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MINIMUM MEANSQUARED ERROR ADAPTIVE ANTENNA ARRAYS FOR DIRECTSEQUENCE CODEDIVISION MULTIPLE ACCESS SYSTEMS By JOHN EARLE MILLER A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1996 UNIVERSITY OF FLORIDA LIBRARIES Copyright 1996 by John Earle Miller This work is dedicated to my wife Kim, my children Marissa and Christopher and my parents Frank and Garnet. ACKNOWLEDGMENTS I would like to thank all of the members of my committee for their involvement, guidance and influence in this research. I would like to extend a special thanks to my chairman, Dr. Scott L. Miller, for his support, many insights and thoughtful suggestions which have influenced this work. The financial support provided by the Department of Electrical and Computer Engineering is also gratefully acknowledged. TABLE OF CONTENTS page ACKNOWLEGMENTS ......................................................................................... iv ABSTRA CT......................................................................................................... vii CHAPTERS 1 INTRODUCTION............................................ .................................... 1 DirectSequence SpreadSpectrum with MultipleAccess ............................ Adaptive Antenna Arrays......................................................................7 Previous W ork .................................................... ................................ 13 Dissertation Outline .................................................................................... 21 2 A MULTIUSER ANTENNA ARRAY PROCESSOR STEADYSTATE PERFORMANCE.................................. ............. 24 The Multiuser MMSE Processor........................................ ...... ........ 25 Spatially Orthogonal Users.............................. ........................................26 Nonorthogonal Users..................................................................................30 SingleUser System....................................................................... 34 One Weak User, One Strong User...................................................34 Two EqualPower Users ....................................................................36 Two Strong Users............................................ ......................... 36 Numerical Results................................................... ............................ 37 3 THE MULTIUSER ARRAY PROCESSOR ADAPTIVE PERFORMANCE................................................................42 MeanBased Performance Measure ........................................................43 VarianceBased Performance Measure ..................................... ............ 47 Sim ulation Results .................................................................................49 4 BASE STATION RECEIVER PERFORMANCE USING A SMART ANTENNA ARRAY IN A DSCDMA SYSTEM WITH IMPERFECT POWER CONTROL ...............................................52 System Description ..................................................................................... 53 A analysis ..................................................................... ............ ................ 54 Simulations and Results............................................................................... 61 Conclusions ........................................................................................ 65 5 BASE STATION RECEIVER PERFORMANCE WITH A SMART ANTENNA ARRAY IN THE PRESENCE OF MULTIPATH FADING AND SHADOW FADING................................... 68 A Single Cell with Signals Subjected to Rayleigh Fading .......................... 69 Multiple Cells with Signals Subjected to Rayleigh Fading and Shadow Fading ............................................................................. 71 A nalysis...................................................................................................... 76 R esults.................................................................................................. 80 Conclusions/Discussion................. .................................................... 88 Summary ............................................................................................ 89 6 BASE STATION ANTENNA ARRAY ADAPTIVE PERFORMANCE................................................................91 Signal and Channel Model............................. ...................................92 Adaptive Receivers .................................................................................. 94 Simulations .........................................................................................97 Conclusions/Summary ............................................................................... 102 7 SUMMARY ............................................................................................. 106 Areas for Future Work ............................................................................. 109 APPENDIX A THE OUTPUT SNR PERFORMANCE SURFACE ................................. 113 LIST OF REFERENCES ..................................................................................... 117 BIOGRAPHICAL SKETCH ............................................................................... 129 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MINIMUM MEANSQUARED ERROR ADAPTIVE ANTENNA ARRAYS FOR DIRECTSEQUENCE CODEDIVISION MULTIPLEACCESS SYSTEMS By John Earle Miller August, 1996 Chairman: Dr. Scott L. Miller Major Department: Electrical and Computer Engineering This dissertation examines the performance of directsequence codedivision multiple access receivers which use a minimum meansquared error adaptive antenna array as a predetection spatial filter. The array attenuates multiaccess interference prior to conventional, directsequence matchedfilter detection. Conventional detectors are vulnerable to heightened levels of multiaccess interference. Minimum meansquared error processing seems like a natural choice for optimization in a codedivision multipleaccess environment since several efficient search algorithms exist which are compatible with decisiondirected equalizer structures. Two array structures are examined. The first structure uses a single set of array weights to equalize more than one desired signal. For two strong incident signals of unequal power levels, the steadystate response gives output signaltonoise ratios that are leveled to a value near the maximum response of the weaker of the two users. The steady state adaptive performance of the multiuser array based on a leastmeansquares algorithm is also examined. It is found that the maximum allowable step size based on stability arguments will also give adequate output signaltonoise ratio performance of the multiuser processor. The second structure uses a single set of weights per multipleaccess signal and the array output feeds a conventional detector. The array/detector performance measure is outage probability and outagebased capacity as a function of the number of array elements and the degree of power control error. A robust, incremental measure of performancethe perelement capacityis defined as the capacity per array element for a given outage probability. Steadystate performance is evaluated for the case of directional signals as well as for the case of signals subjected to multipath fading and shadowing. The adaptive performance of the recursive leastsquares algorithm is also investigated. CHAPTER 1 INTRODUCTION Wireless communication based on directsequence spread spectrum (DSSS) has received considerable attention as an efficient signaling format for codedivision multiple access systems (CDMA). While DSCDMA has a long history of use in defense applications where jamming resistance and security are primary concerns,' some proponents maintain that it will also serve as a highcapacity format for the next generation mobilecellular and wireless systems for commercial use.23'4" System specifications for domestic use have been endorsed by industrial agencies,6 the results of field trials have been published7'8 and it has been studied as a possible format for third generation mobile radio systems in Europe.9'10 Despite a frenzy of activity and interest, DSSS has a serious drawback: the nearfar effect This occurs when strong signals overwhelm a weaker desired signal during the detection process. In a commercial mobilecellular system the nearfar effect can occur at the cellsite basestation receiver. Incident signals originating from multipleaccess, fixed power transmitters geographically distributed throughout a cell can have incident power levels that change drastically as the transmitter positions vary. Signals originating from transmitters near the base station may overwhelm received signals from transmitters on the fringe of the cell. Vigorous proponents of commercial DSCDMA systems have developed transmit powercontrol formats to adjust each user's RF transmit power in real time so that power levels of all multipleaccess signals incident upon the base station are approximately equal. Initially, it was stated that power control would need to be able to adjust the transmit power over a 80 dB dynamic range in a mobile cellular scenario." Field tests indicate that 50 dB is more likely.7'12 Another use for DSSS is in the Global Positioning System (GPS). Originally implemented as a positioning system for the Department of Defense, it has enjoyed commercial applications in mapping, navigation, and surveying." The worldwide GPS uses a DSSS signaling format to obtain relatively accurate estimates of geographical position. An earthbound GPS receiver determines its position by measuring the path delays of several DSSS signals which originate from earthorbiting satellites. The nearfar effect is not a critical issue for many commercial GPS applications, such as aviation or shipping, since the signals are subjected only to freespace path losses. Some commercial applications, such as surveying, suffer other forms of signal losses, such as multipath fading or shadow fading, and these can indirectly lead to nearfar limited performance. This work examines receiver performance when an adaptive antenna array is used at the DSSS receiver. In such a system the array would act as a frontend spatial filter which preserves the integrity of the desired signal or signals, attenuates interferers and supplements the existing power control algorithm. Such an approach might be compatible with existing cellular CDMA standards. The remainder of this chapter is divided into several sections. The next section will give a qualitative review of DSSS systems. The second section will review quantities and expressions used in the analysis of the minimum meansquarederror (MMSE) beamforming antenna arrays. The third section will review previous work germane to the research presented here and the last section will provide an outline of the remainder of this dissertation. DirectSequence SpreadSpectrum with MultipleAccess Generating a DSSS waveform involves multiplying a modulated baseband waveform by a psuedorandom sequence sometimes known as a pseudonoise (PN) code. In an asynchronous multiaccess channel the incident DSSS waveforms may be subjected to random time delays and carrier phase delays and corrupted by noise. For a singlechannel receiver the multipleaccess incident signals may be modeled by the expression: Y(t)=Re Ac,(t T,)b(t T.)exp(j(ct+0,))+n(t) (1.1) where the index m refers to the mth of K signals. The quantity Am is the incident signal level of the mth signal, Cm(t,m) represents the mth PN code waveform which consists of a sequence (of length Nc) of psuedorandom square pulses, (c, (+1,1 }) each with interval Tc. The quantity b.(tm) represents independent binary phaseshift keyed (BPSK) modulation with equallylikely symbols (bi e (+1,1 }) obtained via ideal, square pulses. The time delay Tm is uniformly distributed over a symbol interval [0,Tb). The time delay Tm arises because the channel allows asynchronous access; users may begin transmission at any time. The carrier frequency is designated by ao, and 0, is the random carrier phase uniformly distributed over [0,2x). The quantity n(t) is complex additive white Gaussian noise (AWGN) with a onesided power spectral density of No. Note that in contrast to systems which use frequencydivision multiple access (FDMA), all users in a CDMA system share a common frequency o(. Frequencydivision channels are achieved by separating the carrier frequencies of multiple, bandlimited signals to the point that they do not interfere with one another in the receiver. Codedivision channels in CDMA systems, on the other hand, are achieved through the low crosscorrelation properties of the individual PN sequences. The PN sequences have many interesting properties, including low crosscorrelation14 defined as r (7) = J c.(t)c, (t + )dt<< Nc V m n. The limits of the integral are from zero to NcTc. In a CDMA system the individual users share a common carrier frequency but are assigned distinct PN sequences that permit codedivision channeled links to a base station or other receiver. This work assumes that detection of DSSS signals is accomplished by a correlating detector also referred to as a conventional detector. The ith user's DS signal is multiplied by a synchronized replica of its PN sequence, passed to an integrateanddump filter and t = nTb + bT x fJ( )dt  Figure 1.1 Conventional detector, baseband model. hardlimited to provide an estimate of the modulation symbol, as shown in Figure 1.1. The conventional detector provides optimum performance for a single user in AWGN but gives suboptimum performance in a multiaccess environment The performance of a conventional detector in a multiaccess environment depends roughly on the processing gain which is a measure of the interference rejection capability of the detector and is sometimes defined as the number of code symbols (or "chips") per modulation symbol (Nc = Tb/Tc). For a fixed, finite processing gain it is always possible for strong interferers to overwhelm the detection of a weaker signal and cause poor performance. The complex baseband output of the integrateanddump portion of the conventional detector devoted to user 1 consists of a desired signal component as well as multiaccess interference and noise components: K t, +T T +r so ,,()= Ab, (t)+ A. c (t)c (t )b(t )dt+ fc,(t)n(t)dt =2 T, (1.2) =Ab, i(t)+ I A. [bi,,R,,, (,m) + b,,o.,i (Tm )]+ Jci (t)n(t)dt Rm., (.m ) = Cm(t r. T lc(t)dt where: 0 where bit 1 of user 1 is the detected signal of interest and Tr = 0. The quantities RP,, (r,) and ki,, (r,) are the partial crosscorrelation functions. The quantities b,.l and bo represent the contributions of the mth user's two bits which overlap the current bit interval of the desired user. The values of crosscorrelations depend upon the sequences and their relative time delays. The multiaccess biterror performance of the conventional detector has been studied extensively; the dependence of the detector performance on PNcode parameters has been well characterized.15'6 If one or more crosscorrelations are nonzero and the corresponding input levels are large, those components may overwhelm the desired signal in the detector. This is the nearfar effect. In mobile cellular or personal communication systems, signals incident on the base station (also referred to as the uplink signal path) can take wideranging incident power levels because of the random placement of users about the base station. The current solution to this dilemma is to vary the transmitter power levels in real time via transmit power control. In a system using closedloop power control the base station receiver monitors the detected power level of each multiaccess signal. As the incident power levels of the multiaccess signals vary because of changing channel conditions, instructions are sent from the base station to the mobile units, via an uncoded narrowband side channel, to individually increase or decrease transmit power. The aim of the power control is to force the effective bitenergytonoise spectral density ratio ((E/No),f) of all the incident signals to the same threshold. In practice, however, the power control is unable to perfectly track changing channel conditions because of dopplerinduced multipath fading and finite power control step size. Error in the powercontrolled signal results in an effective bit energy to noise spectral density ratio ((EbNo),r) at the detector output which is approximately lognormal in distribution.17'1 Field trials indicate that highmobility mobile users (such as in fastmoving automobiles) have (Ed/No), standard deviations of oa = 2.5 dB while lower values (or, = 1.5 dB) have been recorded for lowmobility users (pedestrians).7 Existing cellular CDMA specifications also have a contingency for openloop power control.6 In an openloop powercontrolled system the mobile or personal units use the detected basestation carrier power