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Adaptive detection of DS /CDMA signals with reduced-rank multistage Wiener filter

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Adaptive detection of DS /CDMA signals with reduced-rank multistage Wiener filter

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Content INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI flims the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction.. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. ProQuest Information and Learning 300 North Zeeb Road, Ann Arbor, M l 48106-1346 USA 800-521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ADAPTIVE DETECTION OF DS/CDM A SIGNALS WITH REDUCED-RANK MULTISTAGE WIENER FILTER by Dongjun Lee A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Electrical Engineering) May 2000 Copyright 2000 Dongjun Lee Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number 3018012 Copyright 2000 by Lee, Dongjun All rights reserved. ___ ® UMI UMI Microform 3018012 Copyright 2001 by Bell & Howell Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. Bell & Howell Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF SOUTHERN CALIFORNIA T H E GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES. CALIFORNIA 90007 This dissertation, written by . DONGJHH LEE under the direction of hi#. Dissertation Committee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillment of re- quirements for the degree of DOCTOR OF PHILOSOPHY Dean of Graduate Studies A p ril 7 , 2000 DISSERTATION COMMITTEE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To My Family. D edication Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A cknow ledgem ents First, of all I would like to express my profound gratitude toward the chairman of my dissertation committee, Professor Irving S. Reed, for his guidance during my Ph.D. study. It has been an honor for me to work with him, one of the most prestigious scholars in the field of the electrical engineering, even though it has been difficult to keep up with his expectations as his student. My doctoral degree here at the Univer sity of Southern California attributes a great deal to his invaluable encouragement and academic stimulation. Also I would like to recognize Dr. Charles L. Weber, co-chairman of my committee, for his support and understanding. Additionally, I want to thank Dr. Kenneth Alexander from the department of mathematics, Dr. Robert M. Gagliardi and Dr. Antonio Ortega for gladly accepting to serve on my committee. Besides the committee members, there are numerous people who deserve my spe cial thanks. Particularly, Dr. Scott J. Goldstein and Dr. Xiaoli Yu are two front- runners on this list. They did not hesitate to share their valuable knowledge and time with me and have been a tremendous help in refining my dissertation into an integral whole. Also the friendship of my colleagues within Dr. Reed’s group, Ruhua He, Piyapong Thanyasrisung, Greg O. Dubney, James Hu, has been another source of my energy to travel the long journey of the doctoral study. iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. My life in the United States th a t could easily have withered from home-sickness was nourished by many colleagues from my homeland, Republic of Korea: Chan Kyung Park, Junghyun Oh, Young-Min Kim, Dong-.Toon Shin, Sangyoub Lee, Jun Heo, Yongmin Choi, Jun-Sung Park, Jun-Yong Lee, and Won-Seok Baek. It has been a big comfort to have people in my work environment that share the same language with me and have an identical cultural background. Another group of people that I must be grateful to are the CSI staff at EEB 500: Milly Montenegro, Mayumi Thrasher and Gerrielyn Ramos. Their assistance and professionalism in the area of administrative support was an essential part of my life as a graduate student in CSI. Above all nothing could have even come close to the support of my family. My par ents, grandm other and two younger brothers have always been there for me through out the entire period of my study. It would be impossible for me ever to repay what they have done for me, but I hope th at my humble accomplishments can compensate for what little part of their efforts and sacrifices. Finally I am deeply indebted to my beloved wife, Seung Kyung. As a wife, the mother of our daughter, Christie, and also as a full-time doctoral student herself, she has done an unbelievable job that no one could ever match. W ithout her love and emotional support, my odyssey towards the doctoral degree could have wrecked somewhere along the way. She is the one who should take the honor of my accomplishment. Ultimately I would like thank God for his divine guidance in obtaining my doctoral degree. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C o n ten ts D edication ii Acknowledgem ents iii List O f Tables vii List O f Figures viii Abstract x 1 Introduction 1 1.1 CDMA Demodulation A lgorithm s........................................................ 1 1.2 Motivation for Reduced-Rank P ro cessin g ........................................... 3 1.3 Thesis O rganization _ . . . 5 2 System M odel 6 2.1 The Detection Algorithms for DS/CDM A Communication Systems . 6 2.2 The Decentralized MMSE Demodulator for Asynchronous DS/CDM A Communication S y stem s......................................................................... 9 2.2.1 General Structure of the Single-User, Single-Symbol, Receiver 10 2.2.2 Received Waveform Model of an Asynchronous DS-CDMA C om m unications......................................................................... 11 2.2.3 Vector Model of Received-Waveform ........................................... 13 2.3 Derivation of Optimum Receiver........................................................... 16 2.3.1 MMSE Criterion : Wiener Filter ................................................. 16 2.3.2 Maximum-Likelihood (ML) C rite rio n ..................................... 17 3 Algorithm s for Reduced-Rank M M SE D etection 20 3.1 Subspace Approach of the MMSE D e te c tio n .................................... 20 3.2 General Structure of Reduced-Rank Wiener F ilte rin g ................... 27 3.3 Algorithms Based on Eigen-Decomposition of the Covariance Matrix . 28 3.4 Algorithm Based on Truncated Multi-stage Wiener F ilte r............. 31 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 Truncated Multi-stage W iener Filter 32 4.1 Derivation of the Multistage W iener-F ilter.............................................. 32 4.1.1 Orthogonal Decomposition A p p ro ach ........................................... 34 4.1.2 Optimization of the Rank-One Filters in the Multistage Wiener F ilte r..................................................................................................... 43 4.1.3 S u m m a ry ........................................................................................... 58 4.2 Adaptive Realization of the T M W F ........................................................... 59 5 Num erical Results 64 5.1 Capacity S tu d y .............................................................................................. 65 5.2 Convergence Study of Adaptive TMWF D e te c to r ................................. 72 6 Conclusions 78 Reference List 80 Appendix A Equivalence of Transform Domain W iener-Filter.............................................. 85 Appendix B Proof of Lemma 4 .1 .................................................................................................. 87 v i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List O f Tables 4.1 Recursion formulas for the multistage Wiener f i l t e r .............................. 44 4.2 Algorithm for the adaptive realization of the multistage Wiener filter 63 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. L ist Of F igures 2.1 A centralized multi-user detector.................................................................. 8 2.2 A decentralized multi-user detector.............................................................. 9 2.3 The block diagram of a generic single-user, single-symbol, CDMA re ceiver 11 2.4 Equivalent synchronous transmission model............................................... 15 3.1 The reduced-rank MMSE detection of CDMA signal.............................. 28 4.1 A block diagram of the transform-domain Wiener f i l t e r .............. 34 4.2 The first stage of the multistage decomposition........................................ 38 4.3 The realization of the multistage Wiener filter for N = 4....................... 40 4.4 The synthesis filter bank of the multistage Wiener filter........................ 42 4.5 Reduced-rank W iener filtering on the subspace of dimension one. . . . 45 4.6 Reduced-rank W iener filter w ith two taps.......................................... 48 4.7 The reduced-rank multi-stage Wiener filter of dimension two............... 51 5.1 The output SINR of the TM W F CDMA receiver vs. the number of active users. The SNR of the desired user is set to 20dB and the near-far ratio is OdB.................................................................................. 68 5.2 The output SINR of the TM W F CDMA receiver vs. the number of active users. The SNR of the desired user is set to 20dB and the near-far ratio is lOdB.......................................................... 69 5.3 The output SINR of various reduced-rank CDMA receivers vs. the number of active users. The SNR of the desired user is set to 20dB and the near-far ratio is OdB................................................................... 70 5.4 The output SINR of various reduced-rank CDMA receivers vs. the number of active users. The SNR of the desired user is set to 20dB and the near-far ratio is lOdB................................................................ 71 5.5 The output SINR performance of the adaptive TM W F CDMA receiver with 5 taps vs. the number of users. SNRi = 20 dB, NFR = 0 dB. . 74 5.6 The output SINR performance of the adaptive TM W F CDMA receiver with 15 taps vs. the number of users. SNRi = 20 dB, NFR = 10 dB. 75 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.7 The learning curve of the adaptive TMWF CDMA receiver. For com parison, the average SINR’s of the full-rank MMSE detector(known correlation matrix) and the conventional matched-filter detector are 8.78 dB and 0.59 dB, respectively. SNRi = 10 dB, K = 5, near-far ratio = 10 dB.................................................................................................... 76 5.8 Comparison of the learning curves of various adaptive reduced-rank CDMA receivers. SNRI = 10 dB, K = 5, near-far ratio = 5 dB. . . . 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A b stra ct A novel reduced-rank MMSE demodulator for DS-CDMA communication systems is presented, which is based on a new Wiener-filter structure called the truncated multi-stage Wiener filter (TMWF). The conventional full-rank MMSE demodulator requires the inversion of an N x N covariance matrix, where N is the processing gain of the system. If the statistics of the channel vary rapidly, a repeated inversion of the matrix is needed to adjust the filter coefficients. This may impose an exces sive computational burden on the demodulation process. The TMW E structure not only eliminates the task of matrix inversion, but its adaptive version can be real ized from the real data without the need for an explicit estimation of the covariance matrix. Furthermore when the code vector assigned to the desired user is known to the demodulator, the coefficients of the adaptive TMWF detector are determined in a completely blind fashion, i.e., without the help of a known sequence of reference information symbols. In this thesis the performance of the TMWF-based adaptive reduced-rank CDMA demodulator is examined through simulation studies. Numerical results show that the TMWF-based CDMA demodulator can achieve the performance level of the full-rank MMSE demodulator at a rank that is much lower than the di mension of the full-rank system. This implies th at the proposed reduced-rank CDMA detector requires lower sample support for estimating the covariance statistics. Also the TMWF-based demodulator is shown to exhibit rapid adaptivity compared to the full-rank system. x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 1 In trod u ction 1.1 C D M A D em o d u la tio n A lgorithm s A large body of recent communication research is devoted to the code-division mul tiple access (CDMA) technique (see [29, 49] for an extensive reference list). This is due mainly to its potential capability to provide an increased capacity for wireless communication systems when compared to the more well-established multiple-access techniques such as TDMA or FDMA [10, 45]. Also CDMA is quite robust against var ious performance degrading factors such as m ultipath fading and electronic emissions from other devices which occupy the same spectrum [1, 4, 7, 24, 28, 41, 42, 48]. In a seminal paper, Verdii demonstrates that the maximum-likelihood (ML) esti m ate of each user’s information symbol sequence can be obtained jointly via a bank of matched linear filters, followed by a type of dynamic-program algorithm [43]. This work also demonstrates th at the performance of CDMA is not inherently limited by multiple-access interference (MAI), i.e., it exhibits a non-zero limit of the symbol- error rate even when the therm al noise level goes to zero. But, it is well known that 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. this phenomenon, called the near-far problem [10. 27], originates from the conven tional single-user detection algorithm in which the effects of interference from other users are simply ignored. Performance degradation due to the existence of interfering users is often measured by a quantity called asymptotic efficiency [43, 44]. This metric is roughly defined as the ratio of the single user to multiple-user SNR that is required to achieve the same symbol-error probability. It is shown that Verdu’s optimal multi-user detector (VOMD) guarantees a non-zero asymptotic efficiency1 with probability one as long as a set of relatively mild conditions is satisfied [44]. But this optimal behavior of VOMD is gained at the expense of a formidable computational requirement that grows exponentially with the number of users. This latter property has prevented Verdu’s multi-user detection algorithm from practical usage despite its optimal performance in terms of the demodulation-error probability. This fact sparked another stream of research activity on multi-user detection with a lowered complexity [51]. An im portant category in the area of reduced-complexity multi-user detection, consists of the linear multiuser detectors. Two key linear multiuser detectors are the decorrelating detector [3, 5, 6, 26] and the m inim um mean-squared error(MMSE) detector [15, 21, 32, 33, 39, 52]. In addition to the considerable reduction in de modulation complexity, these linear detectors are known to have the same near-far resistance 2 as VOMD [19, 20]. An advantage of using linear detectors over VOMD is that they can be realized adaptively either by exploiting training sequences or by being in the blind mode, that is, without the aid of a training sequence or prior information about other co-existing 1A strictly positive asymptotic efficiency means that the logarithmic decay rate of the bit-error probability of a CDMA receiver is inversely proportional to the noise variance. 