) (5.13) and a single Wiener filter is formed, given by w(i) = R-'(i)p(i), with R(i) and p(i) the correlation matrix and steering vector. The composite correlation matrix R(i) is given by Rl,1(i) R1,2() ... R1,Q(i) R(i) = R2,1(i) R2,2(i) ... R2,Q(i) (5.14) RQg,(i) RQ,2(i) ... RQ,Q(i) and the sub-matrices are given by R,,,q(i) = E [rl,(i)r"(i)] (5.15) 74 The composite steering vector is given by p (i) E [d,(i)r, (i)] p(i) p(i) E [dI (i)r2(i)] p(i) (5.16) pQ(i) E [dl(i)rQ(i)] The bit decision is then made as di (i) = sign(Re[w" (i)r(i)]) for coherent combining of the paths. It makes sense to use coherent combining in this case, as it was previously stated that all of the fading paths of all of the users would be tracked anyway in order to avoid a large performance loss. The task of tracking the fading processes in a dynamic environment is currently an area of active research. In this work, it will be assumed that all of the fading processes on all of the paths are known, in which case we have L/2 Rp,q(i) = y1,p(i)Y,q(i)C1,p,oc1,,0 + 1),P (i)c( ,pmc l,,m + m=-L/2 k=2 m=2 (5.17) where there are M narrowband users present on the carrier for which p = q. Also, IN/QxN/Q is an identity matrix of dimension N/Q. The qth steering vector is P,(i) = 71,,(i)c1,q,o (5.18) If the expression for the qth received vector in equation (5.11) is written as rq(i) = yi,,,(i)di (i)cl,q,o + rq(i) (5.19) where iq(i) represents a composite interference process consisting of MAI, AWGN, NBI, and ISI, it can be shown with the matrix-inversion lemma that the probability 75 of error, conditioned on the fading processes 7yk,q(i), when coherent combining of the paths is used is given by [28] Pr(error/F(i)) = Q (2pH (i)R-l(i)pH(i)) (5.20) where P(i) is a vector representing all of the fading processes of all of the users during the ith bit interval and R(i) = E[i(i)iH(i)] the composite interference correlation matrix. The performance of the CDMA system using an MMSE detector was then simulated. As was done earlier in the paper, a cellular environment with two outer layers of cells beyond the center cell of interest was considered. The mobiles, both CDMA and narrowband, experienced lognormal shadowing with a standard deviation o = 8 dB, propagation loss with an exponent of n = 3, and Rayleigh fading. For a given density of narrowband users/cell, the corresponding density of CDMA users/cell that could be supported by the system was found. This density was based both on the capacity constraints examined in Sections 5.1 and 5.2, and also on the criterion Pr(Pe,cDAIA > 0.05) < 0.02 (5.21) For a given realization of codes, delays, notching, and powers of the CDMA users and of powers and frequency locations of the narrowband users, 300 different realizations of the fading processes were generated, that is 300 realizations of F(i), and the conditional probability of error of the CDMA system was found using equation (5.20). The average probability of error was then found by averaging these 300 values. This process was repeated many times, that is for different realizations of the codes, delays, and so on, in order to give enough values for the average probability of error so that the criterion of equation (5.21) could be tested. Thus if a CDMA user's average probability of error was greater than 5%, it was dropped, and if these drops 76 20 4 carriers 2 carriers 10 :Single callier: 0 0.5 1 1.5 2 Narrowband users/cell Figure 5.4: Two-dimensional capacity curves, taking into account both CDMA receiver performance and results from Figure 5.3. Processing gain is 32 chips/bit. occurred more than 2% of the time, then the CDMA system was above its capacity limits. In order to combine these results with those of Sections 5.1 and 5.2, the notching threshold T and the value of (Eb/No)c must be chosen so as to satisfy the capacity curves shown in Figure 5.3. For example, in the single-carrier case, if there are 2 narrowband users present, the system can support 7 CDMA users if the value of (Eb/NO)b (Eb/N)c = 7 dB, or equivalently (Eb/NO)c = 7 dB. But if the CDMA system is loaded up to this level, it will require much more than (Eb/No)c = 7 dB to satisfy the criterion of equation (5.21). Thus a higher value of (Eb/No), must be used, and hence a lower value of (Eb/No)b (Eb/No)c, which means that we must operate on one of the curves showing a lower joint capacity. In Figure 5.4, the results of combining the criterion of equation (5.21) with the results of Sections 5.1 and 5.2 are shown. Similarly to the example just mentioned, the value of (Eb/No)c was optimized at each loading level. There is a noticeable improvement in capacity over the single-carrier case when 2 carriers are used, and a 77 slight additional increase when 4 carriers are used. Recall that in the multi-carrier scenarios, a smaller value of (Eb/No)c could be used in order to achieve the same performance in terms of CDMA reception. This in turn allows the limits of Sections 5.1 and 5.2 to be relaxed in comparison to the single-carrier case. 5.4 Summary In this chapter, the performance of a cellular overlay system with a fading channel model was evaluated. The effects of the CDMA system on the narrowband system were quantified, and the effects of notch-filtering were found to be beneficial. It was found that the use of multiple carriers allows the CDMA users to transmit less power than in the single-carrier case, and thus more users can be supported without causing interference to the narrowband system. The possibilities for CDMA overlay are strongly reinforced by the results of this section. CHAPTER 6 CONCLUSIONS AND FUTURE WORK In this final chapter, the key contributions of this dissertation to the existing body of research will be summarized. Some areas of future research will also be suggested. 6.1 Conclusions In this dissertation, the possibility of implementing CDMA overlay has been examined in detail. If overlay is to be done in a frequency band in which a narrowband system already exists, then it should be of vital importance to quantify the effects of the overlay on the narrowband system. Surprisingly, this problem had received little attention before this dissertation. On the other hand, the converse problem of the effects of the narrowband signals on the CDMA system had been examined very thoroughly. It was shown first for a Gaussian channel model that the overlay poses quite a problem for the narrowband system. If the CDMA system is to be loaded to a level for which it is worthwhile to even implement it, the narrowband users experience too much degradation in performance. In an effort to alleviate the effects of overlay on the narrowband system, the idea of employing notch filtering in the CDMA users' transmitters, which had been suggested previously by Davis [21], was examined in detail. Four filtering methods were examined in the current work. Two of the methods, the eigenvector filter and the null filter, which were presented by Davis [21], operated with the constraint that the filtered code sequence span only one bit interval, just as the unfiltered code sequence does. Two original methods were presented in this dissertation, which relaxed this 78 79 constraint, and allowed for a small amount of ISI. A digital Butterworth notch filter was used, and a filtering method based on the DFT was also considered. The filtering methods for which the ISI resulted performed significantly better than did the other two methods, in terms of the amount of distortion caused to the CDMA signals' PSDs. They also provided more relief to the narrowband system. The resulting ISI, while an important issue that must be looked at in CDMA receiver design, was certainly manageable. An important point that has been emphasized throughout this research, and has not received enough attention in the literature, is that the use of notch filtering is not simply a way to provide a modest improvement to the performance of an overlay system. Rather, it is absolutely essential if the CDMA system is to be loaded to levels for which the research community is actively striving. Next, the problem of receiving the CDMA signals was considered, and it was shown that the MMSE is a good choice for this purpose. In addition to its ability to reject MAI and NBI, which has been demonstrated before, it was shown here that the MMSE can reject ISI and also that it can adapt to a code sequence which has been filtered, provided that a training sequence is used. The MMSE receiver's functionality in the overlay scenario was demonstrated throughout the dissertation, in both single-cell and cellular scenarios, and with both a Gaussian channel model and a fading channel model. The application for which overlay has perhaps the most potential is the cellular scenario. The implementation of overlay could provide an efficient way for a frequency band which services narrowband users to transition its service to CDMA. For contractual reasons, providers cannot simply discontinue service to narrowband subscribers once it has been decided that the band will be converted to support CDMA. Overlay would allow the new CDMA technology to be introduced while the existing narrowband products are phased out gradually. 80 Issues arising in cellular overlay were considered, such as the necessity once again for CDMA transmitter notching and how much notching must be done. In a cellular environment, the CDMA users might have to notch for narrowband users in other cells. But they cannot notch for every narrowband user that is operational within the system, because that would probably constitute the entire system bandwidth. Thus a criterion must be established, and then evaluated to see how much notching the CDMA users must do. A criterion comparing the received powers of the CDMA and narrowband signals at the receiver of interest was adopted, wherein a notch was placed if the CDMA signal was received at too high a level above that at which the narrowband user was received. The amount of notching required was shown to be reasonable. Another design issue that was addressed was the frequency assignment for the forward and reverse links of the CDMA and narrowband systems. It was shown that the forward links of the two systems should be supported by the same band, as should the reverse links, in contrast to supporting the forward link of one and the reverse link of the other in the same band. In the latter case, the fact that a signal will be transmitted from the base station antenna and would appear at the other system's receiver antenna a short distance away on the same tower would result in an unmanageable near-far interference problem. Another conclusion reached was that the notching is much less of a problem for the CDMA system than is the reception of strong narrowband interference. The signals could be notched so that about 50% of their spectrum was missing, and suffer only a modest amount of degradation. Narrowband interference, however, was shown to cause significant problems if too strong. Finally, cellular overlay was examined in the fading channel. Once again, it was concluded that the CDMA users must employ notching in order to avoid the narrowband system. But it was also found that the CDMA signals required a large amount 81 of notching. This imposed disappointing constraints on the joint capacity of the system, that is the number of CDMA and narrowband users that can simultaneously occupy the spectrum. A solution for this problem is to use multi-carrier CDMA, in which the CDMA signal is transmitted on several carriers, such that the signals on each carrier undergo independent fading. This provides a significant amount of frequency diversity, which allows the CDMA users to transmit at lower power levels. Thus less notching is also necessary, and the joint capacity of the system is improved dramatically. The results presented in this dissertation indicate that CDMA overlay has great potential for providing an increase in user capacity. It is especially valuable in cellular scenarios, in which a transition from narrowband service to CDMA service is in progress. There remains, however, some open issues which will be discussed in the next section. 6.2 Future Work Some topics of future research will be suggested in this section. The results presented in this dissertation have clearly indicated that overlay has significant potential. The next step is to look at some of the implementation issues that were not addressed. Throughout the research, two important tasks were assumed to have been done perfectly, the timing acquisition, that is the knowledge about when the desired CDMA user's code sequence begins, and the estimation of the fading processes of the CDMA users. They are certainly important problems that need to be addressed in future work. The results presented in this dissertation had only the goal of demonstrating that there is indeed a reason to pursue overlay. If the results had demonstrated otherwise, then the effects of imperfect estimation of the timing and the fading parameters would not be interesting or meaningful. The timing acquisition problem has been investigated by many researchers, and there exists an extensive body of literature on the subject [30-35]. Obviously, these 82 works have not considered the effects of the notch-filtering presented in this dissertation. It must be determined what effects the notching will have on algorithms such as these, if used unmodified. It is likely that some modifications to fit the timing estimation problem to the case with filtered codes would be necessary and certainly would be beneficial. Also the possibility of the timing estimator working without knowing the code sequence, as the MMSE receiver can do for detection, might be useful. But it seems that this information would be known by the base stations, and the estimator would probably suffer a good deal of performance loss if the code sequence is not used in the algorithm. Additionally, in a frequency-selective fading channel, the estimator must be able to lock onto several paths with different delays. Another significant area of future research is the estimation of the fading processes of the CDMA users. It was determined by Miller et al. [28] and by Miller and Rainbolt [29] that the MMSE receiver will take a large loss in performance if it is not able to track all of the fading processes of all of the users. This is a fairly undeveloped area, with a few solutions having been proposed. Barbosa and Miller used linear prediction in conjunction with an MMSE receiver to estimate the fading process of the desired user in a flat-fading channel [36]. A subspace-approach was investigated by Wang and Poor [37]. Methods based on a decorrelator idea, that is a transformation on the received signal which removes the data, are presented by Miller and Rainbolt [29] and by Juntti [38] in an effort to track the fading in a frequency-selective fading channel. The estimation of the fading processes is very important, not only for the overlay scenario in this dissertation, but for the use of the MMSE in fading channels in general. This will continue to be a popular topic of research, as there has not been a tremendous amount of success realized. REFERENCES [1] D. V. Sarwate and M. B. Pursley, "Crosscorrelation properties of pseudorandom and related sequences," Proc. IEEE, vol. 68, pp. 593-619, May 1980. [2] M. B. Pursley, "Performance evaluation for phase-coded spread-spectrum multiple-access communication part I: system analysis," IEEE Trans. Commun., vol. COM-25, pp. 795-799, Aug. 1977. [3] K. Yao, "Error probability of asynchronous spread-spectrum multiple-access communication systems," IEEE Trans. Commun., vol. COM-25, pp. 803-809, Aug. 1977. [4] S. Verdii, "Minimum probability of error for asynchronous Gaussian multipleaccess channels," IEEE Trans. Inform. Theory, vol. IT-32, pp. 86-96, Jan. 1986. [5] S. Verdi, "Optimum multi-user asymptotic efficiency," IEEE Trans. Commun., vol. COM-34, pp. 890-897, Sept. 1986. [6] R. Lupas and S. Verdi, "Linear multiuser detectors for synchronous codedivision multiple-access channels," IEEE Trans. Inform. Theory, vol. 35, pp. 123-136, Jan. 1989. [7] R. Lupas and S. Verdi, "Near-far resistance of multiuser detectors in asynchronous channels," IEEE Trans. Commun., vol. 38, pp. 496-508, Apr. 1990. [8] M. K. Varanasi and B. Aazhang, "Multistage detection in asynchronous codediviion multiple-access communications," IEEE Trans. Commun., vol. 38, pp. 509-519, Apr. 1990. [9] P. B. Rapajic and B. S. Vucetic, "Adaptive receiver structures for asynchronous CDMA systems," IEEE J. Select. Areas Commun., vol. 42, pp. 685-697, May 1994. [10] U. Madhow and M. L. Honig, "MMSE interference suppression for directsequence spread-spectrum CDMA," IEEE Trans. Commun., vol. 42, pp. 31783188, Dec. 1994. [11] S. L. Miller, "An adaptive direct-sequence code-division multiple-access receiver for multiuser interference rejection," IEEE Trans. Commun., vol. 43, pp. 17461755, Feb./ Mar./ Apr. 1995. 83 84 [12] M. Abdulrahman, A. Sheikh, and D. Falconer, "Decision feedback equalization for CDMA in indoor wireless communications," IEEE J. Select. Areas Commun., vol. 12, pp. 698-706, May 1994. [13] C. N. Pateros and G. J. Saulnier, "An adaptive correlator receiver for directsequence spread-spectrum communication," IEEE Trans. Commun., vol. 44, pp. 1543-1552, Nov. 1996. [14] S. L. Miller, "Training analysis of adaptive interference suppression for directsequence code-division multiple-access systems," IEEE Trans. Commun., vol. 44, pp. 488-495, Apr. 1996. [15] M. Honig, U. Madhow, and S. Verdii, "Blind adaptive multiuser detection," IEEE Trans. Inform. Theory, vol. 41, pp. 944-960, July 1995. [16] L. B. Milstein, D. L. Schilling, R. L. Pickholtz, V. Erceg, M. Kullback, E. G. Kanterakis, D. S. Fishman, W. H. Biederman, and D. C. Salerno, "On the feasibility of a CDMA overlay for personal communication networks," IEEE J. Select. Areas Commun., vol. 10, pp. 655-668, May 1992. [17] J. W. Ketchum and J. G. Proakis, "Adaptive algorithms for estimating and suppressing narrowband interference in PN spread-spectrum systems," IEEE Trans. Commun., vol. COM-30, pp. 913-924, May 1982. [18] L. Li and L. B. Milstein, "Rejection of narrowband interference in PN spreadspectrum systems using transversal filters," IEEE Trans. Commun., vol. COM30, pp. 925-928, May 1982. [19] R. A. Iltis and L. B. Milstein, "Performance analysis of narrowband interference rejection techniques in DS spread spectrum systems," IEEE Trans. Commun., vol. COM-32, pp. 1169-1177, Nov. 1984. [20] L. B. Milstein, "Interference rejection techniques in spread spectrum communications," Proc. IEEE, vol. 76, pp. 657-671, June 1988. [21] M. E. Davis, Signal processing for interference avoidance and multiple-access noise rejection in DS-CDMA, Ph.D. dissertation, University of California, San Diego, 1993. [22] R. L. Pickholtz, L. B. Milstein, and D. L. Schilling, "Spread spectrum for mobile communications," IEEE Trans. Veh. Technol., vol. 40, pp. 313-321, May 1991. [23] H. V. Poor and X. Wang, "Code-aided interference suppression for DS/CDMA communications part I: interference suppression capability," IEEE Trans. Commun., vol. 45, pp. 1101-1111, Sept. 1997. [24] V. K. Garg, K. F. Smolik, and J. E. Wilkes, Applications of CDMA in Wireless/Personal Communications, Prentice-Hall, Upper Saddle River, NJ, 1997. 85 [25] T. S. Rappaport, Wireless Communications, Prentice-Hall, Upper Saddle River, NJ, 1996. [26] R. Johanneson and E. Paaske, "Further results on binary convolutional codes with an optimum distance profile," IEEE Trans. Inform. Theory, vol. IT-24, no. 2, pp. 264-268, Mar. 1978. [27] S. Kondo and L. B. Milstein, "Performance of multicarrier DS-CDMA systems," IEEE Trans. Commun., vol. 44, no. 2, pp. 238-246, Feb. 1996. [28] S. L. Miller, M. L. Honig, and L. B. Milstein, "Performance analysis of MMSE receivers for DS-CDMA in frequency selective fading channels," submitted to IEEE Trans. Commun. [29] S. L. Miller and B. J. Rainbolt, "MMSE detection of multi-carrier CDMA," submitted to 1999 IEEE International Conference on Communications. [30] E. G. Strim, S. Parkvall, S. L. Miller, and B. E. Ottersten, "Propagation delay estimation in asynchronous direct-sequence code-division multiple access systems," IEEE Trans. Commun., vol. 44, no. 1, pp. 84-93, Jan. 1996. [31] S. E. Bensley and B. Aazhang, "Subspace-based channel estimation for code division multiple access communication systems," IEEE Trans. Commun., vol. 44, pp. 1009-1020, Aug. 1996. [32] U. Madhow, "Blind adaptive interference suppression for the near-far resistant acquisition and demodulation of direct-sequence CDMA signals," IEEE Trans. Signal Processing, vol. 45, no. 1, pp. 124-136, Jan. 1997. [33] D. Zheng, J. Li, S. L. Miller, and E. G. Str6m, "An efficient code-timing estimator for DS-CDMA signals," IEEE Trans. Signal Processing, vol. 45, no. 1, pp. 82-89, Jan. 1997. [34] Z. S. Liu, J. Li, and S. L. Miller, "An efficient code-timing estimator for receiver diversity DS-CDMA systems," IEEE Trans. Commun., vol. 46, no. 6, pp. 826835, June 1998. [35] R. F. Smith and S. L. Miller, "Acquisition performance of an adaptive receiver for DS-CDMA systems," submitted to IEEE Trans. Commun. [36] A. N. Barbosa and S. L. Miller, "Adaptive detection of DS/CDMA signals in fading channels," IEEE Trans. Commun., vol. 46, no. 1, pp. 115-124, Jan. 1998. [37] X. Wang and H. V. Poor, "Blind multiuser detection: a subspace approach," IEEE Trans. Inform. Theory, vol. 44, no. 2, pp. 677-690, Mar. 1998. [38] M. Juntti, Multiuser demodulation for DS-CDMA systems in fading channels, Ph.D. dissertation, University of Oulu, 1997. BIOGRAPHICAL SKETCH Brad J. Rainbolt was born in Normal, IL, in 1972. He received the B.S. and M.E. degrees, both in electrical engineering, in August 1993 and December 1994 from the University of Florida, Gainesville, FL. He was employed by Motorola Land Mobile Products Sector, Plantation, FL, from May 1994 through August 1995 as an engineer in the Applied Research Group, where he worked on the design of digital communication systems. In August 1995, he returned to the University of Florida, and received the Ph.D. degree in electrical engineering in December 1998. 86 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 L. Miller, Chairman 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. Leon W. Couch II 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. Jian Li 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. Ulrich H. Kurzweg Professor of Aerospace Engineering, Mechanics, and Engineering Science 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. December 1998 Winfred M. Phillips Dean, College of Engineering M. J. Ohanian Dean, Graduate School |