as a reference to adjust their transmit power. No side channel is required. Some researchers have proposed estimation algorithms which would allow a mobile to make estimates of the base station carrier power. The researchers present evidence that the carriermeasurement error in terrestrial mobile cellular systems is wellapproximated as lognormally distributed with a qc, 3 dB for vehicle speeds of up to 60 mph and 8 dB of shadow fading.19 Other researchers working in the field of mobile cellular systems based on lowearthorbitingsatellites (LEOS) have also found openloop power control error to be wellapproximated by a lognormally distributed random variable (r.v.).2021 If we generalize the previous work and assume that power control error may be approximated by a lognormal r.v., then equation (1.1) may be rewritten slightly: ) =Re( )b)ep + (1.3) where em is normally distributed with zero mean and variance equal to a,' (N(O, ce,2 )). Note that in previous works the lognormal approximation of power control error applied to the incident power levels for openloop power control, and to the (EVNo)ef for the case of closedloop power control. To simplify analysis, this research will assume that power control error results in lognormally distributed incident signal levels for closedloop or openloop powercontrolled systems. Adaptive Antenna Arrays This research assumes an adaptive antenna array is used as a frontend spatial filter to attenuate directiondependent cochannel interference prior to conventional detection of multiaccess DSCDMA signals. The anticipated role of the adaptive array will be to attenuate the strongest interferers and thereby lessen performance degradations due to the nearfar effect. The antenna arrays explored here will be limited to beamforming arrays  linear combiners which use a minimum meansquared error (MMSE) optimization criterion. This technique is also referred to as Wiener filtering22 or optimum combining.23 A MMSE optimization is well suited to a mobile DSSSCDMA scenario since each user's PN sequence, required for DSSS detection and decoding, can provide a convenient reference waveform. There are efficient MMSE adaptive algorithms which would require no more side information than the code sequence and its timing.22 The array is composed of ideal isotropic sensors with no mutual coupling between the elements. Incident signals are spatially sampled by the sensors as they propagate across the array. The incident signals are assumed to be narrowband. This approximation is valid for spread spectrum signals as long as signal bandwidths are a small percentage of the carrier frequency. Current specifications6 call for bandwidths of 1.2288 MHz at a carrier frequency of 850 MHz. The resulting doublesided signal bandwidth is much less than one percent, so the narrowband approximation should be acceptable. The physical displacements between the array sensors induce relative phase shifts on our spatially sampled, narrowband signals. The phaseshifts are constant over frequency and depend upon the array geometry and the signal's directionofarrival (DOA). The complex signal component outputs of the N discrete antenna elements for the mth incident signal can be described by a vector: exp(j .o) S= g.(t)exp(j(ky. t))= g (t)exp( jot) exp(ij.,) exp(j./ _i) = g8(t)exp( jco)t)u, (1.4) where: = [exp(jo.,) exp(ji,.) ... exp(jo.,_ )]l where the vector y, (see Figure (1.2)) represents the physical displacements between some reference point and the N array elements. An element of y. (yn n = O...N1) represents the physical distance between a reference point and the corresponding array element. The parameter k is the freespace wave number of the incident plane wave ( k = 2 Tx/ ). Carrier phaseshifts have been absorbed into the complex, baseband function g,(t).The vector u. contains electrical phase shifts resulting from the physical displacements ym between the elements and the point of reference. The individual elements of u, ( i.e. ~, ) represent the mth signal's phase shift between the reference point and the nth array element. The vector u. will be referred to as the interelementphaseshift vector or simply the phaseshift vector and will not be considered a function of time unless stated otherwise. For the remainder of the report constants will appear as normal block or script characters, vectors will be represented by lowercase boldface and matrices by uppercase boldface. A diagram illustrating some of the physical quantities is shown in Figure 1.2. An incident signal propagates as a plane wave s(t).The signals and noise are spatially sampled by each element of the array; ignoring an explicit time dependence allows the samples to be represented by x., n = O...N1, and form the input vector x. The inputs are weighted (via the weight vector w = [wo wi ...WNI]T) and summed to form the beamformer output. The quantity ky is the phaseshift of the propagating wave due to physical displacement y. As the wave propagates across the array the phaseshifts due to displacement give rise to interelement phaseshifts which are contained within the vector u. The lower part of the figure illustrates how the directiondependent phaseshift arises between two adjacent elements. The quantity < is the phase shift between any two adjacent elements. A change in array geometry would affect only the functional form of the relative phaseshifts between the array elements. The quantity 0 = 2zd(sin0)/L in Figure 1.2 above is the directiondependent phaseshift between any two adjacent array elements for a linear array. Arrays composed of individual elements arranged in a circle will be examined in later chapters because, unlike linear arrays, they can resolve incident signals over the directionofarrival interval [0, 2z). The phaseshift between any two adjacent elements is given by 0 = (Rdcos[62,/Nl)/(1sin[/N]) for a circular array. y = dsin0 foralinear s=ky='2yl/I array: d = element spacing the phase shift 9p i 2dsin between the two 2 d sinQ elements is then: A Figure 1.2 A directional plane wave incident on a linear array of sensors. The baseband array output vector is the sum of the noise and signal vectors: K K x = n(t)+ st) = n(t)+ g.(t)u. m=l m=l (1.5) where n(t) is the AWGN vector. The noise is spatially and temporally white. The narrowband portion of the array autocorrelation matrix may be expressed as a sum of outer products or as a product of matrices: K R = E x'xr= I+ A, u u, = r I+U'A2UT (1.6) Ml1 where U is the matrix of phaseshift vectors and A is a diagonal matrix of input amplitudes, (*)* represents a complex conjugate and (O)T a vector transpose, respectively. The expectation E[*] averages the AWGN, n(t) which is assumed to be N(O, 2). The normsquared of Um is: iuJ2 =u u. = N (1.7) independent of signal DOA where (O*) represents the hermitian transpose. The MMSE weight vector minimizes the meansquared error between the processor output and the reference signal. The weight vector is given by: wo = R'p (1.8) where p is the steering vector and is given by: p = E[r(t)s;] = AdAu (1.9) and Sd and Ud are the signal vector and phaseshift vector of the desired signal respectively. The quantity r(t) is the time varying reference waveform with a peak amplitude of AR. The MMSE weight vector maximizes the output signaltointerferenceandnoiseratio27 (SINR) which is defined as the quotient of the desired signal power and the sum of the interference and noise power SINR = Ps/(PJ+PN). The output signaltonoise ratio (SNR) for the mth signal is the quotient of the output power of the mth signal and the output noise power. It is given by:27 SNRo = P = SNRi w u N SNRi, (1.10) N, ollII2 The quantity SNRim = Am2/a2 is the incident SNR for the mth signal, Ps,m is the desired signal output power, PI is the interference output power and PN is the output noise power. For a single signal in AWGN, Wo = u*, and the upper bound becomes an equality. The choice of w. = u,* represents the spatial equivalent of a matched filter (to the mth signal) and is referred to as a conventional beamformer24, a classical beamformer2, or a maximal ratio combiner.26 The sum total of the number of beams and nulls an array is capable of directing simultaneously is N1, one less than the number of array elements. This is also referred to as the degreesoffreedom27 of an array. At this point, it may be important to explain the difference between the terms adaptive array and diversity combiner. The term adaptive array implies that the processor uses an array of sensors and exploits the phase and amplitude shifts between the array elements, induced by directional signals, to make processing decisions. A diversity combiner, on the other hand, exploits some degree of statistical independence between the input samples to ensure signal integrity at the summed output. For example, in the case of communication channels which contain multipath fading, an array of sensors might provide statistically independent spatial samples. If a signal is faded at one sensor, it might not be faded at another sensor; the statistical independence of the inputs is exploited by the processor to improve the integrity of the overall output response. There are many diversity combining strategies.28 A popular strategy for analytical purposes, mentioned previously as maximalratio combining, maximizes the output SNR but does not allow adaptive interference suppression. The diversitycombining counterpart to MMSE filtering is referred to as optimum combining. It is sometimes possible to achieve statistically independent samples through time sampling. Although not an issue in this research, a few examples of time diversity will be cited in the following section. Previous Work This section is a survey of previous work in adaptive beamforming antenna arrays and in CDMA that is pertinent to this research. The first part of the survey will give a historical perspective to research in adaptive antenna arrays. Details of the earlier work oriented towards radar and military communications is admittedly sparse. As the timeline and focus become more current the coverage will become more detailed. The second part of the survey will review the most recent work in which adaptive arrays function as an integral component of DSCDMA systems. The last part of the survey will examine previous work in cellular CDMA ( without antenna arrays) as it applies to this research. A beamsteered array was investigated by Applebaum29 but the results were not published in open literature until a decade later.30 Widrow et al.31 reported on an antenna array using a leastmeansquares (LMS) processor in 1967. A special issue of the IEEE Transactions on Antennas and Propagation devoted to active and adaptive antennas was published in 1964,32 197633 and 1986.34 Other processors were investigated by Frost35, Griffiths36 and Schorr.37 Much of the initial adaptive array work focused on the performance of particular processors or radaroriented applications. One exception to this was the maximal ratio diversity combiner investigated by Brennan.26 Subsequent work explored the role of adaptive arrays in communication systems.3"39 In particular, Compton40 presented a qualitative evaluation of an experimental adaptive array in a DS communication system. The evaluation focused on a single desired signal and a limited number of jammers. One of the conclusions reached by Compton is that the array makes appreciable contributions to interference suppression. Winters41 studied the acquisition performance of an LMS adaptive array in a DS system using fourphase modulation with two PN codes, a short code for rapid acquisition and a long code for protection against jammers. Ganz42 evaluated the biterrorrate (BER) performance of a receiver employing an adaptive array and one of several detectors for binaryphaseshift keying (BPSK), quadrature phaseshift keying (QPSK), or differential phaseshift keying (DPSK) modulation. The receiver was subjected to continuouswave (CW) jamming. Several authors have investigated the use of antenna arrays for mobile or personal communications. Bogachev and Kiselev43 evaluated optimumcombining diversity arrays for the case of a single interferer. Winters23 conducted a comparative study of optimum combining and maximal ratio combining base station arrays in a multiuser mobile environment with multipath fading but no shadow fading. The results quantify the possible improvement in SINR if optimum combining is selected over maximal ratio combining. Like optimum combining, maximal ratio combining preserves the integrity of the desired signal, but unlike optimum combining maximal ratio combining has no ability to adaptively suppress interference. Winters did propose the use of psuedonoise codes to generate the LMS reference, but transmit power control was mentioned only for its effect on the convergence properties of the LMS array. He did not investigate in any detail the condition when the number of users greatly exceeds the number of array elements. Winters also explored the use of adaptive arrays on base stations for inbuilding systems using dynamic channel assignment4 He again considered a PNcoded PSK modulation and circumvented power control considerations by assuming interferers were of equal power and much stronger than the desired signal. A more recent study45 investigated the acquisition and tracking performance of LMS and samplematrixinverse (SMI) beamformers in a timedivision multipleaccess system. Yeh and Reudink4 examined the contributions made by spatial diversity combiners to spectral efficiency in FDMA cellular systems when the arrays are located on the base station and on the mobiles. Glance and Greenstein47 also examined the contributions of diversity order (the number of array elements) on average BER in a mobile FDMA system. Vaughan4 discussed the benefit of MMSE combining at the mobile in an FDMA system and concluded with the comment that for MMSE combining to be successful wide bandwidth signals, such as those found in spreadspectrum systems, are necessary. More recently, adaptive array research has examined commerciallyoriented applications as CDMA and nonCDMA wireless systems gain popularity. At this time, antenna arrays are under investigation as a means of providing spacedivision channels in multipleaccess systems. The technique, called spacedivision multiple access (SDMA) by some authors, utilizes the spatial filtering properties of the array to selectively receive signals which share the same time slot and the same frequency band. The SDMA technique might apply to either timedivision multipleaccess systems (TDMA) or CDMA systems. Swales et al.49 established that a steerable, multibeam antenna array can increase the capacity and spectral efficiency of a cellular system. Suard and Kailath50 studied the upper bound of the informationbased capacity of a wireless system which used a base station antenna array in the uplink path. Experimental studies have been conducted. Xu et al.51 and Lin et al.52 have examined algorithms based on directionfinding techniques MUSIC" and ESPRIT.5 They have concluded that SDMA techniques based on DOA estimation will not be effective in multipathrich environments. Xu and Li55 developed an SDMA/TDMA protocol. Ward and Compton56'57 examined the contributions that a receiving array can make to the performance of an ALOHA system. Arrays which include spatial and temporal adaptive processing nested within LMS feedback loops were proposed by Kohno et al.58 and Ko et al.59 Kohno used an LMS array in conjunction with adaptive temporal filtering which successively canceled DS signals due to multiaccess interference. Ko described a nullsteering beamformer nested within an LMS loop, not necessarily restricted to CDMA applications. Both considered limited multiaccess scenarios with deterministic interference parameters. Diversity combining is not necessarily restricted to space diversity. Balaban and Salz60'61 examined in detail the performance of a general multichannel MMSE combiner working in conjunction with a decisionfeedback equalizer. They established an upper bound on BER which is a function of the MMSE. A great deal of work has also been devoted to temporal diversity combiners which consist of a single input to a bank of matched filters, the outputs of which are coherently combined via maximal ratio combining. This is the basis for the RAKE62 receiver as well as variations studied by other authors. Several structures were examined by Turin63 while Lehnert and Pursley64 examined diversity combining in multiuser CDMA system in which successive bits are spread with different PN code subsequences. Wang et al. consider diversity for an indoor DSCDMA system with Rician fading.6 Some authors have studied the possible contributions made by arrays to the uplink path performance in cellular systems. Liberti and Rappaport have studied the effects of a directive, steeredbeam basestation receiving array on uplink performance in a CDMA cellular system