Adaptive Techniques for

Multiuser CDMA Receivers

Enhanced Signal Processing with Short Spreading Codes

prcad-spcctrum systems have a long history in military and civilian wireless comrnunications [1}. As originally conceived, they did not involve elaborate signal processing, nor were they envisioned as a way of arbitrating channel resources among multiple users. The merits of spread-spectrum modulation fill" nl1lltip1cxing voice users (code division multiple access-CDMA) arc now widely accepted for cellular applications [2], Still, existing CDMA systems (like 1S-95) include limited signal processing and interference suppression, namely, the single-user matched filter or RAKE receiver, which treats interference from other users as noise. The statistical averaging of out-of-cell interference and exploitation of silence periods in voice conversations made possible in the CDMA environment provide unique benefits for cellular applications compared with rival TDMA/FDMA options. For this reason, most proposals considered for third-generation wireless networks involve some flavor (lfCDMA r3J. It is expected that the requirements imposed on third-generation systems in terms of capacity and flexibility will necessitate advanced signal processing solutions for interference suppression and joint decoding of multiple users.

I t was observed in the mid-1980s that joint, optimal, maximum-likelihood decoding of all mel'S has significant performance benefits compared with matched filter alternatives [4]. Unfortunately, the solution also involves a joint Vitcrbi processor with exponential complexity in the number of users. The seminal work of [4] and the promised gains of multiuser detection (MD) have initiated much research in the area which continues unabated to this day. A number ofCDMA receivers have been proposed that cover the whole spectrum of perfor-

MAY 1000

Michael Honig ,LOd Michail K. Tsatsanis

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49

The use of relatively small spreading factors combined with a proportionately smaller number of users reduces complexity and facilitates the use of advanced signal processing.

mancc/complexirv from the simple matched filter to the optimal Vircrbi processor. Ada privc solutions, in P" rricular, have the potential of providing the anticipated MD performance gains with a complexity that would be managcabic for third generation systems.

Our goal, in this article, is to provide an overview of recent work in MD with an emphasis Oil adaptive methods. We start with (suboptimal) linear receivers and discuss the data-aided MMSE receiver. Blind (nondata-aided) implementations Me also reviewed 10- gether with techniques that em mitigate possible rnultipath effects and channel dispersion. In anticipation of those developments, appropriate discrete-time (chip rate) CDMA models arc reviewed, which incorporate asynchronisrn anc channel dispersion,

For systems with large spreading factors, the convergence and tracking properties of conventional adaptive filters may be inadequate due to the large number of'coefficienrs which must be estimated. In this context, reduced rank adaptive filtering is discussed. In this approach, the number of parameters is reduced by restricting the receiver tap vector to belong to a carefully chosen subspace. In this way the number of coefficients to be estimated is significantly reduced with minimal perforrnance loss.

It is well known in the context of single-user channel equalization that decision feedback (DF) adaptive scheme.': can provide near-optimal performance with little added complexity. This is true also in the case of multiuser systems; the difference here is that interference docs not only originate from past symbols, but also from current symbols of other (interfering) users. Therefore, tentative decisions for the interfering symbols arc needed to implement such interference cancellation schemes providing further justification for the use of linear receivers (as a preprocessing step). Both adaptive sequential and parallel decision-feedback strategies are possible as explained later. These receivers arc motivated by a briefdiscuss ion of'fundamental limits on performance (capacity) with error control coding.

The receiver design is affected by the type of sprcadi ng sequences used. This article deals with short spreading codes which repeat every symbol period. Some systems (like IS-95) employ long codes (with period much longer than the symbol period) which causes the interference to

50

vary randomly hum symbol to symbol l5J. Usually, systems which employ long codes also employ large spreading factors and ,1 large number of users per CDMA channel, in an effort to randomize the interference further and justify the usc of the marched filter receiver. All alternative approach is to combine short codes with the adaptive receivers discussed here, which exploit thc structure of the interference. The usc of relatively small spreading factors combined with a proportionately smaller Humber of users reduces complexity and facili tares the usc of advanccd signal processing. Of course smaller spreading factors also imply smaller bandwidth expansion, so that the number of users accommodated per Hertz is not reduced.

A linearly modulated digital conununicarions signal is cyclosrarionary with period equal to the symbol period. CDMA signals with short spreading sequences tall into this category. For systems with long codes, rhc chip-sampled interference is stationary, if the codes arc unkown, If the interferers' codes arc known, then the interference can be modeled as a time-varying, cyclostationary pmccss. In general, long codes complicate the development of adaptive signal processing algorithms for multiuser detection [6]-[11Ilnd will not be treated here.