Full Text |

CHAPTER 6
CONCLUSIONS AND FUTURE WORK In this final chapter, the key contributions of this dissertation to the existing body of research will be summarized. Some areas of future research will also be suggested. 6.1 Conclusions In this dissertation, the possibility of implementing CDMA overlay has been ex amined in detail. If overlay is to be done in a frequency band in which a narrowband system already exists, then it should be of vital importance to quantify the effects of the overlay on the narrowband system. Surprisingly, this problem had received little attention before this dissertation. On the other hand, the converse problem of the effects of the narrowband signals on the CDMA system had been examined very thoroughly. It was shown first for a Gaussian channel model that the overlay poses quite a problem for the narrowband system. If the CDMA system is to be loaded to a level for which it is worthwhile to even implement it, the narrowband users experience too much degradation in performance. In an effort to alleviate the effects of overlay on the narrowband system, the idea of employing notch filtering in the CDMA users transmitters, which had been suggested previously by Davis [21], was examined in detail. Four filtering methods were examined in the current work. Two of the methods, the eigenvector filter and the null filter, which were presented by Davis [21], operated with the constraint that the filtered code sequence span only one bit interval, just as the unfiltered code sequence does. Two original methods were presented in this dissertation, which relaxed this 78 65 Without the CDMA notching that has been mentioned, the amount of CDMA loading that the narrowband system can tolerate is practically zero. To implement the notching, when a. CDMA signal is received such that the BPSK users received power is less than T dB above that CDMA signal, a. notch is placed in the CDMA signal, and the near-far ratio, with the PDF given by equation (5.1), is then reduced by about 40 dB. This reduction assumes that the code sequences will be filtered when necessary by using the DFT-based filtering from Section 2.4.4, with the filtered code sequences zero-padded to 8 bits. In Figure 5.1, for a range of values for the notching threshold T and for several values of the difference (Ei,/No)b {Eb/N0)c, the amount of CDMA loading tolerable to the BPSK user was found such that the criterion of equation (5.9) was satisfied. As expected, for larger values of (EiJNq)t (Ei,/No)c, the PDF of (Pk/Pb) tends to shift toward lower values and thus more CDMA users can be tolerated by the BPSK user. And as the notching threshold T increases, and the CDMA users are therefore more likely to place notches, the BPSK user also can tolerate more CDMA users. In order to make use of these results we must next investigate how much notching the CDMA system can handle before its signals become too distorted to be received reliably. 5.2 Limits on Narrowband Capacity In this section, we will look at the effects of notching from the perspective of the CDMA system. As more notches are necessary, there will be a point at which an excessive amount of some of the CDMA signals spectra must be notched out, and hence many of the CDMA users would have to be dropped. This would occur not only in cases for which the notching threshold T is large, but would also occur when the quantity (Eb/No)b {Ei,/Nq)c is decreased, or equivalently the value of (Eb/No)c is increased. 80 Issues arising in cellular overlay were considered, such as the necessity once again for CDMA transmitter notching and how much notching must be done. In a cellular environment, the CDMA users might have to notch for narrowband users in other cells. But they cannot notch for every narrowband user that is operational within the system, because that would probably constitute the entire system bandwidth. Thus a criterion must be established, and then evaluated to see how much notching the CDMA users must do. A criterion comparing the received powers of the CDMA and narrowband signals at the receiver of interest was adopted, wherein a notch was placed if the CDMA signal was received at too high a level above that at which the narrowband user was received. The amount of notching required was shown to be reasonable. Another design issue that was addressed was the frequency assignment for the forward and reverse links of the CDMA and narrowband systems. It was shown that the forward links of the two systems should be supported by the same band, as should the reverse links, in contrast to supporting the forward link of one and the reverse link of the other in the same band. In the latter case, the fact that a signal will be transmitted from the base station antenna and would appear at the other systems receiver antenna a short distance away on the same tower would result in an unmanageable near-far interference problem. Another conclusion reached was that the notching is much less of a problem for the CDMA system than is the reception of strong narrowband interference. The signals could be notched so that about 50% of their spectrum was missing, and suffer only a modest amount of degradation. Narrowband interference, however, was shown to cause significant problems if too strong. Finally, cellular overlay was examined in the fading channel. Once again, it was concluded that the CDMA users must employ notching in order to avoid the narrow- band system. But it was also found that the CDMA signals required a large amount 63 verified by simulation that with respect to the codes and delays, the variance can be well-approximated as an exponential random variable. Assuming coherent detection for the BPSK user, the decision on the th BPSK data bit is db(i) = sign(Re[ZÂ¡ exp{jZ.7/,(?'))]) Invoking a Gaussian approximation on the CDMA interference contribution, the probability of bit error, conditioned on the BPSK users fading process 7/,(/), is Pr(error/76()) = Q |76|2 v\ (2 (^)J + N I7*(0I* cos2(Z7fc(0) (it) Vk (5.6) where (Eb/No)b has been clarified to apply to the BPSK user, not to be confused with the CDMA system, for which (Eb/No)c will be used. Also, Vk is a unit-mean exponential random variable. Note that the probability of bit error for the BPSK user does not depend on the number of carriers used in the multi-carrier CDMA signaling, which checks with intuition. When the number of carriers is doubled, for example, the CDMA users on each carrier are only spread by half of the original processing gain, but in turn, they transmit only half of the power on each carrier. In order to determine how many CDMA users may be tolerated by the BPSK system before the overlay is excessive, consider the probability of error in the presence of no overlay, conditioned again on the BPSK users fading process, which is Pr(error/7t(?)) Q '2 Nr 0/6 MOP (5.7) We denote SNR as the square of the argument of the Q-function of equation (5.7) and SNReff as the same for equation (5.6). Thus with the addition of the overlay, the BPSK user must increase its power by a factor of SNR ( 2 \ SNReff ~ +\n) Eh ay, k l7A-(v')|2cos2(Z7fc(0) 6=1 vk (5.8) 67 Later in this chapter, the performance of the CDMA system employing the MMSE receiver will be investigated. The sampling rate used in the MMSE is 1 /QTC, or 1 /Tc in the single-carrier case. The CDMA signals, with spectrally efficient pulse shapes, will typically be contained within the frequency range (1/TC < / < 1 /Tc). However, the notching can really only be done uniquely within the range (0.5/Tc < f < 0.5/Tc) as a result of the sampling rate. Each notch within this range, therefore, gives rise to a second notch outside of this range, but still within the range (1 /Tc < f < l/Tc). Thus, recalling that the DPSK and CDMA systems have the same data rate, there are only 16 unique notching locations when the processing gain is 32 chips/bit. In the single-carrier case, a CDMA user will be dropped if more than half of its signal must be notched, or 8 unique notching locations. The density of narrowband users/cell for which this criterion can be satisfied was found for a range of values of T ((Eb/No)b~ {Eb/No)c), in dB, and the results appear in Figure 5.2. Notice as expected that as T gets larger, and hence more notching is necessary, fewer narrowband users can be present. Also, as ((Eb/N0)b (Eb/N0)c) gets larger, the CDMA users are less likely to need notches and more narrowband users can be present. In Figure 5.3, a combination of the results of Figures 5.1 and 5.2 is shown for the single-carrier case. Several two-dimensional capacity curves, each with a constant value of ((Eb/No)b (Eb/N0)c). were formed by finding the tolerable densities of CDMA users/cell from Figure 5.1 and narrowband users/cell from Figure 5.2 for a given value of the notching threshold T. This was repeated for a range of values of T. As ((Eb/N0)b (Eb/No)c) gets larger, the CDMA users are less likely to interfere with the BPSK user and are less likely to require notches, and thus more users of each type should be supportable. It is important to keep in mind that we have not yet considered how much self interference the CDMA system can handle, nor the effects of narrowband interference 34 It would also be interesting to observe the transient behavior of the MMSE receiver in the overlay environment, in addition to its steady-state operation reported above. The performance of the RLS algorithm will be looked at for this purpose, as its speed of convergence does not depend on the input power as in the case of the LMS algorithm. For 200 trials, the RLS algorithm was run first with 20 CDMA users and without narrowband users and notches, and then with 2 narrowband users and notches added. The squared error was averaged over the 200 trials in Figure 3.4. Both of the previously described MMSE criteria were used for comparison. In the first case with no narrowband users or notches, the mean-squared error is seen to converge to a steady-state value in about 150 bits. Notice that there is a significant improvement in the mean squared-error when the receiver mini mizes J = E[{Re(Z,- c/i())}2] as opposed to J E[|Z di(i)\2], but the re quired convergence time is about the same. Also notice that in the case when J = E[{Re(Z di(?'))}-] is minimized, the corresponding absolute value of the mean squared-error is very high. As stated before, the separe of the real part of the error, which is used in making the bit decision, is minimized at the cost of allowing a large increase in the imaginary part of the error, which will not be used anyway. The same environment was simulated with 2 narrowband users received at a near-far ratio of 20 dB and with the corresponding notches added. The same amount of time, about 150 bits, was required for convergence to a steady-state value, which is seen to be somewhat higher than when no narrowband users are present. It is worth noting that the computational complexity of the two algorithms is the same, as every com plex computation that is required for minimizing J E[|Z d\ ()|2] requires two real computations in the other case. Thus minimizing J = E[{Re(Z
fers a substantial improvement in performance with no added complexity or requiredtraining time. 8 in Section 3.1. Performance equations for the MMSE receiver are presented in Section 3.2. Simulation results are then presented in Section 3.3, both for a receiver which is able to track the ideal Wiener solution and for one which uses an adaptive algorithm. The MMSE receiver was compared to a conventional matched filter receiver, and was clearly superior, showing a capacity gain of 8-11 times when using the Wiener solution and 6-8 times when using an adaptive algorithm. It is also important to note that the MMSE receiver using an adaptive algorithm was able to perform well although it did not even know that the code sequence was filtered, a very attractive feature in this scenario. The results of these simulations, combined with the results on the performance of the narrowband system presented in Chapter 2, present a strong case for the feasibility of CDMA overlay. The previous results simply demonstrated that overlay can be done if the CDMA users employ transmitter notching. In Chapter 4, a compelling motivation for overlay is presented, as these results are extended to the cellular scenario, where overlay seems to have the most potential from a commercial point of view. If a frequency bandwidth which supports a narrowband cellular system is designated for conversion to a CDMA cellular system, overlay is an ideal way to make this transition gradually. Over the long term, the number of subscribers still using the narrowband products would shrink while the number of subscribers using the new CDMA products would increase. Conclusions similar to those reached in Chapters 2 and 3 will be realized in the cellular case. In Section 4.1. the cellular environment will be characterized. The effects of overlay on the narrowband system will be examined in Section 4.2, and it will be seen again that the narrowband users undergo a large amount of performance degradation as a result of the overlay. In Section 4.3, the idea of notch-filtering the CDMA signals will be applied, and will again provide tremendous benefits to the narrowband system. Simulations and results will be presented in Section 4.4. 68 C/5 u 0) C/5 3 Xl Â£ o & Figure 5.2: Narrowband users/cell tolerable to CDMA system before too much notch ing is required. Shown are single-carrier and multi-carrier cases. The no notching curves represent cases in which a carrier is dropped if even one notch is necessary. NB users/cell (a) NB users/cell (b) NB users/cell (c) Figure 5.3: Two-dimensional capacity curves combining Figures 5.1 and 5.2. Labels on curves indicate the value of (Ei)/Aro)ii (Ei,/Nq)c. Processing gain is 32 chips/bit. (a) Single-carrier case; (b) 2-carrier case; (c) 4-carrier case. 48 to an adjacent cell or even to a cell that is two layers away. But it obviously cannot notch for all of the narrowband users in the system, as that would likely constitute the entire system bandwidth, nor does it need to. As a result of the shadowing described in Section 4.1, there will be in some cases narrowband users near to the CDMA mobile that are not significantly degraded by the CDMA signal, and hence a notch is not necessary for them. This also holds for a significant percentage of the users that are not near to the CDMA mobile. The total CDMA interference seen by a narrowband user depends on the CDMA signals received powers, phases, and on the 14 variables according to equation (4.9). When deciding whether or not a notch is necessary, the power can be estimated and the information exchanged between base stations, but the phases and the 14- obviously will be unknown. Hence a specific criterion that will be used is that a CDMA signal must be notched if the power level at which it arrives at the corresponding narrowband receiver relative to the narrowband signal is higher than a given threshold. From Table 2.3, when the DFT-based filtering method is used with 8 bits of zero padding, the interference contribution is reduced by about 40 dB. The experiment of Section 4.2 was repeated, that is the number of CDMA users for which the ex cessive overlay criterion of equation (4.11) could be met was found, with transmitter filtering employed in those users for which the narrowband-to-CDMA power ratio at the narrowband receiver was less than a threshold T, given in dB. The results are shown in Figure 4.4 for T = 7,9,11 dB. In contrast to the 0.75 CDMA users/cell that was found in the unfiltered case, it is possible that with T = 9 dB, 19 users/cell can be tolerated by the narrowband system when filtering is employed, a tremendous increase in capacity. As the threshold is raised to T = 11 dB, and hence a CDMA user is more likely to need a notch, the amount of CDMA loading that can be tolerated by the narrowband system increases substantially to 29 users/cell, a level at which the CDMA system itself probably cannot function. And if it is reduced to T = 7 dB, the REFERENCES [1] D. V. Sarwate and M. B. Parsley, Crosscorrelation properties of pseudorandom and related sequences, Proc. IEEE, vol. 68, pp. 593-619, May 1980. [2] M. B. Pursley, Performance evaluation for phase-coded spread-spectrum multiple-access communication part I: system analysis, IEEE Trans. Com- mun., vol. COM-25, pp. 795-799, Aug. 1977. [3] K. Yao, Error probability of asynchronous spread-spectrum multiple-access communication systems, IEEE Trans. Commun., vol. COM-25, pp. 803-809, Aug. 1977. [4] S. Verd, Minimum probability of error for asynchronous Gaussian multiple- access channels, IEEE Trans. Inform. Theory, vol. IT-32, pp. 86-96, Jan. 1986. [5] S. Verd, Optimum multi-user asymptotic efficiency, IEEE Trans. Commun., vol. COM-34, pp. 890-897, Sept. 1986. [6] R. Lupas and S. Verd, Linear multiuser detectors for synchronous code division multiple-access channels, IEEE Trans. Inform. Theory, vol. 35, pp. 123-136, Jan. 1989. [7] R. Lupas and S. Verd, Near-far resistance of multiuser detectors in asyn chronous channels, IEEE Trans. Commim., vol. 38, pp. 496-508, Apr. 1990. [8] M. K. Varanasi and B. Aazhang, Multistage detection in asynchronous code- diviion multiple-access communications, IEEE Trans. Commun., vol. 38, pp. 509-519, Apr. 1990. [9] P. B. Rapajic and B. S. Vucetic, Adaptive receiver structures for asynchronous CDMA systems, IEEE J. Select. Areas Commim., vol. 42, pp. 685-697, May 1994. [10] U. Madhow and M. L. Honig, MMSE interference suppression for direct- sequence spread-spectrum CDMA, IEEE Trans. Commun., vol. 42, pp. 3178- 3188, Dec. 1994. [11] S. L. Miller, An adaptive direct-sequence code-division multiple-access receiver for multiuser interference rejection, IEEE Trans. Commun., vol. 43, pp. 1746- 1755, Feb./ Mar./ Apr. 1995. 83 64 As mentioned before, it should be stipulated that the overlay cause only a minor amount of degradation to the existing narrowband system. For a fixed number of CDMA users, the quantity in equation (5.8) is a random variable depending on the CDMA users fading processes and near-far ratios (with respect to the BPSK user) and also on the exponentially-distributed 14 variables. As the criterion here, we will say that the number of CDMA users is excessive if the random variable of equation (5.8) is greater than 3 dB more than 2% of the time. That is, we find the maximum value of K such that Pr (((I) l7(i)|W(ZT(i)) (t)Vk) >1)< '02 (5'9) The capacity limits dictated by equation (5.9) will be found for a system with a composite processing gain of AT = 32 chips/bit and with (Eb/No)b = 14 dB for the BPSK system. This choice results from averaging the conditional probability of error of equation (5.7) over the Rayleigh fading process 7b(i), which gives the well-known result P, 0.25 (5.10) (Eb/N0)b and choosing (Eb/No)b = 14 dB to get a bit error rate of 0.01 in the absence of overlay. The maximum number of users K will be found for a range of values of (Eb/N0)c. Recall that the results will not depend on the number of carriers used. However, the added frequency diversity resulting from multiple carriers allows for a smaller value of (Eb/No)c in order to achieve the same probability of error with the same number of CDMA users. For a CDMA user which is assigned to one of the cells within three layers of the center cell of interest at random, its near-far ratio in dB, (Pk/Pb), will come from the approximate PDF of equation (5.1), with an adjustment to account for the difference in the values of Eb/N0 for a CDMA and a BPSK user assigned to the same cell. 84 [12] M. Abdulrahman, A. Sheikh, and D. Falconer, Decision feedback equalization for CDMA in indoor wireless communications, IEEE J. Select. Areas Commun., vol. 12, pp. 698-706, Maj' 1994. [13] C. N. Pateros and G. J. Saulnier, An adaptive correlator receiver for direct- sequence spread-spectrum communication, IEEE Trans. Commun., vol. 44, pp. 1543-1552, Nov. 1996. [14] S. L. Miller, Training analysis of adaptive interference suppression for direct- sequence code-division multiple-access systems, IEEE Trans. Commun., vol. 44, pp. 488-495, Apr. 1996. [15] M. Honig, U. Madhow, and S. Verd, Blind adaptive multiuser detection, IEEE Trans. Inform. Theory, vol. 41, pp. 944-960, July 1995. [16] L. B. Milstein, D. L. Schilling, R. L. Pickholtz, V. Erceg, M. Kullback, E. G. Kanterakis, D. S. Fishman, W. H. Biederman, and D. C. Salerno, On the feasibility of a CDMA overlay for personal communication networks, IEEE J. Select. Areas Commun., vol. 10, pp. 655-668, May 1992. [17] J. W. Ketchum and J. G. Proakis, Adaptive algorithms for estimating and suppressing narrowband interference in PN spread-spectrum systems, IEEE Trans. Commun., vol. COM-30, pp. 913-924, May 1982. [18] L. Li and L. B. Milstein, Rejection of narrowband interference in PN spread- spectrum systems using transversal filters, IEEE Trans. Commun., vol. COM- 30, pp. 925-928, May 1982. [19] R. A. litis and L. B. Milstein, Performance analysis of narrowband interference rejection techniques in DS spread spectrum systems, IEEE Trans. Commun., vol. COM-32, pp. 1169-1177, Nov. 1984. [20] L. B. Milstein, Interference rejection techniques in spread spectrum communi cations, Proc. IEEE, vol. 76, pp. 657-671, June 1988. [21] M. E. Davis, Signal processing for interference avoidance and multiple-access noise rejection in DS-CDMA, Ph.D. dissertation, University of California, San Diego, 1993. [22] R. L. Pickholtz, L. B. Milstein, and D. L. Schilling, Spread spectrum for mobile communications, IEEE Trans. Veh. Technol, vol. 40, pp. 313-321, May 1991. [23] H. V. Poor and X. Wang, Code-aided interference suppression for DS/CDMA communications part I: interference suppression capability, IEEE Trans. Com mun., vol. 45, pp. 1101-1111, Sept. 1997. [24] V. K. Garg, K. F. Smolik, and J. E. Wilkes, Applications of CDMA in Wire less/Personal Communications, Prentice-Hall, Upper Saddle River, NJ, 1997. 