with perfect power control. The study focused on the effect of beam shape and beam width on the average BER. It was found that beam width has the greatest impact on performance and that adding a threeelement array to the base station can improve BER performance by three orders of magnitude. Tsoulos, Beach and Swales have recently examined the role of adaptive antenna technology in large "umbrella" cells overlaying smaller microcells in cellular CDMA systems ; they also examined the outage based capacity enhancement due to an adaptive antenna array using a recursive least squares (RLS) processor in a multicell CDMA environment.6 The latter study was confined to simulations of the total interference to calculate outage. The authors concluded that an antenna with 6 dB of directivity gain can increase capacity by a factor of five. Winters, Salz and Gitlin6 studied the effects of optimum combining spatial diversity arrays on the capacity of a TDMA system in which the total number of incident signals was less than or equal to the array DOF. They applied previous workw'' which resulted in an upper bound on BER. The assumption of high input SNR allowed a zeroforcing approximation to the optimum combining solution. Using these assumptions, and examining analytical expressions for BER, they found that optimum combining with N antennas and K interferers gives the same results as a maximal ratio combiner with NK+1 elements and no interferers. Their theoretical results, as they point out, no longer apply when the number of interferers exceeds the number of antenna elements. A structure which combines the spatial filtering of an antenna array and the temporal, diversitycombining properties of a RAKE receiver have been proposed and investigated by Khalaj in concert with several other authors.6970'71 The structure is intended for use in channels with frequencyselective multipath fading. The structure allows the resolution of identifiable multipath rays by the timefiltering properties of the RAKE combiner and by the spatial filtering properties of the antenna array. A number of authors have conducted brief simulationbased studies of adaptive arrays for DSCDMA systems. Yoshino et al.72 examine the simulation performance of two RLSbased spatial diversity combiners operating in concert with (Viterbi) sequence estimators. One processor subtracts estimates of the cochannel interference from the output prior to estimation of the desired user's data sequence while the other processor does not. Wang and Cruz73 examine the pattern behavior and BER of a sixelement arrays based on the RLS and ESPRIT algorithms with six active users with wellseparated DOAs. Liu74 examined the performance of an LMS array with a scenariodependent matrix preprocessor which aids in interference cancellation. Hanna et al.75 investigated the BER performance of a twoelement LMS array which operates in conjunction with an adaptive equalizer. Perhaps the most indepth study of the possible contributions of MMSE adaptive beamforming arrays to the performance of mobile cellular CDMA systems using closed loop power control has been made by Naguib in concert with other authors. Initial work76 focused on steadystate performance of an array of sensors, each followed by a DS conventional detector, which functions as a postdetection combiner (termed code filtering by the authors). Analysis resulted in a simple expression for capacity. The signal model was for unfaded signals originating in an isolated single cell. Power control error was modeled by assigning a single interferer an incident signal level 10 dB higher than the other signals. Naguib, Paulraj and Kailath77 extended the steadystate model in order to determine the outage probability in a cellular system with BPSK modulation, perfect power control, shadow fading, multipath fading, and cochannel interference equivalent to two tiers of surrounding cells. Assuming that the array pattern response consisted of a main lobe and no sidelobes resulted in a simple, closedform expression for an upper bound on outage probability which simplified to the singlechannel results of Gilhousen et al." when the array is reduced to a single element. Modeling the MMSE processor as a maximal ratio processor and assuming the interference was Gaussian resulted in a simpler expression for the outage probability upper bound.78 Naguib and Paulraj then modified the simulation model to include DPSKmodulated signals and determined the Erlang capacity.79 The uplink performance with Mary orthogonal modulation was examined as well. 0 A recursive beamforming algorithm was proposed by Naguib and Paulrajs8 and simulated results were presented. The algorithm performed recursive updates on the matrix square root of the inverse of the covariance matrix. The authors claimed that the accumulation of numerical and quantization errors may cause the covariance matrix inverse to cease being hermitian definite and that updating the matrix square root will allow the covariance matrix inverse to remain hermitian definite even when the matrix square root is not. Thus, say the authors, numerical instabilities are avoided. Naguib and Paulraj82 continued their investigation of cellular base station antenna arrays by examining the tradeoffs in coverage area, mobile transmit power, and capacity that are available when an array is used and the users are subjected to perfect power control. Using simplifying assumptions the authors derived expressions which generalized the effect of the antenna array on performance. This research will attempt to extend some of the results to the case of imperfect power control. Later investigations have resulted in detailed, simulationbased studies of an IS95 system which uses orthogonal signaling, forward errorcorrection coding, closedloop power control and a basestation antenna array. Unlike their previous studies Naguib and Paulraj applied a model for imperfect closedloop power control. They studied the standard deviation of power control error, although where the error is defined is not clear in the paper. Using simulation results they show that power control error dependence on the power control step size, the number of array elements, and the maximum doppler frequency and the spread in DOA of the multipath rays.83 The dependence of BER on the same parameters was the topic of a subsequent paper." A multiuser LMS array for use in GPS receivers was examined by Beach et al.85 in a simulation study. The results were confined to plots which show the evolution of the adaptive array pattern over time in the presence of CW jammers; no steadystate results were presented. The near/far effect was not an issue in the study. A variety of authors have investigated the performance of cellular CDMA systems with imperfect power control and no antenna array. Simpsom and Holtzman6 used simplified analytical models in order to provide insight into the interactions between power control, coding and interleaving. Stuber and Kchao87 examined a multiplecell CDMA system and evaluated the dependence of BER as a function of the distance from the base station. Jalali and Mermelstein" conducted a simulation study of a microcellular CDMA system which included imperfect closedloop power control and antenna diversity with squarelaw combining. Milstein et al.89 studied the average BER performance of a multiplecell system in which users where subjected to power control error which was described by a uniform random variable. Newson and Heath90 examined a CDMA system which suffered from imperfect sectoring and lognormallydistributed power control error and made capacity comparisons to TDMA and FDMA systems. Dissertation Outline The remainder of the dissertation will be divided into six chapters. Chapter 2 will examine some aspects of the steadystate performance of a multiuser MMSE array; some limited aspects of an adaptive LMS version were investigated by Beach et al.8 The near/far performance will first be investigated for strong, spatially orthogonal users and will then be extended to the case of two users with any DOA spacing. The analysis will focus on the ability of the array to confine the signal outputs to the same output SNR. The results will show that the array may be suitable when signals are wellseparated in DOA and do not outnumber the array DOFs. The analysis will borrow some of the analytical techniques used by Gupta9" and apply them to evaluate the steadystate performance of the multiuser MMSE array. Unlike either previous study, however, the analysis will focus on the performance of the array in a near/far environment. The third chapter will examine the adaptive performance of the multiuser MMSE array when it is implemented as an LMS processor. Analysis will show that the ruleof thumb for picking step size to ensure stability also serves as a limit to reduce the spread in output signal levels due to LMS misadjustment The fourth chapter will examine the steadystate performance of an MMSE array feeding a conventional detector. Unlike the array of the first two chapters, this array will be dedicated to a single user among many multipleaccess users. The singlecell users will not be subjected to many of the influences normally found in a mobile environment, such as multipath fading, shadow fading or interference which originates from surrounding cells. This will simplify the analysis: the intent is not to provide an analysis fraught with mathematical rigor, but to use simplifying assumptions to aid in the derivation of closed form analytical expressions which accurately predict the uplink performance and which clearly show the dependence of uplink capacity on the system parameters, especially the array and the degree of power control error. The performance measures will be outage probability and outagebased capacity. The analysis of chapter four will result in a simple closedform expression for the capacity which is linear in the product of processing gain and the number of array elements and which decays exponentially with increasing power control error. The slope of the capacity line (the perelement capacity) will serve as a robust measure of the array's incremental contribution to capacity. Although some of the assumptions are highly idealized (i.e. no multipath fading), they will allow some wellunderstood quantities, such as array gain, to be exploited in the analysis. Some of the work in this chapter is related to that of Padovani7 and also to Naguib and Paulraj.82 The details will be discussed during the chapter's derivations and discussions. Chapter 5 will extend the steadystate multiaccess model of Chapter 4 to include the effects of multipath fading, shadow fading and outercell interference. The performance measures will again be the steadystate outage probability and outagebased capacity. The sixth chapter will examine the recursiveleastsquares (RLS) adaptive performance of the MMSE adaptive array in a mobile cellular scenario in a nonstationary channel. The discreteevent simulation model includes timevarying multipath fading, stationary shadow fading and timevarying outercell interference. The last chapter will give a brief summary and some conclusions of this research. It will also identify some areas of future work. This research makes contributions in two areas. First, analysis and numerical solutions provide more insight into the steadystate and the adaptive behavior of the multiuser MMSE array (chapters 2,3) than was provided by previous authors. Analysis shows that the array can remove the nearfar effect and level the output SNRs to nearly the same level, under certain circumstances. Second, the analysis and simulation of a singleuser array operating in conjunction with a conventional detector results in simple expressions for capacity when the signal sources are subjected to imperfect power control (chapters 4,5,6). The results characterize the incremental contributions an adaptive array can make to uplink performance. This is in contrast to previous works which examine the effects of closedloop power control error only in simulation for limited scenarios. CHAPTER 2 A MULTIUSER ANTENNA ARRAY PROCESSOR STEADYSTATE PERFORMANCE This chapter examines the ideal, steadystate performance of the multiuser MMSE array proposed by Beach et al.5 The authors conducted a simulation study of the pattern behavior of a multiuser LMS adaptive array with and without directional interference. They did not provide any analytical results which give general insight into the array's steadystate performance. In related work, Gupta91 examined a multiuser steeredbeam array for nonCDMA applications and simplified his analysis by assuming the special case of spatially orthogonal users. Gupta was interested in using the array's effective aperature to improve the output SNR of the desired signals and in using the adaptive properties to reject interference. Other authors have shown that steeredbeam arrays and LMS arrays give the same output SNR performance as long as the steering vectors differ by a constant.92 Using this rationale it might seem that Gupta's work could provide some insight to this problem. However, he was trying to configure the array processor to avoid the power leveling effect that this work is attempting to exploit. This chapter will focus on the array output SNR performance for strong incident signals. The analysis will deemphasize the role of the spreadspectrum signalling other than to note that it provides a relatively easy way to provide separation and detection of the multiple, desired signals present at the single array output. The remainder of this chapter will be divided into several sections. The first section will provide some qualitative information about the multiuser MMSE array processor and give some refinements to the general array equations given in the introduction. The second section will develop analytical expressions of the output SNR for K _N1 spatially orthogonal users. The third section will present analytical expressions for two users which have arbitrary DOAs and are therefore not necessarily spatially orthogonal. The last section will compare the analytical and the numerical results and discuss their implications. The Multiuser MMSE Processor The multiuser array treats each of the K signals as a component of a single composite desired signal. This is accomplished via a reference signal which is a sum of the modulated PN sequences of the desired incident signals. A block diagram is shown below in Figure 2.1. In the figure all functions of time are ignored. For analytical simplicity we will assume that perfect estimates of the modulation and code waveforms are available for the generation of the reference waveform. This will not usually be the case. Generation of the reference signal can be a challenging issue and it has been investigated by other authors.27.40.41 The array will force the outputs of the individual signals to the levels of that signal's component of the reference waveform. To avoid the near/far effect the individual components comprising the reference waveform have the same peak amplitudes. Since all of the incident signals share the same beamformer weights, the output SNRs will be leveled to approximately the same value. The analysis will show that the array will force the output SNRs to values very close to the maximum output SNR of the weakest user under certain conditions. The steering vector defined by equation (1.9) represents the crosscorrelation between the reference signal and the multiple, independent input signals. It becomes a sum of singleuser steering vectors: K K p= p. = R Au'. (2.1) where A, and Um are the incident amplitude and phaseshift vector of the mth signal, respectively. Figure 2.1 Multiuser MMSE processor Spatially Orthogonal Users Consider the limiting case of multiple users which are spatially orthogonal to one another (uifuk = 0 V ik ). The minimum source separation (in DOA) which allows spatial orthogonality corresponds to the Rayleigh limit.22 The Rayleigh limit is generally To a bank of conventional Output DS detectors K (t) no(t)+ on() + ,=1 rence al ', bI (t TI )c, (t T) + 4'Lb 2 (t 2r )c (t T K ) b( V T )C K (t T K XNl(t) considered to be the minimum amount of source separation which still allows resolution of two sources by a beamformer.24 If the users are orthogonal and the array elements are isotropic the inverse of the autocorrelation matrix is:91 1 ( K NSNRi, ' R'1 I u 'uT (2.2) aO =1 +NSNRi  and the resulting MMSE weight vector, w, = R'p, is given by: wO R i= .u' (2.3) o ,=, 1+N.SNRi, Substituting these quantities into the general expression for the n'th users output SNR (equation (1.10)) gives: f SNRi, K NSNRi SNRo, = 1+NSNRi,,) ^ (1 +N SNRi ,)2 (2.4) Some coarse judgements on performance for multiple users can be made using equation (2.4). If {SNRim >> 1 V m =1 ... K) the output SNRs for all of the users are approximately equal to: SNRo = N S (2.5) ..