COMA Signal Model

In CDMA system s all users transmit simultaneously ill the same trcqucncy band. Therefore if K users arc active, the received, baseband, continuous-time signal is a superposition of all K signals

K

r(t)= L t'k(t)+n(t)

('~I

(1)

where net) is additive Gaussian noise and and each user's signal is

(2)

a superposition of signature waveforms PI' (t) spaced by multiples of the symbol period T, and linearly modulated by the information symbol sequence bk Iii with mTIplitudes .Ak• In the case of asynchronous systems, each user may have a different delay v k'

It is desirable to utilize different signature waveforms tor different users (with sufficient excess bandwidth) to facilitate signal separation at the receiver. Often each user's signature is generated by modulating its low rate symbol W;1VefOl'IIl with a high rate code waveform (sec Fig. 1). While the description of }·ig. 1 is conceptually simple, it docs not lend itself to the development of appropriate discrete-time baseband models useful in receiver design. In an effort to derive chip-rate models for CDMA systems, we may write the spread-spectrum signature as a succession of chip pulses h(t)

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,\'-1

r, (t) = Lh [IJh(t -IT,)

HI

modulated by the usn code c].rtJ, JI",O, ... ,N -I, where N is the number of chips per hit (or processing gain) and T, is the chip period. In the case ofpropagation in a dispersive environment, the waveform h(t) represents tile convolutinn of the chip pulse with d1C Ch,I11- ncl respollsc. If the received signal is sampled at a fraction of the chip period (1' samples per chip), then rhc system is described by the following multiratc convolution:

K

r[n.J=r(t)I 't =LLAA.l.ijPi.!"n-iNJ'1

t=ll- i,- ./.,....,1 i

where

N-J

P,lnl=Pl(t)1 r. "'L"kll.llJrll-1P]

'~1J' .I=()

where h[n J i~ a similarly fractionally sampled version of h(t - v"). Fig. 2 gives a mulrirarc intcrprerarion of (4i and (5). The spectral spn.:adillg operation maybe considered as an upsampling of the information svmbol sequencc by N followed by a tinirc impulse response (FI JI..) filter oflcngth N with impulse respOl1St; equal to the spreading code. Then, rhc chi p-fracrionally sampled channel model is given by one more rnultiratc srn icturc with upsampling factor V 1;he model of Fig. 2 is ,1 most general description ofrhc rcccivcd sigll~ll and makes [10 assumptions on the channel spread or on the amount of possible intcrsymbol intcrfcrcncc (lSI). In rhat rcspccr, it is useful in describing CDMA systcms with porcntially small processing gain and large channel spread like the one considered fix the UMTS third-generation wireless s),srel11s 1121.

Vector models of the received data (within a certain observation window) arc more desirable and GIl1 he derived from Fig. 2 frlllowillg standard mulriratc tools (c.g, polyphase decomposition [13, Ch. 4 l). l'or example, in the case of synchronous users with P = 1 (one .%1.11Iple per chip), and no inrcrchip interference (ICI) the user's signature has IcngthN, P, = IPI.[O), ... ,tJ.IN -1]]' and is a scalar multipic of the user code Pk = A!. C k = Ak Ici. [O], .. "cJ.lN -in' . Then, the received signal call be written in matrix form as

I'ln I = Pb[n) -I- nln I

where P=[PJ,,,,,PK] and blnl=[b1(n), ... ,hK(n)1'. If leI is presellt due to a dispersive channel oforder t], then the mer signature has length N I- if and is given by the convolution of the code C i. [i.l wi til tilt: channel response h! [iJ. Therefore, the vector P (. = T (c I' )11 k is gi veil by the channel vector h , multiplied by rhc Tocplitz filtering rnatrix constructed from the code c (.

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(3)

o l CJ (0) j

cl(N-l)

If the length ofthe vector P, is grcater than N, then this leads to lSI.

The length of the observation window depends on the choice of system p'lmlllcters. If q «: N and lSI is ncgligiblc, then 1'[1/ J has length N. In the case of asynchronism and/or severe lSI, however, a longer window should be considered that spans multiple symbols.

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Minimum Mean Squared (MMSE) Linear Receivers

Given the received vector of samples at the output of'rhc chip matched tilter for symbol i; a lineal' ruulriuscr dctccfor forms the (soft) estimate

(5)

(7)

where W is an N x K matrix. (Here we assume the filter spans a single symbol inrcrval.) '-IVe can select W to minimize mean squared error (MBE), defined as

(8)

The solution i.~ given by Wo:::p(pllP+crlr)-1

(9)

where b'(n[iJn II [i.l)=021.