2Near-far resistance is defined as the minimum asymptotic efficiency over all possible combinations of interfering users’ energies. 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. users [21, 33, 50]. This is im portant for the forward link or the down-link of a CDMA systems since it is practically impossible to use a centralized multiuser detector that requires accurate estimates of the parameters of the interfering users such as the signature waveforms, their amplitudes, and their transmission delays. In contrast to the centralized receivers an adaptive linear detector th at has a single-user structure3 usually uses a sampled version of the received waveform. This is basically the chip-matched-filter output of the continuous received waveform, sampled at the chip rate or higher. This kind of detectors use an adaptive tapped-delay-line or FIR structure and are called the iV-tap detectors [21], where the number of taps N is the processing gain of the system. The optimization of the iV-tap linear filter under the MMSE constraint simply leads to the classical discrete-time W iener filter. In this case the demodulation complexity per binary symbol is independent of the number of users but depends on N . 1.2 M otivation for R educed-R ank P rocessin g Among various algorithms to determine the Wiener-filter4 weights adaptively, the sample covariance-matrix inversion (SMI) technique is found to provide the fastest convergence rate in all cases. As a rule of thumb, at least 2N independently collected data samples are required if one wishes to maintain an average performance differ ence of 3 dB between the optimal Wiener filter and the adaptive filter [34]. But in a non-stationary environment repeated matrix inversions are required to form new 3The single-user receiver is interested in the demodulation of the desired user’ s bit stream only. Also it assumes that the only information that is available to the demodulator is the spreading waveform and the transmission delay of the user of interest. 4 Since the Wiener filter L s equivalent to the matched filter that maximizes the SINR within a scale factor [2], the two terms are often used interchangeably. 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. filter weights. This can be computationally expensive since the complexity associated with the inversion of an N x N covariance matrix is 0 ( N 3). The statistics of the channel may vary significantly; as quickly as every 280 data symbols in a typical radio channel [30] so that for best performance the covariance m atrix needs to be recalculated quite frequently. In order to minimize the com putational requirements associated with the adaptive SMI technique, receivers that operate in the reduced-dimension subspace [37, 39, 46] need to be used. To reduce the number of taps of the linear filter, the received vectors of dimension N are processed by a transform matrix d > of size M x A'- , where M < N is the reduced-dimension. Then, the MMSE solution is sought within this reduced-rank subspace that is spanned by the rows of the transform m atrix. Since the reduced-rank Wiener solution is restricted to be contained in the reduced-rank subspace that is dictated by < !? , the choice of basis vectors for the reduced-rank subspace is the most critical factor in the design of a reduced-rank MMSE filter. Even though the overall amount of computation is somewhat decreased by such a dimension reduction, the linear transform that projects the iV-dimensional vector onto the M-dimensional subspace typically requires an eigen-decomposition of the covariance m atrix [46] or knowledge of all of the interfering users’ spreading sequences and transmission delays [39], which are not available a priori in a single-user receiver. The main objective of the present thesis is to investigate the performance bound of a CDMA communication system that uses an optimal reduced-rank Wiener fil tering algorithm. The truncated multistage Wiener filter (TMWF) [11, 13] provides a systematic procedure to implement a reduced-rank MMSE demodulator th at does not require the inversion or eigen-decomposition of an estimated covariance matrix. Furthermore, it must be emphasized that all of the parameters in the TM W F de modulator can be calculated from the raw data without an explicit estimation of the 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. covariance matrix. This im portant feature contributes to a significant reduction of the computational cost and the numerical stability th at arises from the conventional full-rank approach. It is shown in this thesis that when the dimension of the reduced- rank filter is fixed, the TM W F demodulator outperforms the CDMA demodulators th at are based on other reduced-rank filtering techniques in terms of the output SINR. Also the adaptive TMW F filter weights converge faster than in the full-rank case. 1.3 T hesis O rgan ization In the next chapter a model for the receiver and the received waveforms is presented of a DS/CDM A communication system that uses a full-rank Wiener filter in its demod ulator. In Chapter III the recursive procedure, needed to best obtain a reduced-rank subspace, is outlined. The adaptive realization of the reduced-rank Wiener filter de veloped in Chapter III is described in Chapter IV. The numerical results in Chapter V compare the performance of the proposed scheme with other reduced-rank processing techniques. Concluding remarks are presented in the final chapter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 2 S ystem M od el The structure of the CDMA receiver th a t is considered in this thesis is outlined. Also the discrete-time model for the received waveform is developed along with the corre sponding demodulator that is based on the Wiener-filter output. The Wiener filter solution is shown to be optimal in both the minimum mean-squared error (MMSE) and maximum a posteriori (MAP) sense. 2.1 T h e D etectio n A lgorithm s for D S /C D M A C om m un ication S ystem s D ata detection algorithms for DS/CDMA communication systems can be classified into the following three categories: 1. Conventional matched-filter (MF) detector 2. Centralized detector 3. Decentralized detector 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The conventional detector [47] uses only one matched-filter, matched to the user of interest, assuming that the combined effect of MAI and the thermal noise can be modeled as white Gaussian noise with a increased variance. This assumption turns out to be quite erroneous especially when the thermal noise level is negligible. The performance of conventional detector is MAI-limited unless a stringent power-control scheme is used to maintain the similarity between all users’ power levels and the assigned signature waveforms have low correlations for all possible delays between the data streams transm itted by the asynchronous users. The latter two types of detectors are collectively called multi-user detectors, in the sense that they take the MAI into full account to demodulate each user’s data sequence and ensure that they can operate in non-zero near-far resistance region. The key difference between the centralized and decentralized detectors rests in the content of information that is available to each detector to generate the set of sufficient statistics for the estimate of each user’s data sequence. For the centralized detectors the precise knowledge or the highly reliable estimates of the transmission delays and the spreading codes of all users is assumed to be accessible, which is usually the case for a demodulator that resides in the base station. Consequently a bank of matched filters, each of which is matched to a particular user’ s signal, i.e., spreading code, serves as the front-end processor of this type of detection system. The output of the matched-filter bank constitutes a sufficient statistic for jointly estimating all users’ information sequences. A block diagram of the general structure of a centralized multi-user detector is shown in Fig. 2.1. On the other hand, a decentralized structure is employed when the receiver is interested in the demodulation of the data sequence sent by only one particular user of interest. The front-end of a typical decentralized receiver consists of a chip-matched filter followed by a chip-rate sampler which is synchronized to the desired user’s 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. MFK MF1 MF 2 Vector - Input Vector - Output Decision Device il b 2 b K M atched Filter Bank Figure 2.1: A centralized multi-user detector. transmission. A linear filter, which can be implemented as a tapped-delay line, is applied to the output of receiver front-end to produce the estimate of the transmitted data sequence. The filter coefficients are generally optimized under criteria such as the MMSE and zero-MAI1. A conventional MF detector can be conceived as a degenerate special case of the decentralized detectors, where the coefficients of the linear filter is simply the desired user’s code vector itself. When realized in an adaptive fashion, the decentralized detectors require no ad ditional information than that is needed by a MF detector. Only the code vector and the timing epoch of the desired user is necessary for the demodulation. Also the computational complexity of the decentralized detectors is often comparable to that of the MF detector2. But the design criterion of a decentralized detector is fun damentally different from th at of a conventional detector in a sense that the former maintains the same degree of MAI-suppressing capability as the centralized multi user detector. From the standpoint of system design, the decentralized detector is more suitable for the down-link, or forward link of the channel since it is not feasible for each mobile-user to be equipped with the same amount of information as the base station. However, the bank of decentralized detectors may form a multi-user detector l The zero-MAI design leads to the decorrelating detectors. 2The decentralized detectors and the MF detector constitute the family of single-user detectors. The term ‘single-user’ is used to indicate that they demodulate only one user’s information stream. 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. th at demodulate the multiple users’ information stream simultaneously. A multi-user detector functioning in a decentralized mode is described in Fig. 2.2. 1 r(r)- C M C M F C M F r K Chip-Rate Samplers C M F : chip-matched filter Detector 1 Detector 2 Detector K Bank o< Vector - Input S calar-O jlp u t Decision Devices 1 Figure 2.2: A decentralized multi-user detector. 2.2 T h e D ecen tralized M M S E D em od u lator for A syn ch ron ou s D S /C D M A C om m unication S ystem s The structure of the CDMA receiver that is investigated in this thesis can be clas sified as a family of the single-user, single-symbol (SU-SS) receivers. As explained in the previous section, the receiver of the single-user (SU) type focuses only on the demodulation of the symbols transmitted by a particular user. Also it observes only one symbol duration th at is aligned with the desired user’s transmission in order to obtain a sufficient decision variable. In this section we present a detection scheme for a SU-SS CDMA receiver that is optimized under the MMSE criterion in the context of the well-known W iener filtering problem. 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2.1 General Structure of th e Single-User, Single-Sym bol, Receiver The receiver structure considered henceforth is a single-symbol MMSE detector of a single-user (SU) [21, 38, 39]. Unlike the multi-user (MU) receivers, a SU receiver focuses only on the demodulation of the desired user’s symbol stream. Even though the generic structure of a SU receiver resembles that of the conventional non-adaptive matched-filter (MF) detector, it is possible to design a near-far resistant SU receiver by adaptively suppressing or cancelling the multiple-access interference. The asso ciated filter complexity is often comparable to that of a conventional non-adaptive MF detector. The necessary parameters in the SU receiver family consist of the code-waveform that is assigned to the user of interest and its timing epoch. In situations where the synchronism between each user’s transmission is practi cally impossible to obtain, which is often the case in the reverse link of a CDMA channel, one needs to observe the entire span of all of the active users’ data stream in order to construct the optimal MMSE detector [16, 20]. But concern for the overall system complexity and the need for acceptable demodulation-delay rules out such an extreme design option. Among the various suboptimal techniques that utilize only a finite observation interval [15, 17, 21, 25, 33], attention in this thesis is confined to the single-symbol (SS) receivers that observe only a single symbol interval. The general ization of the proposed receiver structure to multiple-symbol observation receivers is straightforward. The front-end of the SU-SS CDMA receiver usually consists of a conventional chip matched-filter, followed by a chip-rate sampler3. An important function of the 3Chip-rate sampling is sufficient only when a perfect synchronism among the active users is maintained. Otherwise, a higher sampling rate is required, and the overall system complexity is increased. 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. receiver front-end is to convert the continuous version of the received waveform into a discrete waveform. The output from the receiver front-end is then correlated with a vector of filter coefficients that is optimized under a predefined criterion to produce a soft estimate of the transmitted bits. The final symbol decisions are made by comparing the soft estimates against a threshold, which in most cases is set to the zero bias. The generic structure of a SU-SS CDMA receiver is depicted in Fig. 2.3. i-----------------------------------------------------------------------------------1 r(0 Chip-Rate Sampler r. Linear Filter Threshold Comparison Chip - Matched Filter i ____________________________i Receiver Front-End Figure 2.3: The block diagram of a generic single-user, single-symbol, CDMA receiver. 