4 sends information back to the mobile users, telling them to either raise or lower their transmit powers. With perfect power control, all of the signals are received at the same power level at the base station. In response to the near-far problem, researchers have developed a large number of receiver structures with varying degrees of complexity and performance. A key feature of these near-far resistant receivers is that they utilize information about the other users signals, either explicitly or implicitly in their processing. They take advantage of the known form of the MAI and hence they also are referred to as multi-user detectors. The matched-filter receiver, on the other hand, treats the MAI as white noise which simply increases the thermal noise floor. The optimum multiple-user detector [4,5] consists of a bank of matched filters and uses a Viterbi algorithm to demodulate the data streams of all of the CDMA users. While it does minimize the probability of bit error, its complexity increases expo nentially with the number of users, rendering it infeasible in practice. The optimum detector is very important, however, because at the time of its inception, common sentiment was that the near-far problem was an inherent shortcoming of CDMA that could not be overcome. The existence of this near-far resistant detector served as mo tivation for the development of implementable receivers that could outperform the conventional matched-filter receiver. Because the optimum detector has such a high complexity, and requires a large amount of side information about all of the CDMA users, several sub-optimum near- far resistant receivers have been developed which have more manageable complexities. A few notewortlty receivers include the decorrelator [6,7] and the multistage detector [8]. The minimum mean-squared error (MMSE) receiver [9-15] has received a great deal of attention during the past few years as it also offers near-far resistance and is very simple to implement. The MMSE also has an inherent resistance to narrowband interference (NBI), a feature that is very beneficial in an overlay scenario, which will 36 The resulting histograms of the probability of bit error are shown in Figure 3.5 for the three different near-far ratios and each is compared to the case in which no narrowband signal is present. It is seen that when the narrowband signal is received at 0 dB, there is little difference, but the performance does degrade somewhat for the 20 dB case, and degrades significantly for the 40 dB case. Hence it appears that the MMSE receiver is relatively robust to the presence of the narrowband signal, but for large near-far ratios its performance will be degraded. This is consistent with the conclusions reached by Poor and Wang [23], that while the MMSE receiver does outperform conventional narrowband interference rejection schemes, the output signal-to-interference ratio of the MMSE receiver will degrade for high-powered nar rowband interference. Fortunately, the probability that a narrowband user will have such a large near-far ratio is low, as this would occur only in a small geographic region around the CDMA receiver. 3.4 Summary In this chapter, we have shown that with the notch-filtering which was determined necessary in Chapter 2, the MMSE receiver can function quite well in an overlay en vironment. In the simulations performed here, the CDMA system could be loaded up to about 20 users for a processing gain of 32 chips/bit when 5 narrowband users were present, or about 15% of the band was jammed. It offers a substantial performance improvement over a conventional matched-filter receiver. These results, along with those of Chapter 2, are quite encouraging for the prospects of CDMA overlay. 41 dB Figure 4.2: Exact PDFs (solid) and Gaussian approximations (dashed) for Y\ and W, defined in equation (4.3), for a user at d\ = 0.75 and 9\ = 0. where rnyy and ayi are the mean and variance of Yt, which again depend on the users position, and must be found numerically. The variables in equations (4.2) and (4.3) can also be used to describe the effect that power control will have on the power levels at which users will be received at a base station. In practice, the transmitted powers of the mobiles assigned to a cell are adjusted so that the received powers of all of those users are the same at the base station. For both narrowband and CDMA systems, the users employ power control as a means of conserving battery life. It is wasteful for a mobile that has a strong path to the base station to transmit as much power as does a user with a severely attenuated path. In CDMA systems only, there is the additional motivation of alleviating the near-far problem amongst a systems own users, so that one strong user does not disrupt communication for all the rest. Consider again a user located in cell 1 at a position of d\ and 9\ from Figure 4.1. As mentioned before, the user will be assigned to the *th cell, i = 1,2,... ,19, for which G{ in equation (4.2) is maximum. If the minimum acceptable received power 13 where the notation N(m, a2) is used to denote a Gaussian random variable with mean m and variance a2. The probability of error can then be approximated as Pe Q 2 Eu V 2PkT} cos2(e,,.)Var(4i 4- N (2.5) where the Q-function is defined as Equation (2.5) then sim Pe^Q Q(X) = :S tO \/2n exp(-)du (2.6) 2Eb + N ^ V A cos2 (6k) Var (4) -i/2\ (2.7) where Pb = Eb/Tb is the BPSK users average power. The variance of Ik, with the averaging done only with respect to the data bits, is then found as LN Var (Ik) = Tc-Tk)f m i=LN LN + ^ h(ni\Tc Tk)h(m2Tc rfc) m i = L N m > ^ rn j W-H (2.8) X C-Ar.motl(r?xi,A')(ni i.N) Because the variance is dependent on the particular code sequence, it may take on a range of values. If the expectation is further taken over all code sequences, the second term of equation (2.8) drops out, and the variance is equal to the sum of samples of h2(t) spaced at intervals of Tr. Because h(t) was normalized such that the integral of h2(t) on (00,00) is unity, then this variance can be approximated by invoking 33 (a) (b) Figure 3.4: Transient behavior of mean-squared error using the RLS algorithm and a training sequence. (1) is for minimizing J = E[{Re(Zdi(*))}2], (2) is for minimizing J = E[|Zi di(i)|2], (3) is resulting absolute squared-error under criterion of (1). (a) 20 CDMA users, 0 narrowband users and 0 notches; (b) 20 CDMA users, 2 narrowband users with near-far ratio of 20 dB, and 2 notches. algorithm may have stability problems, the training was avoided for now by initial izing the tap weights to the Wiener solution and allowing an LMS algorithm to run with a known preamble for many bits. This allows the filter to reach a steady state solution which takes into account the excess mean-squared error which would result from an adaptive algorithm. For a step size of 0.1 /tip, where tip is the total input power, almost no difference was seen in comparison to the Wiener solution. For a step size of 0.2/tip, some difference was seen as shown in Figure 3.3. Thus if the filter is trained well, results close to the ideal ones with the Wiener solution can be achieved for a small enough step size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n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f RYHUn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n WLDO ORVV LQ SHUIRUPDQFH 7KH XVH RI QRWFKILOWHULQJ LQ WKH &'0$ XVHUVf WUDQVPLWWHUV LQ RUGHU WR DYRLG DFWLYH QDUURZEDQG XVHUV FDQ DOOHYLDWH WKH GHJUDGDWLRQ H[SHULHQFHG E\ WKH QDUURZEDQG V\VWHP ,Q WXUQ WKH HIIHFWV RI WKH QRWFKLQJ RQ WKH &'0$ V\VWHP LWVHOI DUH VHHQ WR EH TXLWH PRGHVW YLLL PAGE 9 2YHUOD\ KDV JUHDW SRWHQWLDO LQ D VLWXDWLRQ LQ ZKLFK D IUHTXHQF\ EDQG ZKLFK FXUn UHQWO\ SURYLGHV QDUURZEDQG FHOOXODU VHUYLFH LV GHVLJQDWHG WR SURYLGH &'0$ FHOOXODU VHUYLFH LQ WKH IXWXUH 7KLV WUDQVLWLRQ FDQ EH PDGH JUDGXDOO\ ZLWK WKH LPSOHPHQWDn WLRQ RI RYHUOD\ 6XFK D VFHQDULR LV LQYHVWLJDWHG LQ WKLV UHVHDUFK DQG SURPLVLQJ UHVXOWV DUH SUHVHQWHG 7KH XVH RI PXOWLFDUULHU &'0$ 0&&'0$f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n WDWLRQ DQG D VWURQJ PRWLYDWLRQ IRU LWV XVH DV D PHWKRG RI WUDQVLWLRQ IURP QDUURZEDQG VHUYLFH WR &'0$ VHUYLFH LV DUJXHG DV ZHOO ,; PAGE 10 &+$37(5 ,1752'8&7,21 ,Q WKLV FKDSWHU WKH FRQFHSW RI FRGHGLYLVLRQ PXOWLSOHDFFHVV &'0$f ZLOO EH SUHVHQWHG 7KH EHQHILWV WKDW &'0$ RIIHUV DV ZHOO DV VRPH RI WKH PDLQ ZRUNV RI UHVHDUFK LQ WKLV ILHOG ZLOO EH VXPPDUL]HG 7KLV FKDSWHU ZLOO DOVR LQWURGXFH &'0$ RYHUOD\ WKH IRFXV RI WKLV GLVVHUWDWLRQ ZKLFK LV WKH VKDULQJ RI IUHTXHQF\ VSHFWUXP E\ D &'0$ V\VWHP DQG D VSDUVHO\SRSXODWHG QDUURZEDQG V\VWHP $Q RYHUYLHZ RI WKH GLVVHUWDWLRQ ZLOO FRQFOXGH WKLV LQWURGXFWRU\ FKDSWHU &RGH'LYLVLRQ 0XOWLSOH$FFHVV &'0$ LV DQ HPHUJLQJ WHFKQRORJ\ ZKLFK HPSOR\V VSUHDGVSHFWUXP 66f VLJQDOLQJ WKDW LV WKH LQWHQWLRQDO VSUHDGLQJ RI D GLJLWDO VLJQDO RYHU D EDQGZLGWK WKDW LV PXFK JUHDWHU WKDQ LWV LQIRUPDWLRQ EDQGZLGWK W\SLFDOO\ RQ WKH RUGHU RI D IROG LQFUHDVH RU PRUH $OWKRXJK VSUHDGVSHFWUXP V\VWHPV DUH JHQHUDOO\ GLYLGHG LQWR WZR FDWHJRULHV GLUHFWVHTXHQFH '6f DQG IUHTXHQF\KRSSHFO )+f GHSHQGLQJ RQ WKH PHWKRG XVHG WR VSUHDG WKH XVHUfV EDQGZLGWK WKLV UHVHDUFK ZLOO RQO\ EH FRQFHUQHG ZLWK GLUHFWVHTXHQFH VSUHDGVSHFWUXP ZKLFK LV LOOXVWUDWHG LQ )LJXUH ,Q WKH GLUHFW VHTXHQFH PHWKRG D GLJLWDO VLJQDO LV PXOWLSOLHG LQ WKH WUDQVPLWWHU E\ D SHULRGLF ZDYHIRUP RI QDUURZ SXOVHV FDOOHG WKH VSUHDGLQJ FRGH ,Q WKH H[DPSOH VKRZQ DQ XQVSUHDG ELQDU\ SKDVHVKLIW NH\LQJ %36.f VLJQDO LV VKRZQ DW EDVHEDQG DV D VHULHV RI XQLWDPSOLWXGH VTXDUH SXOVHV RI ZLGWK 7 WKH ELW WLPH DQG LWV SRZHU VSHFWUDO GHQVLW\ 36'f KDV D QXOO EDQGZLGWK RI 7Â 7KH UHVXOWLQJ ZDYHIRUP DIWHU VSUHDGLQJ DOVR FRQVLVWV RI D VHULHV RI SXOVHV EXW WKH ZLGWK LV UHGXFHG WR 7F 7A1 ZKHUH 7F DQG 1 DUH UHIHUUHG WR DV WKH FKLS WLPH DQG WKH SURFHVVLQJ JDLQ UHVSHFWLYHO\ DQG ,9 LQ WKLV FDVH 7KH 36' RI WKH VSUHDG VLJQDO KDV WKH VDPH IRUP DV WKDW PAGE 11 Df 7LPH )LJXUH ,OOXVWUDWLRQ RI '666 ZDYHIRUPV DQG 36'V FKLSVELW Df 8QVSUHDG VLJQDO ZDYHIRUP Ef 36' RI XQVSUHDG VLJQDO Ff 6SUHDG VLJQDO ZDYHIRUP Gf 36' RI VSUHDG VLJQDO RI WKH XQVSUHDG VLJQDO EXW KDV D QXOO EDQGZLGWK RI O7F 17c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f FDSDELOLW\ PHDQLQJ WKDW LW LV GLIILFXOW IRU DQ DGYHUVDU\ WR UHFHLYH DQG GHPRGXODWH WKH VLJQDO ZLWKRXW NQRZLQJ WKH VSUHDGLQJ FRGH )LQDOO\ EHFDXVH WKH SRZHU LV VSUHDG RYHU VXFK D ODUJH EDQGZLGWK WKH VSHFWUDO KHLJKW RI WKH VSUHDGVSHFWUXP VLJQDO LV UHGXFHG VLJQLILFDQWO\ SRVVLEO\ WR WKH SRLQW ZKHUH DQ DGYHUVDU\ ZRXOG QRW EH DEOH WR GLVWLQJXLVK LW IURP WKH FKDQQHO PAGE 12 QRLVH DQG KHQFH ZRXOG QRW HYHQ NQRZ WKDW D FRPPXQLFDWLRQ LV WDNLQJ SODFH 7KLV LV UHIHUUHG WR DV ORZ SUREDELOLW\ RI GHWHFWLRQ /3'f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n DEOH LQ DV\QFKURQRXV FRPPXQLFDWLRQ V\VWHPV 7KH SUREOHP RI FKRRVLQJ JRRG FRGH VHTXHQFHV KDV EHHQ VWXGLHG DQG D JRRG VXPPDU\ RI PDQ\ RI WKH PDMRU UHVXOWV LV IRXQG LQ 6DUZDWH DQG 3XUVOH\ >@ &'0$ LV D YHU\ SURPLVLQJ WHFKQRORJ\ IRU VHYHUDO UHDVRQV 0RVW LPSRUWDQWO\ LQ D FHOOXODU VFHQDULR WKHUH LV WKH SRWHQWLDO IRU D PDQ\IROG LQFUHDVH LQ XVHU FDSDFLW\ RYHU WUDGLWLRQDO IUHTXHQF\GLYLVLRQ PXOWLSOHDFFHVV )'0$f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f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n QHQWLDOO\ ZLWK WKH QXPEHU RI XVHUV UHQGHULQJ LW LQIHDVLEOH LQ SUDFWLFH 7KH RSWLPXP GHWHFWRU LV YHU\ LPSRUWDQW KRZHYHU EHFDXVH DW WKH WLPH RI LWV LQFHSWLRQ FRPPRQ VHQWLPHQW ZDV WKDW WKH QHDUIDU SUREOHP ZDV DQ LQKHUHQW VKRUWFRPLQJ RI &'0$ WKDW FRXOG QRW EH RYHUFRPH 7KH H[LVWHQFH RI WKLV QHDUIDU UHVLVWDQW GHWHFWRU VHUYHG DV PRn WLYDWLRQ IRU WKH GHYHORSPHQW RI LPSOHPHQWDEOH UHFHLYHUV WKDW FRXOG RXWSHUIRUP WKH FRQYHQWLRQDO PDWFKHGILOWHU UHFHLYHU %HFDXVH WKH RSWLPXP GHWHFWRU KDV VXFK D KLJK FRPSOH[LW\ DQG UHTXLUHV D ODUJH DPRXQW RI VLGH LQIRUPDWLRQ DERXW DOO RI WKH &'0$ XVHUV VHYHUDO VXERSWLPXP QHDU IDU UHVLVWDQW UHFHLYHUV KDYH EHHQ GHYHORSHG ZKLFK KDYH PRUH PDQDJHDEOH FRPSOH[LWLHV $ IHZ QRWHZRUWOW\ UHFHLYHUV LQFOXGH WKH GHFRUUHODWRU >@ DQG WKH PXOWLVWDJH GHWHFWRU >@ 7KH PLQLPXP PHDQVTXDUHG HUURU 006(f UHFHLYHU >@ KDV UHFHLYHG D JUHDW GHDO RI DWWHQWLRQ GXULQJ WKH SDVW IHZ \HDUV DV LW DOVR RIIHUV QHDUIDU UHVLVWDQFH DQG LV YHU\ VLPSOH WR LPSOHPHQW 7KH 006( DOVR KDV DQ LQKHUHQW UHVLVWDQFH WR QDUURZEDQG LQWHUIHUHQFH 1%,f D IHDWXUH WKDW LV YHU\ EHQHILFLDO LQ DQ RYHUOD\ VFHQDULR ZKLFK ZLOO PAGE 14 EH GLVFXVVHG LQ WKH QH[W VHFWLRQ ,W ZLOO DOVR EH VHHQ WKDW WKH 006( FDQ UHMHFW LQWHUV\PERO LQWHUIHUHQFH ,6,f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n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f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fV SHUIRUPDQFH WR WKH SRLQW RI LQn RSHUDELOLW\ 2QH SRVVLEOH VLWXDWLRQ RFFXUV ZKHQ D &'0$ V\VWHP LV RYHUODLG RQ D VSDUVHO\SRSXODWHG QDUURZEDQG V\VWHP WKDW LV DOUHDG\ LQ RSHUDWLRQ 7KH GHVLJQHUV RI WKH &'0$ V\VWHP PXVW ZRUN DURXQG WKH QDUURZEDQG V\VWHP DQG VKRXOG QRW H[SHFW WKH GHVLJQHUV RI WKH QDUURZEDQG V\VWHP WR PDNH VLJQLILFDQW PRGLILFDWLRQV WR WKHLU GHn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fV VLJQDO D URRW UDLVHG FRVLQH SXOVH VKDSH ZLOO EH XVHG LQ WKH WUDQVPLWWHU DQG LQ WKH PDWFKHGILOWHU UHFHLYHU $VVXPLQJ HTXDO FDUULHU IUHTXHQFLHV IRU WKH QDUURZEDQG DQG &'0$ V\VWHPV PAGE 20 WKH UHFHLYHG VLJQDO DW WKH LQSXW WR WKH %36. PDWFKHG ILOWHU DIWHU GRZQFRQYHUVLRQ LV 22 O? 2* UWf \(E GE0W L7Ef < 9P < GNM&NLW a L7E a Qf FRVNf QWf LfÂ§fÂ§RR $ O c fÂ§ f ZKHUH (E LV WKH DYHUDJH HQHUJ\SHUELW RI WKH %36. V\VWHP 7E LV WKH ELW WLPH IRU ERWK V\VWHPV DQG GEÂ DQG GNÂ DUH WKH "nWOL GDWD ELWV RI WKH %36. XVHUfV DQG WKH $WK &'0$ XVHUfV GDWD VWUHDP DQG KWf LV WKH %36. XVHUfV URRWUDLVHG FRVLQH SXOVH VKDSH ZKLFK LV QRUPDOL]HG VR WKDW WKH LQWHJUDO RI KWf RQ fÂ§f LWV HQHUJ\ LV XQLW\ $OVR LV WKH QXPEHU RI &'0$ XVHUV 3N LV WKH DYHUDJH SRZHU RI WKH NWK &'0$ XVHU FNWf LV WKH NWK XVHUfV VSUHDGLQJ ZDYHIRUP FRQVLVWLQJ RI XQLWDPSOLWXGH SRVLWLYH DQG QHJDWLYH SXOVHV RI GXUDWLRQ 7F WKH FKLS WLPH DQG 1 LV WKH SURFHVVLQJ JDLQ RU WKH QXPEHU RI FKLSVELW 7KH NWK &'0$ XVHUfV GHOD\ DQG SKDVH UN DQG N DUH WDNHQ DV FRQVWDQW WKURXJKRXW WUDQVPLVVLRQ DV WKH FKDQQHO LV DVVXPHG WR EH QRQIDGLQJ DQG VWDWLRQDU\ $OVR QWf LV D ZKLWH *DXVVLDQ QRLVH SURFHVV ZLWK VSHFWUDO KHLJKW 1T 7KH MWK VDPSOH RI WKH RXWSXW RI WKH PDWFKHG ILOWHU LV $n =M \(EGEÂ 93N7F FRVNf,NMf 1M f IF L ZKHUH 1M LV D ]HURPHDQ *DXVVLDQ UDQGRP YDULDEOH ZLWK YDULDQFH $UR DQG WKH UDQGRP SURFHVV ,NMf DVVRFLDWHG ZLWK WKH FRQWULEXWLRQ IURP WKH NWK &'0$ XVHU LQ WKH MWK ELW LQWHUYDO LV JLYHQ E\ /1 ,N^Mf fÂ§ n\ @ GNAMAZAA&NnOQRG7Q1fK^"7O7F 7Nf f P fÂ§/1 DQG >]M LV WKH IORRU IXQFWLRQ ,W LV DVVXPHG LQ WKLV H[SUHVVLRQ WKDW KWf LV QHJOLJLEOH IRU _L_ /7EÂ‘ 7KXV WKH LQWHUIHUHQFH FRQWULEXWLRQ IURP HDFK &'0$ XVHU LV WKH VXP RI VDPSOHV RI WKH URRWUDLVHG FRVLQH SXOVH ZHLJKWHG E\ ERWK D GDWD ELW DQG D VSUHDGLQJ FKLS ,W DSSHDUV WKDW WKH FHQWUDO OLPLW WKHRUHP PD\ EH LQYRNHG WR DSSUR[LPDWH ,N DV D *DXVVLDQ SURFHVV VLQFH LW LV WKH VXP RI PDQ\ UDQGRP YDULDEOHV $OWKRXJK WKLV PAGE 21 Ef )LJXUH +LVWRJUDPV IRU &'0$ LQWHUIHUHQFH FDXVHG WR D %36. XVHU Df 6LQJOH &'0$ XVHU %36. XVHU HPSOR\LQJ URRWUDLVHG FRVLQH SXOVHV Ef 6LQJOH &'0$ XVHU %36. XVHU HPSOR\LQJ VTXDUH SXOVHV Ff &'0$ XVHUV %36. XVHU HPSOR\LQJ URRW UDLVHG FRVLQH SXOVHV Gf &'0$ XVHUV %36. XVHU HPSOR\LQJ VTXDUH SXOVHV DSSUR[LPDWLRQ GRHV QRW KROG IRU HDFK LQGLYLGXDO WKH WRWDO &'0$ LQWHUIHUHQFH LV WKH VXP RI RI WKHVH WHUPV DQG IRU UHODWLYHO\ VPDOO YDOXHV RI WKH GLVWULEXWLRQ RI WKH VXP RI WKH r WHUPV GRHV DSSURDFK *DXVVLDQ $Q H[DPSOH LV VKRZQ LQ )LJXUH 7KH KLVWRJUDPV IRU D VLQJOH &'0$ XVHUf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f LV RI ]HUR PHDQ WKH GHFLVLRQ VWDWLVWLF IURP HTXDWLRQ f FDQ EH UHZULWWHQ DV =M \(EGE1 A Â 3;\ FRVf9DUf 1 @ f PAGE 22 ZKHUH WKH QRWDWLRQ 1P Df LV XVHG WR GHQRWH D *DXVVLDQ UDQGRP YDULDEOH ZLWK PHDQ P DQG YDULDQFH D 7KH SUREDELOLW\ RI HUURU FDQ WKHQ EH DSSUR[LPDWHG DV 3H m 4 (X 9 3N7` FRVHf9DUL 1f f ZKHUH WKH 4IXQFWLRQ LV GHILQHG DV (TXDWLRQ f WKHQ VLP 3HA4 4;f 6 W2 ?Q H[SfÂ§fGX f (E 1 A 9 $ FRV Nf 9DU f L? f ZKHUH 3E (E7E LV WKH %36. XVHUfV DYHUDJH SRZHU 7KH YDULDQFH RI ,N ZLWK WKH DYHUDJLQJ GRQH RQO\ ZLWK UHVSHFW WR WKH GDWD ELWV LV WKHQ IRXQG DV /1 9DU ,Nf 7F7NfI P L fÂ§/1 /1 A KQL?7F 7NfKP7F UIFf P L fÂ§ / 1 P A UQ M :+ f ; &$UPRWOU"[L$nfQL L1f %HFDXVH WKH YDULDQFH LV GHSHQGHQW RQ WKH SDUWLFXODU FRGH VHTXHQFH LW PD\ WDNH RQ D UDQJH RI YDOXHV ,I WKH H[SHFWDWLRQ LV IXUWKHU WDNHQ RYHU DOO FRGH VHTXHQFHV WKH VHFRQG WHUP RI HTXDWLRQ f GURSV RXW DQG WKH YDULDQFH LV HTXDO WR WKH VXP RI VDPSOHV RI KWf VSDFHG DW LQWHUYDOV RI 7U %HFDXVH KWf ZDV QRUPDOL]HG VXFK WKDW WKH LQWHJUDO RI KWf RQ fÂ§f LV XQLW\ WKHQ WKLV YDULDQFH FDQ EH DSSUR[LPDWHG E\ LQYRNLQJ PAGE 23 WKH GHILQLWLRQ RI D 5LHPDQ VXP ZKLFK VWDWHV WKDW Wf FO[ -L ED 0 f D 7KXV IRU VXIILFLHQWO\ ODUJH 1 WKH YDULDQFH LV DSSUR[LPDWHO\ O7F &RPELQLQJ WKLV ZLWK HTXDWLRQ f JLYHV f 7KH VDPH H[SUHVVLRQ IRU WKH SUREDELOLW\ RI HUURU ZDV REWDLQHG E\ 'DYLV >@ DQG ZDV DOVR GHULYHG E\ 3LFNKROW] HW DO >@ IRU WKH VLQJOHXVHU FDVH ZKHQ VTXDUH SXOVHV DUH XVHG LQ WKH WUDQVPLWWHU DQG WKH FRUUHVSRQGLQJ PDWFKHG ILOWHU LV DQ LQWHJUDWHDQG GXPS $Q DSSUR[LPDWLRQ IRU WKH FDVH RI VTXDUH SXOVHV ZDV DOVR JLYHQ DV >@ f 7KLV H[SUHVVLRQ LV PXFK HDVLHU WR HYDOXDWH WKDQ LV HTXDWLRQ f ZKLFK UHTXLUHV D .GLPHQVLRQDO QXPHULFDO LQWHJUDWLRQ RYHU WKH &'0$ XVHUVf UDQGRP SKDVHV D YHU\ WLPHFRQVXPLQJ SURFHVV HYHQ IRU D VPDOO QXPEHU RI XVHUV 7KH QXPHULFDO LQWHJUDWLRQ ZDV SHUIRUPHG DQG D SORW RI WKH SUREDELOLW\ RI HUURU YV (A1T IRU WKH %36. V\VWHP LV VKRZQ LQ )LJXUH IRU WKH FDVH RI &'0$ XVHUV RI HTXDO SRZHU UDWLR 3FSf DQG D SURFHVVLQJ JDLQ RI FKLSVELW 7KH GLIIHUHQW FXUYHV UHSUHVHQW GLIIHUHQW YDOXHV RI 3F3f DQG WKH FDVH RI QR RYHUOD\ LV VKRZQ IRU UHIHUHQFH 7KH DSSUR[LPDWLRQ RI HTXDWLRQ f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f )LJXUH 3UREDELOLW\ RI HUURU 3H RI %36. XVHU YV (?c1T ZLWK &'0$ RYHUOD\ RI XVHUV ZLWK FKLSVELW 6ROLG FXUYH LV IRU QXPHULFDO LQWHJUDWLRQ GDVKHG FXUYH LV IRU DSSUR[LPDWLRQ RI HTXDWLRQ f DQG WKH QDUURZEDQG V\VWHP 7KHVH UHVXOWV FDVW FRQVLGHUDEOH GRXEW RQ WKH IHDVLELOLW\ RI &'0$ RYHUOD\ ,W PD\ QRW HYHQ EH ZRUWKZKLOH WR LPSOHPHQW D &'0$ V\VWHP ZLWK D ORDGLQJ RI .1 DQG WKHVH UHVXOWV LQGLFDWH WKDW WKLV YHU\ OLJKWO\ORDGHG V\VWHP VWLOO FDXVHV VHYHUH GHJUDGDWLRQ WR WKH QDUURZEDQG V\VWHP 2EYLRXVO\ VRPHWKLQJ PXVW EH GRQH WR OHVVHQ WKH HIIHFW RI WKH &n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n TXHQFH ZKLFK FRQVLVWV RI SXOVHV PRVW OLNHO\ VTXDUH ZKRVH DPSOLWXGHV DUH QR ORQJHU FRQVWUDLQHG WR WKH YDOXHV s 7KH ILOWHUHG FRGH VHTXHQFH LV WKHQ PRGXODWHG ZLWK WKH GDWD ZDYHIRUP DQG FRQYHUWHG XS WR EDQGSDVV DQG WUDQVPLWWHG MXVW DV WKH XQILOWHUHG FRGH VHTXHQFH ZRXOG EH )RXU PHWKRGV RI ILOWHULQJ WKH FRGH VHTXHQFH ZLOO EH H[DPLQHG DQG ZLOO EH FRPSDUHG EDVHG RQ WKUHH FULWHULD ZKLFK ZLOO EH HODERUDWHG RQ LQ WKH QH[W VHFWLRQ WKH JDLQ UHDOL]HG LQ WKH %36. V\VWHP WKH HIIHFW RQ WKH 36' RI WKH &'0$ VLJQDO DQG WKH HIIHFW RQ WKH &'0$ FRGH VHTXHQFH )LOWHULQJ 3HUIRUPDQFH &ULWHULD ,Q WKLV VHFWLRQ WKUHH FULWHULD ZLOO EH SUHVHQWHG ZKLFK ZLOO PHDVXUH WKH HIIHFWLYHQHVV RI D JLYHQ PHWKRG RI QRWFKLQJ DW WKH &'0$ WUDQVPLWWHU 7KH JDLQ RI WKH %36. V\VWHP WKH HIIHFW RQ WKH 36' RI WKH &'0$ VLJQDO DQG WKH HIIHFW RQ WKH &'0$ FRGH VHTXHQFH ZLOO QRZ EH H[SODLQHG LQ GHWDLO *DLQ RI WKH %36. 6\VWHP 7KH SUREDELOLW\ RI HUURU RI WKH %36. V\VWHP JLYHQ LQ HTXDWLRQ f GHSHQGV RQ WKH YDULDQFH RI ,N JLYHQ LQ HTXDWLRQ f ZKLFK LQ WXUQ GHSHQGV RQ WKH XVHUVf FRGH VHTXHQFHV $V ZLOO EH VHHQ ODWHU WZR RI WKH IRXU ILOWHULQJ PHWKRGV XVHG LQ WKLV ZRUN UHVXOW LQ ,6, DQG KHQFH WKH H[SUHVVLRQ IRU ,N PXVW EH PRGLILHG LQ WKRVH FDVHV WR LQFOXGH WKH DSSURSULDWH FRQWULEXWLRQV ZKLFK UHVXOW IURP WKH VSLOORYHU 7KH HYDOXDWLRQ RI WKH YDULDQFH RI W\ UHTXLUHV VLPXODWLRQ ,W LV WKHQ XVHG LQ HTXDn WLRQ f WR JHW WKH SUREDELOLW\ RI HUURU %HFDXVH WKH YDULDQFH LV PXOWLSOLHG E\ WKH &'0$WR%36. SRZHU UDWLR 3N3Ef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n PLWWHU ,I WKH &'0$ XVHUV DQG WKH %36. XVHU DUH QRW SRZHUFRQWUROOHG E\ WKH VDPH PHFKDQLVP WKHQ ODUJH QHDUIDU UDWLRV DUH YHU\ SRVVLEOH (IIHFW RQ WKH 36' RI WKH &n'0$ 6LJQDO 7R VHH KRZ FHUWDLQ ILOWHUV SHUIRUP LQ QRWFKLQJ WKH &'0$ VLJQDO FRQVLGHU WKH 36' RI D &'0$ XVHUfV VLJQDO ZKLFK LV JLYHQ DV 6VV8f ?4If? 7E $nO TPL b _&f_ f ZKHUH 4If LV WKH )RXULHU WUDQVIRUP RI WKH FKLS SXOVH VKDSH Ff LV WKH QWK FKLS DQG WKH QXPEHU RI FKLSV LQ WKH VSUHDGLQJ VHTXHQFH LV 1 ZKLFK PD\ EH JUHDWHU WKDQ WKH SURFHVVLQJ JDLQ 1 )URP WKLV H[SUHVVLRQ LW LV FOHDU WKDW WKH HIIHFWV RI ILOWHULQJ ZLOO DSSHDU RQO\ LQ WKH ULJKWPRVW PDJQLWXGHVTXDUHG WHUP _&f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fÂ§ 7KH %36. XVHU DQG DQ LQWHUIHULQJ &'0$ XVHU ZHUH DVVXPHG WR KDYH WKH VDPH GDWD UDWH 7Â ZLWK WKH &'0$ XVHU HPSOR\LQJ D UDQGRP FRGH VHTXHQFH RI OHQJWK 1 FKLSVELW RU D ILOWHUHG YHUVLRQ RI WKLV VHTXHQFH DQG D GHOD\ FKRVHQ IURP D XQLIRUP GLVWULEXWLRQ RQ 7&f 7KHQ WULDOV ZHUH UXQ HDFK RI ZKLFK JHQHUDWHG D VDPSOH RI r ZKHQ ILOWHULQJ ZDV HPSOR\HG DQG D VDPSOH ZKHQ QR ILOWHULQJ ZDV HPSOR\HG 7KH UDWLR RI VDPSOH YDULDQFHV RI IRU WKH ILOWHUHG DQG XQILOWHUHG FDVHV ZDV WKHQ FRPSXWHG 7KLV JDLQ DQG D SORW RI _&f_ LQ HTXDWLRQ f ZLOO EH SUHVHQWHG IRU HDFK ILOWHULQJ PHWKRG $ SORW RI WKH ILOWHUHG FRGH VHTXHQFH ZLOO EH SUHVHQWHG IRU WKH WZR PHWKRGV LQ ZKLFK ,6, UHVXOWV (LJHQYHFWRU )LOWHULQJ 7KLV PHWKRG ZDV SURSRVHG E\ 'DYLV >@ DQG ZLOO EH VXPPDUL]HG EULHIO\ 7KH FRGH VHTXHQFH LV FORFNHG WKURXJK D ILQLWH LPSXOVH UHVSRQVH ),5f ILOWHU ZLWK 0 WDSV DQG WKH ILOWHUHG FRGH VHTXHQFH ZLOO EH WDNHQ IURP WKH GHOD\HG RXWSXW 7KH ZHLJKWV DUH FKRVHQ VR DV WR PLQLPL]H WKH HIIHFWV RI WKH &'0$ LQWHUIHUHQFH RQ WKH %36. V\VWHP )RU WKH FDVH ZKHQ WKH %36. V\VWHP XVHV VTXDUH SXOVHV QRW URRW UDLVHG FRVLQH SXOVHV WKH YDULDQFH RI WKH &'0$ V\VWHPfV FRQWULEXWLRQ WR WKH %36. V\VWHPfV PDWFKHG ILOWHU ZDV GHULYHG ZLWK DQG ZLWKRXW WUDQVPLWWHU ILOWHULQJ >@ 7KH WZR TXDQWLWLHV ZHUH IRXQG WR GLIIHU E\ D PXOWLSOLFDWLYH IDFWRU RI D%D7 ZKHUH D LV D PAGE 28 7DEOH *DLQV LQ FO% IRU %36. V\VWHP IRU HLJHQYHFWRU DQG QXOO ILOWHUV 7DSV (LJHQYHFWRU 1XOO URZ YHFWRU RI WDS ZHLJKWV % LV D PDWUL[ ZKRVH LMfWK HOHPHQW LV JLYHQ E\ Â‘RU f ? 