The linear MMSr. receiver has the importanr property that a single user call be detected without having to detect

Bit Waveform

5(1)

ModUlated Signal

(6)

A. J. Spectral spreading: continuous-time model.

Multipath Channel

A. 2. DSjSS signal in multi path: discrete-tim« modeJ.

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all other users. That is, the 11th column ofW, which is used to detect user J?, is given by

(10)

where R== E{t[iJr· [ill is the input covariance matrix. In other words, linear multiuser detection can be implemented as a set of "single-user" interference suppression filters, and is therefore well suited for the DS-CDM A forward link (in addition to the reverse link).

Some important properties oflinear MMSE detection are:

... The filter for each user can be implemented as an adaptive digital FIR. filter, analogous to an adaptive equalizer for a single-user channel,

.. When implemented as an adaptive filter, rhc linear MMSE detector suppresses total interference, independent of'origin. lt therefore suppresses (strong) other-cell interference, in addition to intra-cell interference.

.It.. When implemented as an adaptive filter, the MMSE receiver requires little side information. Specifically, either a training sequence is needed at the start of each transmission or the receiver must know the desired user's spreading code, channel, and associated timing. Amplitudes, phases, and spreading codes of interferers arc not rcquircd for adaptation.

.. In principle, the filter can suppress N -1 interferers for synchronous CDMA. For asynchronous CDMA, a digital filter that spans a single symbol interval can suppress UN -1) /2 J interferers. By increasing the observation window, the filter can suppress up to N -1 users [14], [15]; however, adaptation becomes more difficult.

.. The (coherent) MMSE solution automatically COL11- bines all rnulripath within the window spanned by the filter.

.... The performance of the linear MMSE receiver degrades gracefi.dly with the number of (equal power) users (c.g., sec [16J), although for very large loads J( IN» 1 (K = number of strong users, N = processing gain), the performance of the .MJ.V1SE receiver is close to that of tile matched filter.

Blind Minimum Output Energy Methods MMSE solutions arc typically implemented with the aid of training sequences. Even in the absence of training data, however, (10) indicates that the solution is implcmcnrablc ifthe signature and timing of the user of interest is known. In particular, the covariance matrix needed in (10) can be estim atcd from rhc data, while (in rhc absence of ICI) the user's signature coincides with the spreading code and may be readily available. This problem is analogous to bcamforrning problems appearing in array processing, in which the direction of arrival and signature of the user of interest are known (c.g., [17]). Adaptive implementation of such receiver filters which explicitly take into account the signature of the

52

user of inrcrcst can be developed using constrained optimization approaches [18].

In the context of CDM A and in the absence of dispersive channels, an MMSE receiver can be obtained by minimizing the output energy E{ 1 ~ [i.112 } (let user 1 be the user of interest without loss ofgenerality) r19J. To avoid the trivial solution WI = 0, the response of the reccivcr to the user of interest is constrained to one w ~ p I == I. This constrained optimization problem has been studied extensively in the context of array processing and the solution is termed minimum variance disrorrionlcss bcamforrncr (e.g., 117]). Alternatively, the adaptive filter can be decomposed <IS WI == PI + X I where x I is adapted but is always forced to be orthogonal to the signature PI [19]. Extensions to longer observation intervals and separate treatment of the in-phase and quadrature signals were presented in [20J. Similar approaches in array processing come under generalized sidelobe cancellers. The solution can be shown to be a scalar multiple ofthe MMSE solution and the minimum output energy (MOE) is

MOIi(PI)==-H-1.j-.

PI R PI

(11)

Unfortunately, it was observed in [19], and in prior array processing Iitcraturc, that those constrained optimization approaches are very sensitive to possible signature mismatch created by mulripath effects or timing errors. If the actual user signature differs from the one assumed in the derivation of the receiver, significant signal cancellation can occur resulting in poor performance. In [21J and [22J, the problem is mitigated by constraining the solution to the signal subspace, to reduce signal cancellation. An adaptive implementation based on subspace tracking was shown to improve performance at the expense of more computational complexity.

The method of [19J was later extended by adding more constraints [231. In particular, a solution for the dispersive channel case was attempted in [24 J and later ill [25J, by forcing the receiver response to delayed copies of the signal of interest to zero. Given the structure of the user's signature in multipath PI =Tic; )h" the receiver vector is constrained to satisfy w Iff T(c1 ) == [0, ... ,1, ... ,0J. With these additional constraints, minimum variance techniques are applicable, but have inferior performance since they treat part of the useful signal as interference.

This obstacle was overcome by constrained optimization solutions which combine all mulripath components of the signal of interest and jointly mi nirn ize the interfercncc, while maximizing the signal component at the receiver's output [261, r27J. The idea is again borrowed from array processing, known by the term Capon bcamformcr 128 J, 117 J. If the user signature is parameterized by some unknown parameters, then the Capon solution selects those parameter values which maximize the minimum output energy. In array process-

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A reduced ... rank filter first projects the received signal onto a lower-dimensional subspace before processing. This type of dimension reduction can improve tracking and convergence in time-varying environments.

iug those paraml.:w·s typically refer to directions of arrival while in CDMA to channel tap values, Based on (I I)', the Capon receiver selects l~ 1 such that

A h"h

h oo;argmax--·· --_.,

1 h hIlTIi(cl)r-IT(cl)h

The rational for this minimax optimization setup relates to an effort to maximize the sign;)l component ill the output after the interference has beccn suppressed. Equation (12) represents " Rayleigh quotient and hence the solution corresponds to the principal eigenvector of the matrix TIl(cl)R-IT(cl). Finally, given hi' the receiver vector can be obtained 8S a multiple of the MMSE solution R -I T (c L )h 1 •

Those Capon blind methods exhibit superior perterrnancc which under some circurnxtanccs is close to that ofrhc trained MMSE receiver [261, Furhcrmorc, at high SNR I; can be shown to collverge to the true channel

1 • _

parameters h.. Figure 3 illustrates these claims tor a sys-

tem with ten users, spreading factor N =31

and a severe ncar-far efffcct, Two variations

of the Capon method arc compared with the trained MMSE detector and show dose sign,1l-to-intcrference-plus-noise-ratio (SINK) performance (for details see [261). On the other hand, if the NIOE method is applied with no regard for the multiparh induced signature distortion, a substantial SINH..

penalty is incurred. .

Adaptive implementations arc proposed 111 [291 through two coupled lcasr mean square (LMS) recursions, [ointly updating WI and h]. Blind solutions with performance which is identical to that of the trained MMSE receiver in a dispersive environment arc possible as was demonstrated in [30.1 and 131]. Those techniques make explicit usc of subspace information from the autocorrelation matrix and do not lend themselves to time-recursive implementations. Rclatcd developments also include 132], 191, andr:BJ.

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Reduced-Rank Approximations

Given an adequate number of data samples, the alg()1'ithm~ presented ill the preceding section simultaneously suppress multiple access interference and perform multipath combining. Tracking and convergence may be problems, however, for some wireless systems in which a large numbel' of filter coefficients must be estimated. l-or example, a conventional implementation of a time-domain adaptive filter which spans three symbols for proposed third-generation widcba nd CDM A cellular systems can have over 300 coefficients, Introducing multiple antennas for additional space-time interference suppression capability exacerbates this problem. Adapting such a large number offilter coefficients implies very slow response to changing interference and channel conditions.

1\ reduced-rank filter first projects the received signal onto a lower-dimensional subspace before processing. This type of dimension reduction can improve tracking and convergence in time-varying environments. Reduced-rank linear tittering has been studied primarily tor array processing and radar applications (e.g., s~e ~17L 134]); however, recently it has been proposed for uirerterence suppression ill direct-sequence (DS)-CDMA systems [35J-[38J, [21).

In what follows for simplicity we assume a svnchron01.IS CDMA channel without mulripath, The generalization to asynchronous CDMA is straightforward, where the filter may span multiple symbols. Although there has been some work in applying reduced-rank techniques to frequency-selective channels [2l), this is currently an active area of research.

Let M" be the N x 1) matrix with column vectors which arc an orthonormal basis for aD-dimensional

ill

~ 10

0:

Z iii

"S 5' c.

:;

o

-10

Output SINA versus Input SNR (A~Yl1chronous Case)

25 r--~~-~ .~~~,--,-"

20

15

.... _ : :'

-15 L-----'--1._-~_j__..2..--~.c_.l:..__j====:c:::==~30

o 5 10 15 20 25

Input SNR (dB)

.... J. Performance of blind algorithms (Of Of) asynchronous system.

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subspace, where D < N. The projected received vector corresponding to symbol i is then given hy

r[iJ "" M:~ r[iJ.