= SNRi. If there are K users with the same input SNR = SNRi then equation (2.5) simplifies to: N. SNRi SNRo = N i (2.6) K which shows that the users will all have output SNRs equal to the maximum possible SNR divided by the number of active users. If the array parameters remain constant, the performance will degrade as the number of users increases. If there are S users with the same input SNRi = ki.SNR and KS users with SNRi = SNR then the output SNR for the K users simplifies to: SNRo = N SNR k (2.7) S+k, .(KS) If k(K S) >> S (a possible near/far scenario) the expression for output SNR simplifies further: NSNR K S SNROSSK SSNR (2.8) SNRo . KS (2.8) k,NSNR K K where: S = no. of strong users with input SNRi = ki SNR, k >> S K = total number of users K S = no. of weaker users with input SNRi = SNR From the equation immediately above it is apparent that one weak user (KS = 1) will dominate the overall performance of the array. This might make intuitive sense, even for the general case. A minimum MSE processor will try to force each output signal to have the same value as that signal's component of the reference waveform. This strategy will favor weaker incident signals which need large weight values to cause their array output to match their component of the reference waveform ( i.e. minimum error power). Large incident signals will be attenuated via phase shifts and the noise power the denominator of equation (2.4) will be large thus causing a lower than maximum output SNR even for strong input signals. Equation (2.8) above establishes that the weakest incident signal drastically affects the overall performance of the array. The worst array response towards an arbitrary signal (SNRon) for a different input signal (SNRi,) can be found by finding the value of SNRim which forces dSNRon/dSNRim, the derivative of the nth signal's output SNR with respect to the mth signal's input SNR, to zero: dSNRo, N SNRi, ( N SNRi,. N SNRik) dSNRi, I++NSNRi +N SNR k1 (1+ N SNRi) (2.9) From the equation above its is evident that a minimum occurs for SNRim = 1/N. Substituting SNRim = 1/N into equation (2.4) gives (for SNRin >> 1, n i m): SNRo, 4. N SNRi,, 4 1 + N. SNRi, (2.10) SNRo, = 1 For SNRim = 1/N the value of the mth eigenvalue of R is twice that of the noiseonly eigenvalues. At this input level the array is barely able to resolve the m'th input signal from the input noise. It might be useful to define an output SNR "spread" which bounds the output SNRs for all of the users. From equation (2.4) it can be seen that the largest and smallest input signals result in the largest and smallest output SNRs respectively. We can predict the output spread by the difference between the largest and smallest output SNRs: ASNRo = SNRo, SNRo. = f NSNRi, f NSNRi, V, NSNRij [ +N SNRi) +N SNRi,, (1+N SNRi,2 ) (2.11) If we let SNRimax pass to infinity the output spread becomes: lim I N SNRi Y D A N SNRiA Ri oo:ASNRo = 1 (2.12) SNRi Rii + SNi. + (I + N SNRij)2 The equation above simplifies to: 2+1/NSNRi' ASNRo = 2 + NSNRin (2.13) (1 +NSNRi )2 D NSNRij 1+ 2 N SNRi, ,,m*a. (1+ NSNRi ) An upper bound on the output SNR spread results: ASNRo < 2 + (2.14) N.SNRi,,4 In summary it would appear that for NSNRim >> 1, m=l...D, the user outputs will be near the value of the maximum output SNR for the weakest user and the output SNR spread will be equivalent to 2. Nonorthogonal Users Analyzing the general case of multiple narrowband users which are not spatially orthogonal is difficult because of the matrix inverse in equation (1.8). When the users are not orthogonal the analytical matrix inverse (via the matrix inversion lemma) quickly becomes intractable as the number of users increases beyond unity. The general case of only two users was examined in detail. The approach taken here is to derive expressions for the two output SNRs. The expressions are then interpreted as transfer functions with poles and zeros which are dependent upon the input levels and the DOAs. Critical points of the functions are then evaluated. The autocorrelation matrix R for two independent, narrowband users in AWGN (equation (1.6)), is given by: R = a2 I+ A':u;u + A'u2u (2.15) = 2 [I + SNRi2u u + SNRi2u;u ( The inverse of R (R1) may be calculated via the matrix inversion lemma: R1 1 1 SNRi T R = IU.U a. a2, 1+ SNRi, (2.16) T u u;SNRi uI' uSNRi U SNRi 1+ NSNRi, 1 +NSNRi, o uu;1 SNRi, SNRi2 1 + SNRi,  2 1+SNRi, The steering vector p which results from equation (2.1) is: p = RAu; + RA2u; = aR S uiu + aRR SRu (2.17) Substituting equations (2.16) and (2.17) into equation (1.8), wo = R'1p, gives the MMSE weight vector: w = R =R [pi(1+ SNRi2) uTu;SNRi, SNRi2]u; + (2.18) R [ (NRi2 ( + SNRi) ) uluSNRi, JSNRi ]u; oDo The quantity Do is defined by: D, =1+ N(SNRil + SNRi) + (N uI u;2 SNRiiSNRi2 (2.19) Two vector inner products are required to find the output SNR defined in equation (1.10). If we assume the point of reference for the array is the same as the physical center of the array, the inner products will be real.24 An inner product between two phaseshift vectors may then be expressed in dotproduct form: uTu* = Ncosa 2 where al2 is the angle between the two vectors in signal space. Substituting this relationship into equation (2.18): 11w.112 = N(SNRiI +SNRi2) +2N SNRizSNRi2 cosa,2 + R2 N 2SNRi SNRi, sin2 a2 (4 + NSNRi, + NSNRi, 2N SNRi SNRi2 cosa 2) a ,20,D where: Do = 1+ NSNRi + NSNRi2 + N2SNRiNSNRi, sin2 a,2 (2.20) Expressions for the numerator of equation (1.10), the output SNR, are possible for the two users: loUw. = R2 [NSNRi2l + NSNRi22 sin2 a12)+ SNRi cosa1]2 (2.21) Substituting equations (2.20) and (2.21) into equation (1.10) will result in the steadystate output SNR for the two users at the point of minimum meansquared error. An additional substitution might simplify the expressions further. If we substitute the expression SNRij= a2 SNRi2 into equations (2.20) and (2.21) the output SNR can be expressed as a quotient of polynomials in a with coefficients that are functions of NSNRi2 and a12. The numerators of the quotients are: cosa2 nums = N2SNRi2 ( + NSNRi, sin2 a2,,) a+ cosa12 S+ 12NSNRi2 sin2 a12 (2.22) num2 = N 4SNRi sin a2 a, a+a +cos2_+ num2 = NNSNRi, sin2 a21 NSNRi2 sin2 a12 for user 1 and user 2 respectively. The denominator of both biquadratic terms is: Dens2 = a2N2SNRi2 sin2 a [4 + NSNRi2(a2 2acosa,2 +1)] + (2.23) NSNRi4(a2 + 2acosa,2 +1) and the expressions for the two output SNRs become: SNRo = a2SNRi2 numn SNRo2 = SNRi um2 (2.24) Dens, Densm The expressions for output SNRs may be interpreted as transfer functions in the variable a. The numerators may be treated as products of second order polynomials with easily determined roots which may be interpreted as real zeros of the output SNR "transfer function." The roots of the numerators are: Z cosa12 1 + NSNRi2 sin2 a2 (2.25) = cosa12 [1 +Jl4NSNRi tan2a I Znu 2NSNRi2 sin' a12 The denominator is a fourthorder polynomial in a and analytical expressions for its roots are not particularly useful in the general case. To circumvent this difficulty the analytical expressions for output SNR will be examined and approximated for specific cases of a2 (the nearfar ratio) and a12 (the angle between the phaseshift vectors in weightspace) and results will be compared to numerical solutions. Note that a12 is determined exclusively by the users' DOAs for a given antenna array configuration. The output SNR expressions will be examined for four scenarios of the variable a: 1. a2 = 0 (single user system) 2.0 < a2 << 1 (1 weak, 1 strong user) 3. a2 = 1 (2 strong, equal power users) 4. 1 < a2 < oo (2 strong, unequal powers) Two cases of DOA separation will be examined for each of the four scenarios listed immediately above. One case (referred to as case a) will be that of spatially orthogonal users (cos2k12 = 0, sin2a12 = 1). This will allow comparisons with the the results in the previous section. The second case (case b) will be for DOAs with spacings greater than the Rayleigh limit (0 < cos2a12 << sin2c12 < 1) and will also be referred to as well separated. The following analysis will consider approximations to equations (2.22)(2.25) for some combinations of the intervals of a listed above. Rather than being exhaustive the analysis will focus on critical points which will reveal trends in performance. SingleUser System When a = 0 we have a single user system and : SNRo = a2SNRi2 nun = 0 DensNR (2.26) num, SNRo2 = SNRi2 Dens = NSNRi2 DenSNR One Weak User. One Strong User For the case of one weak user and a moderately strong user which are well separated (case b) the numerators and denominator of equation (2.24) can be approximated by: cosa num, = N 2SNRi 2(+ NSNRi2 sin ao)2 a + cosa12 2 1 + NSNRi, sin2 a,2 (2.27) num2 N4SNRi3 sin4' a 2 + NSNR sin2 1 2 1 NSNRi, sin2 a21 Dens, = N 2 SNRi2 (4+ NSNRi,2)sin' a a2 a2 +N+ N2 2 1 NSNRi 2(4+ NSNRis) Sin 2 a12 If we can further assume that NSNRi2 >> 4 then the output SNRs may be approximated [a + (cosa12 )z2b ]2 by: SNRo, = a2N SNRi2 sin 2 [a2 + P2b 2 [a + Z2b j SNRo2 = N SNRi, sin2 a,2 [a +z b (2.28) [a +P2b] where: 1 1 Z2b where: NSNRi sin2 a P2b N2 SNRi22 sin a,2 N SNRi, Where the notation z2b represents the numerator zero for scenario 2 and DOA case b. Note that P2b < Z2b. Also note that for scenario 2, SNRoI is small since it is multiplied by a second order zero at a = 0. As a increases, SNRoj increases with some possible wrinkles in the magnitude curve added by the numerator zeros (a z2bcosal2) and denominator poles ( a = 4p,). Note that since the cosa12 term in the numerator of equation (2.28) can be less than zero the possibility exists that SNRoj could equal zero for nonzero a. The most interesting behavior is displayed by SNRo2. For very small a, SNRo2 is near the maximum value SNRo2= NSNRi2. As a increases away from zero, the SNRo2 function rolls off at the 3 dB point of a2 = P2b and continues to decrease until a2 = Z2b ( SNRil = [Nsin2a,121 ). At a2 = z2b the SNRo2 rolloff attenuation is halted and SNRo2 begins increasing since the pole at p2b is canceled by the zero at a2 = z2b. Over the range of a specified by this scenario, the point a2 = z2b represents a minimum of SNRo2. Substituting a2 = z2b into equation (2.28) gives SNRo1 = 1 and SNRo2 = 4NSNRi2/(NSNRi2 + 5)= 4. Note that similar expressions for the case of orthogonal users ( case a above) can be readily found by simply setting sin2a12 = 1 and cos2a12 = 0 into equation (2.28). In summary, for most of scenario 2 the array is unable to resolve the lowpower user 1 input signal from the input noise. The array can just begin to resolve the user 1 input signal from the input noise when a2 = z2b ( SNRi1 = [Nsin2al21 ). At this point, SNRoI = 1 and SNRo2 4 which represents a minimum value of SNRo2 over this scenario. Two EqualPower Users If the two users have equal input power levels the output SNRs are equal: NSNRi (1 + cosa2 + NSNRi sin2 a,,)2 SNRo (2.30) 2 1+ cosao2 + NSNRi sin2a 2 (2+ NSNRi(1 cosa12)) and for the case of wellseparated users (case b: 0 < cos2a 2 << sin2a12 < 1) the output SNRs may be approximated as: NSNRi (NSNRi sin'2 a2)2 SNRo  2 NSNRi sin2 a12(NSNRi(l cosa2)) (2.31) NSNRi 2S (l+cosa12) If the users are spatially orthogonal (case a) then sin2a12 = 1 and cos2al2 = 0 and the output SNRs are exactly equal to SNRo = NSNRi/2, which is consistent with equation (2.6). Two Strong Users For this scenario (#4) we have two strong users with unequal input power levels. For case b ( 0 < cos2aI2 << sin2a12 < 1 ) the numerator terms of equation (2.26) may be modified by multiplying all SNRi2 terms by an additional sin2a12 term. The ~ denominator (equation (2.23))changes slightly: 2[ 1 4 1 D4,b = a +a N2NR2 sin2 + 4 + 1 + (2.32) NaSNRgi sin' a,, NSNRi2 N2SNRi sin a12 Terms containing higher powers of a dominate the numerators (equation (2.22)) and denominator (equation (2.32)) of the output SNR expression (2.24). This condition represents the presence of two strong users of unequal power: SNRo, >> SNRo2. The expressions for the (scenario 4, case b) output SNRs may be approximated as: (1+ NSNRi2 sin2 a2)2 NSNRi, sin a12 (2.33) SNRo2 = NSNRi2 sin2 a12 Note that for large input SNR the expressions in equation (2.33) should be nearly equal. The difference between the output SNRs for this scenario is: 1 1 ASNRo = SNRo, SNRo, 2 + NS 2 = 2 + 1 (2.34) NSNRi, sin2 aa2 SNRo, Which is very similar to the expression for output spread for orthogonal users given by equation (2.14). Numerical Results This section will first present numerical solutions to corroborate the twouser analysis of the previous section. The data will be presented in plots of the output SNRs (equation (2.26)) versus the input nearfar ratio (a2). Array performance will be evaluated for spatially orthogonal users and well separated users evaluated over a wide range of a2. Specific comparisons between analytical and numerical solutions will be made for critical points identified in the analysis. Performance for a linear MMSE array was investigated by plotting the output SNRs of two signals while varying one signal input level and holding the other at a fixed value of input SNR. For each point the MMSE weights and corresponding output SNRs were calculated using equations (1.6)(1.10) and equation (2.1). The DOAs were fixed to specified values. The plot below shows the output SNRs for the case of a three element array with two users. User 1 is positioned broadside to the array (DOA = 0 degrees) with varying input SNR, and user 2 is positioned at 41.8 degrees from broadside with an input SNR of 10. This DOA condition gives the least amount of source separation which allows the sources to be spatially orthogonal. Figure 2.2 shows the (N = 3) array output SNRs versus a2 for a twosignal scenario over the range 105 < a2 <105. The users are spatially orthogonal, al2 = 90 degrees and the user 2 input SNR is 10. For a2 = 105 the array is in essence a single user system, SNRo2 NSNRi2 = 30 which is the maximum possible output SNR for user 2 and SNRo1 0. As a (SNRi1) increases, SNRo2 begins its rolloff attenuation: SNRo2 rolls off 3 dB by the time a2 = .0011, as predicted by equation (2.28). As a increases the SNRo2 rolloff continues until it reaches the minimum value of 3.33 at a2 = 0.033. At this point the array is just able to resolve signal 1 from the input noise and SNRol = 0.89. These numbers agree well with equations (2.10) and (2.28). SNo o *. .............. 0 10 F 10d NearFar Ratio: a2 10s Figure 2.2 Output SNRs versus nearfar ratio for spatially orthgonal incident signals For the scenario a2 = 1 the output SNRs are equal to 15 (see eqn. (2.6)). As a2 becomes very large the output SNRs lose their dependence on a2 and approach nearly the same value. This is shown in equations (2.33) and (2.34). At large nearfar ratios (a2 = 105) SNRol = 32.03 and SNRo2 = 30. The difference between their values is ASNRo = 2.03 which agrees precisely with equations (2.14) and (2.34). Figure 2.3 shows the performance for DOA spacings greater than the Rayleigh limit. For this scenario user 1 remains at array broadside and the DOA for user 2 moves to 90 degrees (in line with the array). This case gives the vector quantities UJTU2*= Ncosa12=  1, cos2ai2 = 1/9 and sin2 al2 = 8/9. When a2 is small (a2 = 10"5) SNRo2 = NSNRi2 30 as in the previous case. As a2 increases to 1.18x103 SNRo2 rolls off approximately 3 dB __ to a value of 15 and SNRo, increases from nearly zero to 6.5x1073. Note that equation (2.28) approximates SNRol = 5x103 and a 3 dB rolloff point of a2 = 1.25x103. At a2 = 0.042, SNRoi = 0.798 and SNRo2 takes on a minimum value of 2.807. Equation (2.28) predicts a minimum value of SNRo2= 3.87 will occur for a2 = zzb = 0.0375 and SNRo, = 1.01. From the plot SNRo, = SNRo2 = 9.94 when a2 = 1 which agrees with the approximations given by equation (2.31). When a2 increases beyond unity the outputs quickly level to values near the maximum of SNRo2: SNRol = 28.64 and SNRo2 = 26.61 at a2 = Ix105. This gives an output spread (eqn. (2.34)) of ASNRo = 2.03. Figure 2.4 shows what happens when the DOA spacings are less than the Rayleigh limit In this case, signal source 1 remains at broadside while source 2 moves to DOA2 = Figure 2.3 Output SNRs versus nearfar ratio for nonorthogonal incident signals with DOA spacings greater than the Rayleigh limit. Output 100 NearFar Ratio: a2 10' 20 deg., about half the Rayleigh limit. The outputs are degraded for a2 >> 1, compared to the two previous plots. Output SNR 10D NearFar Ratio: a2 1 Figure 2.4 Output SNR versus nearfar ratio for incident nonorthogonal signals with DOA spacings less than the Rayleigh limit CHAPTER 3 THE MULTIUSER ARRAY PROCESSOR ADAPTIVE PERFORMANCE Consider for a moment that filter weights in an adaptive process are random variables with salient statistical properties. Adaptive processors are unable to perfectly track the corresponding steadystate solution; this induces a penalty known as misadjustment93 The goal of this chapter is determine the effect of the step size of the complex LMS algorithm" on the steadystate SNRleveling