1 a ,r -, FrVr @f$X7Ff %EMf A ZKHUH 1 LV WKH SURFHVVLQJ JDLQ DQG $X! LV WKH GLIIHUHQFH RI WKH WZR V\VWHPVf FDUULHU IUHTXHQFLHV ,I WKH TXDQWLW\ D%D7 LV PLQLPL]HG WKHQ WKH HIIHFW RI WKH &'0$ LQWHUn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f_ IRU DQ HLJHQYHFWRU ILOWHU ZLWK 0 DQG $FQ Uf VKRZV WKDW WKH ILOWHU VHYHUHO\ GLVWRUWV WKH &'0$ VLJQDO 7KH GHVLUHG UHn VSRQVH VKRXOG KDYH D GHHS QXOO RI ILQLWH ZLGWK DURXQG WKH GHVLUHG IUHTXHQF\ DQG DV OLWWOH GLVWRUWLRQ DV SRVVLEOH LQ WKH SDVVEDQG 7KLV ILOWHU GRHV QRW HYHQ KDYH D SDUDPHn WHU WR FRQWURO WKH ZLGWK RI WKH QRWFK $OWKRXJK LW GRHV SURYLGH D SUHWW\ VL]DEOH JDLQ PAGE 29 RI G% IRU WKH %36. V\VWHP WKH HLJHQYHFWRU ILOWHU GRHV QRW VHHP WR EH D SODXVLEOH RSWLRQ EDVHG RQ LWV GHWULPHQWDO HIIHFW RQ WKH &'0$ VLJQDO 1XOO )LOWHULQJ ,Q WKH VHFRQG PHWKRG SURSRVHG E\ 'DYLV >@ WKH ILOWHU SXWV D VSHFWUDO QXOO DW WKH %36. V\VWHPfV FDUULHU IUHTXHQF\ 7KH WDS ZHLJKWV D DUH JLYHQ DV 3R 3P fÂ§ FRV Qf $X7F @ 0 VLQ$L Of$Z7Ff VLQ$XnfFf LQ s s $0 f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f? IRU WKH QXOO ILOWHU ZLWK 0 DQG $X! Wf LV DOVR VKRZQ LQ )LJXUH $JDLQ WKH &'0$ VLJQDO LV VRPHZKDW GLVWRUWHG DQG WKHUH LV RQO\ D PLQRU QRWFK DW WKH GHVLUHG QRWFKHG IUHTXHQF\ DQG QR SDUDPHWHU WR FRQWURO WKLV ZLGWK H[SOLFLWO\ %DVHG RQ WKLV WKH QXOO ILOWHULQJ PHWKRG GRHV QRW VHHP OLNH D SODXVLEOH RSWLRQ HLWKHU %XWWHUZRUWK )LOWHULQJ :H QH[W FRQVLGHU WKH XVH RI D GLJLWDO %XWWHUZRUWK QRWFK ILOWHU ZKLFK LV DQ LQILQLWH LPSXOVH UHVSRQVH +5f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f (LJHQYHFWRU ILOWHU 0 fÂ§ WDSV RQ HDFK VLFN ff 1XOO ILOWHU 0 fÂ§ WDSV RQ HDFK VLGH Ff %XWWHUZRUWK ILOWHU WK RUGHU G% %?< Gf ')7EDVHG ILOWHULQJ QR ]HURSDGGLQJ Hf ')7EDVHG ILOWHULQJ ]HURSDGGHG WR ELWV 6ROLG FXUYHV DUH IRU XQILOWHUHG GDVKHG FXUYHV DUH IRU ILOWHUHG PAGE 31 7DEOH *DLQV LQ ,% IRU %36. V\VWHP IRU %XWWHUZRUWK ILOWHU %: 1 1 1 *DLQ G%f 7K 7E 7E Df )LJXUH 5HDO SDUW RI ILOWHUHG FRGH VHTXHQFHV 3URFHVVLQJ JDLQ LV FKLSVELW Df %XWWHUZRUWK ILOWHU WK RUGHU G% %: Ef ')7EDVHG ILOWHULQJ ]HURSDGGHG WR ELWV ILOWHU 7KH JDLQV UHDOL]HG E\ WKH %XWWHUZRUWK ILOWHU DUH VKRZQ LQ 7DEOH IRU DQ WK RUGHU ILOWHU ZLWK G% EDQGZLGWKV RI $& 1 DQG $& 7KH SORW RI _&f_ LQ )LJXUH IRU D EDQGZLGWK RI 1 VKRZV WKDW WKH QRWFK LV YHU\ SURQRXQFHG DQG WKDW WKH VLJQDO LQ WKH SDVVEDQG LV QRW GLVWRUWHG $V VKRZQ LQ )LJXUH DOVR IRU D ILOWHU RI EDQGZLGWK -9 WKH FRGH VHTXHQFH FRQWLQXHV ZHOO EH\RQG RQH ELW WLPH EXW LV QHJOLJLEOH EH\RQG DERXW ELWV :KHQ D UHFHLYHU LV XVHG WKDW FDQ KDQGOH VHTXHQFHV ZLWK VSLOORYHU WKH %XWWHUn ZRUWK ILOWHU VHHPV OLNH D YHU\ JRRG RSWLRQ ,W JLYHV WKH %36. V\VWHP D ODUJH JDLQ ZLWKRXW GLVWRUWLQJ WKH &'0$ VLJQDOfV IUHTXHQF\ UHVSRQVH DQG ZLWKRXW FDXVLQJ H[n FHVVLYH VSLOORYHU ')7%DVHG )LOWHULQJ $QRWKHU ZD\ WR ILOWHU WKH FRGH VHTXHQFH LV WR XVH D 'LVFUHWH )RXULHU 7UDQVIRUP ')7f ZKLFK SURYLGHV D UHSUHVHQWDWLRQ RI WKH FRGH VHTXHQFH LQ WKH IUHTXHQF\ GRPDLQ %HFDXVH WKH LQYHUVH ')7 LV WKH VXP RI GLVFUHWH IUHTXHQF\ FRPSRQHQWV ZHLJKWHG E\ WKH FRUUHVSRQGLQJ YDOXH RI WKH ')7 FRHIILFLHQW QRWFK ILOWHULQJ RI WKH FRGH VLJQDO FDQ A 0 H Z 2QH ELW 7E r Ef +L IS 2QH ELW PAGE 32 7DEOH *DLQV LQ FO% IRU %36. V\VWHP IRU ')7EDVHG ILOWHU /HQJWK ELWVf *DLQ FO%f EH SHUIRUPHG E\ IRUFLQJ WR ]HUR WKRVH YDOXHV RI WKH ')7 WKDW FRUUHVSRQG WR IUHTXHQn FLHV ZLWKLQ WKH GHVLUHG QRWFKLQJ UDQJH DQG WDNLQJ DQ LQYHUVH ')7 7KH IUHTXHQFLHV UHSUHVHQWHG DUH VSDFHG E\ 1 LQ WKH GLJLWDO IUHTXHQF\ GRPDLQ FRUUHVSRQGLQJ WR D IUHTXHQF\ VSDFLQJ RI 7Â LQ WKH DQDORJ GRPDLQ 7KXV LI D QRWFK RI ZLGWK 7LV GHVLUHG WKHUH DUH WKUHH ')7 FRHIILFLHQWV WZR VXUURXQGLQJ WKH UDQJH DQG RQH LQ WKH PLGGOH RI LW ZKLFK VKRXOG ,UH VHW WR ]HUR $ SORW RI _&f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f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n LQHG :H ZLOO FRQVLGHU D $nXVHU &'0$ V\VWHP ZKLFK XVHV FRGH VHTXHQFHV RI OHQJWK 1 FKLSVELW WKDW DUH ILOWHUHG WR JLYH VHTXHQFHV RI OHQJWK 1 ZKLFK ZLOO EH JUHDWHU WKDQ 1 7KH NWK XVHUfV ILOWHUHG VHTXHQFH ZLOO EH GHQRWHG DV Fr FAR &IFL FN IWBLf7 ZKHUH &IFQ LV WKH QWK ILOWHUHG FKLS 7KHQ Fr LV SDUWLWLRQHG LQWR / ?11@ VHTXHQFHV HDFK RI OHQJWK 1 JLYLQJ f &N FLFLf F,OLf7 &NO ^&N-1 L FN-1? Â‘ Â‘ Â‘ &NO11Of7 ZLWK / fÂ§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f LQ )LJXUH LV JLYHQ E\ FR +Wf YP OAFAI L7E a UNf H[SM2Nf N LfÂ§fÂ§RF ?\Âf Q>Wf H[S^MXFWf M^Wf H[SaMX!FWf ZKHUH 3N DQG N DUH WKH IFWK XVHUfV DYHUDJH SRZHU DQG SKDVH DQG GNLf LV WKH LWK GDWD ELW RI WKH NWK XVHU $OVR LQ WKLV HTXDWLRQ QWf LV D ZKLWH *DXVVLDQ QRLVH SURFHVV ZLWK VSHFWUDO KHLJKW 1T 7KH QRLVH SURFHVV UHVXOWLQJ IURP D QDUURZEDQG XVHU LV JLYHQ E\ MWf ,WV 36' LV PRGHOHG DV D VTXDUH SXOVH FHQWHUHG DW DQ RIIVHW RI $FM IURP WKH &'0$ FDUULHU IUHTXHQF\ DQG KDYLQJ ZLGWK S ZKHUH S LV D QXPEHU EHWZHHQ DQG PRVW OLNHO\ FORVH WR WKDW UHSUHVHQWV WKH IUDFWLRQ RI WKH EDQGZLGWK QXOOWRQXOOf WKDW WKH QDUURZEDQG XVHU RFFXSLHV 7KH UHFHLYHU IRUPV D EDQN RI VDPSOHV E\ LQWHJUDWLQJ UWf RYHU FKLSOHQJWK LQWHUYDOV DV VKRZQ LQ )LJXUH 7KH YHFWRU RI PDWFKHGILOWHU VDPSOHV LV JLYHQ E\ /fÂ§ XWf AH[SMKfULYn fÂ§ OfF?M Qrf MLf R L / H[SM PAGE 37 7KH F\FOLF VKLIWV RI WKH WK SDUWLWLRQ RI WKH $ WK &'0$ XVHUfV FRGH VHTXHQFH DUH JLYHQ E\ INO fÂ§ f &A\ &LFM1L &NO1SNOf7 &NnO1 &NnO1 f Â‘ f &NO1SNf7 f JIF fÂ§ tNf&N-1SN &N-1LLN O Â‘ f f &NO1 fÂ§ O f &IF93OL &NO13N Â‘ Â‘ Â‘ &NOLY f7 ZKHUH WKH $WK XVHUfV GHOD\ KDV EHHQ ZULWWHQ DV Ur Sr7F ZLWK SN DQ LQWHJHU LQ WKH UDQJH 1 fÂ§ f DQG D QRQLQWHJHU LQ WKH UDQJH f $OVR LQ HTXDWLRQ f Q]f LV DQ 1OHQJWK YHFWRU RI LQGHSHQGHQW FRPSOH[ *DXVVLDQ UDQGRP YDULDEOHV ZLWK WKH UHDO DQG WKH LPDJLQDU\ SDUWV LQGHSHQGHQW RI HDFK RWKHU DQG HDFK KDYLQJ ]HUR PHDQ DQG YDULDQFH D 1(cM1Rf 7KH YHFWRU MLf FRQVLVWV RI FRPSOH[ QDUURZEDQG LQWHUIHUHQFH VDPSOHV ZLWK WKH UHDO DQG LPDJLQDU\ SDUWV LQGHSHQGHQW RI HDFK RWKHU 7KH m MfWK HOHPHQW RI WKH PDWUL[ UHSUHVHQWLQJ WKH FRUUHODWLRQ EHWZHHQ WKH UHDO SDUWV RI WZR VDPSOHV LV JLYHQ E\ 5 i6DSUÂMffFRV$Xf7FÂ Mff f L ? ZKHUH 3M LV WKH QDUURZEDQG XVHUfV DYHUDJH SRZHU DQG WKH VDPH UHVXOW KROGV IRU WKH LPDJLQDU\ SDUWV RI WKH WK DQG MnWOL VDPSOHV $OVR LQ HTXDWLRQ 6DDUf VLQDfD 'HQRWLQJ WKH WDS ZHLJKWV DV Z LXT LXc [XALfU WKH RXWSXW RI WKH WDSSHG GHOD\ OLQH LV JLYHQ DV =W fÂ§ ZXnf ZKLFK LV JHQHUDOO\ D FRPSOH[ QXPEHU 7KH UHFHLYHU WKHQ FRPSDUHV = ZLWK WKH GHVLUHG RXWSXW GLLf 7KH WDS ZHLJKWV Z ZKLFK ZLOO PLQLPL]H WKH PHDQVTXDUHG HUURU EHWZHHQ WKH WZR (>_=Â fÂ§ GLÂf_@ DUH WKH VROXWLRQ WR WKH :LHQHU+RSI HTXDWLRQ 5Z S f ZKHUH 5 (>X]fXf@ DQG S (>GL"fXf@ FSR DUH WKH FRUUHODWLRQ PDWUL[ DQG VWHHULQJ YHFWRU UHVSHFWLYHO\ 7KH ELW GHFLVLRQ LV PDGH DV G?Lf VLJQ5H=ff PAGE 38 6LPXODWLRQ 5HVXOWV ,Q RUGHU WR JHW DQ LGHD RI KRZ ILOWHULQJ DIIHFWV WKH SHUIRUPDQFH RI WKH 006( UHFHLYHU DQ RYHUOD\ V\VWHP ZDV VLPXODWHG 7KH &'0$ XVHUV RSHUDWHG ZLWK ILOWHUHG UDQGRP VHTXHQFHV DQG D SURFHVVLQJ JDLQ RI 1 fÂ§ FKLSVELW 7KH YDOXH RI (E1R ZDV IL[HG DW G% 7KH SRZHUV RI WKH LQWHUIHULQJ &'0$ XVHUV ZHUH FKRVHQ IURP D ORJQRUPDO GLVWULEXWLRQ ZLWK D VWDQGDUG GHYLDWLRQ RI G% WR VLPXODWH SRZHU FRQWURO HUURU 7KH QRWFKILOWHULQJ ZDV GRQH XVLQJ D ')7EDVHG ILOWHU ZLWK ELWV RI ]HURSDGGLQJ 7KH QDUURZEDQG XVHUV ZHUH JHQHUDWHG E\ SXWWLQJ ZKLWH *DXVVLDQ QRLVH WKURXJK D %XWWHUZRUWK EDQGSDVV ILOWHU ZLWK D GLJLWDO G% EDQGZLGWK RI 7KHLU FHQWHU IUHTXHQFLHV ZHUH FKRVHQ WR IDOO ZLWKLQ WKH DQDORJ IUHTXHQF\ UDQJH fÂ§7F ÂQE 7Ff ZKLFK FRUUHVSRQGV WR WKH GLJLWDO IUHTXHQF\ UDQJH RI fÂ§ IX%G f DV WKH VDPSOLQJ UDWH LV 7& 7KHLU SRZHUV QHDUIDU UDWLRVf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n^3H&'0$ f f PAGE 39 )LJXUH 7ZRGLPHQVLRQDO FDSDFLW\ SORW IRU ')7EDVHG ILOWHULQJ ]HURSDGGHG WR ELWV 3URFHVVLQJ JDLQ LV FKLSVELW Df 006( UHFHLYHU PLQLPL]LQJ (>_= fÂ§ G[rf_@f Ef 006( UHFHLYHU PLQLPL]LQJ (>^5H=Â fÂ§ GL]ff`@ Ff PDWFKHG ILOWHU ,Q )LJXUH D WZRGLPHQVLRQDO FDSDFLW\ SORW LV VKRZQ IRU D V\VWHP ZKLFK HPSOR\V ')7EDVHG ILOWHULQJ ZLWK ]HURSDGGLQJ XS WR ELWV 7KH FXUYH ODEHOHG Df UHSUHVHQWV SRVVLEOH RSHUDWLQJ SRLQWV IRU WKH V\VWHP LQ WHUPV RI WKH QXPEHU RI &'0$ XVHUV DQG WKH QXPEHU RI QDUURZEDQG XVHUV WKDW FDQ VLPXOWDQHRXVO\ XVH WKH IUHTXHQF\ EDQG DQG VWLOO VDWLVI\ WKH &'0$ V\VWHPfV SHUIRUPDQFH FULWHULRQ JLYHQ LQ HTXDWLRQ f ,W VKRXOG EH QRWHG WKDW LQ WKLV UHFHLYHU WKH ELW GHFLVLRQ LV PDGH E\ ORRNLQJ RQO\ DW WKH UHDO SDUW RI WKH ILOWHU RXWSXW = fÂ§ ZX"nf +HQFH EHWWHU UHVXOWV ZRXOG SUREDEO\ EH REWDLQHG E\ FKRRVLQJ DQ DOJRULWKP WR PLQLPL]H (>^5H=Â fÂ§ GL]ff`@ DV RSSRVHG WR PLQLPL]LQJ (>_=Â fÂ§ FL"nf_@ ,Q WKLV FDVH WKH LPDJLQDU\ SDUW RI WKH HUURU FDQ EH LJQRUHG VLQFH LW ZLOO QRW EH XVHG :LWK WKH VDPH HTXDOL]HU FRQWHQWV DV LQ HTXDWLRQ f WKH ILOWHU RXWSXW FDQ EH ZULWWHQ DV => ZL LZVfXWf MXfLff f PAGE 40 ZKHUH ZL; ZRf X[Âf DQG Q\>Lf DUH DOO UHDO YHFWRUV RI OHQJWK 1 )RU WKH QHZ GHILQLWLRQ RI WKH HUURU ZH KDYH (>^GLLf ZMDXrrf ZAX\Âff`@ f 1RZ LQ FKRRVLQJ WKH WDSV WR PLQLPL]H LW LV FOHDU WKDW WKLV LV HTXLYDOHQW WR D UHDO :LHQHU ILOWHULQJ SUREOHP ZLWK 1 WDSV 7KH ILUVW 1 WDSV DUH ILOOHG ZLWK X[]f ZKLOH WKH ODVW 1 WDSV DUH ILOOHG ZLWK XÂf 7KH :LHQHU+RSI HTXDWLRQ JLYHQ LQ f FDQ WKHQ EH DSSOLHG ZLWK WKH XQGHUVWDQGLQJ WKDW Xf >X[]f8\Lf@7 DQG Z >ZAZA@7 DQG S (>GLÂfX]f@ DUH QRZ UHDO YHFWRUV RI OHQJWK 1 DQG 5 (>XÂfX7Âf@ EHFRPHV D 1 [ 1 PDWUL[ 7KH UHVXOWV IRU WKLV YDULDWLRQ RI WKH 006( UHFHLYHU DUH VKRZQ LQ )LJXUH DV ZHOO LQ WKH FXUYH ODEHOHG Ef 7KHUH LV FOHDUO\ DQ DGYDQWDJH UHDOL]HG RYHU WKH FDVH RI PLQLPL]LQJ WKH VTXDUH RI WKH DEVROXWH HUURU ,W LV VHHQ WKDW EHWZHHQ DQG DGGLWLRQDO QDUURZEDQG XVHUV FDQ EH WROHUDWHG IRU D IL[HG QXPEHU RI &'0$ XVHUV IRU RSHUDWLQJ SRLQWV DW ZKLFK WKHUH DUH RU OHVV QDUURZEDQG XVHUV ,I WKH QDUURZEDQG XVHUV HPSOR\ VSHFWUDOO\ HIILFLHQW GLJLWDO PRGXODWLRQ DW WKH VDPH VLJQDOLQJ UDWH DV WKH &'0$ V\VWHP WKHQ WKLV UDQJH FDQ ILW 1 XVHUV ZLWKRXW JXDUG EDQGV 6R ZLWK QDUURZEDQG XVHUV WKH EDQG LV DERXW b SRSXODWHG DQG WKH V\VWHP FDQ VWLOO VXSSRUW DERXW &'0$ XVHUV 7KH UHVXOWV IRU WKH PDWFKHG ILOWHU LQ ZKLFK WKH UHFHLYHU LV PDWFKHG WR WKH ILOWHUHG FRGH VHTXHQFH DUH DOVR VKRZQ LQ )LJXUH IRU FRPSDULVRQ LQ WKH FXUYH ODEHOHG Ff 1RWH WKDW WKHUH LV D VXEVWDQWLDO SHUIRUPDQFH LPSURYHPHQW UHDOL]HG E\ WKH 006( UHFHLYHU RYHU WKH PDWFKHG ILOWHU :KHQ WKHUH DUH QDUURZEDQG XVHUV SUHVHQW WKH 006( UHFHLYHU SHUIRUPV DV ZHOO DV WKH PDWFKHGILOWHU SHUIRUPV LQ WKH FDVH RI QR RYHUOD\ 7KH SHUIRUPDQFH RI WKH ')7EDVHG ILOWHULQJ PHWKRG ZDV DOVR FRPSDUHG WR WKH %XWWHUZRUWK ILOWHULQJ PHWKRG LQ )LJXUH 7KH ')7EDVHG ILOWHULQJ PHWKRG LV FOHDUO\ PAGE 41 )LJXUH 7ZRGLPHQVLRQDO FDSDFLW\ SORW IRU 006( UHFHLYHU PLQLPL]LQJ (>^5H= fÂ§ GL]fff`@ 6KRZQ IRU ')7EDVHG ILOWHULQJ ]HURSDGGHG WR ELWV DQG %XWWHUZRUWK ILOWHU WK RUGHU G% %: 3URFHVVLQJ JDLQ LV FKLSVELW 6ROLG FXUYHV DUH IRU :LHQHU VROXWLRQV GDVKHG FXUYHV DUH IRU DGDSWLYH VROXWLRQV IRU /06 DOJRULWKP ZLWK VWHS VL]H WRWDO LQSXW SRZHUf VXSHULRU DQG WKLV FDQ EH H[SODLQHG E\ H[DPLQLQJ WKH ILOWHUHG FRGH VHTXHQFHV LQ )LJXUH 7KH FRGH VHTXHQFH UHVXOWLQJ IURP WKH %XWWHUZRUWK ILOWHULQJ KDG PRUH HQHUJ\ WKDW VSLOOHG RYHU LQWR RWKHU ELW LQWHUYDOV WKDQ GLG WKH FRGH VHTXHQFH WKDW XVHG WKH ')7 EDVHG ILOWHULQJ PHWKRG ZKLFK REYLRXVO\ GHJUDGHV WKH &'0$ V\VWHPfV SHUIRUPDQFH $QRWKHU LPSRUWDQW LVVXH LV WKH LPSOHPHQWDWLRQ RI WKHVH DOJRULWKPV 7KH SUHYLRXV UHVXOWV DVVXPH WKDW WKH UHFHLYHU ZLOO EH RSHUDWLQJ DW WKH :LHQHU VROXWLRQ DQG WKXV DUH EHVWFDVH UHVXOWV ,Q SUDFWLFH DQ DGDSWLYH DOJRULWKP VXFK DV D OHDVW PHDQVTXDUHG /06f RU UHFXUVLYH OHDVWVTXDUHV 5/6f ZRXOG EH XVHG 6RPH VLPXODWLRQV ZHUH SHUn IRUPHG WR WDNH WKLV LQWR DFFRXQW ,Q )LJXUH UHVXOWV DUH VKRZQ IRU ERWK %XWWHUn ZRUWK ILOWHULQJ DQG ')7EDVHG ILOWHULQJ IRU WKH FDVH ZKHQ (>^5H=Â fÂ§ ALrff`@ LV PLQLPL]HG $V WKH /06 DOJRULWKP FRQYHUJHV VORZO\ DQG WKH IDVWFRQYHUJLQJ 5/6 PAGE 42 Df Ef )LJXUH 7UDQVLHQW EHKDYLRU RI PHDQVTXDUHG HUURU XVLQJ WKH 5/6 DOJRULWKP DQG D WUDLQLQJ VHTXHQFH f LV IRU PLQLPL]LQJ (>^5H=ÂfÂ§GLrff`@ f LV IRU PLQLPL]LQJ (>_=L fÂ§ GLLf_@ f LV UHVXOWLQJ DEVROXWH VTXDUHGHUURU XQGHU FULWHULRQ RI f Df &'0$ XVHUV QDUURZEDQG XVHUV DQG QRWFKHV Ef &'0$ XVHUV QDUURZEDQG XVHUV ZLWK QHDUIDU UDWLR RI G% DQG QRWFKHV DOJRULWKP PD\ KDYH VWDELOLW\ SUREOHPV WKH WUDLQLQJ ZDV DYRLGHG IRU QRZ E\ LQLWLDOn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n PL]HV (>^5H= fÂ§ FLÂff`@ DV RSSRVHG WR fÂ§ (>_=Â fÂ§ GLLf?@ EXW WKH UHn TXLUHG FRQYHUJHQFH WLPH LV DERXW WKH VDPH $OVR QRWLFH WKDW LQ WKH FDVH ZKHQ (>^5H=Â fÂ§ GL"nff`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n SOH[ FRPSXWDWLRQ WKDW LV UHTXLUHG IRU PLQLPL]LQJ fÂ§ (>_=Â fÂ§ G? mf_@ UHTXLUHV WZR UHDO FRPSXWDWLRQV LQ WKH RWKHU FDVH 7KXV PLQLPL]LQJ (>^5H=Â fÂ§ ÂLÂff`@ RIn IHUV D VXEVWDQWLDO LPSURYHPHQW LQ SHUIRUPDQFH ZLWK QR DGGHG FRPSOH[LW\ RU UHTXLUHG WUDLQLQJ WLPH PAGE 44 Df 2 3H Ef Ff )LJXUH +LVWRJUDPV RI &'0$ SUREDELOLW\ RI ELW HUURU 3H IRU RYHUOD\ V\VWHP ZLWK &'0$ XVHUV DQG QDUURZEDQG XVHU 3URFHVVLQJ JDLQ LV FKLSVELW 006( UHFHLYHU PLQLPL]HV (>^5H= fÂ§ GLYnff`@ ')7EDVHG ILOWHULQJ ]HURSDGGHG WR ELWV 6ROLG FXUYH LV IRU QR QDUURZEDQG VLJQDO SUHVHQW GDVKHG FXUYH LV IRU QDUURZEDQG VLJQDO SUHVHQW ZLWK D JLYHQ QDUURZEDQGWR&'0$ QHDUIDU UDWLR Df 5DWLR LV G% Ef 5DWLR LV G% Ff 5DWLR LV G% ,W LV DOVR LQWHUHVWLQJ WR ORRN PRUH FORVHO\ DW WKH DELOLW\ RI WKH 006( UHFHLYHU WR LQKHUHQWO\ UHMHFW VWURQJ QDUURZEDQG LQWHUIHUHQFH ,W ZDV GHWHUPLQHG SUHYLRXVO\ WKDW ZKHQ (>^5H=c fÂ§GLrff`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n URZEDQG LQWHUIHUHQFH )RUWXQDWHO\ WKH SUREDELOLW\ WKDW D QDUURZEDQG XVHU ZLOO KDYH VXFK D ODUJH QHDUIDU UDWLR LV ORZ DV WKLV ZRXOG RFFXU RQO\ LQ D VPDOO JHRJUDSKLF UHJLRQ DURXQG WKH &'0$ UHFHLYHU 6XPPDU\ ,Q WKLV FKDSWHU ZH KDYH VKRZQ WKDW ZLWK WKH QRWFKILOWHULQJ ZKLFK ZDV GHWHUPLQHG QHFHVVDU\ LQ &KDSWHU WKH 006( UHFHLYHU FDQ IXQFWLRQ TXLWH ZHOO LQ DQ RYHUOD\ HQn YLURQPHQW ,Q WKH VLPXODWLRQV SHUIRUPHG KHUH WKH &'0$ V\VWHP FRXOG EH ORDGHG XS WR DERXW XVHUV IRU D SURFHVVLQJ JDLQ RI FKLSVELW ZKHQ QDUURZEDQG XVHUV ZHUH SUHVHQW RU DERXW b RI WKH EDQG ZDV MDPPHG ,W RIIHUV D VXEVWDQWLDO SHUIRUPDQFH LPSURYHPHQW RYHU D FRQYHQWLRQDO PDWFKHGILOWHU UHFHLYHU 7KHVH UHVXOWV DORQJ ZLWK WKRVH RI &KDSWHU DUH TXLWH HQFRXUDJLQJ IRU WKH SURVSHFWV RI &'0$ RYHUOD\ PAGE 46 &+$37(5 &'0$ 29(5/$< ,1 $ &(//8/$5 6<67(0 :H ZLOO QRZ H[WHQG WKH LGHDV RI WKH SUHYLRXV WZR FKDSWHUV WR WKH FHOOXODU HQYLn URQPHQW DQ DSSOLFDWLRQ IRU ZKLFK &'0$ RYHUOD\ KDV SHUKDSV WKH PRVW SRWHQWLDO IRU LQFUHDVLQJ XVHU FDSDFLW\ %HFDXVH RYHUOD\ LV IHDVLEOH RQO\ ZKHQ WKH H[LVWLQJ QDUn URZEDQG V\VWHP LV VSDUVHO\SRSXODWHG WKH FHOOXODU FRQFHSW VHHPV TXLWH FRQGXFLYH WR RYHUOD\ DV HDFK FHOO LQ D QDUURZEDQG V\VWHP RQO\ XWLOL]HV D IUDFWLRQ RI WKH V\VWHP EDQGZLGWK HYHQ ZKHQ IXOO`n ORDGHG $ FHOOXODU RYHUOD\ V\VWHP ZRXOG EH TXLWH EHQn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nJf >3 QORJGRf@ >1^YJf QORJG3f@ f PAGE 48 Df Ef )LJXUH ,OOXVWUDWLRQ RI D QDUURZEDQG FHOOXODU V\VWHP Df &HOOXODU OD\RXW IRU WKUHH OD\HU FOXVWHU &HOOV QXPEHUHG /HWWHU LV WKH FHOOfV FKDQQHOJURXS DVVLJQPHQW FKRVHQ IURP $* Ef 0DJQLILHG YLHZ RI &HOO :KHQ FRPSDULQJ WKH SDWKV WR GLIIHUHQW EDVH VWDWLRQV LW LV RQO\ QHFHVVDU\ WR FRPn SDUH WKH ULJKWPRVW WHUP LQ WKH VHFRQG OLQH RI HTXDWLRQ f WKDW LV *M >$U FUf UDORJGf@ f ZKHUH GO LV WKH GLVWDQFH IURP WKH PRELOH WR WKH ]nWOL EDVH VWDWLRQ QRUPDOL]HG VXFK WKDW WKH FHOOV DUH RI XQLW UDGLXV $OVR QRWH WKDW WKH TXDQWLW\ LQ HTXDWLRQ f ZLOO EH UHIHUUHG WR DV D FKDQQHO JDLQ DV LW LV WKH VXP RI WKH SDWK ORVV DQG VKDGRZLQJ EXW WKLV LV QRW LQWHQGHG WR LPSO\ WKDW WKHUH LV DQ DPSOLILFDWLRQ RI WKH VLJQDO WKURXJK WKH FKDQQHO 7KLV TXDQWLW\ ZLOO EH XVHG LQ FRPSDULQJ GLIIHUHQW SDWKV RQ D UHODWLYH EDVLV RQO\ :LWKRXW ORVV RI JHQHUDOLW\ LW ZLOO EH DVVXPHG WKDW WKH XVHU RI LQWHUHVW LV SK\VLFDOO\ ORFDWHG LQ FHOO DQG DW D GLVWDQFH G? DQG DQJOH DV VKRZQ LQ )LJXUH 7KH UHVW RI WKH *c TXDQWLWLHV PD\ ER IRXQG LQ WHUPV RI G? DQG %? E\ XVLQJ WKH ODZ RI FRVLQHV :H GHILQH WKH IROORZLQJ TXDQWLWLHV DOVR LQ G% XQLWV : PD[*M *! *ILJf fÂ§ PD[*nL *R Â‘ *f g f PAGE 49 7KH PHDQLQJ RI WKH UDQGRP YDULDEOH : VKRXOG EH FOHDU $QG WKH UDQGRP YDULDEOH < IRU LQVWDQFH LV PD[*L *R *f ZLWK H[FOXGHG IURP WKH DUJXn PHQW RI WKH PD[ IXQFWLRQ 7KH XVH RI WKH < DOORZV WKH DVVLJQPHQW RI D XVHU WR WKH MWK FHOO WR EH FKDUDFWHUL]HG VLPSO\ E\ WZR H[KDXVWLYH HYHQWV QDPHO\ WKDW WKH XVHU LV LQ IDFW DVVLJQHG WR WKH MWK FHOO *M 9ff RU LV QRW *M PAGE 50 G% )LJXUH ([DFW 3')V VROLGf DQG *DXVVLDQ DSSUR[LPDWLRQV GDVKHGf IRU PAGE 51 LQ G% DW WKH EDVH VWDWLRQ LV WKH XVHUfV WUDQVPLWWHG SRZHU ZLOO EH DGMXVWHG WR fÂ§ : ZKHUH : LV GHILQHG LQ HTXDWLRQ f VR WKDW LW ZLOO DUULYH DW D SRZHU DW LWV RZQ EDVH VWDWLRQ $W D EDVH VWDWLRQ WR ZKLFK WKH XVHU LV QRW DVVLJQHG LQ FHOO N WKH UHFHLYHG SRZHU LV *Na PAGE 52 7DEOH &RPSRVLWH PJ P\ DQG (> PAGE 53 )LJXUH +LVWRJUDPV DQG *DXVVLDQ DSSUR[LPDWLRQV IRU LQWHUIHUHQFH FDXVHG WR EDVH VWDWLRQ Df ,QWHUIHUHQFH IURP D XVHU LQ FHOO LQ D QG OD\HU FHOO DQG LQ D UG OD\HU FHOO )RU HDFK WKHUH LV DOVR DQ LPSXOVH DW K ZLWK D ZHLJKW RI S Ef ,QWHUIHUHQFH IURP D XVHU ORFDWHG XQLIRUPO\ WKURXJKRXW WKH OD\HUV 7KHUH LV DQ LPSXOVH DW K ZLWK ZHLJKW SR PAGE 54 XQLW\ EXW FORVH WR LW 7KH IXQFWLRQ LQ HTXDWLRQ f IRU H[DPSOH LQWHJUDWHV WR ,W LV VLPSOH WR JHQHUDWH UDQGRP YDULDEOHV ZLWK 3')V YHU\ FORVH WR WKHVH RQHV KRZHYHU DQG WKHUH LV D VXEVWDQWLDO VDYLQJV LQ WLPH LQ FRPSDULVRQ WR WKH WDVN RI JHQHUDWLQJ D XVHUfV SRVLWLRQ DQG VKDGRZLQJ SURFHVVHV WR HDFK EDVH VWDWLRQ 7KH DSSUR[LPDWH 3') RI HTXDWLRQ f ZLOO DOVR EH XVHIXO LQ WKH QH[W VHFWLRQ LQ LQYHVWLJDWLQJ WKH HIIHFWV RI RYHUOD\ RQ WKH H[LVWLQJ QDUURZEDQG V\VWHP 3HUIRUPDQFH RI WKH 1DUURZEDQG 6\VWHP *LYHQ WKH FKDUDFWHUL]DWLRQ RI WKH PRELOHVf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f ZKLFK ZLOO EH UHSHDWHG KHUH ZLWK D VOLJKW PRGLILFDWLRQ 4 1R 7( n (L 1 N L 9IW FRVNf 9N nn f ZLWK 7F9DU,Nf ZKLFK LV D UDQGRP YDULDEOH ZLWK UHVSHFW WR WKH &'0$ XVHUVf FRGHV DQG GHOD\V $V YHULILHG EY VLPXODWLRQ WKH YDULDEOHV FDQ EH ZHOODSSUR[LPDWHG DV XQLWPHDQ H[SRQHQWLDO UDQGRP YDULDEOHV 1RZ D FULWHULRQ PXVW EH HVWDEOLVKHG WR GHWHUPLQH LI WKH RYHUOD\ FDXVHV WRR PXFK GHJUDGDWLRQ WR WKH QDUURZEDQG XVHU 6XFK D FULWHULRQ LV GHVFULEHG E\ 0LOVWHLQ HW DO >@ ZKLFK VWDWHV WKDW LI WKH %36. XVHUfV SUREDELOLW\ RI HUURU LV LQFUHDVHG IURP PAGE 55 f* WR JUHDWHU WKDQ DV D UHVXOW RQO\ RI WKH RYHUOD\ WKDW LV ZKLOH LWV VLJQDOWR QRLVHUDWLR 615f UHPDLQV FRQVWDQW WKHQ WKH RYHUOD\ LV H[FHVVLYH 7KLV FULWHULRQ ZDV HPSOR\HG LQ FRQMXQFWLRQ ZLWK D FRQYROXWLRQDO FRGH RI UDWH U fÂ§ DQG D FRQVWUDLQW OHQJWK P 7KH RSWLPXP FRGH KDV D IUHH GLVWDQFH FOIUHH DV ZDV IRXQG WKURXJK DQ H[KDXVWLYH FRPSXWHU VHDUFK >@ $ WLJKW ORZHU ERXQG IRU WKH SUREDELOLW\ RI ELW HUURU RI WKH FRGHG V\VWHP LV 3H!a P f 7KH YDOXHV RI (E1R UHTXLUHG IRU WKLV ERXQG WR DFKLHYH ELW HUURU UDWHV RI a DQG GHQRWHG 615* DQG 615 UHVSHFWLYHO\ DUH G% DQG G% ,Q RUGHU WR DSSO\ WKLV WR WKH RYHUOD\ V\VWHP QRWH WKDW WKH VTXDUH RI WKH DUJXPHQW RI WKH 4IXQFWLRQ LQ HTXDWLRQ f PXVW ,UH JUHDWHU WKDQ 615 DQG DOVR QRWH WKDW 615T fÂ§ (E1R WR JLYH WKH QHFHVVDU\ FRQGLWLRQ rerf ^VPVPf RP Qf IRU ZKLFK WKH RYHUOD\ LV QRW H[FHVVLYH 7KH QXPEHU RI &'0$ XVHUV WKDW FDQ VDWLVI\ WKLV FRQGLWLRQ IRU DOO EXW D VSHFLILHG SHUFHQWDJH VWDWLVWLFDOO\ VD\ b ZLOO EH WKH OLPLW RQ WKH &'0$ FDSDFLW\ IURP WKH QDUURZEDQG V\VWHPfV SHUVSHFWLYH 7KLV VKRXOG QRW EH FRQIXVHG ZLWK D GLIIHUHQW OLPLW ZKLFK PD\ RU PD\ QRW EH JUHDWHU WKDQ WKLV RQH ZKLFK LV WKH QXPEHU RI XVHUV IRU ZKLFK WKH 0$, HYHQWXDOO\ EHFRPHV WRR VHYHUH IRU WKH &'0$ V\VWHP LWVHOI 7KDW OLPLW ZLOO EH ORRNHG DW LQ 6HFWLRQ %HIRUH WKH FDSDFLW\ OLPLW GHILQHG E\ HTXDWLRQ f FDQ EH IRXQG LW LV QHFHVVDU\ WR VSHFLI\ WKDW LQ WKLV FKDSWHU LW ZLOO EH DVVXPHG WKDW WKH UHFHLYHG SRZHUV RI WKH &'0$ DQG QDUURZEDQG XVHUV DW WKHLU UHVSHFWLYH DVVLJQHG EDVH VWDWLRQV DUH SRZHU FRQWUROOHG WR WKH VDPH OHYHO 7KH DSSUR[LPDWH 3') RI WKH UHFHLYHG SRZHU JLYHQ LQ HTXDWLRQ f ZDV XVHG WR JHQHUDWH UHDOL]DWLRQV RI WKH &'0$WR%36. QHDU IDU UDWLR 3N3EÂ‘ 7KHQ GLIIHUHQW UHDOL]DWLRQV RI WKH VXP RQ WKH OHIWKDQG VLGH RI HTXDWLRQ PAGE 56 f ZHUH IRUPHG DQG FRPSDUHG ZLWK WKH YDOXH IRU D IL[HG YDOXH RI WKH QXPEHU RI XVHUV DQG IRU D SURFHVVLQJ JDLQ 1 7KHQ ZDV YDULHG XQWLO WKH VXP ZDV JUHDWHU WKDQ IRU PRUH WKDQ b RI WKH UHDOL]DWLRQV ,W ZDV IRXQG WKDW XVHUV ZDV WKH PD[LPXP QXPEHU ZKLFK FRXOG PHHW WKH FULWHULRQ 1RWH WKDW WKHUH DUH XVHUV VSUHDG RYHU WKH WKUHH OD\HUV RU FHOOV ZKLFK LV XVHUVFHOO ZKLFK LV H[WUHPHO\ ORZ IRU D SURFHVVLQJ JDLQ RI $ VLPLODU FRQFOXVLRQ WKDW WKH RYHUOD\ LV H[FHVVLYH HYHQ IRU D OLJKWO\ORDGHG &'0$ V\VWHP ZDV REVHUYHG LQ &KDSWHU IRU WKH VLQJOHFHOO FDVH 2QH ZD\ WR LPSURYH WKLV ORDGLQJ UHODWLYH WR D IL[HG SURFHVVLQJ JDLQ LV WR LQFUHDVH VXEVWDQWLDOO\ WKH SRZHU OHYHO DW ZKLFK WKH QDUURZEDQG XVHUV DUULYH DW WKHLU RZQ EDVH VWDWLRQV 'RLQJ VR ZRXOG QRW OHDG WR LQFUHDVHG FRFKDQQHO LQWHUIHUHQFH DPRQJVW WKH QDUURZEDQG XVHUV EHFDXVH DOO WKH QDUURZEDQG XVHUV LQ WKH V\VWHP ZRXOG LQFUHDVH WKHLU SRZHUV HTXDOO\ ,W ZRXOG KRZHYHU GHFUHDVH WKHLU EDWWHU\ OLIH ZKLFK LV LQFRQVLVWHQW ZLWK WKH LGHD WKDW WKH RYHUOD\ VKRXOG UHTXLUH WKDW WKH QDUURZEDQG V\VWHP FKDQJH DV OLWWOH DV SRVVLEOH $QRWKHU VROXWLRQ LV WR HPSOR\ QRWFKILOWHULQJ LQ WKH &'0$ WUDQVPLWWHUV WR DYRLG FHUWDLQ QDUURZEDQG XVHUV DV ZDV GRQH LQ &KDSWHU IRU WKH VLQJOHFHOO FDVH 7KLV ZLOO EH H[SORUHG LQ WKH QH[W VHFWLRQ (IIHFWV RI &'0$ 7UDQVPLWWHU 1RWFKLQJ ,Q DQ HIIRUW WR DOOHYLDWH WKH VHYHUH