(13)

The sequence of projected received vectors {r[i]} is the input to a tapped-delay line filter, represented by the D~vcctor w[i] for symbol i. The filter onrpur corresponding to the ith transmitted symbol is z[iJ = w H [i]r[i], and the objective is to select w to minimize the reduced-rank MSE

MD "'B{lbj [i]-wHr[llI2}.

(14)

The solution is

(15)

where

- H

R""MJ)RMIJ

( 16)

(17)

In what follows we describe a few different reduced-rank techniques which have been considered. Other related reduced-rank methods have been proposed in [35] and [311]~[40J.

Methods Based on Eigen-Decomposition

TIle reduced -rank technique which has probably received the most attention is "principal components (PC)," which is based on the following cigelHlecomposition of the covariance matrix

(18)

where V is the orthonormal matrix of eigenvectors of R and A is the diagonal matrix of eigenvalues. Given this decomposition, the received vector r is then projected onto the D-dimensional subspace which contains the most energy. Suppose that the eigenvalues are ordered as A I ~ A. 2 ',;;:. ... ~ A. N . For given su bspace di rnension D, the projcction rua trix for PC is then the first 1) columns of V.

VOl' J( < N, the eigenvalues II, I , ... , A. K are associated with the signal subspace, and the remaining eigenvalues an: associated with the noise subspace, i.e., A 0" 0=02 for K <m < N. Consequently, by selecting D> K, I)C retains full-rank MMSB performance (e.g., see [21] and [41]). However, the performance can degrade quite rapidly fix ]) ~ K, since there is no guarantee chat the associated subspace will retain most of the desired signal energy, This is especially troublesome in a ncar-far scenario, since for small D, the subspace which contains mosr ofthc energy wi ll likely correspond to rhc interference, and not the desired signal. Wc remark that in a heavily loaded cellular system, the dimension of the signal subspace may be near, or even exceed the number of dimensions available, in which case PC docs not offer much of an advantage relative to conventional full-rank adaptive techniques.

An alternative to PC is to choose a set of D eigenvectors tor the projection matrix: which minimizes the MSE. Specifically, assuming that the variance of the data symbols is one, we can write the full-rank MSE in terms of projected variables :IS

(19)

The subspace that minimizes the MSE has basis vectors which are the eigenvectors of R associated with the D largest values of I v [r C I ! A k 12, where v. is the kth column of V. (Note the inverse weighting of 1 A k 12 in contrast with PC.)

This technique, called "cross-spccrral (CS)" rcduced-rank filtering, was proposed in [42]. This technique can perform well for lJ < IC since it takes into account the energy in the subspace contributed by the desired user. Unlike PC, the projection subspace for CS rcq uircs knowledge of the desired user's spread ing code c L • Of course, a disadvantage of cigen-dccomposition techniques in general is the complexity associated with cstimarion of the signal subspace.

Portiol Despreoding

In this method, proposed in [431, rhc received DS-CDMA signal is partially dcsprcad over consecutive segments of m chips, where HZ is a par<lmctcr. The p<lL'rially dcsprcad vector has eli rncnsion ]) = r N / Nt laud is rhc input ro the D-tap filter. Consequently, m = 1 corrc-

_______________ ~ __ da_(i)_=_b__c,_(!) ~ ___( +EO(I)~

... 4. Multistage Wiener filter.

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The MMSE DFD has the attractive property that the feedforward filter suppresses other-cell interference, while the feedback filter cancels intra-cell interference.

spends to the full-rank MMSE filter, and m = N corrcspends to the matched filter. The columns ofM j} in this case arc nonoverlapping segments of c I' where each segment is of length m. This allows the selection of pcrformancc between that of the matched and f-i_lII-rallkMMSE filters by simply adjusting the number of adaptive filter coefficients.

Muhistage Wiener Filter

The multistage Wiener filter (MSWl:') was introduced ill [44 J . Pigurc 4 shows a block diagram of a four-stage MSWF, The stages are associated with the sequence of nested filters w L " .. , W Il' where D is the order of the filter. The matrices B 1 , ... , n 1) shown in the figure are blocking matrices, i.e.,

RJ.Jerring to l'ig. 4-, let d '" [i_l denote the output of the filtcr w; and I'", [i] denote the output of the blocking matrix B ,;,' Then the m + Ist multistage filter is determined by correlating the outputs of the preceding stage