2.2.2 Received Waveform M odel of an A synchronous D S- C DM A Com m unications To establish the optimal linear filter in the MMSE sense, one needs to model the waveform that is available at the output of the receiver front-end. The system under consideration is an asynchronous DS-CDMA system, where each user’s signal arrives at the receiver with different transmission delays on an A.WGN channel. Also the binary phase shift keying (BPSK) modulation is assumed. The complex baseband representation of the received waveform for such a CDMA communication system is modeled in the form, r{t) = s{t) + n{t) = R(t) e ^ (i), (2.1) 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where n(t) is an additive white Gaussian-noise(AWGN) process for which the power spectral density is a 2, R (t) is the envelope, < p(£) is the base-band phase, and s(t) is the superposition of the d ata signals of the K users over M symbol periods. Thus s(t) is given by M K = E E s J Z W k i i ) ^ bk{i) sk(t - iTb - T fc), (2 .2 ) 1=1 fc=l '------------ V- ' 0k(i) where the parameters and other quantities are defined as follows:4 K Number of active users. y W k{i) Received am plitude of fc-th user’s signal in the z-th symbol period. M Frame length. bk(i) z-th data symbol of the k-th. user. 4 > k(i) Carrier phase of fc-th user’s signal in the z ’ -th symbol period. rk Relative delay of the k-th user. Tb Symbol period. Here sk(t) is the signature waveform assigned to the fc-th user, given by iV - l Sk(t) = 53 Vk\j\u{t - jT c), (2.3) fc=0 where p*, = ( p*[0], ... , pk[N — 1] )* is the spreading sequence for user k, u(t) is a unit-power rectangular chip-pulse5waveform with finite support in the interval [0, Tc\, 4The subscript index is used to denote a particular user and the index in the parenthesis denotes a particular symbol interval. For example, bk(i) indicates the bit information transmitted by the fc-th user during z-th symbol interval. 5The term ‘chip’, first coined by the engineers at FTL (Federal Telecommunications Laboratories), denotes an elementary pulse which is modulated by a single pseudo-random variable [36]. 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and N = Tf,/Tc is the number of spreading code chips per data symbol, i.e., the spreading ratio or the processing gain of the system. 2.2.3 Vector M odel of Received-W aveform Assume that the output of the chip-matched filter is sampled at the chip rate. Then, the front-end of a SU-SS CDMA receiver for the desired user generates an iV-dimensional vector which consists of N successive chip matched-filter outputs, col lected in the z-th bit period [(z — 1)2),-Fri , z'T 6 + r t] of the first user. The Z-th sample a t the output of the chip-matched filter of duration Tc is denoted by r[Z] = >/2 f r(t)u(t) cos(uict + d> \) dt. (2-4) •/(i-OTc-i-ri Thus, the received waveform vector r(z) for the desired user during the z-th bit is expressed by r(z) 4 ( r[(z - 1 )N], r[(z - 1)N + 1], ... , r[iN - 1] )f. (2.5) After the removal of the second-harmonic term by the low-pass filtering, the integra tion in Eq. (2.4) reduces to the integration-and-dump operation for a rectangular chip waveform6, viz., — == / r(t) dt. (2 .6 ) y/Tc V - i ^ + r . w K ’ Suppose that the relative transmission delay 7% of the fc-th user with respect to the desired user is denoted by • / % = r* — T\ = (mk + p-k) Tc, where symbols and pk 6 For convenience a carrier-synchronized system is assumed in which the carrier phases fc are zero for each i. 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. denote the integral and fractional parts of f k, respectively. Then, it is shown in [22] that the equivalent discrete-time vector for the received-waveform is given by7 r(i) = ^ (i) Pl + £ ; j3k(i - 1 ) [ ( 1 - p k) s[mt) + p k S,(mt+1) ] p k k= 2 7P Zk £ flt(i) [ (1 - W) S<”“ > + l i t S < ™ * +" ] p* + n(i), k= 2 (2.7) where S\p) = 0 ID 0 0 p , s(p> = 0 0 ---- 1 o e x i w 1 (2.8) where the all-zero matrices denoted in Eq. (2.8) are assumed to be of the appropriate dimensions. The application of N x N matrices S|p^ and to a iV-dimensional column vector x = [rro,^ii • • • , results in the acyclic left-shift and right-shift by p given, respectively, by and S[p ) x = [ xp, . . . , rcAr-i, 0, • • •, 0 ]f p zeros Sjft x = [ 0 , . . . , 0 , aq ,. . . , Xff-p-i ]f. p zeros (2.9) (2.10) Also the noise vector n(z) is an A r-dimensional Gaussian random vector with zero mean and covariance matrix T = a2 1at, where In denotes the n x n identity m atrix and a 2 is the power spectral density of the additive white Gaussian-noise process. 7This thesis will consider a short code system only, i.e., the same spreading sequence is repeatedly used. 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Recall th a t the symbols transm it ted in the subsequent sym bol periods are as sumed to be m utually independent. Thus, the vectors z* and in Eq. (2.7) can be regarded as statistically independent interfering vectors of dim ension -V that affect the perform ance of the underlying dem odulation algorithm . This implies that an asyn chronous DS-CDMA system can be interpreted as an equivalent synchronous system with an increased number of’ interferers. The pictorial description of the equivalent synchronous transmission model for a SU-SS receiver is shown in Fig. 2.4. Here, the transm ission delay of the desired user is assum ed to be zero w ithout the loss of gen erality and the users are num bered in the increasing order of the transm ission delay. T he different hatching patterns are used to indicate the different symbol transmission periods of each user. It can be seen th at the portions of interfering vectors due to the (//. — L)th sym bol period falls within the n-th symbol interval [riThAn, -r 1 )T/,i of the desired user. These* vectors: together with the portions of interfering vectors due to the //.-th symbol period, can be regarded as the new interfering users that transm it iri synchronization with the desired user. User 1 User 2 User 3 User K V ////////////A 7ZZK nT, ( n + 1 ) T b Figure 2.4: Equivalent synchronous transm ission model. 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.3 D erivation o f O p tim u m R eceiver 2.3.1 M M SE Criterion : W iener Filter W ith the model developed in the previous section, the MMSE detection of the CDMA signals can be considered to be a binary hypothesis-testing problem in which a decision statistic is generated by the use of a classical Wiener filter with the sufficient statistic r(z'). Thus the estim ate of the z-th symbol that is sent by the desired user is obtained by comparing the Wiener filter output to a fixed threshold. Mathematically, an estimate of the information symbol b is obtained from 8 b = sgn(R e(w t r ) ), (2.11) where Re(-) denotes the real-part of a complex number and sgn(-) denotes the ‘sign’ operation. The problem at hand is to find the MMSE solution, w 6 CN, th at min imizes the mean-squared error between the desired signal b and the filter output, i.e., M M S E = m inE |( w ^ r — b) (w ^r — 6)^ j = min R r w — 2 R e( rT d ) + a 2 b), (2 .1 2 ) where R r = E {r r t } 8The subscript index used for identifying the user and the index in the. parenthesis used for the particular symbol period will be omitted unless it causes any confusion. 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. K K W i Pi p ! + E zj(zj)' + E w * <«)' + ff2 Iw . (2 -1 3 ) k= 2 fc=2 rrr f = E{fer} = y jw x p i, (2.14) and of = E {b2} = 1 . It is shown readily th at the MMSE solution satisfies the Wiener-Hopf equation [31], R r w = rrd, where R r = E | r t - r f r rr f = E {bi r,} and the optimal filter w is provided by (2.15) W = R ,. 1 Trd (2.16) when the covariance matrix, associated with the observation vector R r, is invert ible. It should be noted from Eq. (2.13) that the optimal Wiener filter solution requires the information about the interfering users such as the signature waveforms, their amplitudes, and transmission delays, which are not available to the single-user- type receiver. It is the goal of this thesis to present an adaptive algorithm for the single-user-type CDMA receiver to equivalently solve the Wiener filtering problem in Eq. (2.16) with a substantially reduced amount of computational complexity. 2.3.2 M aximum-Likelihood (ML) Criterion If the source is equi-probable, the maximum-likelihood (ML) criterion is eqidvalent to the maximum a posteriori probability (MAP) criterion, which ensures the minimum 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. bit-error probability [35, 40]. Since the BPSK modulation is being considered, the received vector r is observed under two equally probable hypotheses, namely, Ho : r = s + u , Ht : r = — s + v , (2.17) where s = \JW [ p i is the (known) ‘signal’ vector and u is the colored interference- plus-noise vector whose covariance m atrix is given by K K R , = £ w * z% (z£)t + £ w k 2|(z f)f + < r 2 1„ . (2.18) k=2 k= 2 The probability density of r under each hypothesis follows easily9: / ( r I Ho) = f u i r - s ) Hpt'VTTT exp [ _ ( r " s)t 1 (r “ s ) ] (2-19) |7r||A /'det(R ^) and / ( r I Ht) = /i/(r + s) = iM r d = t(R ^ e x p [ ~ ( r + s ) t R - " '( r + s ) ]- (2 -20) 9It is assumed that i> can be modeled to be Gaussian. 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Thus, the likelihood ratio is given by / ( r | Ho) exp -(r + s)f R „ l (r + s) (2.21) Canceling common terms and taking the logarithm yields the log-likelihood ratio given by L(r) = ln[ A( r ) ] = 2 (st R ; l r + r t R - 1 s) = 2 R e (s t R " 1 r). (2.22) Hence, the likelihood-ratio test is ffo R e ( s t R " l r) > 0. (2.23) Here it should be noted that the final MAP test in Eq. (2.23) is based on the in ner product between the observation vector r and the linear filter wmsir = R-iT 1 s that maximizes the output signal-to-interference-noise ratio (SINR). This observa tion implies that the decision rule based on the maximum output SINR filter w mstr guarantees the minimization of the decision-error probability. Additionally, since the Wiener filter developed in the previous subsection is equivalent to w mstr except for a gain factor [2], [14, Chapter 5], one can conclude th at the MMSE performance measure is equivalent to the maximum output SINR and minimum decision-error probability measures. 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 3 A lgorith m s for R educed-R ank M M SE D etectio n This chapter is devoted to a comprehensive description of the various reduced-rank algorithms that have been published in the literature. As a step prior to the complete explanation of the algorithms, the MMSE solution is represented in terms of the signal subspace parameters. W ith the aid of the signal subspace representation of the Wiener filter it is proved that a lossless reduced-rank Wiener filtering is possible by projecting the received vector onto the subspace that is spanned by the code vectors that are assigned to the active users. The general structure, presented here of the reduced- rank Wiener processor, is inspired by the ideal reduced-rank Wiener filter. Among the numerous reduced-rank techniques, the principal component and the cross-spectral metric methods are introduced. Finally an introduction to the reduced-rank Wiener filtering that is based on the multistage Wiener filter representation is given. 3.1 Subspace A pproach o f the M M SE D etectio n In this subsection attention is restricted to the synchronous transmission model with K , K < N , users to facilitate our understanding of the subspace representation of 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the MMSE solution. Under the assumption of synchronous transmission, Eq. (2.7) is simplified into r = £ \/W k bk Pf c + n. (3.1) Denote C = [ p i,. . . , P k ]l . The covariance matrix of r is given by R r = C W C f + cr2 I^v. (3.2) where W = d\ag(W i, ... , W k )- In general, since is Hermitian symmetric, R r can be decomposed into Hr = V A V f = [Vs VB ] A s i i e < • v t v n (3.3) where A s = diag(Ai , ..., Ak) contains the K largest eigenvalues of R^- and the col umn vectors of V s represent the corresponding orthonormal eigenvectors Vi , ... , v*-. Also the columns of the matrix V n are the N — K orthonormal eigenvectors that cor respond to the eigenvalue cr2. Lemma 3.1 Any N -dimensional vector, that lies in the orthogonal complement of Range(C), is a legitimate eigenvector of R^- and the associated eigenvalue is a2. P r o o f : Suppose th at a unit vector v lies in the orthogonal complement of Range(C), that is, v 6 (Range(C) )x. Then, v should satisfy the relation C f v = 0. (3.4) *It is assumed that the code vectors that are assigned to the users are independent of each other and consequently the matrix C is of full-rank. 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. By combining Eq. (3.4) and Eq. (3.2), the product R r v is calculated to be R r v = C W Cf v + cr2 1,v v = o^v. (3.5) Therefore the unit vector v is an eigenvector of R r and its corresponding eigenvalue is a 2. □ Corollary 3.2 ( Range(C) )x = Range(Vn). P r o o f: Seeing th at the rank of Range(C) is K , the rank of its orthogonal complement space with respect to CjSf, denoted by ( Range(C))-, is N — K. Thus. Lemma 3.1 implies that the the subspace ( Range(C))x is spanned by the columns of V r. = [v*r+ l, .. • , Vtf], viz., ( Range(C) ^ = Range(Vn) (3.6) and that An = [Ar-+1 , ... , A jv] = cr2 Tv-fr- (3.7) □ C o ro llary 3.3 The code vector pi that is assigned to the desired user is orthogonal to Range(V„), i.e., V nfpi = 0 . P r o o f : Since pi G Range(C), the statement follows from Corollary 3.2. □ Since the basis vectors of ( Range(C) )x are the eigenvectors of R r whose corre sponding eigenvalues are all a 2, the column space of V n, Range(V„) , is called the noise subspace. 