performance of the multiuser processor. This concern is motivated by two factors. First, from an intuitive standpoint, it seems that the multiuser processor might cease to level the output SNRs if the misadjustment becomes too large. Second, the multiuser array processor might share some similarities to an LMS automatic line enhancer (ALE) for multiple time signals. Fisher and Bershad97 studied the misadjustment performance of an LMSALE for the case of multiple sinusoids in AWGN and found the equalizer misadjustment to be especially sensitive to the step size. This work will rely heavily of the work of Senne95 who developed expressions for the timedependent and steadystate weight covariance matrix of a real LMS adaptive processor. Other authors have investigated the transient and steadystate behavior of the complex LMS weight covariance matrix.96'97 and have found that the steadystate eigenvalues are very similar to those derived by Senne for the real LMS algorithm. This detail will be applied in a later section of this chapter. Other authors have examined the performance of adaptive processors subjected to hard and soft constraints on SNR as well as optimizations of SNR itself subject to nonlinear constraints.98'9 This chapter will be divided into three sections. The first section will give the development of a performance measure (cost function) which is based on the mean output SNR of the adaptive array. The second section develops a performance measure based on the variance of the array output SNR. The last section presents numerical results which allow comparisons between the analysis and simulations. The cost functions will be used to find acceptable bounds on the LMS step size. It is found that the upper bound on step size arising from stability arguments also gives acceptable output SNR performance for the multiuser processor. MeanBased Performance Measure If the adaptive weights of an LMS processor are considered to be timevarying random variables, w(n) = wo + v(n), the mth user's timedependent output SNR results from substituting the expression for w(n) into equation (1.10): (w + v(n)) u., SNRo(n), = SNRi, (w0 +(3.1) IIwo + v(n)11 The random, timevarying weight component is represented by v(n), the mean of the weight vector is the steadystate MMSE weight vector, wo = R7'p, and the discrete time dependence is introduced by the variable n. Note that the phaseshift vector Um and SNRi, are constants. In order to derive an expression for the performance measure we need to find the mean of the output SNR. Some assumptions facilitate this effort: 1. Ilv(n)+ w.II 1 I1wol ; instantaneous deviations of norm(w) from norm(w.) are negligible because of small step size gt. 2. v(n) are independent from sampletosample. 3. Input signals and noise are i.i.d., stationary and ergodic. 4. The elements of w(n) form a jointly Gaussian process. The resulting mean of the output SNR is: E[SNRo(n), ] = SNRi, I + SNRi uC (3.2) IIW,1I2 .wl3 where the first term on the righthand side is SNRom, the steadystate output SNR defined earlier in equation (1.10). The matrix Cw = cov(v(n)) is the steadystate weight covariance matrix given by Senne.9 The matrix is not a function of time, n, since v(n) components are i.i.d. The last term on the righthand side will cause the average output SNR for user m to increase as the adaptive weight covariance increases. This might have little effect for weak users; the term is not negligible for strong users. The average output SNRs may no longer be leveled if the term becomes too large. This point is clarified in Appendix A which presents some general characteristics of the SNR performance surface. The first measure of performance is the difference between the average and steady state output SNRs divided by the steadystate output SNR: E[SNRo(n)]SNRo. SNRi, uT Cu: SC (3.3) SNRo. SNRo, w0 112 Where SNRo, is the steadystate output SNR for user m and is defined by the first term on the right side of equation (3.2) above. The quantity CI o 1/(NSNRi2) << 1 is a constant which serves as an upper bound which will limit the excursions of the timedependent SNR (resulting from the adaptive process) from the steadystate solution. The constant will be defined in more detail later in the analysis. If the adaptive process is allowed to stray too far from the steadystate solution the output SNRs may no longer be held to nearly the same level and near/far limited performance may result. Using this relationship with equation (3.2) above gives an expression in which step size dependence is expressed indirectly through C,: T "'2 SNRo(3 u:C,,u' cllw SNR (3.4) SNRi, The quantity on the lefthand side may also be lowerbounded by using the maximin theoreom22: 1min mirN(qC q" in (a,) = q e (3.5) q#0 ( q where CN denotes the complex Ndimensional space, q is an Nlength vector and Xw = min{ all eigenvalues of C,}. Let q of equation (3.5) equal Um of equation (3.4) where UmTUm* = N. Equation (3.5) may then be used to form a lower bound for equation (3.4). The lefthand term of equation (3.4) would then be bounded by: min u 2C .112 SNRo. SNRi, (3.6) The next task is to find a useful expression for the lower bound of equation (3.6). Senne95 has shown that the weight covariance matrix is diagonalized by the eigenvector matrix of R, the input autocorrelation matrix. For an N = 3 array, the diagonal of A, the eigenvalue matrix of Cw, is: diagIA, 1 (1pA ^1 I (3.7) 2 (1P r 21 Ip 2 1A 2 1pA2, 2 1pA3J where A1, A2, and A3 are the eigenvalues of R, and p. is the LMS step size. Compton'00 has shown that when SNRil >> SNRi2 > 1, the signal and noise eigenvalues of an NbyN autocorrelation matrix R are approximated by: N' 2 V 1 (3.8) 2 1+( SNRi( +uNu;21) 1, = a m=3...N where the largest eigenvalue A, is established by the strongest incident signal and A, are the noiseonly eigenvalues. If the step size is small (.u < 1/1A)the smallest A, from equation (3.7) may be approximated by: S) Iw ( 1 2 _2 IpA t 2 1pA, 2 1p J, (3.9) 2 The simplification in equation (3.8) results directly from our assumption of small step size. If p. is small the term in brackets and the term in parentheses outside of the brackets are equivalent to unity. This establishes the upper and lower bounds on uC,,u : 1NpE2 < u C,,u < CJwo2 SN (3.10) 2 SNRi, If all the incident signals are wellseparated, strong and perfectly correlated to the reference waveforms the minimum MSE should be wellapproximated by the output noise power: e'. a 1w 11,2. If user m = 1 is the strongest user then SNRil can be determined in terms of the eigenvalues via equation (3.8) (N SNRi = ( x/~ ) 1 = I /a2 ). The constant C, from equation (3.3) establishes a bound on the average excursions from the steadystate output SNR due to misadjustment, so it must be made suitably small. The strongest user user 1 should have a steadystate output SNR leveled to approximately the same value as user 2, therefore SNRol = SNRo2 = NSNRi2. Since even SNRi2 >> 1 then C, = p /(2N*SNRi2) should give acceptable results where go is a constant (0 < pX < 1) and represents the unormalized step size This results in an upper bound on ut: U (3.11) which is similar to the upper bound on the LMS step size derived from stability arguments,22,93 and is consistent with our earlier assumption of a small step size. VarianceBased Performance Measure The second cost function is the variance of the adaptive output SNR divided by the squared mean of the adaptive output SNR. Choosing gt to keep this measure small will limit excursions from the mean of the adaptive process. The second moment of output SNR is: s2] +1(n))T U. 42 E[SNRo(n),] = E(w + v())U m (3.12) Senne assumed that the filter weights, w(n) = w. + v(n), form a jointly Gaussian process. In the following analysis this assumption is used to expand fourthorder joint moments in terms of the secondorder moments via the Gaussian moment factoring theorem.101 If we assume the vector v(n) is multivariate and complex Gaussian we can express the fourth order moments as functions of the secondorder moments.'0' Expanding the bracketed vector term and discarding terms that would give oddordered moments of v(n) results in several terms: EI(w + vf)u = w u,. 4 + 4. lw UI u.C,,u, + E[(w'Um )2(v~u +)2 (Wu )2(TU)2] (3.13) N N N N + Y Ju, u;u,,uq;I E[vv,v;,v,* i] 11,=1, =1 1=114 =1 where we have dropped the timedependent notation from the weights. The third term on the right is zero. The fourthorder moment may be expanded in terms of the secondorder moments: E[v, vvv, v ] = E[v v, ]Ev v +E[v vv ]E[v v, ] (3.14) The summation term becomes: XX auI,,u,u,; E[v v V, V ]= 2 (uTC,,u,)2 (3.15) I=i 4,=4l = 1 The second moment of the output SNR is: 2 (IWOTU.2 C+uC,, 2u lw.U.14 E[SNRo(n)2 ]= SNRi' 2(w (3.16) ilwoi1r and the variance of the output SNR is (from equation (3.2)): 2 i12 I +u(CMr*, V"2w 4u aSNR() = SNRi I 14 (3.17) = E[SNRo,(n)]2 SNRo. where the second line of equation (3.17) follows from (3.2) and SNRo. is the steadystate output SNR of the m'th signal. The performance measure becomes: 2T SNRo02 SNR=o1 < < C2 (3.18) E[SNRo(n), ] E SNRo(n), where C2oc 1/(NSNRi2) << 1 is a constant to be defined later. Substituting quantities from equation (3.2) into equation (3.18) results in the inequality: S .SRo[ 1I uCu: Vu 5 Iw1 2SNRo C 1 (3.19) SNRi. C2 Since C2 < 1 the radical term may be approximated by a twoterm series 1 (1 C2)2 = 1+ C2/2 and the inequality takes the same form as equation (3.4). The same procedure used to determine the step size upper bound for the first performance measure may be followed for this second performance measure with the same results: pt /,/At1. Simulation Results This section will use simulations to corroborate the steadystate analytical results of previous sections. The next two plots show curves of the quantities obtained from equations (3.2) and (3.18). Figure 3.1 shows the average output SNRs versus the step size coefficient for means resulting from simulation and from expression (3.2). The input scenario is SNRi,= 10, SNRi2 = 106, DOAi = 0 deg., DOA2 = 90 deg., N = 3. The initial weight for the LMS simulation was MMSE weight vector from equation (1.8), w, = R1p. Plots of the variancebased performance measure are shown in Figure 3.2. The input scenario is identical to the one described in Figure 3.1.The curves show the ratio of variance divided by the squared mean of the output SNR obtained through analysis Average output SNR vs. stepsize coefficient SNRi2 = 60 dB DOA1 = 0 deg DOA2 = 90 deg CL User2: sim CO S30 User 2: calc User 1: sim. calc 25  20 ..... ... .. ... 10" 10" 10 100 Stepsize coefficient Figure 3.1 Output SNR versus stepsize coefficient go. The step size is normalized by the input power (equation (3.18)) and simulation. The curves show, as we might expect, that the performance will be worst for the strongest user. Selecting step size for acceptable strong user performance will lowerbound the performance for the remaining weaker users. 2nd moment performance measure vs. step size coefficient Figure 3.2 Variancebased estimator versus stepsize coefficient Po. The step size is normalized by the input power. The input scenario is identical to the one described in Figure 3.1 Selecting a small performance measure will limit the pointbypoint excursions of the output SNR from the steadystate value and will give results which are acceptable in the mean. If 1% is selected as an acceptable value of the performance measure then from the curves for user 2 above it would appear that p 0. 1/Tr(R) is the maximum allowable step size. This is in agreement with existing rulesofthumb for selecting step size for stability and convergence. For these conditions, it would appear that upper bounds on LMS step size derived from convergence arguments will give acceptable performance of the multiuser processor. 10 q101 4 S10 10 110 10", 4D 10 10 10 10" Step size coefficient CHAPTER 4 BASE STATION RECEIVER PERFORMANCE USING A SMART ANTENNA ARRAY IN A DSCDMA SYSTEM WITH IMPERFECT POWER CONTROL This chapter examines the steadystate outage probability and outagebased capacity of a single cell containing multiple directional signal sources transmitting to a central base station. The only fluctuation in the signal levels is due to lognormallydistributed power control error in the multiple transmitters. The central receiver consists of an array of N isotropic sensors, K minimum meansquared error singleuser beamforming processors and a bank of conventional detectors. The performance measures are outage probability and outagebased capacity. The goal is to find simple expressions which relate the outage based capacity to the antenna array parameters, the number of active signal sources and the degree of power control error. The analytical results are expressed in terms of the number of users per array element which may be supported for a given outage probability. Analytical results are found to agree closely with those obtained from Monte Carlo simulations. This work is unique in several respects. First, most previous work examined uplink performance for a traditional singlechannel receiver. Second, the authors which considered a base station antenna and imperfect power control presented simulation results for the case of closedloop power control.4'85 As was mentioned in Chapter 1, power control error will be modeled by incident power levels which are lognormally distributed. This model has gained some acceptance for openloop powercontrolled systems. Its suitability as a model for closedloop powercontrolled systems remains an open issue. The remainder of the chapter is divided into several sections. A qualitative description of the array and the receiver is given in the next section. The third section outlines the development of the analytical model and the derivation of the expressions for uplink capacity. The fourth section describes the Monte Carlo simulations and compares the analytical and the simulated results. The last section presents the conclusions. System Description The incident DSCDMA signals are independent and originate from independent transmitters which are arbitrarily placed about the base station receiving antenna. The K active transmitters (located in a single cell) result in incident signals with independent directionsofarrival (DOA) uniformly distributed over the interval [0, 27). The transmitters' output levels are continuously adjustable and have an infinite dynamic range. The output power adjusted by an unspecified power control algorithm results in output power which is independent between sources and which is also lognormally distributed. The modulation format is BPSK. The base station receiving array consists of N ideal isotropic sensors arranged in a circle. Adjacent sensors are separated by a distance with the electrical equivalent of one half wavelength. The sensor array feeds K separate banks of N complex weights controlled by K MMSE beamforming processors. The optimum steadystate weight vector for each beamformer is given by equation (1.8), wo = R'1p. Each of the K beamformer outputs feeds a distinct conventional DSSS detector which in turn provides an estimate of the demodulated output The receiver is assumed to be capable of perfect carrier tracking of the desired signal. Unlike the multipleuser array of the previous chapters, the array/detector combination is devoted to detection of a single desired signal amidst multiaccess interference. A diagram is shown in Figure 4.1. Figure 4.1 A bank of baseband singleuser beamformers and conventional detectors sharing an array of sensors Analysis The stationary, complex baseband output of the sensor array is found by combining equations (1.3) and (1.4): K Y(t) = AI1O" 20 c.(t T.)b(t T,)u, + n(t) (4.1) where A is the peak amplitude and em is N(O, pc2). The sensor outputs are weighted and summed by the beamformer and then processed by the conventional detector which forms an estimate of the current output bit. The signal from source 1 is considered the desired signal. The remaining Kl signals are considered multiaccess interference. The array tends to steer a pattern lobe towards the desired signal and also tends to steer nulls of finite depth towards the N2 strongest interferers when K 2 N1. Because the strongest signals are attenuated, no single signal dominates the output statistics and the output may be wellapproximated as Gaussian. An additional assumption of long codes (codes which span more than one data symbol) allows the PN sequences to be modeled as random codes. The random code model of Pursley'02 will be used here. More refined models exist, but the additional complexity they introduce tends to improve accuracy when only a few users are present. 103,104.05 From Pickholtz et al.' 6 we know that the effective bit energy to noise spectral density ratio may be approximated by (Eb/No)eff= NcSINR. The random code approximation may be modified to account for varying input amplitudes among the incident signals due to their power control error. Using one of the intermediate steps (equation (14) from Pursley'M) allows a convenient expression for (Eb/No)f while retaining the cross correlative properties of the PN codes: E = NcSINR = 1 NcSNRo, (4.2) 1 ./,^ .