For 11l =0, we have do Iii = hi [i_l (the desired input symbot), ro Iii = r[il, and W I is the matched filter WI' The filter output is obtained by linearly combining the outputs of the filters wl'.'" W 1) via the weights a I'" .,a 1).1 • The 1-ISWf has the following properties:

.. At each stage n the filter genera res a "dcsi red" sequence {d" fin and an "obscrvarion" sequence {r, [in, At any stage n, ifw " is replaced by the MMSB filter for cstimating d ,,_I [i} from l' ,,-1 [/:.1, then the resulting filter (with the opti mal combining weights) is the full-rank MMSE filter. Each filter w" can therefore be viewed as the "matched titter" for the associated estimation problem. The MSWl< is constructed iteratively by repeating the same structure, consisting of the matched filter and blocking matrix, at each stage, Continuing this procedure for N iterations gives the full-rank MMSE filter. Terminating after D iterations gives a rank D filter.

.. Computation of the MlvlSE filter coefficients does not require an estimate ofthc signal subspace, as do the cigcndecomposition techniques. Successive filters arc determi ned by "residual correlations" of ~ignals in the prcccd-

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(20)

ing stage. Adaptive algorithms based on this technique were presented in [45] .

.... T t is shown in [441 that the transformed matrix R, given by (16) is tri-diagonal. That is, it has nonzero clcmcnts only along the main diagonal and the adjacent diagonals.

.. The blocking matrix B '" is not unique. Although any rank N - m matrix that satisfies (20) achieves the same performance (MMSE), this choice can affect the performance for a specific data record. In particular, a poor choice of blocking matrix can lead to numerical instability.

It can be shown that theD-dimensional subspace generated by the rank-D MSWF is the same as the subspace spanned by c1 Rei ,Rlel , .. ·,RO-1CI [46]. These vectors are nor orthogonal, whereas it can be shown that the basis vectors generated by the MSWF arc orthogonal.

A large system analysis of reduced-rank filters, including the MSWF, for synchronous DS·CDMA with randomly assigned spreading codes is given in [461. Large system analysis of DS-CDMA with random spreading codes was introduced in [47J-149]. The largc system limit is defined by letting the number of users K and processing gain N tend to infinity with fixed load J( I N. By using results from the mathematics literature on the distribution of eigenvalues of large random matrices [SO I, [51], it is possible to compute the large system limit of the outpllt SINR for the full-rank MMSE filter [49]. It has been observed that this limit accurately predicts the performance with moderate J( and N (c.g. N =32).

In [461, it is shown that the MSWli has the important property that the rank D needed to achieve a target perforrnance le.g., output SINK docs not scale with the system size (IC and N) 1. It is found that D '" 8 achieves essentially full-rank performance over a wide range of loads K I N. This is in contrast to the other reduced-rank methods discussed, which require thatD increase in proportion with J( and N to achieve the target performance.

(21 )

0.25 "

..MF

o~·

o 10

30

40

20

50

60 70

Number of Dimensions D

.... 5. Error rate versus number of dimensions ior reduced-rank adaptive algorithms after rraining with 200 symbols.

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Orthogonal

2.5

0.5

Adaptive Nonlinear Multiuser Detection

is achieved with only eight stages (dimcnsions), which is much smaller than the minimizing order fell" the other reduced-milk tcchniqucs.Turthcr- more, this minimum error rate for the MSWl; algorithm is subsrantiallv lower than the error rate tor the matched filter receiver, and is not very far from the full-rank M MSE crror rate. Additional simulations with only 100 training samples show that the minimum error rate for the MSWF algorithm is again achieved with 1) "" 8, which is consis rene with the large system analysis in [461 mentioned earlier.

Load KIN

o~--------------~--------------~------~----~

o

1.5

0.5

For the reverse lillie of a cellular network, the objective is to demodulate

all users in the corresponding cell in the prcsence of other-cell interference, noise, and channel and receiver impairments. In this section, we discuss 11011- linear decision-feedback receivers for the reverse link which combine multiuser detection of intra-cell users with interference suppression of other-cell users. These techniques arc not appropriate for the forward link of a cellular system, since the objective there is to demodulate asing//! user in the presence of'intcrfcrcncc due to simultanC(lUS transmissions to other users (both inside and outside the cell of interest), noise, and other impairments. Consequently, for rile forward link, the interference JlJPprusion techniques previously discussed arc appropriate, rather than multiuser detection.

... 6. Spectral efficiency versus load for multiuser receivers in synchronous CDMA with E.I No =10 dB.