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. L e m m a 3.4 The column space of C coincides with the column space o f V s. P r o o f : From the decomposition of R,. in Eq. (3.3), it can be seen that the column space of V s spans the orthogonal complement of the noise subspace, i.e., Range(Vs) = ( Range(Vn) )x . (3.8) Applying Corollary 3.2 to Eq. (3.8) yields Range(Vs) = ( ( Range(C) )" L) ± • (3.9) Recalling the fact that (S-L )_ L = S with S being any arbitrary vector space, it is straightforward to show th at the right-hand side of Eq. (3.9) is equal to Range(C), namely, Range(Va) = Range(C). (3.10) □ Thus, one concludes th at the set of orthonormal eigenvectors with the correspond ing eigenvalues other than cr2, th at consists of the columns of V s, spans the column space of C. This subspace is called the signal subspace. Lem ma 3.5 The MMSE solution w must lie in the signal subspace and can be rep resented in terms of signal subspace parameters V s and As. P r o o f : By Eq. (3.8), iV-dimensional full-rank MMSE solution w can be represented as a sum of the signal subspace component w s and the noise subspace component w n, i.e., w = w s wn = V s ks + V n k n, (3.11) 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where ks and k„ are the coordinate unit vectors of w s and wn with re spect to {vt, ... , vr-} and {v/r+x , . . . , v^}, respectively. Then, combining Eqs. (2.12), (2.14), and (3.3), one obtains the MSE, associated with the estimation of the desired signal b with the optimal linear filter w, given by M S E { m v) = E{(wtr - 6) (wtr-fc)*} = w* R^. w — 2 R e ( \/W i w t p j ) + a2 = w f ( v s As Vj + cr2 V„ V^) w — 2 Re( \fw~\ p t ) -F cr2 = kj As ks — 2 R e( y fw [ w | px) + a % + a2 k£ V n V+ k n, (3.12) ' ' = E {(w] r — 6) ( w | r - 6)f } where a% is the variance of the desired signal b. Here the equality V jV n = 0, that follows from Eq. (3.8) and Corollary 3.3, is used to obtain the expression in the last line. Also the equality, E {(wj r - b ) (wj r - 6)f} = kj As k s - 2 R e( P i ) + of, (3-13) follows from the relation, w j R r w s = k\ V l ( V s As V j + V n V t ) V s k s = klvlvsAsvlvsks + kl yly»viv,ks = 0 = k tA s ks. (3.14) The third term in Eq. (3.12) denotes the MSE contribution due to the noise subspace component wn and is strictly positive unless kn is a zero vector. This suggests that the MSE associated with the MMSE solution w is strictly larger than 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the MSE associated with w s, which contradicts the fact that w minimizes the MSE. Hence the noise subspace component w„ of the MMSE solution w must be zero. The substitution of Eqs. (2.14) and (3.3) into Eq. (2.16) results in the signal subspace representation of the MMSE solution w, given by w = s j w i v A - l V fPl = >/wT(VaA7l V tPl + o-2 Vn V+Pl) = ,J w \ V s A j l V \ p u (3.15) where the last equality again follows from Corollary 3.3. □ Above lemma leads us to the following proposition: P ro p o sitio n 3.6 The M M SE filtering in a full-rank N-dimensional space can be done equivalently in a reduced-rank K-dimensional signal subspace by projecting the original observation vector r onto the signal subspace. P r o o f : Consider the problem of finding the reduced-dimension Wiener solution based on the projection of r onto the signal subspace. Denote the A-dimensional projected observation vector by r/c = Vjr. (3.16) Then, the reduced-dimension Wiener solution w k is now provided by w K - = R7# J r rjr<fJ (3.17) 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where the K x K covariance m atrix R rftr associated with the projected vector r k is given by Rr/f = V jR rV * = V t ( v s A s V t + cr 2 V n V t ) V , = A s, (3.18) where the third equality follows by recalling that V j V s = I k and V* V s = 0(n- k)xK- The cross-correlation vector rTKd between Tk and the desired signal b is computed to be TrKd = y J w [ v \p u (3.19) where p i is the code vector th at is assigned to the desired user. Substituting Eqs. (3.18) and (3.19) into Eq. (3.17) yields w ^ = 0 ^ A 7 l V t p i . (3.20) Now the equivalence of the full-rank Wiener filtering on the original observation r with reduced-dimension Wiener filtering on the projected observation r k can be seen by the following equality, w t r = J w \ p \ V^AJ1 V j r = ( y V ^ A .- 'V j p ,) ' (V * r), (3.21) (wat)t where the expression for w given in Eq. (3.15) is used for the first equality. □ 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It is evident from Proposition 3.6 that the resulting MMSE associated with the reduced-rank Wiener filtering is identical to that of full-dimension Wiener filtering, but the dimension of both the observed data vector and the optimal Wiener filter is reduced from N to K . This fact gives us a significant motivation for the reduced- rank MMSE filtering approach for the demodulation of CDMA signals, especially when the processing gain of the system is far larger than the number of active users that communicate simultaneously in a system. 3.2 G eneral S tru ctu re o f R educed-R ank W ien er F ilterin g In the previous section it is demonstrated that the MMSE detection of synchronous CDMA signals can be equivalently accomplished within the signal subspace whose rank is less than the dimension of the observation vector. As described in the proof of Proposition 3.6, the Wiener filtering in the reduced-rank subspace is carried out in two successive steps: dimension-reduction and reduced-rank Wiener filtering. To achieve a reduced-rank Wiener filter it is necessary to transform the N - dimensional observed vector r to an M-dimensional vector r,w for M < N by applying a M x N linear operator. This process is mathematically equivalent to projecting r onto the subspace, which is spanned by the column vectors of the matrix T that rep resents the dimension-reducing linear operation. The next step is to find the MMSE solution with Tm in an M-dimensional subspace of CN. The schematic diagram of the reduced-rank MMSE detection of CDMA signal is depicted in Fig. 3.1. W ith the 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.1: The reduced-rank MMSE detection of CDMA signal, m atrix T given, the reduced-rank Wiener solution w w in CN is obtained as w A / = R ^ l r rM(1 (3.22) where R rw = T R r T t, rTMd = T r rrf and R r and rrd are as defined in Chapter 2 . The resulting MMSE, associated with reduced-rank Wiener filtering, is given by MMSE = a \ - r*„ d rru d, (3.23) where c rd is the variance associated with the desired random signal, which is equal to 1 when the modulation format is binary phase-shift-keying (BPSK). 3.3 A lgorithm s Based on E igen -D ecom p osition o f th e Covariance M atrix It is known [11] that the MMSE that is attainable by the reduced-rank Wiener filtering technique strongly depends on the column space of the covariance matrix associated with the reduced-dimension observation vector ra/. The achievable MMSE by a full- dimension Wiener filter given in Eq. (2.16) is always less than that of a reduced-rank 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. version except when the full-dimension Wiener solution is contained in the reduced- dimension subspace, in which case the two solutions are identical. Thus, the selection of the subspace created by the Unear transform T is the most crucial to the develop ment of reduced-rank Wiener filters in order to achieve a performance comparable to the full-rank case. Among the reduced-rank algorithms described in [1 1 ] the principal-components and the cross-spectral metric methods are based on the eigen-decomposition of the covariance m atrix of the observation vectors. They share a common feature th a t the subset of basis vectors, that span the signal subspace, constitutes the columns of dimension-reducing linear operator T , but they provide different criteria as to which basis vectors are to be selected. For these types of rank-reduction, consider the following decomposition of the observed-data covariance m atrix R r = V A V t, (3.24) where V is the matrix whose columns are the eigenvectors of {vl} ]Y ;l and A is the diagonal m atrix whose entries are the eigenvalues which are arranged in the descending order. The principal-components technique chooses the dimension-reducing operator T in such a manner that its rows consist of the M eigenvectors of R r, which correspond to the largest M eigenvalues. On the other hand the cross-spectral metric method picks out the set of M eigenvectors that corresponds to the largest total cross-spectral energy, where the cross-spectral energy (metric) associated with the z-th eigenvector is defined by /?t - = for i — l, 2,..., N . 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It can be seen th at by selecting the eigenvectors of the covariance matrix in the decreasing order of the cross-spectral metrics one can achieve the best performance among all possible selection rules in terms of the MSE associated with the estimation of the desired signal. Suppose that the columns of a m atrix V m consists of M eigenvectors of the covariance matrix R,. chosen under a certain criterion, where A m is a diagonal m atrix whose entries are the corresponding eigenvalues. Then, from Eq. (3.23), the resulting MMSE associated with a reduced-rank Wiener filter with T = V f A , yields2 M M S E = a2 d - v \ dW M ( v j ^ v ) A " 1 (VfV M) V{rrrd 1m 0 ( N ’- M ) x M — ° ’d r £ r f V A ,/- O m x ( N - M ) ] A = ° d ~ (V lr r rd )t A J V l f Trd _ x2 V ' llvIrrd||2 - ad ~ l ^ T------ fc=l M = &d-Y,Pk, k- 1 V l ,r rd (3.25) which implies that the choice of those eigenvectors, corresponding to the M largest cross-spectral metric, minimizes the MSE as a function of rank. Finally, the M-dimensional Wiener filter, which is applied to tm, is represented by wM = A A f ' Vjw rrd, (3.26) 2It is assumed that V is column-permuted in such a way the first M columns of V coincides with those of Vm- The diagonal entries of A is also permuted in a similar manner. 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where the N x M matrix V/^ is composed of the eigenvectors associated with the M largest eigenvalues (or cross-spectral energies), and the M x M diagonal matrix is composed of the corresponding eigenvalues. 3.4 A lgorith m B ased on Truncated M u lti-stage W ien er Filter The final reduced-rank Wiener-filtering technique, introduced in [1 1 ], is called the multi-stage Wiener filter. The multi-stage Wiener filter is obtained by a decomposi tion of the observation vector by a sequence of orthogonal projections, which has the form of a multi-resolution analysis filter bank. The resulting output vector from the analysis filter bank is processed by a nested chain of scalar Wiener filters. Rank reduction is achieved by simply stopping the decomposition process at the M -th stage. In other words, the first M decomposition stages represent a truncated version of the full-dimension multi-stage Wiener filter, and it serves as the reduced- rank Wiener filter3. One of the most im portant advantages of this new technique over the eigen-decomposition-based technique is that it does not require an estimate of the covariance matrix or its inverse when it is implemented adaptively. A more detailed description of this technique is relegated to the following chapter along with its adaptive implementation. 3 This is termed the truncated multi-stage W iener filter. 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 4 T runcated M u lti-stage W ien er Filter In this chapter a new approach is developed to solve the classical coiored-noise Wiener filter problem presented in Chapter 2 for an asynchronous DS/CDM A system. This new method, called the multistage Wiener filter [13], converts the Wiener filter into a nested chain of one-dimensional matched filters in which no covariance m atrix inver sion is required for realization. By cutting off this chain of one-dimensional filters at a certain level, a reduced-rank Wiener filter can be realized in a very efficient manner. 4.1 D erivation o f th e M u ltistage W ien er-F ilter Two different methods are presented to derive the multistage Wiener-filter structure. The first approach utilizes the transformation of the coordinate system in such a way that the Wiener filter in the original coordinate system can be realized via a nested chain of one-dimensional Wiener filters in transformed coordinates. At each stage, the matrix, that is used for the coordinate transformation, projects the received vector onto two orthogonal subspaces; one subspace is one-dimensional in the direction of the cross-correlation vector at the previous stage and the other subspace is simply 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the orthogonal complement of this one-dimensional space. The dimension of the latter subspace is one less than th at of the subspace at the previous stage. This decomposition is repeated until the dimension of the subspace reduces to one. Then, the coordinate projections of the original observation vector that are generated in the course of successive subspace decompositions, which are all scalar, are combined into the nested chain of scalar Wiener filters. The second derivation of the multistage Wiener filter is based on the optimiza tion of the dimension-reduction linear transform and the associated W iener filter in the reduced-rank subspace induced by the transform. For this purpose, the data- dependent basis vector and corresponding scalar Wiener filter coefficient are recur sively optimized in the mean squared-error (MSE) sense at each stage. Most im portantly, in order to remove the involvement of the received vector’s statistics in the optimization process, the covariance statistics associated with the data vector are assumed to be uncorrelated at each stage. This assumption leads to the so-called max imum correlation criterion for the selection of optimum basis vectors, which coincides with the choice of the steering vectors used in the first approach. The proof of optimality, regarding the choice of each rank-one basis vector, is pre sented and this gives a logical basis for the first approach with regard to the selection of the coordinate transform. Since the optimization in the MSE sense is carried out in a stage-by-stage manner, the resulting reduced-rank W iener filter structure guar antees that the minimization of the MSE, or equivalently, the maximization of the output SINR when the second-order statistics of the received vector are not available. 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.1.1 Orthogonal D ecom position Approach A basic building block for the multistage representation of the Wiener filter is a recursive direct-sum decomposition of the subspace spanned by the columns of the covariance m atrix into two subspaces: One is a one-dimensional space that lies in the direction of the cross-correlation vector, called the steering vector, between the observed vector process and the desired signal and the other is the orthogonal com plement of this steering vector with respect to the subspace spanned by the columns of the covariance matrix. To see how this orthogonal decomposition of a subspace gives rise to a new equivalent Wiener filter representation, we investigate a modified Wiener filtering problem that is based on the transformed version of the original data vector ro, formed by a non-singular matrix A diagram of this problem setting is shown in Fig. 4.1. Figure 4.1: A block diagram of the transform-domain Wiener filter It is fairly easy to see that transform-domain Wiener filtering is equivalent to one that is based on the original observation vector r 0 as long as the linear transformation lThe received vector ro is collected during a certain symbol interval and is defined in Eq. (2.7). The subscripts are used to indicate a particular stage in the decomposition process. 