GHJUDGDWLRQ FDXVHG WR WKH QDUURZEDQG V\VWHP DV D UHVXOW RI WKH &'0$ RYHUOD\ WKH LGHD RI QRWFKLQJ WKH &'0$ VLJQDOV DW WKHLU WUDQVPLWWHUV LQ RUGHU WR DYRLG QDUURZEDQG XVHUV ZDV LQYHVWLJDWHG LQ &KDSWHU IRU WKH VLQJOHFHOO FDVH 7KH ')7EDVHG ILOWHULQJ PHWKRG GHVFULEHG LQ 6HFWLRQ ZLOO EH HPSOR\HG KHUH LQ WKH FHOOXODU FDVH 7KH XVH RI QRWFKLQJ LV PRUH LQYROYHG WKDQ LQ WKH VLQJOHFHOO FDVH ZKHUH WKH &'0$ VLJQDOV ZHUH QRWFKHG WR DYRLG DOO RI WKH QDUURZEDQG XVHUV SUHVHQW ,Q WKH FHOOXODU FDVH LW LV YHU\ SRVVLEOH WKDW LQ DGGLWLRQ WR WKRVH QDUURZEDQG XVHUV ORFDWHG QHDU WR LW SK\VLFDOO\ D &'0$ PRELOH PD\ KDYH WR DYRLG D QDUURZEDQG XVHU WKDW LV DVVLJQHG PAGE 57 WR DQ DGMDFHQW FHOO RU HYHQ WR D FHOO WKDW LV WZR OD\HUV DZD\ %XW LW REYLRXVO\ FDQQRW QRWFK IRU DOO RI WKH QDUURZEDQG XVHUV LQ WKH V\VWHP DV WKDW ZRXOG OLNHO\ FRQVWLWXWH WKH HQWLUH V\VWHP EDQGZLGWK QRU GRHV LW QHHG WR $V D UHVXOW RI WKH VKDGRZLQJ GHVFULEHG LQ 6HFWLRQ WKHUH ZLOO EH LQ VRPH FDVHV QDUURZEDQG XVHUV QHDU WR WKH &'0$ PRELOH WKDW DUH QRW VLJQLILFDQWO\ GHJUDGHG E\ WKH &'0$ VLJQDO DQG KHQFH D QRWFK LV QRW QHFHVVDU\ IRU WKHP 7KLV DOVR KROGV IRU D VLJQLILFDQW SHUFHQWDJH RI WKH XVHUV WKDW DUH QRW QHDU WR WKH &'0$ PRELOH 7KH WRWDO &'0$ LQWHUIHUHQFH VHHQ E\ D QDUURZEDQG XVHU GHSHQGV RQ WKH &'0$ VLJQDOVf UHFHLYHG SRZHUV SKDVHV DQG RQ WKH YDULDEOHV DFFRUGLQJ WR HTXDWLRQ f :KHQ GHFLGLQJ ZKHWKHU RU QRW D QRWFK LV QHFHVVDU\ WKH SRZHU FDQ EH HVWLPDWHG DQG WKH LQIRUPDWLRQ H[FKDQJHG EHWZHHQ EDVH VWDWLRQV EXW WKH SKDVHV DQG WKH REYLRXVO\ ZLOO EH XQNQRZQ +HQFH D VSHFLILF FULWHULRQ WKDW ZLOO EH XVHG LV WKDW D &'0$ VLJQDO PXVW EH QRWFKHG LI WKH SRZHU OHYHO DW ZKLFK LW DUULYHV DW WKH FRUUHVSRQGLQJ QDUURZEDQG UHFHLYHU UHODWLYH WR WKH QDUURZEDQG VLJQDO LV KLJKHU WKDQ D JLYHQ WKUHVKROG )URP 7DEOH ZKHQ WKH ')7EDVHG ILOWHULQJ PHWKRG LV XVHG ZLWK ELWV RI ]HURn SDGGLQJ WKH LQWHUIHUHQFH FRQWULEXWLRQ LV UHGXFHG E\ DERXW G% 7KH H[SHULPHQW RI 6HFWLRQ ZDV UHSHDWHG WKDW LV WKH QXPEHU RI &'0$ XVHUV IRU ZKLFK WKH H[n FHVVLYH RYHUOD\ FULWHULRQ RI HTXDWLRQ f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n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Â WKH YDOXHV RI P\Â DQG FUM$ ZKLFK DUH QXPHULFDOO\ FDOFXODWHG DV ZHOO DV WKH ZKLFK DUH HDVLO\ IRXQG XVLQJ WKH ODZ RI FRVLQHV PAGE 60 )LUVW ZH GHILQH 5HFHLYHG QDUURZEDQG SRZHUf 5HFHLYHG &'0$ SRZHUf ZKHUH WKH UHFHLYHG SRZHU UHIHUV WR WKH SRZHU LQ G% DW WKH EDVH VWDWLRQ WR ZKLFK WKH QDUURZEDQG XVHU LV DVVLJQHG $ QRWFK ZLOO WKHQ EH UHTXLUHG LI 7 ZLWK 7 JLYHQ LQ G% 5HFDOO WKDW DW WKLV OLDVH VWDWLRQ WKH QDUURZEDQG XVHUfV UHFHLYHG SRZHU ZLOO EH D VSHFLILHG YDOXH GHQRWHG ,W ZDV VWDWHG HDUOLHU WKDW WKH &'0$ DQG QDUURZEDQG XVHUV ZRXOG EH SRZHUFRQWUROOHG WR WKH VDPH OHYHO LI DVVLJQHG WR WKH VDPH EDVH VWDWLRQ ,Q WKDW FDVH fÂ§ DQG WKHUH LV FHUWDLQO\ D QRWFK UHTXLUHG )RU D &'0$ XVHU WKDW LV QRW DVVLJQHG WR WKH VDPH FHOO DV WKH QDUURZEDQG XVHU GHQRWHG FHOO N WKH FRQGLWLRQDO 3') RI WKH &'0$ UHFHLYHG SRZHU DW WKH QDUURZEDQG XVHUf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fV WUDQVPLWWHU DQWHQQD DQG WKH PRELOHfV UHFHLYHU DQWHQQD ZKLOH WKH LQWHUIHUHQFH IURP D &'0$ PRELOH WR WKH QDUURZEDQG PRELOH ZLOO RQO\ EH DPSOLILHG E\ LWV RZQ WUDQVPLW DQWHQQD DQG WKH QDUURZEDQG PRELOHf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n UDWLQJ WKH FHOOXODU OD\RXW RI )LJXUH DQG ORJQRUPDO VKDGRZLQJ DQG SRZHU FRQWURO DV GHVFULEHG HDUOLHU 7KH &'0$ V\VWHP KDG D SURFHVVLQJ JDLQ RI $U FKLSVELW DQ (E1R YDOXH RI G% DQG XVHG WKH ')7EDVHG ILOWHULQJ PHWKRG ZLWK ELWV RI ]HURSDGGLQJ ZKHQ ILOWHULQJ ZDV QHFHVVDU\ )RU D IUHTXHQF\ UHXVH V\VWHP HDFK FHOO ZRXOG KDYH DW PRVW QDUURZEDQG XVHUV ,Q D fEUXWHIRUFHf VLPXODWLRQ RI WKH FHOOXODU RYHUOD\ V\VWHP WKH FRYHUDJH DUHD WKDW PXVW EH FRQVLGHUHG ZRXOG FRQVLVW RI OD\HUV RI FHOOV VXUURXQGLQJ WKH FHQWHU FHOO RI LQWHUHVW ,W ZDV REVHUYHG WKDW D &'0$ XVHU PLJKW KDYH WR QRWFK IRU D QDUURZEDQG XVHU WKDW LV OD\HUV DZD\ IURP LWV JHRJUDSKLF ORFDWLRQ $QG DQ\ XVHU FRXOG EH DVVLJQHG WR D FHOO DW PRVW XVXDOO\ OD\HUV DZD\ IURP LWV JHRJUDSKLF ORFDWLRQ 6R D &'0$ XVHU LQ WKH VHFRQG OD\HU RXW PLJKW KDYH WR QRWFK IRU D QDUURZEDQG XVHU LQ WKH IRXUWK OD\HU RXW ZKRVH EDVH VWDWLRQ FRXOG EH LQ WKH VL[WK OD\HU RXW 7KH WRWDO QXPEHU RI FHOOV LQ WKH JHRJUDSKLF FRYHUDJH DUHD PXOWLSOLHG E\ WKH GHQVLW\ RI XVHUVFHOO JLYHV WKH WRWDO QXPEHU RI XVHUV HLWKHU &'0$ RU QDUURZEDQG ZKLFK ZRXOG WKHQ EH GLVWULEXWHG XQLIRUPO\ DERXW WKH FRYHUDJH DUHD (DFK XVHU ZRXOG WKHQ EH DVVLJQHG WR WKH FHOO DW ZKLFK LWV UHFHLYHG SRZHU LV PD[LPXP DQG SRZHU FRQWURO ZRXOG EH LPSOHPHQWHG $IWHU LW LV GHWHUPLQHG KRZ PXFK RI DQ LQFUHDVH RU GHFUHDVH PAGE 63 LQ WUDQVPLWWHG SRZHU IURP WKH PRELOHV LV UHTXLUHG WKH QHFHVVDU\ QRWFKLQJ IRU HDFK &'0$ PRELOH ZRXOG EH IRXQG E\ FRPSDULQJ LWV UHFHLYHG SRZHU ZLWK WKDW RI HDFK RI WKH QDUURZEDQG PRELOHV DVVLJQHG WR RQH RI WKH FHOOV ZLWKLQ D WKUHHOD\HU FOXVWHU RI WKH &'0$ PRELOHfV EDVH VWDWLRQ 7KLV SURFHGXUH ZRXOG REYLRXVO\ UHTXLUH D JUHDW GHDO RI VLPXODWLRQ WLPH ZKLFK FDQ EH GHFUHDVHG VLJQLILFDQWO\ ZLWK VRPH VLPSOLILFDWLRQV 7KH ILUVW VLPSOLILFDWLRQ UHVXOWV IURP WKH REVHUYDWLRQ ZKLFK ZDV YHULILHG E\ VLPn XODWLRQ WKDW LI DQ LQWHUIHULQJ &'0$ XVHU LV OHIW XQILOWHUHG WKHUH LV QR QRWLFHDEOH GLIIHUHQFH LQ LWV HIIHFW RQ WKH GHVLUHG &'0$ XVHU 7KH QRWFKLQJ SUREDELOLW\ IRU D &'0$ XVHU GHSHQGV MRLQWO\ RQ LWV SRVLWLRQ DQG LWV WUDQVPLWWHG SRZHU %XW VLQFH LW GRHV QRW QHHG WR EH NQRZQ LQ WKH VLPXODWLRQV LW LV VXIILFLHQW RQO\ WR NQRZ WKH LQ WHUIHUHUfV UHFHLYHG SRZHU UHODWLYH WR WKH GHVLUHG XVHU DQG LWV SRVLWLRQ LV XQLPSRUWDQW 6R ZH FDQ IXUWKHU VLPSOLI\ WKH VLPXODWLRQV E\ XVLQJ HTXDWLRQ f ZKLFK JLYHV WKH DSSUR[LPDWH 3') RI WKH UHFHLYHG SRZHU IRU D XVHU WKDW LV ORFDWHG XQLIRUPO\ WKURXJKn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f 8 QRWFKHV ff f ,Q WKLV FULWHULRQ D &'0$ XVHU ZLOO EH EORFNHG LI LWV ELW HUURU UDWH LV WRR KLJK RU LI LW QHHGV PRUH WKDQ QRWFKHV $V WKH VDPSOLQJ UDWH RI WKH V\VWHP LV 7& WKH UDQJH LQ ZKLFK QRWFKLQJ FDQ EH GRQH LV DFWXDOO\ fÂ§7F I 7Ff $Q\ IUHTXHQF\ ORFDWLRQV RXWVLGH RI WKLV UDQJH PXVW EH QRWFKHG LQ WKH DSSURSULDWH ORFDWLRQ PLUURUHG DURXQG s7F 7KXV ZLWK D SURFHVVLQJ JDLQ RI WKHUH DUH HIIHFWLYHO\ RQO\ QRWFKLQJ ORFDWLRQV :LWK PRUH WKDQ b RI WKH &'0$ VLJQDO QRWFKHG RXW WKH XVHU PXVW EH GURSSHG $ WZRGLPHQVLRQDO FDSDFLW\ SORW LV VKRZQ LQ )LJXUH IRU WKH VDPHOLQN FDVH ZLWK QRWFKLQJ WKUHVKROGV RI 7 G% 7KH QRWFKLQJ WKUHVKROG PD\ DIIHFW WKH FDSDFLW\ FULWHULRQ RI HTXDWLRQ f LQ WZR ZD\V )LUVW DV PRUH QRWFKLQJ EHFRPHV QHFHVVDU\ LW LV PRUH OLNHO\ WKDW WKH GHVLUHG XVHU ZLOO KDYH PRUH WKDQ QRWFKHV DQG ZLOO EH GURSSHG $OVR LW VKRXOG EH PRUH GLIILFXOW WR GHPRGXODWH WKH GHVLUHG XVHUf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fÂ§ G% WKH HIIHFWV RI GURSSLQJ XVHUV VKRZ XS IRU KLJKHU YDOXHV RI QDUURZEDQG ORDGLQJ PAGE 65 )LJXUH &DSDFLW\ SORW IRU VDPHOLQN FDVH 3URFHVVLQJ JDLQ LV FKLSVELW (E1 RI &'0$ V\VWHP LV G% 1RWFKLQJ WKUHVKROGV DUH 7 G% )RU DQG 1% XVHUV UHVSHFWLYHO\ WKH &'0$ V\VWHP UHDFKHV FDSDFLW\ LQ WHUPV RI WKRVH GURSV ZKLFK UHVXOW IURP QHHGLQJ WRR PDQ\ QRWFKHV 1RWLFH WKDW WKH &'0$ V\VWHP LV FRQVWUDLQHG E\ WKH IDFW WKDW IRU D KLJK HQRXJK QDUURZEDQG ORDGLQJ WKH SUREDELOLW\ WKDW D &'0$ XVHU LV GURSSHG LV JUHDWHU WKDQ b DQG WKH FULWHULRQ RI HTXDWLRQ f FDQQRW EH VDWLVILHG IRU DQ\ DPRXQW RI &'0$ ORDGLQJ )URP )LJXUH WKLV RFFXUV DW DQG QDUURZEDQG XVHUVFHOO UHVSHFWLYHO\ IRU 7 fÂ§ G% 7KH UHVXOWV RI )LJXUH FKDUDFWHUL]H WKH MRLQW FDSDFLW\ OLPLWV RI WKH V\VWHP DV GLFWDWHG E\ WKH &'0$ V\VWHP ,Q FRQMXQFWLRQ ZLWK WKH OLPLWV UHVXOWLQJ IURP FRQVLGn HUDWLRQV RI WKH QDUURZEDQG V\VWHP DV VKRZQ LQ )LJXUHV DQG LW LV FOHDU WKDW WKH QRWFKLQJ WKUHVKROG 7 fÂ§ G% LV WKH EHVW FKRLFH DV LWV OLPLWV DFFRUGLQJ WR HDFK RI WKH WZR V\VWHPV DUH UHODWLYHO\ FORVH )URP )LJXUH LI WKH SUREDELOLW\ RI PLVVLQJ D QRWFK LV b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fV FHOO ZDV KHOG FRQVWDQW DW RU ZKLOH WKH QXPEHU RI QRWFKHV LQ WKH GHVLUHG XVHU ZDV YDULHG DQG WKH &'0$ GHQVLW\ ZDV IRXQG DFFRUGLQJ WR RQO\ WKH ELW HUURU UDWH SDUW RI WKH FULWHULRQ RI HTXDWLRQ f 5HFDOO WKDW LQ WKLV FRQILJXUDWLRQ WKH VLJQDOV RQ WKH QDUURZEDQG V\VWHPfV IRUZDUG OLQN ZLOO DSSHDU RQ WKH &'0$ UHYHUVH OLQN DV LQWHUIHUHQFH DW D YHU\ KLJK QHDUIDU UDWLR $VVXPLQJ WKDW WKH SUREOHP FDQ EH DOOHYLDWHG WR VRPH H[WHQW E\ VKLHOGLQJ WKH WZR DQWHQQDV WKH QHDUIDU UDWLR ZDV IL[HG DW G% ,Q JRLQJ GRZQZDUG IURP FXUYH WR FXUYH LW LV VHHQ WKDW WKH HIIHFW RI DQ DGGLWLRQDO QDUURZEDQG XVHU LV WR GHFUHDVH WKH &'0$ GHQVLW\ E\ DERXW XVHUVFHOO ZKLFK LV VLJQLILFDQW )RU WKH VDPHOLQN FDVH ZKHQ WKHUH DUH QR QDUURZEDQG XVHUV DVVLJQHG WR WKH GHVLUHG &'0$ XVHUfV FHOO WKH FDSDFLW\ FXUYH LV WKH VDPH DV LQ WKH VWDJJHUHGOLQN FDVH 7KH QDUURZEDQG XVHUV WKDW DUH DVVLJQHG WR WKH FHOO ZLOO DUULYH DW G% ZLWK UHVSHFW WR WKH PAGE 67 )LJXUH &'0$ XVHUVFHOO YV QXPEHU RI QRWFKHV LQ WKH GHVLUHG XVHU 6ROLG OLQH LV VWDJJHUHGOLQN FDVH ZLWK QDUURZEDQGWR&'0$ QHDU IDU UDWLR DW &'0$ EDVH VWDWLRQ RI G% DQG RU QDUURZEDQG XVHUV DVVLJQHG WR FHOO RI LQWHUHVW 'DVKHG OLQH LV VDPHOLQN FDVH ZLWK QDUURZEDQG XVHUV ZLWK QHDUIDU UDWLR RI G% DVVLJQHG WR FHOO RI LQWHUHVW Â‘(nW$nR RI &'0$ V\VWHP LV G% &'0$ XVHUf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f DV f :H ZLOO QRZ ORRN DW WKH SHUIRUPDQFH RI D %36. XVHU UHFHLYHG LQ WKH SUHVHQFH RI RYHUOD\ LQ D IDGLQJ FKDQQHO $ JHQHUDO PXOWLFDUULHU &'0$ IRUPDW ZLOO EH XVHG ZKHUH WKHUH DUH 4 FDUULHUV DQG WKH WRWDO SRZHU RI HDFK &'0$ XVHU LV GLYLGHG HTXDOO\ DPRQJVW WKH 4 FDUULHUV 7KH LGHQWLFDO &'0$ VLJQDO ZLOO EH WUDQVPLWWHG VLPXOWDQHn RXVO\ RQ HDFK FDUULHU 7KH IDGLQJ SURFHVV RQ HDFK FDUULHU ZLOO EH WDNHQ DV IUHTXHQF\ QRQVHOHFWLYH DQG LQGHSHQGHQW RI WKH IDGLQJ SURFHVVHV RQ RWKHU SDWKV 7KH &'0$ EDQGZLGWK LQ WKH VLQJOHFDUULHU FDVH PXVW EH VPDOOHU WKDQ RU RQ WKH RUGHU RI WKH FRn KHUHQFH EDQGZLGWK RI WKH FKDQQHO LQ RUGHU WKDW WKH IDGLQJ EH IUHTXHQF\QRQVHOHFWLYH 7KXV LQ WKH UHDOL]DWLRQ RI PXOWLFDUULHU &'0$ WKH FDUULHUV PXVW KDYH VXIILFLHQW IUHn TXHQF\ VSDFLQJ EHWZHHQ WKHPVHOYHV LQ RUGHU IRU WKH IDGLQJ RQ HDFK FDUULHU WR EH LQGHSHQGHQW ,W ZLOO EH DVVXPHG WKDW RQO\ WKH SDUWLFXODU &'0$ VLJQDO ZKLFK RYHUODSV WKH %36. VLJQDO DW EDQGSDVV ZLOO SDVV WKURXJK WKH %36. UHFHLYHU DQG WKXV ZH ZLOO FRQFHQWUDWH RQO\ RQ WKDW VLJQDO LQ WKH HTXDWLRQV ZKLFK IROORZ 7KH %36. XVHU HPSOR\V VTXDUH SXOVHV RI ZLGWK 7E WKH ELW WLPH DQG D PDWFKHGILOWHU UHFHLYHU 7KH FRPSOH[ EDVHEDQG UHFHLYHG VLJQDO FDQ EH ZULWWHQ DV 22 UWf ; ?=3E\ELfGE^LfQ^^W L7Ef7Ef QWf H[SfÂ§MX!FWf RR (( 9PM4ON^LfGNLfFNW L7E 7Nf f PAGE 71 ZKHUH QI7cf LV D XQLWDPSOLWXGH VTXDUH SXOVH RI ZLGWK 7Â 3E LV WKH %36. XVHUfV DYHUDJH SRZHU 3E LV WKH IFWK &'0$ XVHUfV FRPSRVLWH DYHUDJH SRZHU IURP DOO RI LWV FDUULHUV ÂÂf DQG ccLf DUH WKH %36. XVHUfV DQG NWK &'0$ XVHUfV IDGLQJ SURFHVV GXULQJ WKH ÂWK ELW LQWHUYDO ZLWK HDFK D ]HURPHDQ FRPSOH[ *DXVVLDQ UDQGRP YDULDEOH 7KH +K XVHUfV VSUHDGLQJ ZDYHIRUP KDV D SHULRG RI 7E DQG FRQVLVWV RI XQLW DPSOLWXGH VTXDUH SXOVHV RI ZLGWK 47& ZKHUH WKH VLQJOHFDUULHU FKLSWLPH LV 7F 7E1 DQG 1 LV WKH FRPSRVLWH SURFHVVLQJ JDLQ 7KH RXWSXW RI WKH %36. XVHUfV PDWFKHGILOWHU LQ WKH ÂWK ELW LQWHUYDO LV LOf7E N ]Lf P 7^IfGW 7}ZIFR Â MYf} L7E W 9 f ZKHUH 1[Lf DQG 1\Lf DUH ]HURPHDQ *DXVVLDQ UDQGRP YDULDEOHV HDFK ZLWK YDULDQFH FU (E1Rfa (E LV WKH DYHUDJH HQHUJ\SHUELW RI WKH %36. V\VWHP DQG WKH WHUP GXH WR LQWHUIHUHQFH IURP WKH NWK &'0$ XVHU GXULQJ WKH ÂWK ELW LQWHUYDO rrf LV 7L 7IF KLf GNL f &NWfGW GN^Lf &NWfGW f 7N 'HQRWLQJ WKH QWK FKLS RI WKH NWK XVHUfV VSUHDGLQJ FRGH DV FÂLQ DQG DVVXPLQJ WKDW WKH SHUFDUULHU SURFHVVLQJ JDLQ 14 LV DQ LQWHJHU ZH KDYH KLf 7E 14 ? Â f Â 6N&N14SNL A &NM 9 M 143N GN^Lf 14fÂ§SLWfÂ§ &Nn Â a WLNf&N14SNO L f ZKHUH WKH GHOD\ KDV EHHQ ZULWWHQ DV SE 6Ef47F ZLWK SE DQ LQWHJHU LQ WKH UDQJH ^14f fÂ§ f DQG 6E D QRQLQWHJHU LQ WKH UDQJH f 7KH YDULDQFH RI WKLV WHUP IRU IL[HG FRGHV DQG GHOD\V DQG KHQFH ZLWK WKH H[SHFWDWLRQ WDNHQ RQO\ ZLWK UHVSHFW WR WKH GDWD ELWV LV LWVHOI D UDQGRP YDULDEOH )URP 6HFWLRQ LW FDQ EH PAGE 72 YHULILHG E\ VLPXODWLRQ WKDW ZLWK UHVSHFW WR WKH FRGHV DQG GHOD\V WKH YDULDQFH FDQ EH ZHOODSSUR[LPDWHG DV DQ H[SRQHQWLDO UDQGRP YDULDEOH $VVXPLQJ FRKHUHQW GHWHFWLRQ IRU WKH %36. XVHU WKH GHFLVLRQ RQ WKH ÂWK %36. GDWD ELW LV GELf VLJQ5H>=c H[S^fÂ§M="nff@ff ,QYRNLQJ D *DXVVLDQ DSSUR[LPDWLRQ RQ WKH &'0$ LQWHUIHUHQFH FRQWULEXWLRQ WKH SUREDELOLW\ RI ELW HUURU FRQGLWLRQHG RQ WKH %36. XVHUfV IDGLQJ SURFHVV f LV 3UHUURUÂff 4 _f_ Y? Af1 Â ,r,r FRV=IFf LWf 9N f ZKHUH (E1RfE KDV EHHQ FODULILHG WR DSSO\ WR WKH %36. XVHU QRW WR EH FRQIXVHG ZLWK WKH &'0$ V\VWHP IRU ZKLFK (E1Rf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fV IDGLQJ SURFHVV ZKLFK LV 3UHUURUW"ff m 4 n 1U 023 f :H GHQRWH 615 DV WKH VTXDUH RI WKH DUJXPHQW RI WKH 4IXQFWLRQ RI HTXDWLRQ f DQG 615HII DV WKH VDPH IRU HTXDWLRQ f 7KXV ZLWK WKH DGGLWLRQ RI WKH RYHUOD\ WKH %36. XVHU PXVW LQFUHDVH LWV SRZHU E\ D IDFWRU RI 615 ? 615HII a ?Qf (K D\ N O$Ynf_FRV=IFf YN f PAGE 73 $V PHQWLRQHG EHIRUH LW VKRXOG EH VWLSXODWHG WKDW WKH RYHUOD\ FDXVH RQO\ D PLQRU DPRXQW RI GHJUDGDWLRQ WR WKH H[LVWLQJ QDUURZEDQG V\VWHP )RU D IL[HG QXPEHU RI &'0$ XVHUV WKH TXDQWLW\ LQ HTXDWLRQ f LV D UDQGRP YDULDEOH GHSHQGLQJ RQ WKH &'0$ XVHUVf IDGLQJ SURFHVVHV DQG QHDUIDU UDWLRV ZLWK UHVSHFW WR WKH %36. XVHUf DQG DOVR RQ WKH H[SRQHQWLDOO\GLVWULEXWHG YDULDEOHV $V WKH FULWHULRQ KHUH ZH ZLOO VD\ WKDW WKH QXPEHU RI &'0$ XVHUV LV H[FHVVLYH LI WKH UDQGRP YDULDEOH RI HTXDWLRQ f LV JUHDWHU WKDQ G% PRUH WKDQ b RI WKH WLPH 7KDW LV ZH ILQG WKH PD[LPXP YDOXH RI VXFK WKDW 3U ,f OfLf_:=7fLff Wf9Nf !f rn nf 7KH FDSDFLW\ OLPLWV GLFWDWHG E\ HTXDWLRQ f ZLOO EH IRXQG IRU D V\VWHP ZLWK D FRPSRVLWH SURFHVVLQJ JDLQ RI $7 FKLSVELW DQG ZLWK (E1RfE G% IRU WKH %36. V\VWHP 7KLV FKRLFH UHVXOWV IURP DYHUDJLQJ WKH FRQGLWLRQDO SUREDELOLW\ RI HUURU RI HTXDWLRQ f RYHU WKH 5D\OHLJK IDGLQJ SURFHVV ELf ZKLFK JLYHV WKH ZHOONQRZQ UHVXOW 3 f (E1fE DQG FKRRVLQJ (E1RfE G% WR JHW D ELW HUURU UDWH RI LQ WKH DEVHQFH RI RYHUOD\ 7KH PD[LPXP QXPEHU RI XVHUV ZLOO EH IRXQG IRU D UDQJH RI YDOXHV RI (E1fF 5HFDOO WKDW WKH UHVXOWV ZLOO QRW GHSHQG RQ WKH QXPEHU RI FDUULHUV XVHG +RZHYHU WKH DGGHG IUHTXHQF\ GLYHUVLW\ UHVXOWLQJ IURP PXOWLSOH FDUULHUV DOORZV IRU D VPDOOHU YDOXH RI (E1RfF LQ RUGHU WR DFKLHYH WKH VDPH SUREDELOLW\ RI HUURU ZLWK WKH VDPH QXPEHU RI &'0$ XVHUV )RU D &'0$ XVHU ZKLFK LV DVVLJQHG WR RQH RI WKH FHOOV ZLWKLQ WKUHH OD\HUV RI WKH FHQWHU FHOO RI LQWHUHVW DW UDQGRP LWV QHDUIDU UDWLR LQ G% 3N3Ef ZLOO FRPH IURP WKH DSSUR[LPDWH 3') RI HTXDWLRQ f ZLWK DQ DGMXVWPHQW WR DFFRXQW IRU WKH GLIIHUHQFH LQ WKH YDOXHV RI (E1 IRU D &'0$ DQG D %36. XVHU DVVLJQHG WR WKH VDPH FHOO PAGE 74 :LWKRXW WKH &'0$ QRWFKLQJ WKDW KDV EHHQ PHQWLRQHG WKH DPRXQW RI &'0$ ORDGLQJ WKDW WKH QDUURZEDQG V\VWHP FDQ WROHUDWH LV SUDFWLFDOO\ ]HUR 7R LPSOHPHQW WKH QRWFKLQJ ZKHQ D &'0$ VLJQDO LV UHFHLYHG VXFK WKDW WKH %36. XVHUfV UHFHLYHG SRZHU LV OHVV WKDQ 7 G% DERYH WKDW &'0$ VLJQDO D QRWFK LV SODFHG LQ WKH &'0$ VLJQDO DQG WKH QHDUIDU UDWLR ZLWK WKH 3') JLYHQ E\ HTXDWLRQ f LV WKHQ UHGXFHG E\ DERXW G% 7KLV UHGXFWLRQ DVVXPHV WKDW WKH FRGH VHTXHQFHV ZLOO EH ILOWHUHG ZKHQ QHFHVVDU\ E\ XVLQJ WKH ')7EDVHG ILOWHULQJ IURP 6HFWLRQ ZLWK WKH ILOWHUHG FRGH VHTXHQFHV ]HURSDGGHG WR ELWV ,Q )LJXUH IRU D UDQJH RI YDOXHV IRU WKH QRWFKLQJ WKUHVKROG 7 DQG IRU VHYHUDO YDOXHV RI WKH GLIIHUHQFH (L1RfE fÂ§ ^(E1fF WKH DPRXQW RI &'0$ ORDGLQJ WROHUDEOH WR WKH %36. XVHU ZDV IRXQG VXFK WKDW WKH FULWHULRQ RI HTXDWLRQ f ZDV VDWLVILHG $V H[SHFWHG IRU ODUJHU YDOXHV RI (L-1TfW fÂ§ (L1RfF WKH 3') RI 3N3Ef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f VSHFWUD PXVW EH QRWFKHG RXW DQG KHQFH PDQ\ RI WKH &'0$ XVHUV ZRXOG KDYH WR EH GURSSHG 7KLV ZRXOG RFFXU QRW RQO\ LQ FDVHV IRU ZKLFK WKH QRWFKLQJ WKUHVKROG 7 LV ODUJH EXW ZRXOG DOVR RFFXU ZKHQ WKH TXDQWLW\ (E1RfE fÂ§ ^(L1TfF LV GHFUHDVHG RU HTXLYDOHQWO\ WKH YDOXH RI (E1RfF LV LQFUHDVHG PAGE 75 G% G% )LJXUH &'0$ XVHUVFHOO WROHUDEOH WR WKH %36. V\VWHP &'0$ XVHU LV QRWFKHG LI %36. SRZHU LV OHVV WKDQ 7 G% DERYH &'0$ SRZHU &XUYHV DUH IRU (E1RfE fÂ§ (E1RfF YDOXHV RI fÂ§ G% (LM1RfA DQG (E1Rf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fÂ§ (-1TfÂ fÂ§ (L-1TfFf :H ZLOO QRZ ORRN DW VHYHUDO GLIIHUHQW PXOWLFDUULHU &'0$ VFKHPHV DQG IRU HDFK WKH QXPEHU RI QDUURZEDQG XVHUV ZKLFK FDQ EH SUHVHQW VR WKDW WKH &'0$ XVHUV GR QRW UHTXLUH DQ H[FHVVLYH DPRXQW RI QRWFKLQJ ZLOO EH IRXQG $ GLIIHUHQW TXDQWLILFDWLRQ RI KRZ PXFK QRWFKLQJ LV H[FHVVLYH ZLOO EH JLYHQ IRU HDFK FDVH PAGE 76 /DWHU LQ WKLV FKDSWHU WKH SHUIRUPDQFH RI WKH &'0$ V\VWHP HPSOR\LQJ WKH 006( UHFHLYHU ZLOO EH LQYHVWLJDWHG 7KH VDPSOLQJ UDWH XVHG LQ WKH 006( LV 47& RU 7F LQ WKH VLQJOHFDUULHU FDVH 7KH &'0$ VLJQDOV ZLWK VSHFWUDOO\ HIILFLHQW SXOVH VKDSHV ZLOO W\SLFDOO\ EH FRQWDLQHG ZLWKLQ WKH IUHTXHQF\ UDQJH fÂ§7& 7Ff +RZHYHU WKH QRWFKLQJ FDQ UHDOO\ RQO\ EH GRQH XQLTXHO\ ZLWKLQ WKH UDQJH fÂ§7F I 7Ff DV D UHVXOW RI WKH VDPSOLQJ UDWH (DFK QRWFK ZLWKLQ WKLV UDQJH WKHUHIRUH JLYHV ULVH WR D VHFRQG QRWFK RXWVLGH RI WKLV UDQJH EXW VWLOO ZLWKLQ WKH UDQJH fÂ§ 7F I O7Ff 7KXV UHFDOOLQJ WKDW WKH '36. DQG &'0$ V\VWHPV KDYH WKH VDPH GDWD UDWH WKHUH DUH RQO\ XQLTXH QRWFKLQJ ORFDWLRQV ZKHQ WKH SURFHVVLQJ JDLQ LV FKLSVELW ,Q WKH VLQJOHFDUULHU FDVH D &'0$ XVHU ZLOO EH GURSSHG LI PRUH WKDQ KDOI RI LWV VLJQDO PXVW EH QRWFKHG RU XQLTXH QRWFKLQJ ORFDWLRQV 7KH GHQVLW\ RI QDUURZEDQG XVHUVFHOO IRU ZKLFK WKLV FULWHULRQ FDQ EH VDWLVILHG ZDV IRXQG IRU D UDQJH RI YDOXHV RI 7fÂ§ (E1RfEa ^(E1RfFf LQ G% DQG WKH UHVXOWV DSSHDU LQ )LJXUH 1RWLFH DV H[SHFWHG WKDW DV 7 JHWV ODUJHU DQG KHQFH PRUH QRWFKLQJ LV QHFHVVDU\ IHZHU QDUURZEDQG XVHUV FDQ EH SUHVHQW $OVR DV (E1fE fÂ§ (E1fFf JHWV ODUJHU WKH &'0$ XVHUV DUH OHVV OLNHO\ WR QHHG QRWFKHV DQG PRUH QDUURZEDQG XVHUV FDQ EH SUHVHQW ,Q )LJXUH D FRPELQDWLRQ RI WKH UHVXOWV RI )LJXUHV DQG LV VKRZQ IRU WKH VLQJOHFDUULHU FDVH 6HYHUDO WZRGLPHQVLRQDO FDSDFLW\ FXUYHV HDFK ZLWK D FRQVWDQW YDOXH RI (E1RfE fÂ§ (E1fFf ZHUH IRUPHG E\ ILQGLQJ WKH WROHUDEOH GHQVLWLHV RI &'0$ XVHUVFHOO IURP )LJXUH DQG QDUURZEDQG XVHUVFHOO IURP )LJXUH IRU D JLYHQ YDOXH RI WKH QRWFKLQJ WKUHVKROG 7 7KLV ZDV UHSHDWHG IRU D UDQJH RI YDOXHV RI 7 $V (E1fE fÂ§ (E1RfFf JHWV ODUJHU WKH &'0$ XVHUV DUH OHVV OLNHO\ WR LQWHUIHUH ZLWK WKH %36. XVHU DQG DUH OHVV OLNHO\ WR UHTXLUH QRWFKHV DQG WKXV PRUH XVHUV RI HDFK W\SH VKRXOG EH VXSSRUWDEOH ,W LV LPSRUWDQW WR NHHS LQ PLQG WKDW ZH KDYH QRW \HW FRQVLGHUHG KRZ PXFK VHOIn LQWHUIHUHQFH WKH &'0$ V\VWHP FDQ KDQGOH QRU WKH HIIHFWV RI QDUURZEDQG LQWHUIHUHQFH PAGE 77 ' 2 & X f & ;O e R Â‘fÂ§ t )LJXUH 1DUURZEDQG XVHUVFHOO WROHUDEOH WR &'0$ V\VWHP EHIRUH WRR PXFK QRWFKn LQJ LV UHTXLUHG 6KRZQ DUH VLQJOHFDUULHU DQG PXOWLFDUULHU FDVHV 7KH fQR QRWFKLQJf FXUYHV UHSUHVHQW FDVHV LQ ZKLFK D FDUULHU LV GURSSHG LI HYHQ RQH QRWFK LV QHFHVVDU\ 1% XVHUVFHOO Df 1% XVHUVFHOO Ef 1% XVHUVFHOO Ff )LJXUH 7ZRGLPHQVLRQDO FDSDFLW\ FXUYHV FRPELQLQJ )LJXUHV DQG /DEHOV RQ FXUYHV LQGLFDWH WKH YDOXH RI (Lf$URfLL fÂ§ (L1TfF 3URFHVVLQJ JDLQ LV FKLSVELW Df 6LQJOHFDUULHU FDVH Ef FDUULHU FDVH Ff FDUULHU FDVH PAGE 78 DQG QRWFKLQJ RQ &'0$ SHUIRUPDQFH :H KDYH RQO\ FRQVLGHUHG WZR WKLQJV ZKLFK QRQHWKHOHVV GR LPSRVH VRPH OLPLWDWLRQV RQ XVHU FDSDFLW\ WKH QXPEHU RI &'0$ XVHUV IRU ZKLFK WKH QDUURZEDQG V\VWHPfV SHUIRUPDQFH LV VHYHUHO\ GHJUDGHG DFFRUGLQJ WR WKH FULWHULRQ RI HTXDWLRQ f DQG WKH QXPEHU RI QDUURZEDQG XVHUV IRU ZKLFK WKH &'0$ XVHUV ZRXOG VLPSO\ UHTXLUH WRR PXFK QRWFKLQJ ,Q WKH QH[W VHFWLRQ ZH ZLOO ORRN DW WKH &'0$ SHUIRUPDQFH LQ PRUH GHWDLO :H QH[W FRQVLGHU XVLQJ PXOWLFDUULHU &'0$ ZLWK FDUULHUV DQG KHQFH D SURFHVVn LQJ JDLQ SHU SDWK RI FKLSVELW ,Q GHFLGLQJ KRZ PXFK QRWFKLQJ LV H[FHVVLYH KHUH ZH PXVW QRWH WKDW RQO\ XQLTXH QRWFKLQJ ORFDWLRQV DUH DYDLODEOH LQ HDFK FDUULHU DQG KHQFH LI WKHUH DUH PRUH WKDQ QRWFKHV UHTXLUHG LQ HLWKHU FDUULHU WKH &'0$ XVHU ZLOO EH GURSSHG :LWK RQO\ RQH RI WKH WZR FDUULHUV LQ XVH WKLV LV HTXLYDOHQW WR WKH VLQJOHn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fE fÂ§ (E1RfFf +RZHYHU WKHUH LV QR VLJQLILFDQW FRQn FOXVLRQ WR EH GUDZQ IURP WKLV DQG WKHVH UHVXOWV DUH RQO\ SUHVHQWHG IRU IXWXUH XVH ,W ZLOO EH VHHQ ODWHU WKDW ZKHQ &'0$ UHFHLYHU SHUIRUPDQFH LV WDNHQ LQWR DFFRXQW WKH PXOWLFDUULHU V\VWHPV DUH DEOH WR RSHUDWH DW PXFK ORZHU YDOXHV RI (E1RfF WKDQ FDQ WKH VLQJOHFDUULHU FDVH DV D UHVXOW RI WKH DGGHG IUHTXHQF\ GLYHUVLW\ 7KXV LW ZRXOG EH PRUH IDLU WR FRPSDUH D FXUYH LQ WKH VLQJOHFDUULHU FDVH ZLWK RQH RI WKRVH IURP WKH PXOWLFDUULHU FDVH ZLWK D VLJQLILFDQWO\ ODUJHU YDOXH RI (E1RfE fÂ§ (E1fFf ZKLFK ZRXOG OHDQ PRUH IDYRUDEO\ WRZDUG WKH PXOWLFDUULHU FDVH )LQDOO\ ZH FRQVLGHU XVLQJ FDUULHUV ZKLFK PHDQV WKDW WKH &'0$ VLJQDOV ZLOO KDYH D SURFHVVLQJ JDLQ SHU SDWK RI FOLLSVELW 7KXV D FDUULHU ZLOO EH GURSSHG LI PAGE 79 PRUH WKDQ QRWFKHV DUH QHFHVVDU\ ,Q FRQWUDVW WR WKH FDUULHU FDVH LI RQO\ RQH RI WKH FDUULHUV PXVW EH GURSSHG RI WKH FDUULHUV ZRXOG VWLOO UHPDLQ DQG LW PLJKW EH SRVVLEOH WKDW WKH &'0$ UHFHLYHU FRXOG VWLOO IXQFWLRQ UHOLDEO\ 7KLV ZLOO EH ORRNHG DW LQ 6HFWLRQ EXW IRU QRZ WKH FULWHULRQ WKDW ZH ZLOO HPSOR\ LV WKDW D &'0$ XVHU ZLOO EH GURSSHG LI DW OHDVW RI WKH FDUULHUV KDYH PRUH WKDQ QRWFKHV 7KHVH UHVXOWV DUH VKRZQ LQ )LJXUHV DQG DV ZHOO ,Q )LJXUH WKHUH LV DQ LPSURYHPHQW RYHU WKH FDUULHU FDVH LQ ZKLFK WKH &'0$ XVHU ZDV GURSSHG HYHQ LI RQO\ RQH RI WKH FDUULHUV QHHGHG VLJQLILFDQW QRWFKLQJ ,W KDV EHHQ VXJJHVWHG SUHYLRXVO\ WKDW ZLWK WKH XVH RI PXOWLFDUULHU &'0$ LW LV SRVVLEOH WR DYRLG WKH QDUURZEDQG XVHUV LQ DQ RYHUOD\ VFHQDULR E\ VLPSO\ QRW WUDQVn PLWWLQJ RQ WKRVH FDUULHUV ZKLFK PLJKW LQWHUIHUH ZLWK D QDUURZEDQG XVHU >@ 7KLV SRVVLELOLW\ ZDV DOVR H[DPLQHG KHUH IRU FRPSDULVRQ :H ZLOO ILUVW XWLOL]H WKH SUHYLRXV QRWFKLQJ FULWHULD DQG GHFODUH WKDW LI WKH &'0$ VLJQDO RQ D JLYHQ FDUULHU LV UHFHLYHG DW D KLJK SRZHU OHYHO LQ WHUPV RI WKH QRWFKLQJ WKUHVKROG 7 WKH FDUULHU ZLOO QRW EH XVHG DQDORJRXV WR VLPSO\ SODFLQJ D QRWFK LQ WKH SUHYLRXV VFHQDULRV :H ZLOO FRQVLGHU ERWK DQG FDUULHUV LQ WKLV W\SH RI V\VWHP ,W GRHV QRW VHHP IUXLWIXO WR UDLVH WKH QXPEHU RI FDUULHUV EH\RQG ZLWK D FRPSRVLWH SURFHVVLQJ JDLQ RI FKLSVELW IRU VHYHUDO UHDVRQV )LUVW HYHQ DVVXPLQJ WKDW WKH &'0$ VLJQDO RQ HDFK FDUULHU ZRXOG H[SHULHQFH LQGHSHQGHQW IDGLQJ WKH LQFUHPHQWDO GLYHUVLW\ DGYDQWDJH UHn DOL]HG E\ XVLQJ PRUH FDUULHUV GLPLQLVKHV ZLWK VXFK D KLJK QXPEHU RI FDUULHUV 6HFRQG LW KDV EHHQ DVVXPHG WKDW WKH &'0$ V\VWHP ZKHQ XVHG ZLWK PXOWLSOH FDUULHUV FDQ EH VSOLW VXFK WKDW WUDQVPLVVLRQ WDNHV SODFH RQ GLVMRLQW IUHTXHQF\ EDQGV VR WKDW LQGHn SHQGHQW IDGLQJ FDQ EH UHDOL]HG :KLOH WKLV VHHPV SODXVLEOH ZLWK RU HYHQ FDUULHUV WKH SRVVLELOLW\ EHFRPHV OHVV OLNHO\n ZLWK D ODUJH QXPEHU RI FDUULHUV 7KXV VRPH RI WKH FDUULHUV ZLOO XQGHUJR FRUUHODWHG IDGLQJ ZKLFK ZLOO QRW JLYH QHDUO\ DV PXFK GLYHUVLW\ DV ZLOO LQGHSHQGHQW IDGLQJ )LQDOO\ WKH SURFHVV RI WUDFNLQJ WKH IDGLQJ SURFHVVHV ZKLFK LV PAGE 80 QRW FRQVLGHUHG LQ WKLV SDSHU ZRXOG EHFRPH WUHPHQGRXVO\ FRPSOLFDWHG ZLWK VR PDQ\ FDUULHUV ,W ZDV REVHUYHG LQ ODWHU VLPXODWLRQV WKDW WKH &'0$ V\VWHP VXIIHUHG D ODUJH SHUn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n WKHU 5HVXOWV IURP 6HFWLRQV DQG ZLOO EH H[WHQGHG WR LQFOXGH WKH HIIHFWV RQ &'0$ SHUIRUPDQFH RI 0$, 1%, QRWFKLQJ DQG WKH SRVVLELOLW\ RI RSHUDWLQJ RQ IHZHU FDUULHUV WKDQ WKH PD[LPXP :H FRQVLGHU WKH XVH RI WKH 006( UHFHLYHU ZKLFK LV ZHOOVXLWHG WR WKH RYHUOD\ HQYLURQPHQW 6HYHUDO RI LWV GHVLUDEOH SURSHUWLHV ZHUH PHQn WLRQHG LQ 6HFWLRQ VXFK DV LWV DELOLW\ WR UHMHFW 0$, 1%, DQG ,6, DV ZHOO DV LWV DELOLW\ WR DGDSW WR WKH GHVLUHG XVHUfV VLJQDO ZLWKRXW HYHQ NQRZLQJ WKDW LWV FRGH KDV EHHQ ILOWHUHG ,W ZDV VKRZQ E\ 0LOOHU HW DO >@ WKDW WKH 006( FDQ VXFFHVVIXOO\ UHn DOL]H GLYHUVLW\ LQ D IUHTXHQF\VHOHFWLYH IDGLQJ FKDQQHO DQG WKDW WKHUH LV D VXEVWDQWLDO SHUIRUPDQFH ORVV ZKHQ DOO RI WKH SDWKV RI DOO RI WKH LQWHUIHULQJ XVHUV DUH QRW WUDFNHG H[SOLFLWO\ LQ IRUPLQJ WKH :LHQHU VROXWLRQ 7KH SHUIRUPDQFH RI WKH 006( ZDV HYDOXn DWHG LQ WKH PXOWLFDUULHU FDVH E\ 0LOOHU DQG 5DLQEROW >@ DQG LW ZDV UHDIILUPHG WKDW PAGE 81 DOO RI WKH SDWKV RI DOO RI WKH &'0$ XVHUV PXVW EH WUDFNHG LQ RUGHU WR DYRLG D VL]DEOH SHUIRUPDQFH ORVV 7KH SHUIRUPDQFH LQ WKH RYHUOD\ HQYLURQPHQW LV D UHODWLYHO\ VWUDLJKWIRUZDUG H[n WHQVLRQ RI WKH DQDO\VHV JLYHQ E\ 0LOOHU HW DO >@ DQG E\ 0LOOHU DQG 5DLQEROW >@ 7KH GHVLUHG &'0$ XVHU QRZ PD\ RSHUDWH ZLWK D ILOWHUHG FRGH VHTXHQFH DQG QDUn URZEDQG QRLVH PXVW EH DGGHG ZKHQ QHFHVVDU\ :H FRQVLGHU D JHQHUDO PXOWLFDUULHU V\VWHP ZLWK 4 FDUULHUV DQG D SURFHVVLQJ JDLQ SHU SDWK RI 14 ,W LV DOVR DVn VXPHG WKDW WKH GHVLUHG XVHUfV FRGH VHTXHQFH LQ WKH TWK FDUULHU ZLOO EH ILOWHUHG ZKHQ QHFHVVDU\f XVLQJ WKH ')7EDVHG ILOWHULQJ PHWKRG IURP 6HFWLRQ ZKLFK UHVXOWV LQ D FRGH VHTXHQFH RI OHQJWK JUHDWHU WKDQ WKH SURFHVVLQJ JDLQ 14 LQ WKLV FDVH ,W ZLOO EH H[SUHVVHG DV WKH FDVFDGH RI / LQGLYLGXDO VHTXHQFHV RI OHQJWK 14 DV FL &-/! f f f F/L!F/R &8! f f f f FOT/f7 ZLWK FLR LQ WKH PLGGOH DVVRFLn DWHG ZLWK WKH GHVLUHG FRPSRQHQW DQG WKH RWKHU VHTXHQFHV FRUUHVSRQGLQJ WR ,6, 1RWH WKDW RQ HDFK FDUULHU WKH ILOWHUHG FRGH VHTXHQFH ZLOO EH GLIIHUHQW DV WKH QRWFKLQJ QHFn HVVDU\ RQ HDFK FDUULHU LV JHQHUDOO\ QRW WKH VDPH $OVR IRU VLPSOLFLW\ LW ZDV YHULILHG LQ 6HFWLRQ WKDW WKHUH LV QR VLJQLILFDQW GLIIHUHQFH LQ WKH UHFHSWLRQ RI WKH GHVLUHG XVHU LI WKH LQWHUIHULQJ &'0$ XVHUV DUH OHIW XQILOWHUHG )RU WKH JWK FDUULHU WKH VDPSOHV RI D FKLSPDWFKHGILOWHU EDQN ZLOO EH FROOHFWHG GXULQJ WKH ]WK ELW LQWHUYDO UHVXOWLQJ LQ D FROXPQYHFWRU RI 14 VDPSOHV JLYHQ E\ / Urf nKT^LfGL^LfFKT LrfGLr r1fFLP QT^Lf MLf P fÂ§/ Â -UUnIN$Lf GNLfIN GNL LfJIF@ N n f ZKHUH IFIf LV WKH IDGLQJ SURFHVV RQ WKH TWK FDUULHU RI WKH NWK XVHU GXULQJ WKH LWK ELW LQWHUYDO DQG WKH IDGLQJ SURFHVVHV IRU WKH VDPH XVHU RQ HDFK FDUULHU DUH LQGHSHQGHQW $OVR DQG JN DUH WKH HYHQ DQG RGG F\FOLF VKLIWV RI WKH NWK XVHUfV FRGH VHTXHQFH ZKLFK ZHUH GHILQHG LQ HTXDWLRQ f DQG QTLf LV D YHFWRU RI OHQJWK 14 RI LQGHSHQGHQW PAGE 82 FRPSOH[ *DXVVLDQ QRLVH VDPSOHV ZLWK WKH UHDO DQG LPDJLQDU\ SDUWV HDFK KDYLQJ YDULDQFH RI Dn 1(E1Rf 7KH YHFWRU MÂf FRQVLVWV RI WKH VXP RI VDPSOHV RI DOO RI WKH QDUURZEDQG QRLVH SURFHVVHV SUHVHQW LQ WKH \WK FDUULHU LI DQ\ (DFK SURFHVV LV FRPSOH[ ZLWK WKH UHDO DQG LPDJLQDU\ SDUWV LQGHSHQGHQW DQG HDFK ZLWK D FRUUHODWLRQ PDWUL[ JLYHQ E\ IRU WKH PWOL QDUURZEDQG XVHUf 6D FRV47F$X!L fÂ§ Mff f IRU WKH ÂMfWK HOHPHQW ZKHUH 3c3?f LV WKH QDUURZEDQGWR&'0$ QHDUIDU UDWLR DQG $X LV WKH IUHTXHQF\ GLIIHUHQFH EHWZHHQ WKH ORFDWLRQ RI WKH QDUURZEDQG XVHU DQG WKH &'0$ FDUULHU IUHTXHQF\ ,W ZDV VKRZQ E\ 0LOOHU DQG 5DLQEROW >@ WKDW WKH UHFHLYHU ZLOO ZRUN EHVW LI WKH 4 GLIIHUHQW UHFHLYHG YHFWRUV RI HTXDWLRQ f DUH FDVFDGHG LQWR D VLQJOH FRPSRVLWH YHFWRU RI OHQJWK 1 JLYHQ E\ Lnnf U_ Lf U Lf 7QLf\ f DQG D VLQJOH :LHQHU ILOWHU LV IRUPHG JLYHQ E\ ZÂf 5B]fSÂf ZLWK 5Âf DQG SIf WKH FRUUHODWLRQ PDWUL[ DQG VWHHULQJ YHFWRU 7KH FRPSRVLWH FRUUHODWLRQ PDWUL[ 5]f LV JLYHQ E\ 5 5XIf 5O fff 584 5O 5Lrf 5L4L 5TL 5Trf Â‘ Â‘ Â‘ 5T4 DQG WKH VXEPDWULFHV DUH JLYHQ E\ f 50 ( >USLfULf@ f PAGE 83 7KH FRPSRVLWH VWHHULQJ YHFWRU LV JLYHQ E\ 3 3L (>GL r!rnf@ 3Â ( >G? "fUrf@ B3J B( >Ar! rf@ f 7KH ELW GHFLVLRQ LV WKHQ PDGH DV G?Lf VLJQ5H>ZÂfU]f@f IRU FRKHUHQW FRPELQLQJ RI WKH SDWKV ,W PDNHV VHQVH WR XVH FRKHUHQW FRPELQLQJ LQ WKLV FDVH DV LW ZDV SUHYLRXVO\ VWDWHG WKDW DOO RI WKH IDGLQJ SDWKV RI DOO RI WKH XVHUV ZRXOG EH WUDFNHG DQ\ZD\ LQ RUGHU WR DYRLG D ODUJH SHUIRUPDQFH ORVV 7KH WDVN RI WUDFNLQJ WKH IDGLQJ SURFHVVHV LQ D G\QDPLF HQYLURQPHQW LV FXUUHQWO\ DQ DUHD RI DFWLYH UHVHDUFK ,Q WKLV ZRUN LW ZLOO EH DVVXPHG WKDW DOO RI WKH IDGLQJ SURFHVVHV RQ DOO RI WKH SDWKV DUH NQRZQ LQ ZKLFK FDVH ZH KDYH / OS:Orf&OSR&OLUf n\ @ OSrfOrf&OSP&OLLP 7 P fÂ§ / "O Â MUf IFSrfÂÂf >WIM JIFJr@ FW,Q[ Â 5-LP@ 6ST N n ?P f ZKHUH WKHUH DUH 0 QDUURZEDQG XVHUV SUHVHQW RQ WKH FDUULHU IRU ZKLFK S T $OVR ,Q4[1T LV DQ LGHQWLW\ PDWUL[ RI GLPHQVLRQ 14 7KH TWK VWHHULQJ YHFWRU LV 3 LUrfFLR f ,I WKH H[SUHVVLRQ IRU WKH TWK UHFHLYHG YHFWRU LQ HTXDWLRQ f LV ZULWWHQ DV U LLGLFLLR UWf f 0 ZKHUH "TLf UHSUHVHQWV D FRPSRVLWH LQWHUIHUHQFH SURFHVV FRQVLVWLQJ RI 0$, $:*1 1%, DQG ,6, LW FDQ EH VKRZQ ZLWK WKH PDWUL[LQYHUVLRQ OHPPD WKDW WKH SUREDELOLW\ PAGE 84 RI HUURU FRQGLWLRQHG RQ WKH IDGLQJ SURFHVVHV ZKHQ FRKHUHQW FRPELQLQJ RI WKH SDWKV LV XVHG LV JLYHQ E\ >@ 3UHUURUUrff 4 S+ LfS+ Lf? f ZKHUH 7Lf LV D YHFWRU UHSUHVHQWLQJ DOO RI WKH IDGLQJ SURFHVVHV RI DOO RI WKH XVHUV GXULQJ WKH LWK ELW LQWHUYDO DQG 5Lf (>UÂfI"nf@ WKH FRPSRVLWH LQWHUIHUHQFH FRUUHODWLRQ PDWUL[ 7KH SHUIRUPDQFH RI WKH &'0$ V\VWHP XVLQJ DQ 006( GHWHFWRU ZDV WKHQ VLPXn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f f )RU D JLYHQ UHDOL]DWLRQ RI FRGHV GHOD\V QRWFKLQJ DQG SRZHUV RI WKH &'0$ XVHUV DQG RI SRZHUV DQG IUHTXHQF\ ORFDWLRQV RI WKH QDUURZEDQG XVHUV GLIIHUHQW UHDOn L]DWLRQV RI WKH IDGLQJ SURFHVVHV ZHUH JHQHUDWHG WKDW LV UHDOL]DWLRQV RI UIf DQG WKH FRQGLWLRQDO SUREDELOLW\ RI HUURU RI WKH &'0$ V\VWHP ZDV IRXQG XVLQJ HTXDWLRQ f 7KH DYHUDJH SUREDELOLW\ RI HUURU ZDV WKHQ IRXQG E\ DYHUDJLQJ WKHVH YDOn XHV 7KLV SURFHVV ZDV UHSHDWHG PDQ\ WLPHV WKDW LV IRU GLIIHUHQW UHDOL]DWLRQV RI WKH FRGHV GHOD\V DQG VR RQ LQ RUGHU WR JLYH HQRXJK YDOXHV IRU WKH DYHUDJH SUREDELOLW\ RI HUURU VR WKDW WKH FULWHULRQ RI HTXDWLRQ f FRXOG EH WHVWHG 7KXV LI D &'0$ XVHUfV DYHUDJH SUREDELOLW\ RI HUURU ZDV JUHDWHU WKDQ b LW ZDV GURSSHG DQG LI WKHVH GURSV PAGE 85 1DUURZEDQG XVHUVFHOO )LJXUH 7ZRGLPHQVLRQDO FDSDFLW\ FXUYHV WDNLQJ LQWR DFFRXQW ERWK &'0$ UHn FHLYHU SHUIRUPDQFH DQG UHVXOWV IURP )LJXUH 3URFHVVLQJ JDLQ LV FKLSVELW RFFXUUHG PRUH WKDQ b RI WKH WLPH WKHQ WKH &'0$ V\VWHP ZDV DERYH LWV FDSDFLW\ OLPLWV ,Q RUGHU WR FRPELQH WKHVH UHVXOWV ZLWK WKRVH RI 6HFWLRQV DQG WKH QRWFKLQJ WKUHVKROG 7 DQG WKH YDOXH RI (L1TfF PXVW EH FKRVHQ VR DV WR VDWLVI\ WKH FDSDFLW\ FXUYHV VKRZQ LQ )LJXUH )RU H[DPSOH LQ WKH VLQJOHFDUULHU FDVH LI WKHUH DUH QDUURZEDQG XVHUV SUHVHQW WKH V\VWHP FDQ VXSSRUW &'0$ XVHUV LI WKH YDOXH RI (E1RfE fÂ§ (E1RfF G% RU HTXLYDOHQWO\ (Lf1RfF G% %XW LI WKH &'0$ V\VWHP LV ORDGHG XS WR WKLV OHYHO LW ZLOO UHTXLUH PXFK PRUH WKDQ (E1RfF G% WR VDWLVI\ WKH FULWHULRQ RI HTXDWLRQ f 7KXV D KLJKHU YDOXH RI eÂ$URfF PXVW EH XVHG DQG KHQFH D ORZHU YDOXH RI (E1RfL fÂ§ (E1fF ZKLFK PHDQV WKDW ZH PXVW RSHUDWH RQ RQH RI WKH FXUYHV VKRZLQJ D ORZHU MRLQW FDSDFLW\ ,Q )LJXUH WKH UHVXOWV RI FRPELQLQJ WKH FULWHULRQ RI HTXDWLRQ f ZLWK WKH UHVXOWV RI 6HFWLRQV DQG DUH VKRZQ 6LPLODUO\ WR WKH H[DPSOH MXVW PHQWLRQHG WKH YDOXH RI (E1RfF ZDV RSWLPL]HG DW HDFK ORDGLQJ OHYHO 7KHUH LV D QRWLFHDEOH LPSURYHPHQW LQ FDSDFLW\ RYHU WKH VLQJOHFDUULHU FDVH ZKHQ FDUULHUV DUH XVHG DQG D PAGE 86 VOLJKW DGGLWLRQDO LQFUHDVH ZKHQ FDUULHUV DUH XVHG 5HFDOO WKDW LQ WKH PXOWLFDUULHU VFHQDULRV D VPDOOHU YDOXH RI (E1Rf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n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f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f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fV IXQFWLRQDOLW\ LQ WKH RYHUOD\ VFHQDULR ZDV GHPRQVWUDWHG WKURXJKRXW WKH GLVVHUWDWLRQ LQ ERWK VLQJOHFHOO DQG FHOOXODU VFHQDULRV DQG ZLWK ERWK D *DXVVLDQ FKDQQHO PRGHO DQG D IDGLQJ FKDQQHO PRGHO 7KH DSSOLFDWLRQ IRU ZKLFK RYHU OD`n KDV SHUKDSV WKH PRVW SRWHQWLDO LV WKH FHOOXODU VFHQDULR 7KH LPSOHPHQWDWLRQ RI RYHUOD\ FRXOG SURYLGH DQ HIILFLHQW ZD\ IRU D IUHn TXHQF\ EDQG ZKLFK VHUYLFHV QDUURZEDQG XVHUV WR WUDQVLWLRQ LWV VHUYLFH WR &'0$ )RU FRQWUDFWXDO UHDVRQV SURYLGHUV FDQQRW VLPSO\ GLVFRQWLQXH VHUYLFH WR QDUURZEDQG VXEn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fV UHFHLYHU DQWHQQD D VKRUW GLVWDQFH DZD\ RQ WKH VDPH WRZHU ZRXOG UHVXOW LQ DQ XQPDQDJHDEOH QHDUIDU LQWHUIHUHQFH SUREOHP $QRWKHU FRQFOXVLRQ UHDFKHG ZDV WKDW WKH QRWFKLQJ LV PXFK OHVV RI D SUREOHP IRU WKH &'0$ V\VWHP WKDQ LV WKH UHFHSWLRQ RI VWURQJ QDUURZEDQG LQWHUIHUHQFH 7KH VLJQDOV FRXOG EH QRWFKHG VR WKDW DERXW b RI WKHLU VSHFWUXP ZDV PLVVLQJ DQG VXIIHU RQO\ D PRGHVW DPRXQW RI GHJUDGDWLRQ 1DUURZEDQG LQWHUIHUHQFH KRZHYHU ZDV VKRZQ WR FDXVH VLJQLILFDQW SUREOHPV LI WRR VWURQJ )LQDOO\ FHOOXODU RYHUOD\ ZDV H[DPLQHG LQ WKH IDGLQJ FKDQQHO 2QFH DJDLQ LW ZDV FRQFOXGHG WKDW WKH &'0$ XVHUV PXVW HPSOR\ QRWFKLQJ LQ RUGHU WR DYRLG WKH QDUURZ EDQG V\VWHP %XW LW ZDV DOVR IRXQG WKDW WKH &'0$ VLJQDOV UHTXLUHG D ODUJH DPRXQW PAGE 90 RI QRWFKLQJ 7KLV LPSRVHG GLVDSSRLQWLQJ FRQVWUDLQWV RQ WKH MRLQW FDSDFLW\ RI WKH V\Vn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n VHQWHG LQ WKLV GLVVHUWDWLRQ KDYH FOHDUO\ LQGLFDWHG WKDW RYHUOD\ KDV VLJQLILFDQW SRWHQWLDO 7KH QH[W VWHS LV WR ORRN DW VRPH RI WKH LPSOHPHQWDWLRQ LVVXHV WKDW ZHUH QRW DGGUHVVHG 7KURXJKRXW WKH UHVHDUFK WZR LPSRUWDQW WDVNV ZHUH DVVXPHG WR KDYH EHHQ GRQH SHUIHFWO\ WKH WLPLQJ DFTXLVLWLRQ WKDW LV WKH NQRZOHGJH DERXW ZKHQ WKH GHVLUHG &'0$ XVHUfV FRGH VHTXHQFH EHJLQV DQG WKH HVWLPDWLRQ RI WKH IDGLQJ SURFHVVHV RI WKH &'0$ XVHUV 7KH\ DUH FHUWDLQO\ LPSRUWDQW SUREOHPV WKDW QHHG WR EH DGGUHVVHG LQ IXWXUH ZRUN 7KH UHVXOWV SUHVHQWHG LQ WKLV GLVVHUWDWLRQ KDG RQO\ WKH JRDO RI GHPRQVWUDWLQJ WKDW WKHUH LV LQGHHG D UHDVRQ WR SXUVXH RYHUOD\ ,I WKH UHVXOWV KDG GHPRQVWUDWHG RWKHUn ZLVH WKHQ WKH HIIHFWV RI LPSHUIHFW HVWLPDWLRQ RI WKH WLPLQJ DQG WKH IDGLQJ SDUDPHWHUV ZRXOG QRW EH LQWHUHVWLQJ RU PHDQLQJIXO 7KH WLPLQJ DFTXLVLWLRQ SUREOHP KDV EHHQ LQYHVWLJDWHG E\ PDQ\ UHVHDUFKHUV DQG WKHUH H[LVWV DQ H[WHQVLYH ERG\ RI OLWHUDWXUH RQ WKH VXEMHFW >@ 2EYLRXVO\ WKHVH PAGE 91 ZRUNV KDYH QRW FRQVLGHUHG WKH HIIHFWV RI WKH QRWFKILOWHULQJ SUHVHQWHG LQ WKLV GLVVHUWDn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f&URVVFRUUHODWLRQ SURSHUWLHV RI SVHXGRUDQGRP DQG UHODWHG VHTXHQFHVf 3URF ,((( YRO SS 0D\ >@ 0 % 3XUVOH\ f3HUIRUPDQFH HYDOXDWLRQ IRU SKDVHFRGHG VSUHDGVSHFWUXP PXOWLSOHDFFHVV FRPPXQLFDWLRQ SDUW V\VWHP DQDO\VLVf ,((( 7UDQV &RP PXQ YRO &20 SS $XJ >@ PAGE 93 >@ 0 $EGXOUDKPDQ $ 6KHLNK DQG )DOFRQHU f'HFLVLRQ IHHGEDFN HTXDOL]DWLRQ IRU &'0$ LQ LQGRRU ZLUHOHVV FRPPXQLFDWLRQVf ,((( 6HOHFW $UHDV &RPPXQ YRO SS 0DMn >@ & 1 3DWHURV DQG 6DXOQLHU f$Q DGDSWLYH FRUUHODWRU UHFHLYHU IRU GLUHFW VHTXHQFH VSUHDGVSHFWUXP FRPPXQLFDWLRQf ,((( 7UDQV &RPPXQ YRO SS 1RY >@ 6 / 0LOOHU f7UDLQLQJ DQDO\VLV RI DGDSWLYH LQWHUIHUHQFH VXSSUHVVLRQ IRU GLUHFW VHTXHQFH FRGHGLYLVLRQ PXOWLSOHDFFHVV V\VWHPVf ,((( 7UDQV &RPPXQ YRO SS $SU >@ 0 +RQLJ 8 0DGKRZ DQG 6 9HUG f%OLQG DGDSWLYH PXOWLXVHU GHWHFWLRQf ,((( 7UDQV ,QIRUP 7KHRU\ YRO SS -XO\ >@ / % 0LOVWHLQ / 6FKLOOLQJ 5 / 3LFNKROW] 9 (UFHJ 0 .XOOEDFN ( .DQWHUDNLV 6 )LVKPDQ : + %LHGHUPDQ DQG & 6DOHUQR f2Q WKH IHDVLELOLW\ RI D &'0$ RYHUOD\ IRU SHUVRQDO FRPPXQLFDWLRQ QHWZRUNVf ,((( 6HOHFW $UHDV &RPPXQ YRO SS 0D\ >@ : .HWFKXP DQG 3URDNLV f$GDSWLYH DOJRULWKPV IRU HVWLPDWLQJ DQG VXSSUHVVLQJ QDUURZEDQG LQWHUIHUHQFH LQ 31 VSUHDGVSHFWUXP V\VWHPVf ,((( 7UDQV &RPPXQ YRO &20 SS 0D\ >@ / /L DQG / % 0LOVWHLQ f5HMHFWLRQ RI QDUURZEDQG LQWHUIHUHQFH LQ 31 VSUHDG VSHFWUXP V\VWHPV XVLQJ WUDQVYHUVDO ILOWHUVf ,((( 7UDQV &RPPXQ YRO &20 SS 0D\ >@ 5 $ OLWLV DQG / % 0LOVWHLQ f3HUIRUPDQFH DQDO\VLV RI QDUURZEDQG LQWHUIHUHQFH UHMHFWLRQ WHFKQLTXHV LQ '6 VSUHDG VSHFWUXP V\VWHPVf ,((( 7UDQV &RPPXQ YRO &20 SS 1RY >@ / % 0LOVWHLQ f,QWHUIHUHQFH UHMHFWLRQ WHFKQLTXHV LQ VSUHDG VSHFWUXP FRPPXQLn FDWLRQVf 3URF ,((( YRO SS -XQH >@ 0 ( 'DYLV 6LJQDO SURFHVVLQJ IRU LQWHUIHUHQFH DYRLGDQFH DQG PXOWLSOHDFFHVV QRLVH UHMHFWLRQ LQ '6&'0$ 3K' GLVVHUWDWLRQ 8QLYHUVLW\ RI &DOLIRUQLD 6DQ 'LHJR >@ 5 / 3LFNKROW] / % 0LOVWHLQ DQG / 6FKLOOLQJ f6SUHDG VSHFWUXP IRU PRELOH FRPPXQLFDWLRQVf ,((( 7UDQV 9HK 7HFKQRO YRO SS 0D\ >@ + 9 3RRU DQG ; :DQJ f&RGHDLGHG LQWHUIHUHQFH VXSSUHVVLRQ IRU '6&'0$ FRPPXQLFDWLRQV SDUW LQWHUIHUHQFH VXSSUHVVLRQ FDSDELOLW\f ,((( 7UDQV &RPn PXQ YRO SS 6HSW >@ 9 *DUJ ) 6PROLN DQG ( :LONHV $SSOLFDWLRQV RI &'0$ LQ :LUHn OHVV3HUVRQDO &RPPXQLFDWLRQV 3UHQWLFH+DOO 8SSHU 6DGGOH 5LYHU 1 PAGE 94 >@ 7 6 5DSSDSRUW :LUHOHVV &RPPXQLFDWLRQV 3UHQWLFH+DOO 8SSHU 6DGGOH 5LYHU 1>@ 5 -RKDQQHVRQ DQG ( 3DDVNH f)XUWKHU UHVXOWV RQ ELQDU\ FRQYROXWLRQDO FRGHV ZLWK DQ RSWLPXP GLVWDQFH SURILOHf ,((( 7UDQV ,QIRUP 7KHRU\ YRO ,7 QR SS 0DU >@ 6 .RQGR DQG / % 0LOVWHLQ f3HUIRUPDQFH RI PXOWLFDUULHU '6&'0$ V\VWHPVf ,((( 7UDQV &RPPXQ YRO QR SS )HE >@ 6 / 0LOOHU 0 / +RQLJ DQG / % 0LOVWHLQ f3HUIRUPDQFH DQDO\VLV RI 006( UHFHLYHUV IRU '6&'0$ LQ IUHTXHQF\ VHOHFWLYH IDGLQJ FKDQQHOVf VXEPLWWHG WR ,((( 7UDQV &RPPXQ >@ 6 / 0LOOHU DQG % 5DLQEROW f006( GHWHFWLRQ RI PXOWLFDUULHU &'0$f VXEPLWWHG WR ,((( ,QWHUQDWLRQDO &RQIHUHQFH RQ &RPPXQLFDWLRQV >@ ( 6WURP 6 3DUNYDOO 6 / 0LOOHU DQG % ( 2WWHUVWHQ f3URSDJDWLRQ GHOD\ HVWLPDWLRQ LQ DV\QFKURQRXV GLUHFWVHTXHQFH FRGHGLYLVLRQ PXOWLSOH DFFHVV V\Vn WHPVf ,((( 7UDQV &RPPXQ YRO QR SS -DQ >@ 6 ( %HQVOH\ DQG % $D]KDQJ f6XEVSDFHEDVHG FKDQQHO HVWLPDWLRQ IRU FRGH GLYLVLRQ PXOWLSOH DFFHVV FRPPXQLFDWLRQ V\VWHPVf ,((( 7UDQV &RPPXQ YRO SS $XJ >@ 8 0DGKRZ f%OLQG DGDSWLYH LQWHUIHUHQFH VXSSUHVVLRQ IRU WKH QHDUIDU UHVLVWDQW DFTXLVLWLRQ DQG GHPRGXODWLRQ RI GLUHFWVHTXHQFH &'0$ VLJQDOVf ,((( 7UDQV 6LJQDO 3URFHVVLQJ YRO QR SS -DQ >@ =KHQJ /L 6 / 0LOOHU DQG ( 6WURP f$Q HIILFLHQW FRGHWLPLQJ HVWLPDWRU IRU '6&'0$ VLJQDOVf ,((( 7UDQV 6LJQDO 3URFHVVLQJ YRO QR SS -DQ >@ = 6 /LX /L DQG 6 / 0LOOHU f$Q HIILFLHQW FRGHWLPLQJ HVWLPDWRU IRU UHFHLYHU GLYHUVLW\ '6&'0$ V\VWHPVf ,((( 7UDQV &RPPXQ YRO QR SS -XQH >@ 5 ) 6PLWK DQG 6 / 0LOOHU f$FTXLVLWLRQ SHUIRUPDQFH RI DQ DGDSWLYH UHFHLYHU IRU '6&'0$ V\VWHPVf VXEPLWWHG WR ,((( 7UDQV &RPPXQ >@ $ 1 %DUERVD DQG 6 / 0LOOHU f$GDSWLYH GHWHFWLRQ RI '6&'0$ VLJQDOV LQ IDGLQJ FKDQQHOVf ,((( 7UDQV &RPPXQ YRO QR SS -DQ >@ ; :DQJ DQG + 9 3RRU f%OLQG PXOWLXVHU GHWHFWLRQ D VXEVSDFH DSSURDFKf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n DEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 6FRWW / 0LOOHU &KDLUPDQ $VVRFLDWH 3URIHVVRU RI (OHFWULFDO DQG &RPSXWHU (QJLQHHULQJ FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWn DEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ /HRQ : &RXFK ,, 3URIHVVRU RI (OHFWULFDO DQG &RPSXWHU (QJLQHHULQJ FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWn DEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ +DQLSK $ /DWFKPDQ $VVRFLDWH 3URIHVVRU RI (OHFWULFDO DQG &RPSXWHU (QJLQHHULQJ FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWn DEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ -LDQ /L $VVRFLDWH 3URIHVVRU RI (OHFWULFDO DQG &RPSXWHU (QJLQHHULQJ FHUWLI\ WKDW KDYH UHDG WKLV VWXG`nn DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWn DEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 8OULFK + .XU]ZHJ 3URIHVVRU RI $HURVSDFH (QJLQHHULQJ 0HFKDQLFV DQG (QJLQHHULQJ 6FLHQFH PAGE 97 7KLV GLVVHUWDWLRQ ZDV VXEPLWWHG WR WKH *UDGXDWH )DFXOW\ RI WKH &ROOHJH RI (Qn JLQHHULQJ DQG WR WKH *UDGXDWH 6FKRRO DQG ZDV DFFHSWHG DV SDUWLDO IXOILOOPHQW RI WKH UHTXLUHPHQWV IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 'HFHPEHU :LQIUHG 0 3KLOOLSV 'HDQ &ROOHJH RI (QJLQHHULQJ 0 2KDQLDQ 'HDQ *UDGXDWH 6FKRRO TABLE OF CONTENTS ACKNOWLEDGMENTS iii LIST OF ABBREVIATIONS vi ABSTRACT viii CHAPTERS 1 INTRODUCTION 1 1.1 Code-Division Multiple-Access 1 1.2 CDMA Overlay 5 1.3 Overview of the Dissertation 6 2 NARROWBAND SYSTEM PERFORMANCE IN THE AWGN CHANNEL 10 2.1 Effects of Overlay on a Narrowband User 10 2.2 CDMA Transmitter Filtering 15 2.3 Filtering Performance Criteria 16 2.3.1 Gain of the BPSK System 16 2.3.2 Effect on the PSD of the CDMA Signal 17 2.3.3 Effect on the CDMA Code Sequence 17 2.4 Filtering Methods and Results 18 2.4.1 Eigenvector Filtering 18 2.4.2 Null Filtering 20 2.4.3 Butterworth Filtering 20 2.4.4 DFT-Based Filtering 22 2.5 Summary 24 3 MMSE DETECTION OF FILTERED CDMA SIGNALS 25 3.1 The MMSE Receiver 25 3.2 MMSE Detection of Filtered CDMA Signals 26 3.3 Simulation Results 29 3.4 Summary 36 4 CDMA OVERLAY IN A CELLULAR SYSTEM 37 4.1 Characterization of the Cellular Environment 38 IV 12 (b) Figure 2.1: Histograms for CDMA interference caused to a BPSK user, (a) Single CDMA user, BPSK user employing root-raised cosine pulses; (b) Single CDMA user, BPSK user employing square pulses; (c) 3 CDMA users, BPSK user employing root- raised cosine pulses; (d) 3 CDMA users, BPSK user employing square pulses. approximation does not hold for each individual //,, the total CDMA interference is the sum of K of these terms, and for relatively small values of K, the distribution of the sum of the /*. terms does approach Gaussian. An example is shown in Figure 2.1. The histograms for a single CDMA users interference contribution are shown for the case when the BPSK signal uses root-raised cosine pulses, and also for the case when square pulses are used. In both cases, the interference is clearly non- Gaussian. However, when the root-raised cosine filter is used, the interference does take on somewhat of a continuum of values concentrated in three areas, in contrast to the discrete 4-valued variable resulting in the square pulse case. As a result of this continuum, the sum of only 3 such variables is seen to be well-approximated by a Gaussian distribution, while in the square pulse case, the sum of 3 interference terms is still not close to Gaussian. Using the Gaussian approximation, and the fact that Ik{j) is of zero mean, the decision statistic from equation (2.2) can be rewritten as Zj = y/2EbdbJ + N ^0, 2P/,Xy cos2(0,)Var(4) + N0 ] (2.4) Overlay has great potential in a situation in which a frequency band which cur rently provides narrowband cellular service is designated to provide CDMA cellular service in the future. This transition can be made gradually with the implementa tion of overlay. Such a scenario is investigated in this research and promising results are presented. The use of multi-carrier CDMA (MC-CDMA) is investigated for this purpose, and is seen to perform quite well. In a fading channel, diversity such as that offered by the use of multiple carriers improves the performance of a system operating at a given power level. In the overlay scenario in particular, there is the additional benefit to overall system performance, in terms of both CDMA and narrowband, that the CDMA users can lower their power in a multi-carrier scenario and thus can reduce the amount of interference on the narrowband system. The feasibility of CDMA overlay is bolstered by the results presented in this disser tation, and a strong motivation for its use as a method of transition from narrowband service to CDMA service is argued as well. IX 55 was satisfied: Pv((Pe,cDMA > 0.05) U (#notches > 10)) < 0.02 (4.15) In this criterion, a CDMA user will be blocked if its bit error rate is too high, or if it needs more than 10 notches. As the sampling rate of the system is 1/TC, the range in which notching can be done is actually (0.5/Tc < f < 0.5/Tc). Any frequency locations outside of this range must be notched in the appropriate location mirrored around 0.5/Tc. Thus with a processing gain of 32, there are effectively only 16 notching locations. With more than 60% of the CDMA signal notched out, the user must be dropped. A two-dimensional capacity plot is shown in Figure 4.7 for the same-link case, with notching thresholds of T = 7,9,11 dB. The notching threshold may affect the capacity criterion of equation (4.15) in two ways. First, as more notching becomes necessary, it is more likely that the desired user will have more than 10 notches and will be dropped. Also, it should be more difficult to demodulate the desired users signal as more notches are added, and hence the probability of error should be higher on average. Notice that when the narrowband system is lightly-loaded, there is little difference in the amount of CDMA loading possible for each threshold. It was observed that when the loading was at most 1.5 narrowband users/cell, no drops occurred as a result of excessive notching. The fact that the threshold had little effect at these loading levels suggests that the notching has a minimal effect on the bit error rate, an idea which later will be investigated further. As the narrowband loading increases beyond 1.5 users/cell, the system in which a notch is placed if the narrowband-to-CDMA power ratio is less than T = 11 dB immediately shows the effects of having users dropped as there now are some instances in which the desired user requires more than 10 notches. For the other values T 7, 9 dB, the effects of dropping users show up for higher values of narrowband loading. I certify that I have read this study and that in my opinion it conforms to accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Scott L. Miller, Chairman Associate Professor of Electrical and Computer Engineering I certify that I have read this study and that in my opinion it conforms to accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Leon W. Couch II Professor of Electrical and Computer Engineering I certify that I have read this study and that in my opinion