Performance Comparison

Here we indicate how the different reduced-rank techniques in the preceding section perform when used with a finite-length training sequence, Details on the adaptive algorithms arc given in l36] and r 45.1. Essentially, statistical expectations which OCCI11" in the MMSE representations arc replaced by sample averages, so that when D= N, all algorithms reduce to a full-rank least squares algorith rn.

l-igurc 5 shows error rate versus number of dimensions tor reduced-rank adaptive algorithms after training with 200 symbols. In this plot N = 128, K = 42, 'Hill the received powers arc log-normal with standard deviation 6 dB, which models received pmver variations with loose powcr control. The background SNR is 10 dB. The hit error rate P, is computed by assuminj; that the residual intcrfcrcncc plus noise at the Olltpu t of the filter is Gaussi all. Results arc averaged over random spreading codes, delays, and powers. Curves arc shown for the following algorithms: adaptive MSvVF, cross-spectral (CS), and the matched filter (Ml-), I n addition three Cl11VCS arc shown for partial dcsprcading with different adaptive estimation methods lor the combining coefficients: stochastic gradicur (SG-PD), least sL1LLares (LS-PD), and MMSE (MMSE-PD). The principal components algorithm performs worse than the cross-spectral method and the corresponding results arc omitted from this plot.

figure 5 shows that the adaptive reduced-rank techniques generally achieve optimum performance when j) < N. Namely, when }) is large, insufficient training data is available to obtain accurate estimates of rhc filter coefficients, whereas tor small D, the filter has insufficient degrees of'frccdom with which to .~~Ippress interference. The minimum error rate for the MSWF algorirhm

4.5 ....

:g: £ 3.5 @

4

3 ..

G .~ 2.5 tg w

2

~

[J 1.5

~

(/)

... 7. Spectral efficiency versus Eo I No for multiuser receivers in synchronous CDMA with load K / N ~ 1.

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Fundamental Limits

To understand the potential benefits of nonlinear multiuser detection for the reverse link, we first present some fundamental limits 011 the performance associated 'with linear and nonlinear receivers. The large-system Shannon capacity for synchronous CDMA with random spreading, additive white Gaussian noise, and optimal (maximum-likelihood) detection W,\S evaluated in [47J and r 4R I. The.l7lm capacity, or capacity summed over all users, is computed, assuming single-user coders and decoders. The "channel" in this analysis consists ofthe combined synchronous CDMA channel and receiver filter which produces soft outputs. The soft outputs arc passed to the single-user decoders. As explained, large-system analysis lets T( and N rend to infinity with fixed K / N. Linear receivers arc also considered in 14B], and rhc extension to multiuser decision-feedback receivers has been presented in [52}-154}.

~igl1res 6 and 7 show spectral efficiency versus load KIN and spectral efficiency versus R/, J N n for the matched filter, the linear MMSE filter, and the MMSE DfD to be described. In the former plot, 1':,; ! N n = 10 dB, and in tile latter plot, J( / N = I. These plots were gmcrated usiog the results in [48], 153], and l54_1. Spectral efficiency refers to the total l111m bel' of bits pCI' chip, summed over all users, which can be reliably transmitted, Also shown is the analogous curve corresponding to orthogonal multiple access. The latter corresponds to the single-user bound since there is no MAL An important property of the MMSE decision-feed hack detector (DrD), to be described, is that it achieves the same slim capacity as rill: optimal multiuser receiver 152].

These results indicate that:

... The linear MMSE detector is nearly optimal t!.Jr a wide range of loads (]( / N < 70%).

.. At v(;ry high loads, <Inc! with sufficient Er, / No, the MMSE decision-feedback receiver otters a significant performance improvement relative to the linear MMSE receiver.

... The capacity of DS-CDMA with the MMSE-DFD (equivalently, maximum-likelihood detection) is close to the capacity of

all orthogonal multiple access scheme .

... TI\e spectral efficiency of the matched tiltel' reaches an asymptotic limit as

Jib I N" ~ co, whereas the spectral efficiencies of the linear and MMSE-DFD multiuser detectors increase WIt! IOu r h lund,

We conclude that for low to moderate loads and power constra i nrs (r;" / N o ), linear MMSE detection can achieve most of the av a ilahlc gain due to multiuser detection. Given sufficient pow(;r (H" / No)' however, significantly higher spectral cffi-

cicncics arc achievable with nonlinear

niqucs enable a significant power savings relative to linear techniques.