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. # 1 is invertible (see Appendix A). A unitary m atrix with the following specific structure is now considered, namely, the N x N matrix, * 1 = ut (4.1) where Ui is the unit vector in the direction of the ‘steering’ vector of the Wiener filter, S o = r rg tig , i.e., ui = 7i~~rr = „ rrorf° „ • (4.2) so || || r ro rfo || Here in Eq. (4.1) the (N — 1) x N m atrix B i is known as the blocking matrix whose range space conforms with the null space of u t in such a manner that B t Ut = 0 . (4.3) See Ref. [13] for different methods to construct a blocking matrix. The transformed data vector Z\ is given by ta t _ zt = $ t r 0 = U[ r 0 A d, Bi r0 T l (4.4) its two entries represent the coordinate projections of r 0 in the vector spaces, Range(ut) and Range(Bi), respectively. The direct sum of these two subspaces con stitutes the entire iV-dimensional Euclidean space. Then, by a use of Appendix A the optimal Wiener filter to estimate the desired signal dQ , that is based on z lt is given by w*, R ZI r»[d0. (4.5) 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Here, the covariance m atrix R Zl, associated with the transformed vector process z }, can be expressed in the form of the partitioned matrix2, R Z l = ud, r ri^i r f ndi Rr, (4.6) W ith the aid of the m atrix inversion lemma [23], the inverse of Rz, is computed to be = _i_ _ r t T5 - 1 ridi -^ * T | (4.7) *ridi (jli T -N — i -F rndi rridl Rri ) where r]i = ad x — rj R ~ lr rid,. Note th at rj\ is exactly the error variance which results when finding the optimal linear estimate of the scalar dx from the ( N — l)-dimensional vector rj. Also the cross-correlation vector rzldo is ^z\do ^ 1 r ro d o where is the norm of vector s0, i.e., < 5 , 0 (4.8) *1 = I I S O I I = I I T ro rfo (4.9) 2 Unless otherwise specified R a hereafter denotes the covariance matrix of vector process a. Simi larly rab is the short-hand notation for the cross-correlation between the two vector processes a and b. 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Next by Eqs. (4.7) and (4.8) one obtains the equation for the transform-domain Wiener filter, viz., *1 1 1 w ;i = — V\ — R j- j r n<fi . _W ri . (4.10) where by Fig. 4.2 the new (iV — l)-dimensional weight vector w ri = R “ l r r,d, turns out to be the Wiener filter needed to estim ate the scalar di from the vector r t. Here rji is the variance of the estimation error eL of this new (iV — l)-dimensional Wiener filter, given by e x = di — w j, rt. (4.11) Also from Fig. 4.2 it is seen that A —l r — T J i O i (4.12) is, indeed, the one-dimensional Wiener filter needed to optimally estim ate the desired signal d0 from the error signal ex. This is demonstrated by showing th at the cross correlation r e iC [0 between ei and dQ is given by reid o = E {M o} = E {(uj r 0 - wj, B, r 0) d0} = uj rro d o - w*, B , rro d o = , (4.13) where the last equality follows from Eqs. (4.2) and (4.3). From Eqs. (4.4) and (4.10) the output of the now two-level Wiener filter is calculated to be < ^ o = zx = vlldo R^1 zx = w x (di - rx) = u t (dt - r ^ R ^ r , ) , (4.14) V > i 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. r0 w . X r 1 1 1 . + £ i w J - Figure 4.2: The first stage of the multistage decomposition. where 951 is the output of the Wiener filter th at operates on ri in a subspace whose dimension is reduced by one. Thus, the transform-domain Wiener filter can be realized as a cascade of two Wiener filters wr, and u > i as shown in Fig. 4.2. This completes the decomposition at the first stage of the multi-stage Wiener filter. Now define a new steering vector Si in the reduced-(iV — l)-dimensional subspace by (4.1-5) and the coordinate transform 4*2 by # 2 = u | S1 list || 1 C O to I B 2 (4.16) where the rows of the (iV — 2)xN matrix B 2 is the nullspace of U 2 , i.e., B 2 U 2 = 0. Also, the new coordinate vector of ri under the transform $ 2 is represented by A A y z2 = ‘ f>2 r 1 = u |r . A d2 B 2 ri r 2 (4.17) in accordance with the definition in Eq. (4.4) for the first stage decomposition. 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. W ith this transformation 3> 2 operating on r i , the output of the W iener filter is obtained equivalently in the transform domain by = r LrfI Z2- (4.18) A comparison of Eq. (4.18) with Eq. (4.14) implies that cpi is computed via exactly the same equations as y ? 0 except that the indices representing the decomposition stage are all increased by one. Thus, the procedure for the decomposition of the Wiener filter in the first stage is repeated in the second stage. By following the same arguments th at lead to Eqs. (4.7) and (4.8), the cross-correlation vector rZ 2 d x and the inverse of the covariance m atrix Rz, are obtained by where 6 2 is the norm of (N — l)-vector r rid, and 772 is the MMSE associated with the Wiener filter w r2 rr2d2, which gives the optimal linear estimate of d ,2 from r 2. The substitution of Eqs. (4.17),(4.19) and (4.20) into Eq. (4.18) yields 0 ^ z2di ^ 2 (4.19) 0 7/2 - R t 21 r r2d2 1 1 (4.20) (4.21) where a/2 = ^ is the optimal scalar Wiener-filter coefficient needed to estimate the scalar di from the error signal e2 = d2 — R^;1 r2. This result for u 2 is similarly 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. obtained by applying the first-stage identity given in Eq. (4.13). Hence, the overall three stage Wiener-filter output is calculated from Eq. (4.14) by the relation, < P o = ux (dx - uj2 (d2 - rj.2d2 r 2)). (4.22) Continuing this decomposition process in a similar m anner until all Wiener filters in the chain reduce to scalar coefficients, yields finally the Ar-stage Wiener filter output c p o in the recursive form, given by ipo — ui{(di — u 2 (d2 — ujz(- ■ -o;jv-i(dA/-i ~ cujvd/v) •••)))• (4.23) An example of the complete realization of the W iener filter, based on Eq. (4.23), is provided in Fig. 4.3 for the case of N = 4. Rank reduction can be viewed as Analysis Filter Synthesis Filter Figure 4.3: The realization of the multistage Wiener filter for N = 4. a truncation of the full-scale structure or formula in Eq. (4.23) at a certain stage. The resulting reduced-rank structure is called a truncated multistage Wiener filter (TMWF) representation [13]. Before presenting the complete recursion relation for the multistage Wiener filter representation, it should be noted th at the multistage decomposition obtained in 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Eq. (4.23) or Fig. 4.3 for a Wiener filter, can be partitioned conceptually into two main functional blocks: the analysis filter bank and the synthesis filter bank. The recursion relations between the parameters of this new Wiener filter structure are presented next in accordance with this partition. The analysis filter bank recursively generates an iV-dimensional vector d = [d i, . . . , th at consists of the “desired signals”, each of which serves as a ref erence signal for each of the N scalar Wiener filters. At the z-th stage of the analysis filter bank, where i runs from 1 to N — 1 , the linear transform which applies to the ( N —i + l)-dimensional vector r 2 -_i, is defined in an manner analogous to Eq. (4.1) as follows: ul B; (4.24) r d’ r a where ut - = [|r7 '‘~[tfr~11 | = an<^ m atr^ x t^ ie blocking matrix associated with u t -. Then, the scalar signal dt - and the (N — z)-dimensional vector r* for the z-th stage are calculated by di = u |rj_ i, for 1 < i < N — 1, (4.25) r t - = Bfr,-.!, for 1 < i < N — 1. (4.26) The synthesis filter bank, which is composed of N nested scalar Wiener filters, forms the vector e that consists of a sequence of scalar error signals e as follows:3 e = [ £ ! ,... , 6^ (4.27) 3For the purpose of notational convenience, the scalar output rjv_i of the last blocking matrix in the recursion is set equal to the Ar -th component of both d and e. 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. N - 1 Figure 4.4: The synthesis filter bank of the multistage Wiener filter. with the output vector of analysis filter bank d as its input. The error signal at the i-th stage or component of e in Eq. (4.27) is given by €i = d{ — uii+\ et -_ H . (4.28) The operation of the synthesis filter bank is illustrated in Fig. 4.4. It can be recognized that the operational flow of the synthesis filter bank is in a bottom -to-top fashion as opposed to that of the analysis filter bank. The values of the scalar Wiener filters are found from the Wiener-Hopf equation to be = VF1 rei+ldi = V f 1 <$i+i. (4-29) 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where the second equality is established in analogy with Eq. (4.13) so that the variance T ] i of the scalar error signal et - is computed recursively by Vi = 0 % ~ V T + 1 sf+i (4-30) with the definition, rj^ = a^N. The complete recursion procedure for the optimization of the reduced-rank Wiener filter is summarized in Table 4.1. 4.1.2 O ptim ization of the Rank-O ne Filters in the M ultistage W iener Filter Generally speaking, a reduced-rank Wiener processor consists of an M x N dimension- reducing linear transform T and an M x 1 W iener filter of M components. Since this reduced-dimension Wiener filter is constrained to belong to the subspace th at is spanned by the rows of T , it is apparent th at the performance of the reduced- dimension Wiener filter relies on the choice of basis vectors that span the row space of T . As it is pointed out in [39], the optimal rank-reducing transform that does not compromise the performance of the full-rank filter exists whenever the dimension of the reduced-rank subspace spanned by T is larger than the dimension of the signal subspace. But in order to build such an ideal transform, knowledge of the signal subspace is required, th at is usually not available to the single-user demodulator. In what follows, the recursive algorithm for the optimization of the data-dependent basis vectors [1 2 ] is presented under the assumption that no prior information about the correlation m atrix is utilized, and that if the underlying noise process were to be white noise, the recursion would last only one stage, i.e., the filter is a matched-filter. Consider now the problem of estimating the desired signal d0 = b on a subspace of 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.1: Recursion formulas for the multistage Wiener filter F o rw ard R ecu rsio n Initialization ._ _ _ S p _ rr p r f o 1 “ I I S o || ~ || r ro rfo I I Bi = null(ui) For i = 1 to N — U j- Ti-i = BiTi-! r r.<ft __ r r,-rf, ~ llr r,d £ll _ fc + l = BiRp;., B f = Bj R r,_l U i B ackw ard R e c u rsio n For i = N to 1 a-% = u f R ,.., Ui rji = Var(ei) = o i - r £ 1 ^ 1 oji = Tj~l r£i+ idi = T]~x Si+l a = di — a;I+l eI+i End di r i U £ +i R r , T ridi End Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. dimension one. This situation is depicted in Fig. 4.5. First the scalar Wiener filter h\ n h Figure 4.5: Reduced-rank Wiener filtering on the subspace of dimension one. is optimized on the assumption that the first basis vector Ui is fixed or known. The resulting MMSE, due to the dimension-one filter, is given by = V a r(e a) = EUdo-KldMdo-Kidtf} = rfo -^ iE {d id S } - h i E { d [ d v ] + ||/ii||2o2,. (4-31) where d\ = u | r 0. Completing the square in Eq. (4.31) yields the inequality, _2 { i _ E {d 0dl} \ f L _ E C d o d JV . _2 llE{d0d i} | | 2 ^eo I I I fllO'di I - r <7^ „ \ &d\ } \ Orfi / &di > a l - l|E{df l} J f . (4.32) Evidently in this inequality the lower bound is achieved if and only if fix = = h ? 1. (4.33) a di 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Now turn attention to the optimization of the unit vector Ui. The minimization of the lower bound on of0 is equivalent to the maximization of the term following the minus sign in Eq. (4.32). Denote this term by / , i.e., let J A HE W i l l 2 (4 34) Then, the numerator of J is expressible in the following form: || E { d o d \ } ||2 = ||E { u ld 0r0} ||2 = || v l \ rdoro ||2, (4.35) where ro = E {do r0} is the cross-correlation vector between the desired signal d0 and the observation vector r0. Also the denominator of J in Eq. (4.34) is given by <y\ - E {d?} = E {uj r0 r$ U!} = u[ E { r 0 rj} Ui = u | R ro m , (4.36) where R ro is the covariance m atrix associated with the vector random process r0. From these calculations J can be expressed as a function of the unit vector u t so th at4 J [ui) = (4.37) u[ R ro u t It is shown in the Appendix C that the unconstrained maximization of Eq. (4.37) as a whole leads to the Wiener filter solution, namely, “ >■>" = (4'38) II ■ **'ro da r o II 4Unless ofcherwise specified Ra hereafter denotes the correlation matrix of the vector process a. Similarly ra( , is the short-hand notation for the cross-correlation between the two vector processes a and b. 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. However, since initially R ,.0 in Eq. (4.38) is not known a priori, this Wiener filter solution is not acceptable. Thus, to avoid the W iener solution and eliminate the involvement of the covariance m atrix Rr0 in the basis selection process, the unit vector Ui in Fig- 4.5 must be chosen in such a manner that only the num erator of J(u i) is maximized. To accomplish this maximization apply the Schwarz inequality to the num erator of J(u i) in Eq. (4.37) to obtain the inequality, I I u{ r * ro ||2 < || ui ||2 || ro ||2, (4.39) where the maximum is achieved if and only if Ui and r^Q rQ are parallel. Hence the optimal choice for the unit vector Ui under the constraint, th at no correlation statistics are available, is given by u crt = r *»r° (44Q) T rforo L em m a 4.1 The evaluation o f J at the Wiener solution U i^ p , / ( u ^ - f ) , in Eq. (4..37), is greater than or equal to J(u^pi), where is given in Eq. (4-4-0). Equality holds if and only if the eigenvalues of the correlation matrix R ro are all identical. P r o o f : see Appendix B. The above lemma shows th at the unit vector in Eq. (4.40) provides the optim al solution only when the sequence of input observation vectors turns out to be a white- noise random process. But, except for this special case, there always remains room for the enhancement of the MMSE performance by expanding the dimension of the reduced-rank filter. The selection of the second basis vector u.2 is carried out in a manner similar to that described in Fig. 4.5 (page 46). But at this stage, to extract the information 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. O D t t — W r opt opt Figure 4.6: Reduced-rank Wiener filter with two taps. about the desired signal that is not contained in the direction of Ui, a constraint is placed on the new basis vector T 2 : it must lie in the orthogonal complement of span(ui) with respect to . To this end the iV-dimensional observation vector ro is preprocessed by the blocking matrix Bi = uj in order to generate the modified observation vector ri, the projection of r n onto the (iV — l)-dimensional subspace Vj = (span(ui) ) ■ * • , produced by B i. Also consider the error-residue signal e\ that results from the optimized, rank-one, filter in Fig. 4.5 by ei = do — ti? 1 dx, (4.41) where dx is given by d x t ( u r ) fro, (4.42) the projection of the vector ro onto the direction of the unit vector u ^ £ . The second stage of these operations is illustrated in Fig. 4.6. 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Once the optimum basis vector Ui and the corresponding scalar coefficient h\ for the first stage are determined, the overall estimation error e2 th at is associated with this rank-two filter is given by 2 e2 = do - X) h* *4 r *'-i = “ /li?