___ ^  (4.2) 3N2_ SNiRo. r.i + where SNRo. is the array output SNR of the mth incident signal and r,j is a crosscorrelation parameter between the first and mth PN codes. If SNRo, is replaced by its sample mean and the remaining sum of ri terms is replaced by the random code approximation (equation (16) from Pursley102) the expression becomes: (E = NSNRo, (43) SNo X SNRo, +1l 3 m=2 The quantity SNRok may be rewritten as: SNRo, = 10e1"0 Gp, SNRi (4.4) where Gp. = Iwu, 2AwoE is the normalized power gain of the array towards the mth signal and SNRi = A 2/c2. The power gain is a complicated function of the input scenario and the array geometry. In order to simplify the analysis we will use a simplifying assumption that the array power gain with respect to the first user, Gpl = Gpd, will be approximately equal to its upper bound N, the number of array elements. The MMSE array gain towards the interferers (Gpm,m = 2 ... K) is difficult to characterize. Intuitively, we might expect that the avg{Gpm } = Gpi might be wellapproximated by the average (over DOA) normalized power gain (Gpg) of a classical beamformer (wo = ul) with no adaptive nullsteering capability. The value of Gpg ranges from unity for a single element to 1.6 for a 30 element circular array. Simulated results presented shortly will show that these assumptions with regard to the individual quantities (Gpd, Gpi) are not always valid over the conditions of interest. The average of the ratio GpdGpi ,however, does provide a reasonably good fit to the simulated results when the above assumptions (Gpd N, Gpi N) are used. If there are many interfering signals the sum in the denominator of equation (4.3) may be approximated by averages: (E, = NNSNRilOx(o =1,o, (4.5) N\ 2 ~ 1. v (K1)SNRiGpIO102W +1 Further explanation is required. The sum in the denominator of equation (4.3) is almost a sample mean and may be approximated by an ensemble average. Assume the power gain and the exponential of equation (4.4) are independent: the expectations apply separately. This results in the base10 exponential term, the (K1) term and Gpi in the denominator of (4.5). This approximation causes a slight overestimation of interference: by breaking a sum of products into a product of sums we have invoked Schwartz's inequality twice in succession to get from (4.2) to (4.3) and to get from (4.3) to (4.5). Our simplifications have equated sample means and ensemble averages and also ignored the complicated nature of the array response by treating the power gain as an averaged quantity. Simulations of the power gain show our assumptions regarding Gp may give acceptable results. Figure 4.2 shows the results of Monte Carlo simulations of averaged values of (MMSE) Gp for a desired signal (Gpd), an interferer (Gpi) and their quotient for a varying number of users, 3 dB of power control error and a 30element array. The curves show that as the number of users exceeds the degreesoffreedom of the array our simulated values for the individual gains are not very close to approximated values of Gpd  N = 30 and Gpi, = Gpav, = 1.6 but their quotient Gpd/Gp, = N = 30. So, for this circumstance it might be useful to approximate the gains in equation (4.5) as Gpd = N and Gpi, = 1. Why will this work? As the number of users increases and the system becomes interferencelimited the "1" term in the denominator of equation (4.5) becomes negligible leaving a good approximation of the quotient of the gains as Gpd/Gpi N. When there are few users the multiaccess interference is negligible, Gpd = N and the approximation will still hold. These gain approximations do not hold separately to predict the desired signal output power or the interference output power, but their combination might prove useful in calculating outage. What about more extreme scenarios? Figure 4.3 shows the gains and the gain ratio for a power control error of 10 dB. The individual approximations Gpd N and Gpi, Gp, = 1.6 are even worse than before. However, the simulated gain ratio Gpd/Gpi N/1.3 for K > N. This is a little closer to the ratio of the individual gains Gpd IGpag. = N/1.6. This, GpGp=N 30 Cpd 10' Gain idf Gp, power control error = 3 dB N =30 101  , , 0 20 40 60 80 100 120 Number of Incident Signals Figure 4.2 Gain versus the number of incident signals. Power control error is 3dB. .................... ................... ..................... Gpd/ Gi 101 Gpd Gain 100 Gpi power control error = 10 dB N=30 10"1 ,,, 0 20 40 60 80 100 120 Number of Active Users Figure 4.3 Gain versus the number of incident signals. Power control error is 10 dB. and other simulations, indicate that choosing Gpd = N and Gpi = Gpa, = 1.3 for this model is a good approximation for outage calculations over the ranges of power control error examined here (0 5 O < 10). Note that Gpi = Gpav = 1.3 also arises from a sample mean (over N) of the average power gain (over DOA) for N = 1,2,4,8,15 and 30 element arrays for a classical beamformer. Approximating the interference as an average quantity in equation (4.5) eliminates complex scenariobyscenario interactions between the desired signal and the interference in the analytical model. It also leaves the lognormally distributed desired signal as the only random variable in the model. The resulting distribution of the (Eb/No),y in equation (4.5) is therefore lognormal and results in simple expressions for the outage probability and the capacity. Outage occurs when the (Eb/No)e, is less than some threshold. Converting equation (4.5) to dB and noting that ej is N(O,&p,.) results in a simple expression for the outage probability: Pr., = Pr((< Eb ZdB d (4.6) where Q(x) is the complementary Gaussian CDF, a is the desired threshold in dB and: N " ZdB = 10 logo (4.7) The quantity D is the denominator of equation (4.5). The quantity E/No (Ti/c)SNRi =NcSNRi represents the equivalent bit energy to noise density ratio for a single incident signal and a single array element. Note that Naguib et al.78 has also developed a Q function upper bound for outage probability for the case of perfect power control. For the case of imperfect closedloop power control Naguib et al.81 presented the simulated means and variances of (EI/No),.f Some simple algebraic manipulations result in the average capacity as a function of desired outage probability: K = N 1 (E +1 where: (4.8) 3,1 : (I10a 10 C= 23M 10(.) 2Gp, where for high EANo the capacity is approximately linear in processing gain, Nc, or the number of array elements N but decays exponentially as the power control error increases. Note also that the threshold is no longer in dB. Note that if we interpret equation (4.8) as being linear in N, we can define a perelement capacity by noting the slope of the line. Note also that the capacity asymptotically approaches a finite maximum as Eb/No increases. A similar effect was noted by Naguib and Paulraj82 for the case of a 2D RAKE combiner at the base station receiver and perfect power control in the mobiles. They examined the capacity as the equal incident signal levels went to infinity and named the parameter asymptotic capacity. A simple expression for uplink capacity was also formulated by Suard et. al.76 for a postdetection combiner. The model for power control error was restricted to a single user with an incident power level 10 dB higher than the other users. Simulations and Results MonteCarlo simulations were used to corroborate the analytical results given by equation (4.6). Autocorrelation matrices for the desired signal, interferers and noise were generated by ensemble averages as in equation (1.6). Signal DOAs were uniformly distributed over [0, 27). The processing gain was 127 and Eb/No = 7 for a single antenna element and no power control error. For a single trial the MMSE weights were calculated via equation (1.8) and the desired signal, interference and noise power out of the array were then used to calculate (Eb/No),f via equation (4.3). An outage condition was judged to exist for that trial if (Eb/No)ef < 7 dB (i.e. from equation (4.6), '5 = 7 dB). The quantities were averaged over 20,000 trials for each combination of power control error, number of users and number of array elements. Curves of the outage probability Pr((Eb/No)4ff < 7 dB) versus the number of users are shown in Figure 4.4. The figure contains curves from equation (4.6) as well as the Monte Carlo simulations. Power control error is 4 dB. Note that for 20,000 trials, curves in Fig. 4.4 might be inaccurate for outage less than 102. Figure 4.5 shows the total array/receiver capacity versus the number of array elements with outage probability as a parameter. The curves are formed by plotting constant contours of a threedimensional surface formed by Pr((Eb/No)ff < 7 dB) as it varies over K and N. The solid curves show simulated results; dashed lines show the constantvalue contours of equation (4.6) for power control error equal to 4 dB. The curves of Figure 4.5 show the user capacity of the array/detector is roughly linear in N. We may therefore use as a performance measure perelement capacity, the number of users per array element which may be supported for the given values of outage probability and power control error. As noted earlier the analytical expression for the per element capacity may be obtained by noting the slope of the total capacity line in equation (4.8) with N as the independent variable. A point of note: the average power gain (Gpg) in equation (4.8) has a slight dependence on N: it is equal to unity for a single element and is equal to 1.6 for a 30element array. For the sake of simplicity, this slight dependence is ignored and a mean value of Gpa, = 1.3 is assumed, which was noted earlier when comparing simulated Gpd Gpi curves in figures 4.2 and 4.3. The perelement capacity versus the power control error for Pr((Eb/No)ff < 7 dB) = 0.02 is shown in the upper plot of Figure 4.6. For simulated data, the normalized capacity was determined by extracting the approximate slopes of the capacity curves (as shown in Figure 4.4) via a linear leastsquares curve fit The analytical results were calculated from Figure 4.4 Outage probability versus the number of incident signals. Power control error is 4 dB. The number of array elements N is a parameter 110 ....... ..................... .. .. .. ...... 100 ........... .......... ..... ...... ... ........ 90 .. ... ... .. ""  . .... .......... Sr(Z< dB)= 0.07 / ....... 80  *.. .. z. .o.o Capacity .... .'. .'""'Pr(Z < 7dB)= 0.03 0 ... ........... Equati.......on .......... 140 '. "..  i..*. .   .. ........... 50 ........ /7 _.,. .......... .......... ........... .'//" Simulation 20  .. V...... .......... .... ....... ...................... 20 .... Equation (8) 10 .. / ......... ....... .............. ......... ........... 5 10 15 20 25 30 Number of Array Elements Figure 4.5 Capacity versus the number of array elements with outage probability as a parameter. The power control error is 3 dB. 10' Outage Prob. 120 140 0 20 40 60 80 100 Number of Incident Signals 64 Per 30 Element Simulations Capacity  Analysis, eq. (4.8) 20 ....................* ..........  0   0 2 4 6 8 10 Intercept 20 ......... ..... .......... ........ ...... 10   10 ...... a 0  Aniysis, eq. (4.8) 0 0 2 4 6 8 10 (a) Power Control Error (dB) lower plot of Igure 4.6 shows the intercept point of the line in equation (4.8). Obviously 10     '.  S the analytical and simulated results are not in as close agreement as the upper plot(4.8) Note 0 2 4 6 8 10 (1) Power Control Error (dB) Figure 4.6 Normalized array capacity versus power control error in dB. the slope of the system capacity line given in equation (4.8). As the curves show, the analytical and simulated results for the perelement capacity are in close agreement. The lower plot of Figure 4.6 shows the intercept point of the line in equation (4.8). Obviously the analytical and simulated results are not in as close agreement as the upper plot. Note that to determine the overall system capacity given an array of N elements it would be necessary to know the perelement capacity as well as the intercept. Conclusions In this chapter we attempted to develop accurate, simple analytical expressions for the outagebased uplink capacity in an idealized singlecell DSCDMA system with multiple, possibly near/far, signals incident on a basestation receiving array. Power control error was assumed to be lognormally distributed. Some simplifying assumptions regarding the interference and the array response allowed an approximate expression for the uplink capacity that was compared with results from Monte Carlo simulations. The approximate expression given in equation (4.8) shows that a roughly linear relationship exists between the capacity and the number of array elements. The overall capacity K consists of two components. The first component is the slope of the line in N and has been defined as the perelement capacity, the number of users per array element which may be supported for a given level of outage probability and power control error. Equation (4.8) indicates that the perelement capacity is not dependent on the nominal input level (i.e. the input level without power control error: SNRi) but decreases exponentially with increasing power control error. For the levels of power control error examined here the term which dominates the exponential rolloff is opeQ'l(Prouage)/10 of equation (4.8). The decrease in the perelement capacity is therefore dependent on the outage requirements and the standard deviation in dB of the power control error. Since the perelement capacity is insensitive to the nominal input levels and the array size, but is keenly dependent on the degree of power control error and the outage, it might serve as a useful asymptotic measure of the performance for this array/receiver. The second component of equation (4.8) is the intercept term equal to 1C(Eb/No)' where C is defined in (4.8). The intercept is inversely proportional to the nominal input levels. As the input levels decrease this term becomes larger, diminishing the overall capacity. In a plot of capacity (K) versus array elements (N), the capacity line moves away from the origin along the horizontal Naxis as the nominal input levels decrease. The intercept is weakly dependent on the power control error (via C) and also on the number of array elements (via Gpa,,) and is not dependent on the outage probability. Analytical and simulated results do not agree as closely as those for perelement capacity. The agreement or lack of it between the analytical and simulated results for per element capacity and the intercept can be interpreted in terms of outage probability curves shown in Figure (4.4). The analytical model can predict well the horizontal spacing between the continuous outage probability curves. The per element capacity predicts the incremental increase in capacity with increasing array elements and is a measure of the horizontal displacements of the outage curves relative to oneanother. The less reliable prediction made by the model is the horizontal placement of the outage curves relative to a point on the horizontal axis. This enters into the model via the intercept parameter described above. The simple model presented here was based on some simplifying assumptions regarding the array response towards the incident signals. In spite of this, the analytical model accurately predicts some aspects of the array/receiver performance when directional signals are employed with lognormally distributed power control error. The directional signals originated in a single cell and were not subjected to any kind of environmentally 67 induced fading. The system performance in the presence of fading and interference resulting from outer cells is the topic of the next chapter. CHAPTER 5 BASE STATION RECEIVER PERFORMANCE WITH A SMART ANTENNA ARRAY IN THE PRESENCE OF MULTIPATH FADING AND SHADOW FADING This chapter examines base station receiver performance when incident signals are subjected to frequencynonselective Rayleigh multipath fading and lognormally distributed shadow fading. The effects of multiaccess interference from outer cells subjected to shadow fading will also be included in the incident signal model. The power control error model will continue to be described by a lognormallydistributed random variable and the receiver still consists of an array of N ideal isotropic sensors, K minimum mean squared error singleuser array processors and a bank of conventional detectors (see Figure 4.1). The goal of this chapter is to determine the performance dependence on the number of active signal sources, the number of array elements and the degree of power control error. As before, the performance will be expressed by outage probability and perelement capacity. Unfortunately