it conforms to accept able 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 accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Jian Li Associate Professor of Electrical and Computer Engineering I certify that I have read this stud}'' and that in my opinion it conforms to accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Ulrich H. Kurzweg Professor of Aerospace Engineering, Mechanics, and Engineering Science 3 noise and hence would not even know that a communication is taking place. This is referred to as low probability of detection (LPD) capability. In the past two decades, the research in this area has shifted from military to commercial applications, particularly in the study of CDMA systems, in which many direct-sequence spread-spectrum signals are transmitted in the same bandwidth and the code sequences are used as a means of providing separation between them. When the code sequences have some degree of orthogonality to each other, the job of the receiver is made easier, although exact orthogonality is not necessary, nor is it achiev able in asynchronous communication systems. The problem of choosing good code sequences has been studied, and a good summary of many of the major results is found in Sarwate and Pursley [1], CDMA is a very promising technology for several reasons. Most importantly, in a cellular scenario, there is the potential for a many-fold increase in user capacity over traditional frequency-division multiple-access (FDMA) systems. The cellular scenario will be looked at in detail in Chapters 4 and 5. Another advantage of using CDMA over FDMA is the inherent ability of a wideband signal, such as a CDMA signal, to realize diversity due to the frequency selectivity of the fading channel. An additional attribute of CDMA is the potential for privacy through the unique spreading codes of the different users. A survey of the receiver structures that have been proposed in the literature begins with the conventional matched filter receiver, which was analyzed by Pursley [2] and by Yao [3] for an additive white Gaussian noise (AWGN) channel. One of the principal shortcomings of this receiver is its susceptibility to the near-far problem, a situation in which one or more users are physically located much closer to the CDMA receiver than is the desired user, and the desired users signal is thus overwhelmed by this multi-access interference (MAI). In such an environment, the system must use power control, a process in which the base station, a centralized control center, 5 be discussed in the next section. It will also be seen that the MMSE can reject inter-symbol interference (ISI). A significant portion of the research presented in this dissertation focuses on the MMSE. 1.2 CDMA Overlay In communication systems, a major design factor involves the limited amount of frequency spectrum that is available. With this as motivation, the overlay of a CDMA system on a frequency band that is also populated by narrowband users from another system has been examined bv Milstein et al. [16]. An actual CDMA overlay system was simulated in this field study, and gave preliminary indications that it is an attainable goal, as both the CDMA users and the narrowband users were able to communicate reliably. It would be very beneficial if some of the frequency bands which are currently occupied by sparsely-populated narrowband systems could increase their overall capacity by adding CDMA technology. Of particular interest is the possibility of implementing overlay in a cellular scenario as a means of gradually phasing-out an existing narrowband cellular system in a frequency band which ultimately will support exclusively a CDMA cellular system. This will be looked at in detail in Chapters 4 and 5. In theory, the two systems, CDMA and narrowband, can coexist in the same frequency band a,s a result of their intrinsic properties. Consider first the effect of interference from a CDMA system on a narrowband system. Using the same idea as that used in low probability of detection systems, CDMA users that are spread will have only a fraction of their power affecting the narrowband system which has a relatively small bandwidth. The effect of the CDMA system on the narrowband system may be tolerable if the processing gain is large enough and if the ratio of CDMA users to processing gain is small enough. Conversely, when narrowband interference appears at a CDMA receiver, its effect can be lessened to some extent by using various interference rejection techniques 52 Figure 4.6: Notching probability as a function of dx for same-link case. Notching thresholds are T = 1, 3,... 15 dB. the number of cells, the number of notches required for the CDMA signal is then a binomial random variable with success probability pn. 4.3.2 Staggered-Link Assignment In contrast to the previous case, the CDMA mobiles and the narrowband base station are now transmitting in the same band. It is not necessary to perform a detailed analysis in this case. The signal on the narrowband forward link will have a composite gain from the base stations transmitter antenna and the mobiles receiver antenna, while the interference from a CDMA mobile to the narrowband mobile will only be amplified by its own transmit antenna and the narrowband mobiles receiver antenna. Obviously, the base station antennas will be much larger in effective area than will a size-limited mobile antenna. So the narrowband forward-link signal is likely to arrive at the mobile at a much higher level than would the interference from the CDMA mobile. Hence it is possible that notching is not required of the CDMA mobiles in the staggered-link case. SNR: signal-to-noise ratio SS: spread-spectrum BIOGRAPHICAL SKETCH Brad J. Rainbolt was born in Normal, IL, in 1972. He received the B.S. and M.E. degrees, both in electrical engineering, in August 1993 and December 1994 from the University of Florida, Gainesville, FL. He was employed by Motorola Land Mobile Products Sector, Plantation, FL, from May 1994 through August 1995 as an engineer in the Applied Research Group, where he worked on the design of digital communication systems. In August 1995, he returned to the University of Florida, and received the Ph.D. degree in electrical engineering in December 1998. 86 CHAPTER 4 CDMA OVERLAY IN A CELLULAR SYSTEM We will now extend the ideas of the previous two chapters to the cellular envi ronment, an application for which CDMA overlay has perhaps the most potential for increasing user capacity. Because overlay is feasible only when the existing nar rowband system is sparsely-populated, the cellular concept seems quite conducive to overlay, as each cell in a narrowband system only utilizes a fraction of the system bandwidth, even when full}' loaded. A cellular overlay system would be quite ben eficial in a situation in which a narrowband cellular system is to be phased out in favor of a CDMA system [24], With overlay, the transition could be gradual, as a new CDMA product could be introduced while the system still provides support to the existing narrowband products. In this chapter, it will be shown that overlay can be realized in the cellular scenario. The total system bandwidth of the narrowband system is divided into several frequency groups, each consisting of a number of narrowband channels separated in frequency so as to minimize adjacent-channel interference. The groups are assigned to the cells in an intelligent manner which minimizes the co-channel interference, that is interference caused by users in different cells using the same channel. Thus even when fully-loaded, each cell only utilizes a small fraction of the total system bandwidth, as would be necessary for overlay. A CDMA system could be implemented using the same cellular layout, with the CDMA users in each cell spread in frequency over the whole system bandwidth. 37 23 Table 2.3: Gains in clB for BPSK system for DFT-based filter. Length (bits) 1 2 4 8 16 Gain (clB) 11.2 22.8 31.4 39.2 46.3 be performed by forcing to zero those values of the DFT that correspond to frequen cies within the desired notching range and taking an inverse DFT. The frequencies represented are spaced by 1 /N in the digital frequency domain, corresponding to a frequency spacing of 1/T in the analog domain. Thus if a notch of width 2/TJ, is desired, there are three DFT coefficients, two surrounding the range and one in the middle of it, which should Ire set to zero. A plot of |C(/)|2 is shown in Figure 2.3, in the graph for no zero-padding. Notice that there are deep nulls, but there are also high peaks between the nulls. When an IV-point DFT of the unfiltered code sequence is used, no more resolution is available for notching. However, if the code sequence is padded with zeros so that an IV-point DFT with N > N is performed, there will be more resolution. As shown again in Figure 2.3 for the case when the sequence is padded to cover 8 bits, the resulting notch is much deeper. The DFT-based method was tested for an unpadded sequence and for sequences padded with zeros to cover 2, 4, 8. and 16 bits. The gains are shown in Table 2.3. To achieve a gain comparable to that of the Butterworth filtering method, a sequence padded with zeros up to 16 bits must be used. In Figure 2.4, the code sequence resulting from zero-padding up to 8 bits dies off pretty quickly, with much less spillover than was seen in the Butterworth case. The DFT-based filtering method also seems like a viable filtering method, perhaps in situations where the large gains given to the narrowband system by the Butterworth filtering are not needed. It will be seen later that the DFT-based filtering method is the better choice in terms of performance of the CDMA system. 57 This is a significant capacity improvement which would be very useful as mentioned in the transition from a narrowband cellular system to a CDMA cellular system. It would also be of interest to investigate separately the effects of notching and the effects of narrowband noise on the CDMA system. This was done by comparing results for the staggered-link case, which naturally has a very high level of narrowband interference, and the same-link case explicitly in terms of the number of notches in the desired user and in the number of narrowband users which interfere with the CDMA signal. Recall that these will likely be different as the CDMA mobiles often must notch for narrowband users that are assigned to other cells. Also, although it was stated earlier that notching will not likely be required in the staggered-link case, it will be done here only in an effort to look specifically at the difference in the effects of notching and narrowband noise. In the capacity plot shown in Figure 4.8, results for the staggered-link case are shown as solid lines while the one dashed line applies to the same-link case. For the staggered-link results, the number of narrowband users assigned to the desired users cell was held constant at 0, 1, 2, 3. or 4, while the number of notches in the desired user was varied, and the CDMA density was found according to only the bit error rate part of the criterion of equation (4.15). Recall that in this configuration, the signals on the narrowband systems forward link will appear on the CDMA reverse link as interference at a very high near-far ratio. Assuming that the problem can be alleviated to some extent by shielding the two antennas, the near-far ratio was fixed at 20 dB. In going downward from curve to curve, it is seen that the effect of an additional narrowband user is to decrease the CDMA density by about 4 users/cell, which is significant. For the same-link case, when there are no narrowband users assigned to the desired CDMA users cell, the capacity curve is the same as in the staggered-link case. The narrowband users that are assigned to the cell will arrive at 0 dB with respect to the 21 O 0.25 0.5 Digital frequency Figure 2.3: PSDs for notch-filtered signals. Processing gain is 32 chips/bit. (a) Eigenvector filter, M 7 taps on each sick1: (1)) Null filter, M 8 taps on each side; (c) Butterworth filter, 8th order, 3-dB B\Y 1/32; (d) DFT-based filtering, no zero-padding; (e) DFT-based filtering, zero-padded to 8 bits. Solid curves are for unfiltered, dashed curves are for filtered. 70 more than 2 notches are necessary. In contrast to the 2-carrier case, if only one of the carriers must be dropped, 3 of the 4 carriers would still remain, and it might be possible that the CDMA receiver could still function reliably. This will be looked at in Section 5.3, but for now, the criterion that we will employ is that a CDMA user will be dropped if at least 2 of the 4 carriers have more than 2 notches. These results are shown in Figures 5.2 and 5.3 as well. In Figure 5.3, there is an improvement over the 2-carrier case, in which the CDMA user was dropped even if only one of the carriers needed significant notching. It has been suggested previously that with the use of multi-carrier CDMA, it is possible to avoid the narrowband users in an overlay scenario by simply not trans mitting on those carriers which might interfere with a narrowband user [27]. This possibility was also examined here for comparison. We will first utilize the previous notching criteria, and declare that if the CDMA signal on a given carrier is received at a high power level in terms of the notching threshold T, the carrier will not be used, analogous to simply placing a notch in the previous scenarios. We will consider both 4 and 8 carriers in this type of system. It does not seem fruitful to raise the number of carriers beyond 8, with a composite processing gain of 32 chips/bit, for several reasons. First, even assuming that the CDMA signal on each carrier would experience independent fading, the incremental diversity advantage re alized by using more carriers diminishes with such a high number of carriers. Second, it has been assumed that the CDMA system, when used with multiple carriers, can be split such that transmission takes place on disjoint frequency bands, so that inde pendent fading can be realized. While this seems plausible with 2 or even 4 carriers, the possibility becomes less likely'- with a large number of carriers. Thus some of the carriers will undergo correlated fading, which will not give nearly as much diversity as will independent fading. Finally, the process of tracking the fading processes, which is 15 -2 10 -10 dB -14 dB -2 dB -6 dB 2 dB 7 8 9 10 11 12 Eb/N0 (dB) Figure 2.2: Probability of error, Pe of BPSK user vs. E\Â¡/Nq with CDMA overlay of 3 users with 31 chips/bit. Solid curve is for numerical integration, dashed curve is for approximation of equation (2.11). and the narrowband system. These results cast considerable doubt on the feasibility of CDMA overlay. It may not even be worthwhile to implement a CDMA system with a loading of K/N = 3/31, and these results indicate that this very lightly-loaded system still causes severe degradation to the narrowband system. Obviously, something must be done to lessen the effect of the C'DMA interference on the narrowband system, and one such method will be investigated in the next section. 2.2 CDMA Transmitter Filtering In an effort to improve the previous results, which seem to preclude the chance of CDMA overlay, an idea which was suggested both by Milstein et al. [16] and by Davis [21] will be investigated. The energy from a CDMA signal that does appear in the same bandwidth of a narrowband user apparently requires more attenuation than that which results solely from the processing gain. The attractive properties of a CDMA signal, such as its inherent separation from other CDMA signals as a result of the spreading codes, its inherent separation from narrowband signals as a result of the processing gain, and its robustness to multipath, may not be sacrificed too much if only a small notch is placed in its spectrum. And if the frequency ranges occupied 30 Figure 3.2: Two-dimensional capacity plot for DFT-based filtering, zero-padded to 8 bits. Processing gain is 32 chips/bit. (a) MMSE receiver, minimizing J = E[|Z, dx(*)|2]) (b) MMSE receiver, minimizing J = E[{Re(Z di(z))}2]; (c) matched filter. In Figure 3.2, a two-dimensional capacity plot is shown for a system which employs DFT-based filtering, with zero-padding up to 8 bits. The curve labeled (a) represents possible operating points for the system, in terms of the number of CDMA users and the number of narrowband users that can simultaneously use the frequency band and still satisfy the CDMA systems performance criterion given in equation (3.7). It should be noted that in this receiver, the bit decision is made by looking only at the real part of the filter output Z, w;/u(?'). Hence better results would probably be obtained by choosing an algorithm to minimize J = E[{Re(Z di(z))}2] as opposed to minimizing J = E[|Z c/i(?')|2]. In this case, the imaginary part of the error can be ignored since it will not be used. With the same equalizer contents as in equation (3.3), the filter output can be written as Z[ = (w0i, + iw0s)"(u,(t) + ju(i)) (3.8) 53 The major drawback to this configuration, however, is the severe near-far problem that results at the CDMA base station as it must receive signals from its mobiles in the presence of the signals sent from the narrowband base station, located a short distance away on the same tower, to its own mobiles. It will be seen later that this disadvantage outweighs by far the advantage of needing few, if any, notches in the CDMA mobiles, and that the same-link assignment is the better of the two configurations. 4.4 Simulations and Results In order to investigate the performance of the CDMA system, an environment similar to that used in the single-cell results of Chapter 3 was constructed, incorpo rating the cellular layout of Figure 4.1, and log-normal shadowing and power control as described earlier. The CDMA system had a processing gain of Ar = 32 chips/bit, an Eb/No value of 10 dB, and used the DFT-based filtering method with 8 bits of zero-padding when filtering was necessary. For a 1/7 frequency reuse system, each cell would have at most 4-5 narrowband users. In a brute-force simulation of the cellular overlay system, the coverage area that must be considered would consist of 6 layers of cells surrounding the center cell of interest. It was observed that a CDMA user might have to notch for a narrowband user that is 2 layers away from its geographic location. And any user could be assigned to a cell at most usually 2 layers away from its geographic location. So a CDMA user in the second layer out might have to notch for a narrowband user in the fourth layer out whose base station could be in the sixth layer out. The total number of cells in the geographic coverage area multiplied by the density of users/cell gives the total number of users, either CDMA or narrowband, which would then be distributed uniformly about the coverage area. Each user would then be assigned to the cell at which its received power is maximum, and power control would be implemented. After it is determined how much of an increase or decrease 51 First we define / = (Received narrowband power) (Received CDMA power), where the received power refers to the power in dB at the base station to which the narrowband user is assigned. A notch will then be required if / < T, with T given in dB. Recall that at this liase station, the narrowband users received power will be a specified value, denoted 7. It was stated earlier that the CDMA and narrowband users would be power-controlled to the same level if assigned to the same base station. In that case, I 0 and there is certainly a notch required. For a CDMA user that is not assigned to the same cell as the narrowband user, denoted cell k, the conditional PDF of the CDMA received power at the narrowband users base station is exp f - fHt(h/(Gk < Yk)) = ~1 u(-h) (4.12) /27r(^ + 4,)(Pr(G, that the CDMA user is not assigned to the A th cell. The notching probability, pn, is then found as 19 r Pn = I] k= 1 L Pr(/ < T, NB to cell k, CDMA to cell k) + Pr(I < T, NB to cell k, CDMA not to cell k) which simplifies to 19 1 1 y ' Pn ~ 19 + 19 ^ fc=1 Q -T (mgik myik) a\ + ab QI img,k mVtk) + aU (4.13) (4.14) The notching probability is plotted in Figure 4.6, as a function of d\, for T 1,3,... ,15 dB. As expected, it is highest at the edge of the cell, as those users are more likely to be transmitting at a higher power than are those nearer to the base station, as a result of the power control. For a number of narrowband users located throughout the region, that is the density of users/cell multiplied by 19 for 19 Table 2.1: Gains in clB for BPSK system for eigenvector and null filters. Taps 6 7 8 9 10 Eigenvector 32.8 36.8 35.5 24.0 20.0 Null 9.7 14.9 20.9 15.5 10.6 row vector of tap weights, B is a matrix whose (i,j)th element is given by or \ N ~ I* JI cs((* ])AuTc) B(bj) = ^ where N is the processing gain and Au> is the difference of the two systems carrier frequencies. If the quantity aBaT is minimized, then the effect of the CDMA inter ference on the BPSK system will also lie minimized. This minimization is performed by letting a be the eigenvector of B corresponding to its smallest eigenvalue. Then the quantity aBaT is equal to this eigenvalue. It will be the convention throughout this dissertation to denote vector and matrix quantities in boldface type. In Table 2.1, the gains achieved by the eigenvector filter are shown when the number of taps, M, on each side of the center tap varies from 6 to 10, and the processing gain is 32 chips/bit. Although it seems that the gain should increase and eventually level off as the number of taps is increased, the maximum gain is 37 dB, occurring for M = 7. An explanation for this behavior might be that the eigenvector filter was derived for the case when the BPSK system uses