£ u iP i ro) - h 2 u£ r t = ex — h 2 u£ r t. (4.43) 1 = 1 Since the error-residue e\ is already calculated from the first stage, it can be inferred from Eq. (4.43) that the optimization problem for u 2 and h2 at the second stage reduces to another rank-one filter optimization. More precisely. u 2 and h 2 need to be determined in such a way that the the vector h2 u 2 turns out to be the optimal linear filter that is required to estimate ei from the observation vector ri. Mathematically the optimum pair, (u2, h2), is found by solving the following minimization problem : min E {(ei — h2 u 2r i)2}. (4.44) (u2,/l2) L J Though not stated explicitly in Eq. (4.44), any basis vector, that requires informa tion about the correlation statistics, is excluded in order to guarantee the complete blindness of the selection procedure to the second-order statistics of the received vec tor . Here, it should be noticed that the optimization task specified in Eq. (4.44) is conceptually the same as the one shown in Fig. 4.5 except that the desired signal do and the observation ro are now replaced by et and r l5 respectively. Thus, u 2 is chosen by the maximum correlation criterion th at is used to obtain Ui in Eq. (4.40); th at is, n t = „- rei-rt „ , (4.45) 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and the corresponding scalar Wiener coefficient h^ is determ ined by h ? = (4.46) ad 2 which is the optimum scalar needed to estimate e\ from d 2 = u | r t. However, the selection of u 2 described in Eq. (4.45) can be done without the actual calculation of the error residue e\ by observing that vector r e,ri is actually parallel to rrfiri. This fact is verified from the following relation : = E { d o B x r0} = Bx r rfo ro = B t (dxUx) = 0, (4.47) where £1 = ||E{doro}||- If in Eq. (4.47) is connected with r t, then the use of Eq. (4.41) gives rise to the desired result, namely, r ei ri = r rfori i f f T rf, r, = h!^ T rf, r, . (4.48) Another im portant fact obtained from Eq. (4.47) is th at the two random variables d0 and d2 are orthogonal. This is shown, using Eq. (4.42) with u t replaced by u 2, by E [dod-z] = E {dot^rxj = u 2r dori = 0. (4.49) Thus, a substitution of Eqs. (4.48) and (4.49) into Eqs. (4.45) and (4.46), respectively, yields the optimal solution pair of the second stage, viz., < = (4.50) I I rdiri || 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and h ? = E{dod , - h * d A } = ^ T dida_ = _ h ? t^ °d2 (4.51) > d z where u> 2 = rd^ is the scalar Wiener filter needed to estimate d x from d2. Thus, the 'd2 optimal estimate d0 of dQ based on the rank-two Wiener-filter is given by d0 = ' t ^ d i = hT(dy-U} 2 d2 ), i=l ei (4.52) where ex is the error signal when optimally estimating the scalar d\ from d2 with the Wiener coefficient u 2. The realization of the rank-two Wiener filter based on Eq. (4.52) is depicted in Fig. 4.7. dn 8 ° i i mm U 1 B i d \ = “ i tr o + W , Figure 4.7: The reduced-rank multi-stage Wiener filter of dimension two. Eq. (4.52) leads to a significant interpretation about how an optimal reduced-rank filter is constructed. Eq. (4.50) indicates that, at the second stage of the recursion, the optimization problem is to estimate the new desired signal d x, which results from the first stage using the filtered observation ri. Thus, after r t is projected onto the basis vector 112, that is chosen by the maximal correlation criterion, the projection d2 is combined with d\ via a nested chain of two scalar Wiener filters, ui2 and hr x> t, to produce the optimal estimate of d0. This is in agreement with the expression for h, 2Pt in Eq. (4.51). Also in Fig. 4.7, to guarantee the minimization of the overall 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. mean-squared error, the scalar Wiener coefficient h^ 1 is replaced by where ex = dx — u 2 d2. This argument explains why the computational flow of the optimal scalar filters should be from the bottom to the top of the filter structure. Then, as shown in Fig. 4.7, the error-residue signal e2, associated with the optimal rank-two Wiener filter, is calculated to be e2 = dQ - (u\{di -c u 2 d2)). (4.53) Now for the third stage of the recursion, define the observation vector r 2 by r 2 = B 2 ri, (4.54) where B 2 spans the subspace V 2 = = (span(u2) )x, that is, Range(B2) = (span(u2) )x. (4.55) Then, the optimal pair ( u ^ 4 , h % ,t) at the third stage is the solution of the minimiza tion problem given by min E {(e2 - h3 u£r2)2}, (4.56) (U 3.A 3) L where the error-residue e2, that is carried over from the second-stage, is given in Eq. (4.53). Following the same arguments that are made for the second stage op timization, the third basis vector u 3 is obtained again by the maximum correlation criterion, namely, < = (4.57) I I r « 2 r2 I I 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and the optimal scalar filter coefficient h % )t is given by h % 1 = (4.58) where d3 is the projection of r 2 in the direction of U 3 , viz., d 3 = U3 r2. (4.59) Once again, the determination of the optimal basis vector shown in Eq. (4.57) can be decoupled from the calculation of the error signal e2 by verifying that the vector r e2t.2 is colinear with the direction of rd2 C 2 . Combining Eqs. (4.53) and (4.54), the correlation vector rR 2 r2 between e2 and r2 yields re2r2 = E { ( d a - u ) l di+L}icj 2 d2 )r2 } = E { B 2 d0 r t — u)X B 2 dx ri -f- a > i uj2 d2 r 2 } = B 2 r < * 0 r[ — Ui B 2 ri + ujt a;2 r < /2 r 2 = u i cj2 r< i2r2 , (4.60) where the first term and the second term in the third line vanish using Eq. (4.47) and Eq. (4.55), respectively. This reasoning modifies the optimal solution for the third basis vector in Eq. (4.57) to I I ^rf2 r2 || Also in order to recalculate the optimal scalar coefficient h as in Eq. (4.51), one needs to show that the covariance matrix, associated with the vector d = [do,. .. , d3 ], is tri-diagonal, i.e., E {dt -dy} = 0 when | i — j \ > 1 . Since it is verified that rd Q d 2 = 0 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in Eq (4.49), it is enough to prove th at r^dz and rdld 3 are zero. This latter fact is shown as follows: E {do d3} = E {do u | r2} = u | E (d0 B2 rt} = u£ B 2 ri = 0, (4.62) where Eq. (4.47) is used in the last step. Finally E {di d3} = E {di B 2 rt} = u | B2 rdl r, = 0, (4.63) where the last equality follows from Eq. (4.55). A substitution of Eqs. (4.53), (4.59), (4.62), and (4.63) into Eq. (4.58) reduces to uopt E (do d3 — u i di d3 + u > \ u/2 d2 d3} Td 2 d 3 ,. d3 = ----------------------5 ----------------------=CUiU2 ■ = CJiC02^3, (4.64) O 3 ^3 where u 3 = ^ r 3 - is the scalar W iener filter needed to estim ate d2 from d3. Then, the error-residue signal e3 based on the three-tap optimal filter is obtained by e3 = e2 — d3 = do — ( nq( d\ — 0J2 d2 )) — nq u > 2 c * > 3 d3 = do — ui\ ( d\ — u>2 ( d2 — w3 d3 ) ). (4.65) V V ' e 2 As it is pointed out in the optimization of the rank-two optimal filter, in order to minimize the overall MSE the coefficients u\ and u >2 need to be modified to wi = T - ^ (4.66) 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and (4.67) respectively. Here it should be noted that the determination of the scalar coefficient C c / j is carried out only after the set of the coefficients {uj} with j > i is identified. The generalization of the recursive optimization procedure that is detailed above is now presented. A couple of im portant observations should be mentioned in order to outline the derivation of the multistage Wiener filter more concisely. First, the determination of the scalar coefficients is deferred until the optimization of the basis vectors is completed for a given dimension. This implies that the optimization of the reduced-rank filter is partitioned into two parts: the optimization of the reduced-rank subspaces and the optimization of the filter coefficient within the selected subspace. This idea is embodied in Eqs. (4.48) and (4.60) by which the decoupling of the basis optimization and the scalar coefficient optimization is made possible. Secondly, the procedural flow of the coefficient optimization is in the opposite direction to that of the basis vector optimization. In this regard, the recursion for the basis vector optimization and the scalar coefficients optimization are called a forward recursion and backward recursion, respectively. As it is explained earlier, the basis vectors for the optimal dimension-reduction transform are selected recursively by way of the maximum correlation criterion with out an explicit identification of the optimum scalar coefficients. At the z-th stage the optimum basis vector is given by (4.68) 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where dt - is the norm of the cross-correlation vector and the z-th coordinate di in the reduced-dimension subspace is simply the inner-product between rj_i and lit. The sequence of observation vectors rz is recursively generated by the relation, rt - = Bjr,-_1 } (4.69) where the column space of the blocking matrix Bi, denoted by Vi, is the orthogonal complement subspace of span(ui) with respect to V*_t. In vector-space notation the sequence of linear spaces {Vi} satisfies the recursive formula, K--i = span(ui) ® Vit (4.70) with V o = CjSf, the N-dimensional complex Euclidean space. Suppose that M basis vectors are chosen, and that the corresponding coordinate vector d = [di, ... , dM\ is obtained. Then, the coefficients of the reduced-rank Wiener filter are also determined in a recursive manner, but in reverse order to the basis vector selection procedure. At the first stage of this backward recursion, the scalar coefficient uim, that is required to optimally estimate the reference signal d w -i from dM, is calculated by U m = U u , i u . , (4.71) Vm where is the variance of the scalar observation d ^ . As it is demonstrated in the rank-two example, the error signal e^-x = d ^ -i —(*>\rd\r, that is associated with the scalar Wiener filter u m , serves as the observation signal in the second stage of the recursion to form a nested chain of scalar coefficients. 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Before presenting the complete description of the backward recursion, it is con venient to make the definition em = dm for notational consistency. The general recursion formula for the error signal in the backward recursion is given by (4.72) and the value of the optimum scalar Wiener filter is calculated to be W A T .,- = - f- , i = 0 ,..., M , (4.73) T]M-i where r]M-i = F ar(e/ V / -_l). Finally, the need to calculate the cross-correlation between eAf-i and dM-i- i in Eq. (4.73) can be eliminated by establishing the following identity: t 1 ~ E {(di 1 ) dj— \ } for Uf. The generalization of Eq. (4.49) for E {di+i di-!}, is demonstrated as follows: E {di+i (k - 1} = E {u}+ 1 Vi di- 1} = u}+ 1 E (Bi d i- i} = u}+1 B, (diu*) = 0. (4.75) W rrt - - 1 di- 1 ^> i+1 r di+i di (4.74) The fourth equality in the above derivation is due to the substitution of Eq. (4.68) 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Since the norm < 5 * of the cross-correlation vector is evaluated in the course of the basis vector optimization, the only computational tasks associated with the back ward recursion are the generation of the error signals e£ and the recursive calculation of the variance of e£ - given by t * - i = E{e?_i} = E { (d i_ i-W je,)2} = E {d-_i + u f e l- 2 Ui ei di-t} = V i - 2 w j E {ef 1} = o-Li ~ nT1 (4-76) where Eqs. (4.73) and (4.74) are used to obtain the last equality. By setting the number of basis vectors M equal to N, a new filter structure, th a t equivalently solves the full-rank Wiener filtering problem, can be established. The result is the tree-structured multi-stage representation where N scalar Wiener filters form a nested chain to produce the desired signal. An example of the complete realization of the multi-stage Wiener filter [13] for N = 4 is provided in Fig. 4.3. Rank reduction can be viewed as a truncation of the full-scale structure at a certain stage. At a given rank it minimizes the variance of the error signal e0 and, equivalently, maximizes the output SINR observed at the filter output. 4.1.3 Sum m ary It is worthwhile to make some brief comments on the desirable traits that the TM W F structure possesses from the viewpoint of implementation. First, the components in the TM W F representation are characterized completely by the set of cross-correlation 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vectors in the forward recursion and the variances of the error signals in the back ward recursion. Even though these quantities can be evaluated exactly as a function of the correlation matrix R r„, they also can be estimated directly from finite snap shots of the observation vectors without any explicit estimation of R ro. The adaptive implementation of the TMWF from the set of auxiliary data is presented in the next section. Finally, it is shown [12] that the recursions which lead to the m ulti stage Wiener filter structure can be viewed as a generalized implementation of the Karhunen-Loeve transform which can be solved by means of a series of Householder reductions, followed by a non-unitary diagonalization. Since this procedure does not entail a matrix inversion or an eigen-decomposition of the covariance matrix, the stability of the numerical computation is enhanced considerably. 