the introduction of fading and outer cell interference further complicates the development of simple analytical models. In spite of these complications the simulated results closely follow some general trends established by the model in Chapter 4. In particular, curves of outagebased capacity continue to be linear in N with slopes that decrease exponentially with increasing power control error. Previous authors have investigated the performance of optimum combining from the standpoint of interference rejection. Winters23'44 and other authors60o61 have examined contributions to TDMA system performance while Naguib et al.7684 have studied CDMA systems. Unlike previous work, this research examines the outagebased capacity for the case of imperfect power control and attempts to provide some analytical models which would allow easy assessment of performance. The remainder of this chapter is divided into six sections. The first section will present the receiver model for singlecell signals subjected to Rayleigh fading. The second section will extend the model to the multicell case. The third section will quickly revisit the model of chapter four and introduce a second analytical technique. The fourth section will give a brief description of the simulation parameters and compare the simulated results with the analytical results.. The fifth section entitled Conclusions and Discussion will review the results in some detail. The last section gives a summary. A Single Cell with Signals Subjected to Rayleigh Fading Multipath fading arises when propagating electromagnetic waves originating from a single signal source arrive at a receiver via different propagation paths. The individual paths may include lineofsight propagation as well as paths resulting from reflection off of one or more surfaces. The individual waves as well as their sum are highly dependent upon the frequency of the propagating waves, their path lengths as well as the reflective properties and geometric arrangement of the encountered surfaces. This research will exploit the assumption that multipath fading of a signal from a single source results from the sum of many reflected waves. The individual waves have roughly equal power as well as independent amplitudes and independent phases. This allows the fading component of the incident signal to be modeled as a complex Gaussian random variable with uniformly distributed phase and an envelope which is Rayleigh distributed.107 If the channel is wellapproximated by a constant frequency response characteristic then relative time delays between arriving wave fronts are negligible. This is known asfrequencynonselective Rayleigh fading or flat Rayleigh fading. The flatfading condition is probably an accurate approximation for indoor communication systems with large path losses and mobile systems with scatterers located in close proximity to the mobile. It represents a worstcase condition from the standpoint of detection since it will not allow the use of a RAKE receiver62 to provide resolution of individual timedelayed paths. The complex baseband output of a sensor array outwardly resembles the expression given in equation (4.1): y(t) = AClOe'20. c(t zm)bm(t r)u, + n() (5.1) m=1 where the elements of the phase shift vector u. are complex Gaussian random variables which result from Rayleigh fading. This is in contrast to equations (1.4) and (4.1) in which the components of um are complex exponentials resulting from directional, unfaded signals. The components of Um may have any permissible degree of correlation. The study of spatial diversity combiners is dedicated to antenna array structures which force the correlation of fading components between array elements to be low (ideally zero).'08 A base station diversity array must have larger element spacings than the customary half wavelength spacings used in a mobile radio receiver.28 Lee conducted an empirical study of fading correlations in a twoelement base station array. He concluded that, for low correlation, the interelement spacings must be 15X 20X if the signal arrives from broadside and 70k 80X if the signal arrives along the axial direction. Salz and Winters'9 examined a linear array and developed closedform expressions for the direction dependent fading correlations between array elements when multipath rays are "dense" throughout a range of DOAs. Raleigh et al."0 proposed an analytical model which describes the spatiallydependent correlations of the fading process. Verification of the latter two models through experiment remains an open issue, as does more refined spatiallydependent channel models. Naguib and Paulraj3 examined the effects of the Salz and Winters fading model on a basestation diversity array in an IS95 system using closedloop power control. Because spatial channel models remain an open issue this research will exploit the assumption that the fading process is independent between antenna elements and the elements of u= in equation (5.1) are complex i.i.d. N(0,1). Spatial dependence of the incident signals via the interelement phaseshifts no longer exists and the array geometry is critical only in that it results in independent fading between elements. Winters23 has shown that even when directional information is not used by the processor (i.e. the array functions as a diversity combiner) an optimum combiner will outperform, in steady state, a maximal ratio combiner because it is able to adaptively attenuate cochannel interference. Multiple Cells with Signals Subjected to Rayleigh Fading and Shadow Fading The models and results from the last subsection will be extended here to include the effects of outercell interference. Like the previous section of this chapter the incident signals will be subjected to flat Rayleigh fading. Unlike the previous section, however, the cell which contains the desired user will be surrounded by several layers of cells containing sources of multiaccess interference (i.e. other DS CDMA users). The interference will be subjected to multipath fading and shadow fading. Shadow fading occurs when structures (such as buildings, hills or mountains) attenuate propagating signals. Shadow fading varies more slowly than the multipath fading component of the signal and is interpreted as the timevarying mean of the rapid, multipathinduced signal fluctuations.2 The incident signal model may be modified slightly: K y(t) = A XOe'20 c(t Z)b,(t rz)u, + m=l KNO 10'.20 (5.2) Y A,,10''20 o c,(t.)b,(t )u. +n(t) n1 rY,0 ) where the first summation is for centercell and the second summation (with index n) results from the outercell interference. All users have lognormally distributed power control error where em is N(0, pce2). The quantity No is the number of outer cells while K remains the number of users/cell. The variable s, is N(0,64) which in turn specifies lognormally distributed shadow fading with 8 dB standard deviation. The quantity rn, is the distancedependent, fourthorder propagation loss between the n'th interferer and the center cell base station. The phaseshift vectors u. and u, are, as in the previous subsection of this chapter, composed of i.i.d. random variables which are N(0,1). Note that the incident amplitude An is an indexed quantity unlike the first summation representing the centercell users. This is because the outercell amplitudes are determined by a hand off to the outercell base station with the least path loss. This will be discussed in more detail shortly. The quantity n(t) is a vector of complex AWGN with power o2. Figure 5.1 Spatial region for simulation of outercell interference. Figure 5.6 below shows a diagram of the arrangement of the hexagonal cells, each with a unit radius ( cell area = 3 13/2). The region of interest is circular and contains the equivalent area of N, = 21.67 cells, excluding the center cell. The small circle at the center of each cell designates the position of each cell's base station. The wedgeshaped region within the larger circle contains the equivalent area of 3.61 cells and it is over this region that the spatial distribution of outer cell interferers is uniform, excluding the portion containing the center cell. The interference from this wedgeshaped region will be determined for each scenario and the results replicated to generate the interference components for the remainder of the circular region. The path loss which occurs between an outercell user and an outercell base station contains a deterministic propagation loss (o r4) as well as lognormally distributed shadowing. The random components complicate the handoff or membership of a user to a cell: a user is not necessarily serviced by the closest base station. Viterbi et al."' investigated the properties of outer cell interference for the case of perfect power control and lognormal shadowing while Lee et al.12 investigated the effects of imperfect power control and lognormal shadowing. These previous works examined the case of up to 4 base stations involved in the handoff. Increasing the number of base stations in the hand off beyond Ns = 4 will only slightly decrease the outer cell interference for a given degree of shadowing and power control error. The handoff was computed by selecting the base station path with the minimum basestationtomobile path loss. For the mth outercell user, this results minimizing a convenient ratio of path losses between the outer cell mobile and the outer cell base station and the outer cell mobile and the centercell base station: min r4 10 10 L n.= N[ ~ where: m = ...K N (5.3) n = l... N, 2  r. 10 10 10 10 where m represents the index over the outercell users in the wedgeshaped region and n is the index over the NB = 11 base stations denoted by the solid black circles in Figure 5.6 above. The quantities r, and 10'~.0 are the propagation loss and lognormal shadowing respectively between the mth mobile and the nth base station. The quantities r4 10'.po and are the propagation loss and shadowing between the mth mobile and the center cell at the origin. The quantity 10'/I0 is the power control error of the mth user with respect to the base station chosen for handoff. As the figure shows, base stations just outside of the wedgeshaped region are utilized in the handoff calculations for the outer cell interferers. The shadowing and power control error components are assumed to be independent from pathtopath. Some authors"3'"4 maintain that spatial correlations exist in shadowing components, but those effects are not incorporated into this model. The incident levels (A.'s of equation (5.3)) resulting from handoffs in the wedgeshaped region of Figure 5.6 were replicated to generate interference levels for the remaining portions of the large circular region. The model used in this research agrees precisely with the results of Viterbi et al."1 which reported that for NB = 3 the ratio of the outer cell to inner cell interference is 0.57. It was assumed that power control error correlates perfectly between handoff base stations and does not enter into interference calculations until handoff is chosen based on minimum path loss. This is in contrast to the work of Lee et al."2 which assumed closed loop power control error was uncorrelated between handoff base stations. The authors minimized the quantity: min r4 0 10 .10 10 4 = I .= ...N . where: m = ...K N, (5.4) r4, 10 10 where 10 10 is the power control error between the mth user and the nth handoff base station. They assumed that NB = 3; handoff occurred to one of the three closest base stations. Analysis This section will briefly revisit the analytical results of chapter four which gave an asymptotic expression for capacity when signals were directional. Another analytical model will then be introduced which exploits the assumption of lognormal interferers. The expression given in equation (4.3) for the (Eb/No),g may be rewritten slightly using equation (4.4) for a single cell: (Eb N SNRiGpO 100(5.5) N ef SNRi Gp. 0le+1( m m where SNRi, = (AZ/o ).10'/10 is the input SNR and Gp, = Iw u. 12llw.2 is the normalized array gain towards the mth signal. In chapter four some assumptions (via equation (4.4)) allowed the sum term in the denominator of the expression for (Eb/No),ff to simplify into products of average terms, some of which are not explicit functions of the input parameters (i.e. avg(Gpm ) = Gpi, m 1, the interference signal gain). In addition, the gain towards the desired user Gpj = Gpd was modeled by its upper bound N, the number of array elements. While these approximations were not accurate when considered separately, their quotient resulted in a convenient form and gave acceptable results ( see Figures 4.2 and 4.3). The perelement capacity was the slope of equation (4.8): ( 3NNc In(10)Q'(Pro,) In210 ) KE = exp a (5.6) 2 Gp, 10 200 where the term containing the Qfunction is dominant for the range of power control error considered here. The term analagous to a singlepole rolloff resulted from modeling the multiaccess interference as an averaged quantity. The averaging operation left the desired signal as the only random variable and resulted in a lognormal distribution of (Eb/No)rf in which the standard deviation (in dB) is determined solely by the power control error standard deviation o, (see equations (4.5), (4.6)). Under this simplified analytical model, the number of users K and the number of array elements N affect the mean of (Eb/No)4 but not the standard deviation. This series of assumptions gave acceptable results for the asymptotic case of strong nominal input levels if the signals were directional and had a moderate degree of power control error. With the addition of multipath fading the results of chapter four are less precise. The rolloff of a diversity combiner's perelement capacity is slower for low values of power control error because the standard deviation of (Eb/No)rf is not determined solely by the power control error op. A different analysis technique which might approximate the standard deviation of (Eb/No)f more accuately has been examined by other authors. The model assumes that a sum of lognormally distributed random variables is also lognormal. Along this line, Schwartz and Yeh developed an iterative version of Wilkinson's method15 and then used their model to evaluate the outage probability of a multicell AMPS system.16 Beaulieu et al. examined several methods for approximating a sum of lognormal random variables and concluded that the "best" choice of model depends upon the system parameters (i.e. the degree of shadowing and the magnitude of the outage probability). This research will supplement the analysis of previous sections by approximating the multiaccess interference as a sum of lognormal r.v.'s. This method will result in a closed form expression for outage probability. An explicit equation for the perelement capacity will not be possible. If the array gain towards the interference may be modeled by a constant, the resulting noise and multiaccess interference is approximated by: K x. K Y. IuA =.110 + F 10, + 3 (5.7) m2 Gp, ,.I 2GpSNRi where the leftmost summation represents the innercell interference and the second summation represents the outercell interference. The rightmost term results from the presence of noise. The outercell interference has been modelled as a quantity normalized by the innercell interference: the outercell summation is over K rather than KNoc. This approach has been used for incident signals by previous authors"'11" and will be extended to the diversity array output via the gain constant F. Wilkinson's method begins with the assumption that the multiaccess interference is a lognormal r.v. The first two moments of the In are then matched to the first two moments of the sums of lognormal random variables. The mean and second moment of In are given in equation (5.8) below: El,] = E exp ln(10) = exp(3mz + 22 o/2) =(K1)exp(p82 C/2)+ F Kexp(2 /2)+ 3 = Gp; 2Gpi SNRi E[ E exp 21n(10)[j = exp(2fi +2p j) SE[(. exp(xe.)