square pulses, not root raised cosine pulses. In Figure 2.3, a plot of |C(/)|2 for an eigenvector filter with M = 7 and Acn = 27r(0.125) shows that the filter severely distorts the CDMA signal. The desired re sponse should have a deep null of finite width around the desired frequency, and as little distortion as possible in the passband. This filter does not even have a parame ter to control the width of the notch. Although it does provide a pretty sizable gain 69 and notching on CDMA performance. We have only considered two things, which nonetheless do impose some limitations on user capacity: the number of CDMA users for which the narrowband systems performance is severely degraded according to the criterion of equation (5.9) and the number of narrowband users for which the CDMA users would simply require too much notching. In the next section, we will look at the CDMA performance in more detail. We next consider using multi-carrier CDMA with 2 carriers, and hence a process ing gain per path of 16 chips/bit. In deciding how much notching is excessive here, we must note that only 8 unique notching locations are available in each carrier, and hence if there are more than 4 notches required in either carrier, the CDMA user will be dropped. With only one of the two carriers in use, this is equivalent to the single carrier case with only half of the processing gain, but the same amount of loading. The results for this case are also shown in Figure 5.2. For a given value of T, the probability that at least one of the two carriers will have more than 4 notches out of a possible 8 is greater than the probability that there will be 8 notches out of a possible 16 in the single-cell case, which agrees with intuition. In Figure 5.3, two-dimensional capacity curves for the 2-carrier case indicate performance inferior to the single-carrier case for a given value of {{Eb/No)b (Eb/No)c). However, there is no significant con clusion to be drawn from this, and these results are only presented for future use. It will be seen later that when CDMA receiver performance is taken into account, the multi-carrier systems are able to operate at much lower values of (Eb/No)c than can the single-carrier case, as a result of the added frequency diversity. Thus it would be more fair to compare a curve in the single-carrier case with one of those from the multi-carrier case with a significantly larger value of ((Eb/No)b (Eb/N0)c), which would lean more favorably toward the multi-carrier case. Finally, we consider using 4 carriers, which means that the CDMA signals will have a processing gain per path of 8 cliips/bit. Thus a carrier will be dropped if 82 works have not considered the effects of the notch-filtering presented in this disserta tion. It must be determined what effects the notching will have on algorithms such as these, if used unmodified. It is likely that some modifications to fit the timing estimation problem to the case with filtered codes would be necessary and certainly would be beneficial. Also the possibility of the timing estimator working without knowing the code sequence, as the MMSE receiver can do for detection, might be useful. But it seems that this information would be known by the base stations, and the estimator would probably suffer a good deal of performance loss if the code sequence is not used in the algorithm. Additionally, in a frequency-selective fading channel, the estimator must be able to lock onto several paths with different delays. Another significant area of future research is the estimation of the fading processes of the CDMA users. It was determined by Miller et al. [28] and by Miller and Rainbolt [29] that the MMSE receiver will take a large loss in performance if it is not able to track all of the fading processes of all of the users. This is a fairly undeveloped area, with a few solutions having been proposed. Barbosa and Miller used linear prediction in conjunction with an MMSE receiver to estimate the fading process of the desired user in a flat-fading channel [36], A subspace-approach was investigated by Wang and Poor [37], Methods based on a decorrelator idea, that is a transformation on the received signal which removes the data, are presented by Miller and Rainbolt [29] and by Juntti [38] in an effort to track the fading in a frequency-selective fading channel. The estimation of the fading processes is very important, not only for the overlay scenario in this dissertation, but for the use of the MMSE in fading channels in general. This will continue to be a popular topic of research, as there has not been a tremendous amount of success realized. 50 Figure 4.5: Effects of missing notches on CDMA loading that narrowband system can tolerate. Processing gain is 32 chips/bit. CDMA user notched if NB-to-CDMA power ratio is less than the notching threshold, T = 7,9,11 dB. two systems should use the same frequency band for the forward link, and corre spondingly for the reverse link, or if they should be staggered such that one band covers the forward link of one system and the reverse link of the other. These two scenarios will be referred to as the same-link and staggered-link cases, and will be compared now based on the reverse link of the CDMA system. 4.3.1 Same-Link Assignment In this case, the CDMA mobiles and the narrowband mobiles are transmitting in the same band, and thus the narrowband base station is taken to be the receiver. To determine the required notching, consider again a CDMA user located at a distance d\ from the first base station as shown in Figure 4.1. We will find the probability that a CDMA user located at this position must notch for a single narrowband user that is located uniformly within the hexagonal enclosure of the three layers of cells. This will require for = 1,2,... ,19, the values of my and crjA, which are numerically calculated, as well as the which are easily found using the law of cosines. 42 in dB at the base station is 7, the users transmitted power will be adjusted to 7 W, where W is defined in equation (4.3), so that it will arrive at a power 7 at its own base station. At a base station to which the user is not assigned, in cell k, the received power is 7+ (Gk~Yk). Note that it is necessary to know YÂ¡. under this condition, that is (Y*, > Gk)i as it determines the users transmitted power. But it is not necessary to know which of the base stations other than the fcth that the user is assigned to. We must then find the PDF of Hk, which is the interference power level at the fcth base station relative to those users assigned to the Atth base station. Under the condition that the user is not assigned to the kth. base station, HÂ¡Â¡ will always be less than 0 dB. The PDF of H/, will have a discrete part, which is an impulse at 0 dB, with a weight equal to Pr(G. > Y), the probability that the user is assigned to the kth cell. It will also have a continuous part, resulting from the event that it is not assigned to the Ath cell, and appears as outer-cell interference. To find this part of the PDF, we first find the joint cumulative distribution function (CDF) (4.6) The derivative of equation (4.6) with respect to h gives the continuous part of the PDF of Hk, and the complete PDF is (47) where u(h) is the unit-step function. The continuous part of this PDF is just a Gaussian PDF that is truncated at h = 0. The expression in equation (4.7) is dependent on the exact position of the user. This will be useful in some cases, but it would also be helpful to have a PDF for 54 in transmitted power from the mobiles is required, the necessary notching for each CDMA mobile would be found by comparing its received power with that of each of the narrowband mobiles assigned to one of the cells within a three-layer cluster of the CDMA mobiles base station. This procedure would obviously require a great deal of simulation time, which can be decreased significantly with some simplifications. The first simplification results from the observation, which was verified by sim ulation, that if an interfering CDMA user is left unfiltered, there is no noticeable difference in its effect on the desired CDMA user. The notching probability for a CDMA user depends jointly on its position and its transmitted power. But since it does not need to be known in the simulations, it is sufficient only to know the in- terferers received power relative to the desired user, and its position is unimportant. So we can further simplify the simulations by using equation (4.8), which gives the approximate PDF of the received power for a user that is located uniformly through out a three-layer cluster around the cell of interest, taking into account both cell assignment and power control. This was used to generate the near-far ratios of both the interfering CDMA users and the narrowband users. For the desired user, it was assigned to the center cell, and based on its position, its notching probability was found from a numerically-evaluated table. The number of notches required was then a binomial random variable dependent on the density of narrowband users. Each CDMA user was given a random code sequence, delay, and phase, and each narrowband user was assigned to a random frequency location. The desired user was demodulated using an MMSE receiver, with the true Wiener solution. As explained in Section 3.3, the best performance is obtained by choosing the tap weights to minimize in expected value the square of the real part of the error as opposed to the square of its absolute value. The system capacity in CDMA users/cell was determined to be the maximum density for which the following blocking criterion 2 (a) Time Figure 1.1: Illustration of DS-SS waveforms and PSDs, 7 chips/bit. (a) Unspread signal waveform; (b) PSD of unspread signal; (c) Spread signal waveform; (d) PSD of spread signal. of the unspread signal, but has a null bandwidth of l/Tc = N/TÂ¡,, which is N times larger than the null bandwidth of the unspread signal. Also note that since the total power is the same in the two signals, the height of the PSD is reduced. For many years, spread spectrum has been used in military applications, due mainly to three of its features. First, it has a resistance to jamming, a process in which an adversary transmits an interference signal, which is usually narrowband, in an attempt to destroy communication, but not to intercept and make sense of it. Spread spectrum also has low probability of intercept (LPI) capability, meaning that it is difficult for an adversary to receive and demodulate the signal without knowing the spreading code. Finally, because the power is spread over such a large bandwidth, the spectral height of the spread-spectrum signal is reduced significantly, possibly to the point where an adversary would not be able to distinguish it from the channel xml version 1.0 encoding UTF-8 REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID EKQX66XE1_TYH944 INGEST_TIME 2014-12-08T22:18:28Z PACKAGE AA00024491_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES 58 Figure 4.8: CDMA users/cell vs. number of notches in the desired user. Solid line is staggered-link case, with narrowband-to-CDMA near far ratio at CDMA base station of 20 dB, and 0, 1, 2, 3, or 4 narrowband users assigned to cell of interest. Dashed line is same-link case, with 4 narrowband users with near-far ratio of 0 dB assigned to cell of interest. E't/A'o of CDMA system is 10 dB. CDMA users power level, as mentioned before. The plot when there are 4 narrowband users assigned to the cell of interest is shown, and the plots for 1,2, and 3 users will naturally fall between it and the plot for no narrowband users present. It is seen that 4 narrowband users received at a near-far ratio of 0 dB have the effect of decreasing the CDMA capacity by only about 4 users/cell, as compared to a degradation of about 4 CDMA users/cell for each narrowband user in the staggered-link case. These results are not surprising, but they do give useful insight as to how much effect the narrowband interference has. It is interesting that in all of the curves shown, the CDMA capacity degrades slowly as the number of notches is increased to around 7 or 8, and then degrades more rapidly. Clearly, the presence of narrowband interference has more effect on CDMA capacity than does the notching. 81 of notching. This imposed disappointing constraints on the joint capacity of the sys tem, that is the number of CDMA and narrowband users that can simultaneously occupy the spectrum. A solution for this problem is to use multi-carrier CDMA, in which the CDMA signal is transmitted on several carriers, such that the signals on each carrier undergo independent fading. This provides a significant amount of frequency diversity, which allows the CDMA users to transmit at lower power levels. Thus less notching is also necessary, and the joint capacity of the system is improved dramatically. The results presented in this dissertation indicate that CDMA overlay has great potential for providing an increase in user capacity. It is especially valuable in cellular scenarios, in which a transition from narrowband service to CDMA service is in progress. There remains, however, some open issues which will be discussed in the next section. 6.2 Future Work Some topics of future research will Ire suggested in this section. The results pre sented in this dissertation have clearly indicated that overlay has significant potential. The next step is to look at some of the implementation issues that were not addressed. Throughout the research, two important tasks were assumed to have been done perfectly, the timing acquisition, that is the knowledge about when the desired CDMA users code sequence begins, and the estimation of the fading processes of the CDMA users. They are certainly important problems that need to be addressed in future work. The results presented in this dissertation had only the goal of demonstrating that there is indeed a reason to pursue overlay. If the results had demonstrated other wise, then the effects of imperfect estimation of the timing and the fading parameters would not be interesting or meaningful. The timing acquisition problem has been investigated by many researchers, and there exists an extensive body of literature on the subject [30-35]. Obviously, these 17 performance will be the same as in the unfilterecl case. Thus the ratio of the variance without filtering to the variance with filtering should be found, and it will indicate how much additional near-far effect the BPSK system can tolerate. The amount of gain required depends strongly upon environmental factors such as the cell geometry and path-loss exponent [16]. It is conceivable that gains on the order of 50 dB may be required. This could be the case when many CDMA users are present and/or a severe near-far problem results from one or more of these users transmitting from a location much closer to the narrowband receiver than that of the narrowband trans mitter. If the CDMA users and the BPSK user are not power-controlled by the same mechanism, then large near-far ratios are very possible. 2.3.2 Effect on the PSD of the C'DMA Signal To see how certain filters perform in notching the CDMA signal, consider the PSD of a CDMA users signal which is given as SssU) \Q(f)\2 Tb A'-l 11=0 I qmi % |C(/)|2 (2.12) where Q(f) is the Fourier transform of the chip pulse shape, c is the nth chip, and the number of chips in the spreading sequence is N, which may be greater than the processing gain N. From this expression, it is clear that the effects of filtering will appear only in the rightmost magnitude-squared term, |C(/)|2 which can be plotted as a function of frequency to show these effects. 2.3.3 Effect on the CDMA Code Sequence In two of the four filtering methods examined in this dissertation, the output code sequences are of length greater than N, which means that they span more than a single bit. While this spillover greatly alleviates the effect of CDMA interference on 20 of 37 dB for the BPSK system, the eigenvector filter does not seem to be a plausible option based on its detrimental effect on the CDMA signal. 2.4.2 Null Filtering In the second method proposed by Davis [21], the filter puts a spectral null at the BPSK systems carrier frequency. The tap weights a are given as Po = 1 Pm = 2 cos( n) AuTc ] 2 M 1 + sin((2Ai + l)AwTc) sin(Au.'7c) in = 1, 2,... AM (2.14) a = 0/\0\ and the filtered code sequence is formed again from the delayed output of the FIR filter with tap weights given by a. The gains achieved by the null filter are shown in Table 2.1 for the same values of M as for the eigenvalue filter and also for a processing gain of 32 chips/bit. A maximum gain of 21 dB. significantly lower than that of the eigenvalue filter, occurs for M = 8, and again an explanation for this behavior could be that this filter was derived for the case of square pulses in the BPSK system. A plot of \C(f)\2 for the null filter with M = 8 and Au> = 27t(0.125) is also shown in Figure 2.3. Again, the CDMA signal is somewhat distorted, and there is only a minor notch at the desired notched frequency, and no parameter to control this width explicitly. Based on this, the null filtering method does not seem like a plausible option, either. 2.4.3 Butterworth Filtering We next consider the use of a digital Butterworth notch filter, which is an infinite impulse response (HR) filter. The response when one period of the code sequence appears at the input lasts forever in theory, although it is negligible beyond a few bits. Nonetheless, the filtered code sequence will spill over into other bit intervals, resulting in ISI. To form the output code sequence, the unfiltered code sequence, which is one bit long and has N chips, is padded with several bits of zeros and clocked into the HR 39 (a) (b) Figure 4.1: Illustration of a narrowband cellular system, (a) Cellular layout for three- layer cluster. Cells numbered 1-19. Letter is the cells channel-group assignment, chosen from A-G; (b) Magnified view of Cell 1. When comparing the paths to different base stations, it is only necessary to com pare the rightmost term in the second line of equation (4.1), that is Gj = [Ar(0, cr) 10ralog10(d,-)] (4.2) where dl is the distance from the mobile to the z'tli base station, normalized such that the cells are of unit radius. Also note that the quantity G, in equation (4.2) will be referred to as a channel gain, as it is the sum of the path loss and shadowing, but this is not intended to imply that there is an amplification of the signal through the channel. This quantity will be used in comparing different paths on a relative basis only. Without loss of generality, it will be assumed that the user of interest is physically located in cell 1, and at a distance d\ and angle as shown in Figure 4.1. The rest of the GÂ¡ quantities may bo found in terms of d\ and B\ by using the law of cosines. We define the following quantities, also in dB units: W = max(Gj, G>,... Gfig) I,- max(G'i, Go, .. G19) Â¥=> (4.3) 79 constraint, and allowed for a small amount of ISI. A digital Butterworth notch filter was used, and a filtering method based on the DFT was also considered. The filtering methods for which the ISI resulted performed significantly better than did the other two methods, in terms of the amount of distortion caused to the CDMA signals PSDs. They also provided more relief to the narrowband system. The resulting ISI, while an important issue that must be looked at in CDMA receiver design, was certainly manageable. An important point that has been emphasized throughout this research, and has not received enough attention in the literature, is that the use of notch filtering is not simply a way to provide a modest improvement to the performance of an overlay system. Rather, it is absolutely essential if the CDMA system is to be loaded to levels for which the research community is actively striving. Next, the problem of receiving the CDMA signals was considered, and it was shown that the MMSE is a good choice for this purpose. In addition to its ability to reject MAI and NBI, which has been demonstrated before, it was shown here that the MMSE can reject ISI and also that it can adapt to a code sequence which has been filtered, provided that a training sequence is used. The MMSE receivers functionality in the overlay scenario was demonstrated throughout the dissertation, in both single-cell and cellular scenarios, and with both a Gaussian channel model and a fading channel model. The application for which over la}' has perhaps the most potential is the cellular scenario. The implementation of overlay could provide an efficient way for a fre quency band which services narrowband users to transition its service to CDMA. For contractual reasons, providers cannot simply discontinue service to narrowband sub scribers once it has been decided that the band will be converted to support CDMA. Overlay would allow the new CDMA technology to be introduced while the existing narrowband products are phased out gradually. 74 The composite steering vector is given by P(0 = Pi (0 E[di (*>!(*')] P20 = E [d\ (?)r-2(*)] _Pg(0. _E [^(*>0 (*)]. (5.16) The bit decision is then made as d\(i) = sign(Re[w/()r(z)]) for coherent combining of the paths. It makes sense to use coherent combining in this case, as it was previously stated that all of the fading paths of all of the users would be tracked anyway in order to avoid a large performance loss. The task of tracking the fading processes in a dynamic environment is currently an area of active research. In this work, it will be assumed that all of the fading processes on all of the paths are known, in which case we have L/2 = 7l,pW7l,9(*)Cl,p,oCli(r/)0 + 'y ] 7l,p(*)7l1,(*)Cl,p,mCli,im T m.= L/2 ?7l/0 (jr) 7fc,p(*)7,9() [tfj + gfcg*] + 2ct2Inx 2RJim] Sp k=2 1 2 \m=1 / |