4.2 A d ap tive R ealization o f the T M W F In the previous section it is pointed out that one of the most salient features about a realization of the truncated multistage Wiener filter (TMWF) is that all filter parameters can be estimated from the raw data without requiring any numerically expensive matrix inversions or eigen-decompositions. The only quantities that are needed to implement a TMWF of dimension M are the normalized cross-correlation vectors u,-, 1 < i < M , the scalar cross-correlations 5 ,- = E{ejd,-_i} for 1 < i < M and the standard deviations, a and rji, associated with the random variables di and et - , respectively, for 1 < i < M. Assume that L approximately independent snapshots (samples) of the observation vector ro and the corresponding data symbols for the desired user are available for 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the initial acquisition of the TMWF parameters. This data are given in m atrix form by Xo = [r0(l), . .. , r 0(L)] (4.77) and b = [6(1), • - • , b(L) ]r , (4.78) where ro(n) and b{n) denote the received vector and the desired user’s bit information during the n-th symbol interval, respectively. The block of training symbols b can be exploited as the reference signals when the desired user’s (unnormalized) code vector s is unknown to the demodulator. However, in most of the CDMA communications systems such as IS-95 and its variants, the desired user’s code-vector is identified by listening to the synchronization and paging channel before the dem odulation of the actual data stream starts [18]. Therefore, it is assumed that the desired user’s code vector is known throughout the whole communication process. This assumption about the availability of the desired user’s code vector is maintained throughout the rest of this thesis. As it is illustrated in [34], to speed up the convergence of an adaptive system, the Wiener filter coefficients need to be formed in the absence of the desired signal component. In the classical adaptive array application, it is normally possible to collect signal-free snapshots of the received waveform. As in the CDMA communica tion systems like other cellular-based mobile communications, a voice activity detector (VAD) is usually employed to detect the exact times of the start and end of the speech bursts [45, 9]. Thus, as long as the VAD is operating in a sufficiently accurate mode to sift out the silent periods of the desired user, the Wiener-filter weights can be obtained with the signal components being absent during those periods. Otherwise, 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a sequence of known information symbols such as the block of training data is needed to subtract the signal vector from the received vector. Here, one needs to verify th at the filter weights and the basis vectors, chosen with the signal-free observations, do not have to be modified after the receiver operation is switched from the filter estimation period to the demodulation mode. Suppose th at ro represents the ^-dim ensional d ata vector with the signal component present. Then, the covariance m atrix associated w ith ro is expressed by R ro = R „ + s0 si, (4.79) where Rj, is the correlation m atrix of the signal-free observation vector. From the definition of the transformed vector in Eq. (4.4), it is shown , using Eq. (4.3), that R ri = B i R ro B | = Bi (Rt/ + So sj) B{ = Bi R„ B{ (4.80) and r r,rf, = B i R ro Ui = B t (R „ + s0 sj) U! = B l R t/u i. (4.81) This implies that the (Ar — l)-dimensional Wiener filter w r, in Fig. 4.2 remains un changed regardless of the presence of the signal component in the observation vector r 0, and the adjustment factors of the filter coefficients due to the absence (or presence) of the signal vector is reflected only in the scalar coefficient u i. Hence, considering that the detection criterion in Eq. (2.11) is concerned only with the polarity of the Wiener filter output uif.i and with ui\ being strictly positive, one can conclude that the new decision criterion b = sgn(ei), which is based on this fact, does not affect the performance of the MMSE CDMA demodulator as far as the bit-error probability is concerned. 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Now the recursive algorithm for the adaptive realization of the TM W F of dimen sion M is described under the assumption that L independent realizations of the signal-free observation vector are available. Below is an outline of the algorithm in a pseudo-code format. As it is described in Table 4.2., the adaptive algorithm to realize the TMWF structure is carried out by estimating the parameters such as r ^ , crt - and 7 7, directly from the real data. As a consequence, the need for explicit estimates of the covariance matrix and its inversion are removed. 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.2: Algorithm for the adaptive realization of the multistage Wiener filter B asis V ecto r O p tim iz atio n (F orw ard R ecursion) Initialization : Let ui = c and Bi = I — Ui u{ For i = 1 to (M — I) Calculate dt - and rt - d| = [dj(l), ..., di(L) ] = uf Xj-! ...,rf(L)] = Bi Xi_1 Calculate the (i + l)-th stage basis vector, r rf d i = Sn=l r i d -i = £ Xj d{ S i+ i = I I rrid; || ui+1 = O i-f-l Determine the {i + l)-th blocking matrix Bi+1 B i+1 = I - ui+i u|+ 1 End Compute dt4 and set it equal to e,vr eL = [eA'r(l)i • • • > £A f(L) ] = dj^ = n l f XA /_i Scalar W eights Optim ization (Backward Recursion) For i — M to 2 Estim ate the variance of et - Vi = = z Z n = i I I ei ( n ) I I 2 = z e i e* Calculate the i-th scalar Wiener filter u>i by Vi Calculate the (i — l)-th error signal ej_i by ej_! = [e * -i(l), ■ • • , £ i - i ( L ) ] = d|_! - e | End 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 5 N u m erical R esu lts In this chapter the performance of a CDMA demodulator, that is based on the TMWF representation, is measured in terms of the output SINR. An asynchronous BPSK DS-CDMA system is considered, and the relative delay fk = — n as defined in Chapter 2 of each interfering user is assumed to be uniformly distributed over [0, T & ]. All numerical results in this chapter are obtained by the Monte Carlo technique and they show output SINR performances that are obtained by averaging over ensembles of randomly-selected relative delays. Gold sequences of length 31, whose generator polynomials are taken from [ 8 ], are assigned to both the desired user and the interfer ing users. All interfering users are assumed to have the same power advantage with respect to the desired user. This power advantage is denoted by a quantity called the near-far ratio (NFR), i.e., N F R = W , ' (JU ) where with the subscript T ’ denoting the first user or the user of interest. 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.1 C ap acity S tu d y First the performance of the TM W F-based CDMA receiver is examined when the co- variance m atrix of the interference signal is known to the demodulator. Even though the exact estim ation of the covariance statistics is not possible to achieve practically in a real communication system, the performance is investigated under the assump tion th a t the covariance matrix is known. This provides an upper bound on the STNR performance of a reduced-rank CDMA receiver. Furthermore it is of interest to deter mine the performance of reduced-rank Wiener filters and to evaluate the performance upper bound set by the full-rank Wiener filter under the known covariance condition. In Fig. 5.1, the output SINR of several TMWF CDMA receivers is presented as a function of the number of active users under the perfect power-control scenario, i.e., all users have the same signal power. The dimension of the full-rank filter is equal to N = 31. Also it should be noted that the output SINR of the conventional ‘matched-filter’ detector1 decreases quite rapidly as the number of users increases, even when the interfering users have the same power as the desired user. If the acceptable output SINR level for the user of interest is set to 10 dB, the conventional detector can support only 6 users, while the full-rank MMSE detector accommodates 23 users. The performance gap between the full-rank MMSE detector and the TM W F detector also becomes wider as more users participate in the communication, but the performance of the TMWF detector with taps as small as 5 reaches within 3 dB of that of the full-rank MMSE detector for all values a of the system load.2. l This detector uses the linear filter that is matched to the spreading code of the desired user. It should be noted that it does not actually maximize the output SINR as the term ‘matched filter’ normally indicates. 2The system load a of a cellular communication network is defined as the ratio of the number of users to the spreading gain, i.e., a = ^ . 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 5.2 demonstrates the output SINR performance of the TMWF detector in a severe near-far situation, where the near-far ratio is 10 dB. All other parameters are identical to those used in Fig. 5.1, and an a priori knowledge of the covariance matrix is assumed. For this case, the TMWF detector must be of rank 15 to obtain an SINR within 3 dB of the full-rank detector SINR for any user population. For a fixed system load, the minimum rank that guarantees the SINR of the reduced-rank detector falls within 3 dB of the full-rank SINR bound is termed the critical dimension of the reduced-rank detector. By examining Figs. 5.1 and 5.2, one can note that the critical dimension of the TMWF detector depends on the power distribution of the interfering users. For example, when the NFR is 0 dB and the number of users is 10, the critical dimension of the TM W F detector is 4. But for 10 interfering users with a 10 dB power advantage, it increases sharply to 10. In terms of the system capacity3 if the minimum SINR performance requirement for the desired user is 10 dB, the TM W F detector of rank 15 can allow 13 interfering users with a 10 dB power advantage to communicate simultaneously w ith the desired user, which is only 2 users short of the capacity of the full-rank MMSE detector. In this extreme situation of power disparity, the conventional matched-filter detector nearly fails to suppress the interference even when the number of strong interfering users is merely 3. In Figs. 5.3 and 5.4 the output SINR curves are plotted for two eigen-analysis- based reduced-rank detectors, the principal component (PC) detector and the cross- spectral metric (CSM) detector [11]. These curves illustrate the superior interference- suppression capability of the TMWF detector over eigen-analysis-based detectors. In 3It is defined as the maximum allowable number of users that can be accommodated in a cell while maintaining the pre-defined performance threshold for the desired user. 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. both cases, when the dimension of the reduced-rank detector is fixed, the TM W F de tector rejects the multiple-access interference most effectively, and reveals the quickest convergence to the full-rank performance bound as the rank of the detector increases. 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Output SINR 20 18 16 14 12 10 ^ 9 o * + x o 4- x O * + * ' + * * + x O □ TMWF (M=2) TMWF (M=3) TMWF (M=4) TMWF (M=5) Full-R ank MMSE] M atched Filter 4 - - X * + X * * + + O □ □ + X X X O O * 4 " + + X X O O o * + X X o * ■ * 4 - 4 - 4 - X • • . * * * * 4 - 4 - 4 - X 4 - G D • . * * * * □ □ a n • . • ° O x ° ° + * * x + + + * u . * * a □ 10 15 20 Number of Users 25 30 35 Figure 5.1: The output SINR of the TM W F CDMA receiver vs. the number of active users. The SNR of the desired user is set to 20dB and the near-far ratio is OdB. 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Output SINR 20 15 10 - 5 - -10 • • • t o * : * * ..............................* ....................... TMWF (M=5) * * TMWF (M=10) + + TMWF (M=15) * TMWF (M=20) o o TMWF (M=25) o o Full-Rank MMSE ° ° Matched Filter * + $ * 9 □ # . + < * + * D ' * i □ : □ > * 6 h x © ~ u + * o + x ^ % + + $ * * * + $ $ * $ - ................... * * * .1 a ................... I c 0 □ □ o □ □ * $ * < • • • > • □ □. D D a D □ □ □ □ 10 15 20 Number of Users 25 30 35 Figure 5.2: The output SINR of the TMWF CDMA receiver vs. the number of active users. The SNR of the desired user is set to 20dB and the near-far ratio is lOdB. 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 18 16 14 Z 12 w 3 a. 8 10 I * S 3 8 * i □ ° « o o ■ H -° O : ° 0- % * + X * - X X TMWF (M=2) * * CSM (M=2) ■ ~ 1 • PC (M=2) O O TMWF (M=5) a □ CSM (M=5) + + PC (M=5) 0 0 Full-Rank MMSE3 t x * X □ t * D ★ : v O 3(£ X . * * □ □ X D ★ X X • * * □ □ x a 0 0 o - 0 O° o o ! % 0 * " O ' “ ± x W * X X O □ a • • . * * . X ° x □ □ .......................... . . * T. X * . . . * * * □ .p i. g * * * * x • • * * * 10 15 20 Number of Users 25 30 35 Figure 5.3: The output SINR of various reduced-rank CDMA receivers vs. the number of active users. The SNR of the desired user is set to 20dB and the near-far ratio is OdB. 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 15 - © <g * □ < < 3 < o * X u : a a A A 10 - X X TMWF (M=5) * * CSM (M=5) • PC (M=5) □ □ TMWF (M=10) o o CSM (M=10) + + PC (M=10) < < Full-R ank MMSE O a ± * o + : O * x : 8 D □ + * x . o □ < + .* .* .* .............0 . - ° D Q < 15 20 Number of Users Figure 5.4: The output SINR of various reduced-rank CDMA receivers vs. the number of active users. The SNR of the desired user is set to 20dB and the near-far ratio is lOdB. 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.2 C on vergen ce Study o f A d ap tive T M W F D etecto r If the full-rank MMSE detector were to be implemented to adaptively track the variations of the covariance statistic, repeated inversions of N x N m atrix would be needed whenever the interference signals changed significantly. When the processing gain N of the CDMA system, which is normally in the range of 64 - 256, is large, the am ount of computation required to realize the adaptive full-rank processor is formidable. However, by exploiting the TMWF structure, one can not only avoid the computationally expensive m atrix inversion operation, but also maintain the SINR performance close to th a t of its full-rank counterpart with a much smaller number of data samples, i.e., much less sample support. The performance of the adaptive TMWF (A-TMWF) detector, th at is realized via the batch algorithm outlined in the previous chapter, is shown in Figs. 5.5 and 5.6. The dimensions of the adaptive TMWF receivers in each figure are 5 and 15, respec tively. In Fig. 5.5, where all users communicate with equal signal power, it is shown that 30 snapshots (samples) of the received vector are enough for the adaptive TM W F to achieve an average SINR performance which is within 3 dB of the known-covariance bound for all values of user population. On the other hand, the sample m atrix inver sion (SMI) technique requires at least N = 31 samples to avoid the singularity of the sample covariance m atrix, and the adaptive version of the full-rank MMSE detector th at is approximated w ith 40 data samples shows a worse performance than the 5-tap adaptive TMW F detector that is computed with only 30 data samples. Also it can be observed from Fig. 5.5 that the system load is a key factor for determining the convergence rate of the adaptive algorithm for the reduced-rank TM W F detector. In a lightly loaded system in which the number of users is less than 7, roughly 10 data 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. samples, or 2M samples, appear to be sufficient to form accurate filter weights while more than 30 samples are required for a heavily loaded system. Similar arguments can be applied to the system with the near-far problem shown in Fig. 5.6. Fig. 5.7 illustrates the rapid convergence of the adaptive TMWF demodulator as a function of the required number of samples when the number of users is fixed. The SNR of the desired user is 10 dB, and the channel is shared with 4 interferers which have a 10 dB power advantage over the desired user’s transmission. While the adaptive full-rank demodulator with the SMI technique requires roughly 60 samples to reach within 3 dB of the known-covariance bound (8.78 dB in this case), the adaptive TMWF receiver with 3 taps, for example, requires only 15-20 samples. Finally the convergence behaviors of the other adaptive reduced-rank filters are compared with that of the adaptive TMWF in Fig. 5.8. 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 A-TMWF (10 sam ples) A-TMWF (30 sam ples) A-TMWF (50 sam ples) SMI (40 sam ples) SMI (60 sam ples) M atched filter Known Covariance Matrix 16 14 12 3 10 35 15 0 5 10 20 30 25 Number of Users Figure 5.5: The output SINR performance of the adaptive TMWF CDMA receiver with 5 taps vs. the number of users. SNR[ = 20 dB, NFR = 0 dB. 