+ F exp(Iy)+ 3 )2] m2 Gpi X1 2Gp, SNRi (K 1)exp(2ga'2u) + (K lXK 2)exp(p/a2) + ( J (Kexp(2P2) + K(K 1)exp(I 2F)) E[I = + 2( )K(K 1)exp(2 ( + 2)/2)+ (2G NRi2 =b2 (5.8 3 + 3 (K l)exp(/f2 of/2) + 3F Kexp 2 /2) GpSNRi where the variable z is N(m,, o2) and f = ln(10)/10. Note that the exponential terms of the inner and outer cell interference are specified as N(O, 2) and N(0, 02) respectively. The logarithms of the moments allow linear solutions of m, and %o2 in terms of the moments of the sums (bi,b2): m = 21n(b,) ln(b,) (5.9) a = ln(b,)21n(b)) Once the interference moments are determined then (Eb/No)ff may be expressed as: Sx10log(eCxp(1))z E N 3 P 10 t (5.10) N, 2 Gp and the outage probability is given by equation (5.11) below where log(e) denotes the base10 logarithm. This method does not allow an explicit closedform expression for the perelement capacity. It will, however, allow that quantity to be extracted from outage probability contours as was done for the simulated results. The expression for the outage probability from the approximated moments: E Pr f where: (3 Gp 3 N 7 1= 10*log3 N, d 10*log 3 NN (5.11) (2 Gpi 2 Gpi = + 100log2(exp(1))a. Results Most of the details of the simulation procedure are outlined in the section Simulations and Results of Chapter 4 with one difference: the faded, incident signals were no longer directional and a DOA was not specified. Each signal was specified by its power control error (and shadowing for the outer cell interferers), and its phaseshift vector which consisted of complex, i.i.d. unitvariance Gaussian random variables. For each trial the (Eb/No),f was determined via equations (1.8), (1.10),(5.3) and (5.5) and was then compared to an outage threshold of 7 dB. A sample mean was tabulated over the trials for each combination of users and array elements. Even with fading the resulting outage contours were roughly linear in N with slopes which decrease with increasing power control error. Figure 5.2 shows Pr((Eb/No),f < 7 dB) versus the number of users with the power control error equal to 4 dB and with no outer cell interference. As in previous plots, the curves are plotted parametrically. The different curves represent outage probability for different numbers array elements N. For this value of power control error the performance is not drastically different than for the nonfaded case shown in Figure 4.4. The nominal input signal levels (without power control error) are such that Eb/No = 7 ( or SNRi = 7/Nc = 7/127 = 12.6 dB) for a single signal incident on a single array element feeding a 10 Outage Prob. 102 0 20 40 60 80 100 Number of Incident Signals 120 140 Figure 5.2 Outage probability versus the number of incident signals with power control equal to 4 dB. The number of array elements N is a parameter. conventional detector. When comparing Figures 4.4 and 5.2 it is easy to see the faded and unfaded systems have roughly the same outage performance. This similarity ceases for other values of power control error. Figure 5.3 shows simulated outage contours as the power control error varied. The top two plots show contours of Pr((Eb/No)ff < 7 dB)= 0.02 as the power control error varied from 0 to 7 dB with no outer cell interference. The upper plot (a) is for a nominal input condition (without power control error) of SNRi = 7/Nc = 12.6 dB while plot (b) shows contours for a nominal input condition of SNRi = 108/Nc = 59 dB. Plots (c) and (d) show the same conditions but include outer cell interference. The plots show that, even with fading, outagebased capacity continues to be linear in N. Note also that as the nominal input levels increase, the intercepts of the approximately linear contours move towards the origin. This general behavior was predicted for the nonfaded case by equation (4.8). The slopes of the lines in Figure 5.3 were extracted using a leastsquares curve fit and plotted as the perelement capacity in Figure 5.4. Note that the perelement capacity is less than that for the case of no Rayleigh fading as long as the power control error is less than approximately 1.5 dB. As long as the power control error is greater than 1.5 dB the receiver employing a diversity array combiner in the presence of fading outperforms a receiver using a beamformer to equalize signals which are not subjected to multipath fading. Capacity 100 80 60 40 20 10 15 Capacity 50 40 30 20 10 1 2 3 20 25 30 Number of Array Elements 5 6 7 Number of Array Elements Capacity 25 20 15 10 5 i 2dB .3dB :4rB SNRi=:12.6 dB . .....*.........5 .. .................................... ....... ....... .. ............ .pce.=i. dB 5 10 15 20 25 30 (c) Number of Array Elements Capacity SNRi =59 dB dB / 1 dB 2 dB 3 dB :pce = 0 d . 10 ....... ... ..... .. ... ...... . 5  .. ...:pce  1 2 3 4 5 6 7 (d) Number of Array Elements Figure 5.3 Capacity versus the number of array elements with power control error as a parameter. Outage probability = 0.02. Plots (a), (b) are without outer cell interference. Plots (c), (d) include outer cell interference. SNRi = 12.6 dB pee=0 :1 2dB: 3dB 4dB .. .. .. . . SNRi = 59dB ......... .    '  Note that in Figure 5.4 there is degradation in the perelement capacity as the nominal input levels increase and outer cell interference is present (plot (b)). When the nominal signal level is small (SNRi = 12.6 dB) the individual outer cell signals incident on the inner cell are very weak (compared to ambient noise and the inner cell interferers) due to the path loss. In this circumstance the outer cell interference probably just adds to the ambient AWGN. When the nominal input signals are large (SNRi = 59 dB), outer cell interferers can overcome the outercell to centercell path loss and can have signal levels much higher than the ambient noise, even at the center cell. In this case some percentage of individual outer cell signals compete with the inner cell interference for attention from the array processor, and performance suffers. Per Element ' Capacity Cap......... SNRi = 12.6 dB  SNRi = 59.6 dB 0  0 1 2 3 4 5 6 7 (a) Per 1 Element SNRi =12.6 dB Capacity 10 ::* *........ ... ...  ___ C!. An 0 1 2 3 4 5 6 7 S(b) Power Control Error (dB) Figure 5.4 Perelement capacity (users/array element) versus power control error (dB) for an outage probability = 0.02. The upper plot is for the case of no outer cell interference. The lower plot includes outer cell interference. Figure 5.5 below shows the perelement capacity from the simulated results of Figure 5.4b (with outer cell interference) and the approximation given in equation (5.2) with < = 7 dB and Gp, = 2. Agreement between (5.2) and the simulated results for the case no outer cell interference was poor, and the results are not presented here. The value of Gpi accounts for the fact that the output power of the outer cell interference is about equal to the output power of the inner cell interference when averaged over most of the possible combinations of user population, array elements and power control error. The rapid roll off of the curve from equation (5.2) is due to the use of averaged multiaccess interference when computing (Eb/No)', as discussed in a previous section. Figure 5.6 shows the perelement capacity resulting from the application of Wilkinson's method to model the multiaccess interference as a sum of lognormal variables (equations (5.7) through 5.(11)). The lognormal statistics of the inner cell interferers are assumed to be due to power control error only while the outer cell interference combines power control error and 8 dB of shadowing (x, of equation (5.7) is N(0, pc2)) while ym is N(0, oc,2 + 64)). Simulations show that the average interference gain Gpi for the inner cell interferers can vary from zero to unity depending on the number of elements, the number of users and the nominal input level. Interestingly enough, Gpi does not seem sensitive to the power control error. Averaging over these conditions results in Gp, = 0.7. The gain constant for the outer cell interferers F of equation (5.7)  is taken to be unity. Simulations have shown that the array attenuates the interference so that the average output power due to the inner and outer cell interferers is about equal 0 1 2 3 4 5 6 7 Power Control Error (dB) Figure 5.5 Perelement capacity versus power control error (dB) with outer cell interference. The capacity is from equation (5.2). over many scenarios of interest. Interestingly enough, simulations show the ratio of the incident signal power from the outer cell and inner cell sources is 0.57. The upper two plots of Figure 5.6 show that Wilkinson's method does not provide a particularly useful approximation when there is no outer cell interference. The lower two plots include the effects of outer cell interference and tend to agree more closely with simulated results for a range of power control error from 1 to 6 dB. For power control error = 7 dB the simulated perelement capacity is almost an order of magnitude greater than the analytical results. 87 Per 30 Element : SNRi= 12.6 dB Capacity  Analysis 20 %; .......... :................ ................ ................ A 20 4..... ,8 e Simulation 10 10. l....  1 2 3 4 5 6 7 (a) Element SNRi = 59 dB Capacity ... Analysis 20 5 ......... ..................... .. 8 8 Simulation 10 ... .. ..... .. ....... .......  ........... ......... o0 . 1 2 3 4 5 6 7 (b) Per 15 Element SNRi = 12.6 dB Capacity  Analysis 10 .. .. .......... ............ ............ .ee Simulation 5 ........... ........ ........ .......... ........... ........... 1 2 3 4 5 6 7 (c) Per 15 , Element : SNRi = 59 dB Capacity  Analysis S10 .. .. .. ..... "... ...........i ........ ... y ?.......... ~ee Simulation 1 2 3 4 5 6 7 (d) Power Control Error (dB) Figure 5.6 Perelement capacity versus power control error (dB) for outage probability = 0.02. Plots (a) and (b) are without outer cell interference. Plots (c) and (d) include outer cell interference. Curves from analysis result from equations (5.8) (5.11). Conclusions/Discussion Simulations showed that even in the presence of multipath fading, the perelement outagebased capacity might serve as a useful performance measure for a wide range of conditions when considering a singlecell with multiaccess interferers. As the nominal input levels increase for the multiplecell case the perelement capacity deteriorates for pee < 3 dB compared to the singlecell case. The reason: for low nominal input levels the array treats the outer cell signals like AWGN. When the nominal input levels are high all interferers incident on the base station are well above the noise and the array must dedicate some processing to attenuate the outer cell interferers, at a loss in performance. The simulated results presented in this chapter show that for low nominal input levels (SNRi = 7/127 = 12.90 dB) and power control error 2 1.5 dB a singlecell system with multipath faded signals will have a higher perelement capacity than a system with directional, unfaded signals. The addition of outercell cochannel interference degrades the perelement capacity by about 8090% for p.c.e. 2 dB and 6070% for p.c.e. > 2 dB. For strong nominal input levels (SNRi = 59.6 dB) the addition of outer cell interference degrades the perelement capacity by a factor of 2/3 to 3/4 for power control error 5 2.5 dB. The two analytical models for perelement capacity gave mixed results. The model developed in Chapter 4 which averaged multiaccess interference degrades too rapidly with increasing power control error to be of much use in a singlecell scenario. In a multi cell scenario with low input levels the model gives a good fit to simulated results for power control error 2 1 dB. When input levels are high, the array output power due to the outer cell interferers can be as much as twice the power of the inner cell interferers (there are 21.67 times more signals originating in the outer cells than the inner cells). This accounts for the decrease in the capacity in Figure 5.5 as the nominal input levels increase. This model is limited in several ways. First, we are modeling the interference as an averaged quantity so the complex trialtotrial interactions between the interference, the desired signal and the noise are lost. Second, we are attempting to absorb the relatively complicated behavior of the processor into two parameters, Gpi.and N, which do not vary with the number of users or the power control scenario. The model based on Wilkinson's method of evaluating sums of lognormal random variables was introduced in an attempt to resolve the first of these two issues. It was hoped that the accuracy of the analytical model might be improved by approximating the interference as a sum of variables rather than an average. The plots in Figure 5.6 show that this model might offer some improvement in accuracy compared to simulations for the select case of multiple cells. If more accuracy is required for the case of a singlecell system or power control > 6 dB, then refinements of the models would be necessary. Summary This chapter presented simulation results of the outagebased capacity for incident signals subjected to frequencynonselective Rayleigh multipath fading and lognormally distributed power control error. The models included singlecell and multicell scenarios where outer cell interferers were subjected to lognormal shadow fading. The multipath fading model assumed that the spatial interactions between the faded incident signals and the antenna array allowed the fading process to be fully decorrelated between array elements. Under these conditions the array functioned as a diversity combiner rather than a beamforming antenna array. The perelement capacity the number of users/array element which may be supported for a required outage probability was evaluated via simulations. The motivation behind the use of perelement capacity was to find a performance measure which was robust to the variations in user population and array size and which would reflect the possible contributions a receiving array would make to the capacity of a power controlled CDMA system.. An additional goal was to formulate approximations which would allow easy assessments of the perelement capacity. Physical arguments in conjunction with semiempirical curve fits resulted in two analytical models. The first model was adapted from the model in Chapter 4 in which the signal, the interference and the noise were averaged. The complex interactions between even these quantities was overlooked for the sake of simplicity. Array outputs were modeled as averages of input quantities and gain constants. Values of these constants were derived empirically from simulations. The second model exploited Wilkinson's method of approximating lognormal variables so that averaging of the interference could be avoided. This model resulted in slightly improved accuracy compared to the first model for power control error 5 2 dB but was somewhat worse for power control error 2 6 dB. CHAPTER 6 BASE STATION ANTENNA ARRAY ADAPTIVE PERFORMANCE This chapter examines the steadystate performance of the Recursive Least Squares (RLS) adaptive algorithms for a base station diversity array receiving faded incident signals with power control error. A simulation approach is used since the scenarios are too complicated to allow useful analytical solutions. The simulations were discreteevent simulations of transmitted bits through a nonstationary AWGN channel with timevarying multipath fading as well as stationary power control error and shadow fading. The simulation results will be given in terms of histograms of the effective Eb/No which can facilitate outage calculations. The performance of an RLS array in the presence of cochannel interference has been examined by several authors,67'72'73 but of these only Tsoulos et al.67 considered near/far scenarios with fading and outer cell interference. They used simulations to determine the outage probability when the (N = 8) array functioned as a beamformer, not a diversity combiner. Simulation results show that the RLS algorithm can track the timevarying solution effectively and that the adaptive solution is close to the steadystate solution. The structure of this chapter is similar to the two previous chapters. The first section revisits the nowfamiliar signal model from previous chapters and also introduces the time dependent fading process. Approximations of the fading process used in this research are also presented. The second section outlines the RLS adaptive algorithms and the receiver structure used in the simulations. The third section gives the results of the discreteevent simulations and provides some discussion. The last section gives the conclusions. Signal and Channel Model The signal model is the same as in the previous chapter: K Y(t) = A 10 lO' c.(t T.)b.(t T.)u + m=l KNoc f(6.1) A,. 10''20 10 c,(t ,)b.(t,)u, +n(t) where the first summation is for centercell and the second summation (with index n) results from the outercell interference when it is taken into account. Power control error and shadow fading are lognormally distributed (e, is N(O,pc2), sx is N(0,64)). The quantity Noc is the number of outer cells while K remains the number of users/cell. The quantity r, is the distancedependent, fourthorder propagation loss between the nth interferer and the center cell base station. As in Chapter five the amplitude A, accounts for the handoff with the least path loss when outer cell interference is taken into account. The quantities c( ) and b( ) represent the spreading code and the BPSK modulation with ideal, square pulses. We assume the beginning and end of a PN sequence corresponds to the beginning and end of an information bit. In this chapter the PN sequences, no longer random, are Gold codes of length 127. The quantity n(t) is a vector of complex AWGN with power On2 which is temporally and spatially white. The power control error and shadow fading are assumed to be constant over the observed interval. 
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