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Output SINR 2 0 , A-TMW F (20 sam ples) A-TMW F (40 sam ples) A-TMW F (60 sam ples) SMI (40 sam ples) SMI (60 sam ples) M atched filter Known Covariance Matrix - 5 -1 0 - -1 5 25 35 15 30 Number of Users Figure 5.6: The output SINR performance of the adaptive TM W F CDMA receiver with 15 taps vs. the number of users. SNRi = 20 dB, N FR = 10 dB. 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. cc 6 z C O 3 Q. 3 O 5 + □ x -F 3 3 X V & X ❖ * * X + □ V A * V A d l f f l 9 A-TMW F (M = 2) A-TM W F (M = 3) A-TM W F (M = 4) A-TM W F (M = 5) A-TMW F (M = 6) A-TMW F (M = 7) Full-R ank SMI 20 40 60 80 Number of Training Symbols 100 120 Figure 5.7: The learning curve of the adaptive TMWF CDMA receiver. For compar ison, the average SINR’s of the full-rank MMSE detector(known correlation matrix) and the conventional matched-filter detector are 8.78 dB and 0.59 dB, respectively. SNRi = 10 dB, K = 5, near-far ratio = 10 dB. 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i 5 C O Adaptive TMWF (M = 3) Adaptive PC (M = 3) Adaptive CSM (M = 3) Full-Rank SMI 20 30 25 40 45 55 35 50 60 Number of Training Symbols Figure 5.8: Comparison of the learning curves of various adaptive reduced-rank CDMA receivers. SNR.l = 10 dB, K = 5, near-far ratio = 5 dB. 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C h apter 6 C onclusions In this thesis the realization and the performance of the TM W F-based adaptive reduced-rank MMSE demodulator is considered for DS-CDMA communications. Though the conventional full-rank MMSE demodulator perfectly suppresses the multiple-access interference coming from the co-existing users, its practical imple mentation in many cases is not feasible due to the lack of sample support to estimate a full-rank covariance m atrix and the excessive computations needed to invert this matrix. Adaptation techniques that are based on the stochastic gradient algorithm have been proposed to avoid such matrix inversions, but the convergence rate of these algorithms can be too slow to be used under an often very-rapidly changing communication environment. One of the chief advantages of the TMWF-based demodulator is th at the adaptive adjustment of the filter coefficients can be done quite rapidly. Also the size of the training data, needed to initially estimate the filter coefficients, can be considerably decreased. This fact directly translates into an enhancement of the bandwidth ef ficiency of the underlying communication system or the potential performance gain when the size of the sample support for the adaptation of the filter coefficients is 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. limited. One of the simulation studies shows that the size of the sample support can be reduced by 3 to 4 times compared to a full-rank system th at uses the sample matrix-inversion (SMI) technique. Secondly one can design a lower-complexity receiver without a significant loss in performance as compared with a full-rank system. When the dimension of the reduced-rank filter is fixed, the TM W F technique provides a recursive subspace selec tion procedure th a t results in a maximized output signal-to-interference-noise ratio (SINR) [12]. In fact this new TM W F technique is an optimal reduced-rank Wiener filter solution particularly when the covariance matrix information is not available a priori. Finally the TM W F structure is especially suitable for the adaptive realization. It is shown th at the TM W F equivalently solves the colored-noise matched-filtering problem, while at the same time it eliminates the need for a m atrix inversion. Further more, the adaptation of the filter parameters can be carried out directly by averaging the real data th a t are available to the demodulator without explicitly estimating the covariance matrix. 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. R eference List [1] A. N. Barbosa and S. L. Miller. Adaptive Detection of DS/CDMA Signals in Fading Channels. IEEE Trans, on Comm., vol. 46, pp. 115-124, January 1998. [2] L. W. Brooks and I. S. Reed. Equivalence of the Likelihood Ratio Processor, the Maximum Signal-to-Noise Ratio Filter, and the Wiener Filter. IEEE Trans. Aerospace and Electronic Systems, vol. AES-8, pp. 690-692, Sep. 1972. [3] D. S. Chen and S. Roy. An Adaptive Multiuser Receiver for CDMA Systems. IEEE J. Select. Areas Commun., vol. 12-5, pp. 808-816, June 1994. [4] M. Davis, A. Monk, and L. B. Milstein. A Noise-Whitening Approach to Multiple Access Noise Rejection - Part II: Implementation Issues. IEEE Trans, on Comm., vol. 14, pp. 1488-1498, October 1996. [5] A. Duel-Hallen. Decorrelating Decision-Feedback Multiuser Detector for Syn chronous Code-Division Multiple-Access Channel. IEEE Trans, on Comm., vol. 41-2, pp. 285-298, Feb 1993. [6] A. Duel-Hallen. A Family of Multiuser Decision-Feedback Detectors for Asyn chronous Code-Division Multiple-Access Channels. IEEE Trans, on Comm., pages 421-434, Feb.-Mar.-Apr. 1995. [7] U. Fawer and B. Aazhang. A Multiuser Receiver for Code Division Multiple Access Communications over Multipath Channels. IEEE Trans, on Comm., vol. 43, pp. 1556-1565, Feb.-Mar.-Apr. 1995. [8] F. D. Garber and M. B. Pursley. Optimal Phases of Maximal Sequences for Asynchronous Spread-Spectrum Multiplexing. IEE Electron. Lett., vol. 16, pp. 756-757, Sep. 1980. [9] V. K. Garg and J. E. Wilkes. Principles and Applications of GSM. Prentice Hall, Upper Saddle River, NJ, 1999. [10] K. S. Gilhousen, I. M. Jacobs, R. Padovani, A. J. Viterbi, L. A. Weaver Jr., and C. E. Wheatley III. On the Capacity of a Cellular CDMA System. IEEE Trans, on Vehicular Technology, vol. 40, pp. 303-312, May 1991. 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [11] J. S. Goldstein. Optimal Reduced-Rank Statistical Signal Processing : Detection and Estimation Theory. PhD thesis, Dept, of Electrical Engineering, University of Southern California, Los Angeles, Dec. 1997. [12] J. S. Goldstein, J. R. Guerci, D. E. Dungeon, and I. S. Reed. Theory of Signal Representation: Vector W iener Filtering. IEEE Trans, on Signal Processing, subm itted 1999. [13] J. S. Goldstein, I. S. Reed, and L. L. Scharf. A Multistage Representation of the Wiener Filter Based on Orthogonal Projections. IEE E Trans, on Info. Th., vol. 44, pp. 2943-2959, Nov. 1998. [14] S. Haykin. Adaptive filter theory. Prentice Hall, Upper Saddle River , NJ., 1996. [15] M. Honig, U. Madhow, and S. Verdu. Blind Adaptive Multiuser Detection. IEEE Trans. Info. Th., vol. 41, pp. 944-960, Jul. 1995. [16] M. L. Honig, P. Crespo, and K. Steiglitz. Suppression of Near- and Far-end Crosstalk by Linear Pre- and Post-Filtering. IE E E J. Select. Areas Commun., vol. 10, pp. 614-629, Apr. 1992. [17] M. J. Juntti and B. Aazhang. Finite Memory-Length Linear Multiuser Detection for Asynchronous CDMA Communications. IEEE Trans, on Comm., vol. 45, pp. 611-622, May 1997. [18] J. S. Lee and L. E. Miller. CDMA Systems Engineering Handbook. Artech House Publishers, Boston, 2nd edition, 1998. [19] R. Lupas and S. Verdu. Linear Multiuser Detectors for Synchronous Code- Division Multiple-Access Channels. IEEE Trans, on Info. Th., vol. 35, pp. 123- 136, Jan. 1989. [20] R. Lupas and S. Verdu. Near-Far Resistance of Multiuser Detectors in Asyn chronous Channels. IEEE Trans, on Comm., vol. 38, pp. 496-508, Apr. 1990. [21] U. Madhow and M. L. Honig. MMSE Interference Suppression for Direct- Sequence Spread-Spectrum CDMA. IEEE Trans, on Comm., vol. 42, pp. 3178- 3188, Dec. 1994. [22] U. Madhow and M. L. Honig. On the Average Near-Far Resistance for MMSE Detection of Direct Sequence CDMA Signals with Random Spreading. IEEE Trans, on Information Theory, vol. IT-45, pp. 2039-2045, Sep. 1999. [23] J. M. Mendel. Lessons in Estimation Theory for Signal Processing, Communi cations, and Control. Prentice Hall, Englewood Cliffs, NJ, 1995. 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [24] S. L. Miller. An Adaptive Direct-Sequence Code-Division Multiple-Access Re ceiver for M ultiuser Interference Rejection. IEEE Trans.on Comm., vol. 43, pp. 1746-1755, Feb.-Mar.-Apr. 1995. [25] S. L. Miller. An Adaptive Direct-Sequence Code-Division Multiple-Access Re ceiver for M ultiuser Interference Rejection. IEEE Trans, on Comm., vol. 43, pp. 1746-1755, Feb./M ar./Apr. 1995. [26] U. M itra and H. V. Poor. Analysis of an Adaptive Decorrelating Detector for Synchronous CDMA Channels. IEEE Trans, on Comm., vol. 44-2, pp. 257-268, Feb 1990. [27] A. F. Mohammed. Near-Far Problem in Direct-Sequence Code-Division Multiple-Access Systems. In Seventh IE E European Conference on Mobile and Personal Communications, 1993, pages 151-154, 1993. [28] A. M. Monk, M. Davis, L. B. Milstein, and C. W. Helstrom. A Noise-Whitening Approach to Multiple Access Noise Rejection - P art I: Theory and Background. IEEE Trans, on Comm., vol. 12, pp. 817-827, June 1994. [29] S. Moshavi. Multi-User Detection for DS-CDMA Communications. IEE E Com munications Magazine, vol. 34 10, pp. 124-136, Oct. 1996. [30] D. A. Pados and S. N. Batalama. Joint Space-Time Auxiliary-Vector Filtering for DS/CDM A Systems with Antenna Arrays. IE E E Trans, on Comm., vol. 47, pp. 1406-1415, September 1999. [31] A. Papoulis. Probability, random variables, and stochastic processes. McGraw- Hill Inc., New York, 3rd edition, 1991. [32] H. V. Poor and S. Verdu. Probability of Error in MMSE Multiuser Detection. IEEE Trans, on Information Theory, vol. IT-43, pp. 858-871, May 1997. [33] P. B. Rapajic and B. S. Vucetic. Adaptive Receiver Structures for Asynchronous CDMA Systems. IEEE J. Select. Area Commun., vol. 12, pp. 685-697, May 1994. [34] I. S. Reed, J. D. Mallett, and L. E. Brennan. Rapid Convergence Rate in Adap tive Arrays. IEEE Trans. Aerospace and Electronic Systems, vol. AES-10, pp. 853-863, Nov. 1974. [35] M. K. Simon, S. M. Hinedi, and W. C. Lindsey. Digital Communication Te- chiniques: Signal design and detection. Prentice Hall, Englewood Cliffs, NJ, 1994. 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [36] M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt. Spread. Spectrum Communications. Computer Science Press, Rockville, MD, 1985. [37] R. Singh and L. B. Milstein. Interference Suppression for DS/CDMA. 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Minimum Probability of Error for Asynchronous Gaussian Multiple- Access Channels. IEEE Trans, on Information Theory, vol. IT-32, pp. 85-96, Jan. 1986. [44] S. Verdu. Optimum Multiuser Asymptotic Efficiency. IEEE Trans, on Commu nications, vol. COM-34, pp. 890-897, Sep. 1986. [45] A. J. Viterbi. CDMA : Principles of Spread Spectrum Communication. Addison- Wesley Publishing Company, Reading, MA, 1995. [46] X. Wang and V. Poor. Blind Multiuser Detection : A Subspace Approach. IEE E Trans, on Information Theory, vol. IT-44, pp. 677-690, Mar. 1998. [47] C. L. Weber, G. K. Huth, and B. H. Batson. Performance Considerations of Code Division Multiple-Access Systems. IEEE Trans, on Vehicular Technology, vol. 30, pp. 3-10, Feb 1981. [48] P. Wei, J. R. Zeidler, and W. H. Ku. Adaptive Interference Suppression for CDMA Overlay Systems. IEEE J. Select. Areas Commun., vol. 12, pp. 1510- 1523, December 1994. 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [49] G. Woodward and B. S. Vucetic. Adaptive Detection for DS-CDMA. Proceedings of IEEE, vol. 86 7, pp. 1413-1434, July 1998. [50] G. Woodward and B. S. Vucetic. Adaptive Detection for DS-CDMA. Proceedings of the IEEE, vol. 86, pp. 1413-1433, July 1998. [51] Z. Xie, R. T. Short, and C. K. Rushforth. A Family of Suboptimum Detectors for Coherent Multiuser Communications. IEEE J. Select. Areas Commun., vol. 8, pp. 683-690, May 1990. [52] Y. C. Yoon and H. Leib. Matched Filters with Interference Suppression Capa bilities for DS-CDMA. IEEE J. Select. Areas Comm., vol. 14-8, pp. 1510-1521, Oct 1996. 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p en d ix A E quivalence o f Transform D om ain W ien er-F ilter The output of the classical Wiener filter w ro that is based on the original observation vector r0 is given by Vo = w to r ° = s 0 R ro l ro > (A.l) where S o = r cod o is the steering vector of the first stage of the decomposition. If the matrix # x is invertible, there exists an inverse matrix 1 such th a t < l> x = 1 ^ . A substitution of the identity matrix in terms of # x and its inverse yields vo = sUl(^rl)tRr 0 l^rl^iro = (^lEO rodo})1 [ # i E { r 0r£} $j] * ( # t r0) = ( E l ^ i r o J d o D ^ E ^ i r o ) ^ , ^ } ] " 1 (#i r 0). (A.2) Now let zx denote the transformed coordinate of r 0 under the m atrix d>i, i.e., let zx = $ x r 0. (A.3) 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Then, Eq. (A.2) reduces to < A ) = r£lr fb Rz/ z, = w*, Z[, (A.4) where wZ l = rz 1 ( * 0 is the Wiener filter to estimate the desired scalar signal d0 from the transformed vector zi. This completes the proof. 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A p p en d ix B P ro o f o f L em m a 4.1 The substitution of Eqs. (4.38), (4.40) into Eq. (4.37) yields rrf0roR ro l rd0ro £ do ro ^ T Q 1 rr f o r o _t D -l do r o r o d o r o = l l r ^ f u r X ' u r (B.1) and j ( uopi) = . . . H ^ .ro .ll2. (B.2) u r ' R r o u r Let the correlation m atrix R ro have the eigen-decomposition R r0 = E A E t where E = [4 > x ... ,(f> N ) is the eigenvector m atrix and A = diag[Ai,... , A jv] is the cor responding eigenvalue matrix. Also let k = E^u^4 = [ki, ... denote the coordinate vector of u ^ £ in the eigenvectors in E of R ro. 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. W ith this coordinate transform, the quantities in Eqs. (B .l) and (B.2) , when normalized with respect to || r^ro ||2> simplify to ■ W r j = K , A _ . K ^ H * d li (B 3) 1 1 r r f o ro I [ i ^ and w > _ i _ i (B.4) I I T rforo fl2 E i l K I|2Ai’ where A j-’s are the diagonal entries of the m atrix A. The function f(x) = A is convex when x is positive, since the second derivative f"(x) = A y > 0 for all positive x. Hence by Jensen’s inequality, we have ! > / ( * ,• ) (b .s) for Q fj > 0, ti > 0 and E i &i = T and equality holds when fyi} is the uniform probability mass function, i.e., t* = t for all i. If we set a* = || Ki |[2 and ti = Aj, it is verified that A t - > 0 for all i due to the positive-definiteness of R ro- Additionally, it can be seen that || Ki |[2 > 0 for all i, and 1 | Ki ||2 = ( u ^ ^ E E l u ^ u"pi = k ^k = 1 . Thus, we obtain the desired inequality, Y > I (B.6) T ^ “ E ,-K II2V where equality holds if and only if Ai = A for all i. Q.E.D. 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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Creator Lee, Dongjun (author)

Core Title Adaptive detection of DS /CDMA signals with reduced-rank multistage Wiener filter

School Graduate School

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, electronics and electrical,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Reed, Irving S. (committee chair), Alexander, Kenneth S. (committee member), Weber, Charles L. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-62971

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