MMSE Multiuser Decision-Feedback Detection Multiuser decision-feedback tor DS-CDMA was first proposed in 155] and [56} and was motivated by earlier work on multichannel (multi-input/multi-output) dccision-fccdback equalizers [57} -159]. To simplify the preseutation, for now we assume synchronous CDMA with all ideal (1\ WC;N) channel, and defer the extension to asynchronous CDMA with multipath until later in this section. Figure R shows a block diagram of a multiuser \WD 1:01' synchronous CDM 1\. The input to rhe decision device at time i is

(22)

where 1'[il is the N x l reccivcd vector of chip matchcd-filtel' 011tPUtS corresponding to symbol i, and bllJ is the J( x 1 vector of decisions at the output of the decision device, The fccdforward matrix F is N x K, and the feedback matrix B is K x J( and is typically constrained to have zeros along the diagonal to avoid cancelling the desired symbols.

Dcrcrrn i nation of the matrices F and B depends on the constraints and cost criterion. Namely, a lower diagonal matrix B corresponds to successive decision-feedback, whereas a full B, except for the diagonal, corresponds to para llcl decision-feedback. In the former G1S(; (Succcssivc-, or S- DFD), the users arc demodulated successively, ideally in order of decreasing power_ In this way for each user, only interference from stronger users is cancelled. l-or the parallel-decision feedback detector (P-DFD), all decisions arc fed b<)ck simultaneously so that the initial tentative estimates b must be obtained without caucclla-

rO)

/I

r-----,---- b(1)

... 8. Multiuser decision-feedback detector for synchronous CDMA.

Error Whitening or Interpolation Flltar L

/I

I----..,.-~b(i)

techniques. Conversely, at very high spectral efficiencies and loads, nonlinear tech- ... 9, Multiuser decision-feedback detector with error whitening or interpolation filter.

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Significant progress has been made during the past decade in overcoming obstacles which have prevented the introduction of multiuser detection in commercial COMA systems.

tion (c.g., hom a linear MMSE receiver). Of course, this procedure can he iterated.

Subject to the preced ing constraints on B, the matrices Band B can be selected to minimize the MSE

M=B{llb-yW}

(23)

where b[iJ is the vector of transmitted symbols at time i. Performing this minimization, assuming perfect feedback (b = b), gives the filter structure shown in Fig. 9. Namely, the feedforward filter has the form F = W/i"L, where W/i" is the linear MMSE filter, and L depends 011 the constraints on B. Namely, for the S-DFD, L is an error whitening filter (sec [60], [61, Sec. 7.5]), whereas for the P-DFD, the columns ofL can be interpreted as error interpolation filters [62].

With perfect feedback it is easily shown that the MMSE feedback filter perfectly cancels interference from the associated users. Consequently, the feedforward filter for user II turns out to be the linear MMSE filter for that user with tile "cancelled" users removed. This implies that tor an isolated cell, F for the MMSE P-DFD consists of a ban k of matched filters, and B is the cross -corrclarion marrix. In other words, the MMSE p-DFn for an isolated cell reduces to the conventional interference canceller proposed and analyzed in [631, r60I, and 164J. We also remark that other cost criteria give a structure analogous to that shown in Pig, 9. For example, the fccdforward filter for the zero-forcing, or dccorrclating DB]) consists of the zero-forcing linear filter (decorrdator) followed by an error whitening or error interpolation filrer, Finally, other related DFD structures have been considered in [65]-[67].

To summarize, the MMS1': DFD has the attractive property that the fecdforward filter suppresses other-cell interference, while the feedback: f Iter cancels intra-cell interference. furthermore, it does not require rcmodulation for interference cancellation. The main drawback of the DFD is error propagation, which can significantly compromise performance at high error rates.

Adaptive Decision-Feedback Detection

In analogy with linear interference suppression, with short (repeated) spreading codes, the MMSE filters F and n can be estimated given only a training sequence.

Knowledge of spreading codes is not required. Adaptive least squares and stochastic gradient algorithms for accomplishing this have been presented in [68]-[70]. Examples of convergence curves, taken from [70], arc shown in Fig. 10. The filters arc trained with the number ofbits shown. The bit error rate (BER) is then measured over an additional 150 bits, and the results are averaged over many runs. Although the DFDs require more samples to train than the linear receiver, the asymptotic bit error rate is lower. These results take into account error propagation and assume three-path Rayleigh fading. In the absence of error propagation, the asymptotic performance for the P-DH) is the single-user bound.

Ifthe receiver has knowledge of spreading codes of the intra-cell users, <IS would be expected at the base station, then the DFD filters can be computed directly using this information. This approach is combined with channel esrimation in l711 to estimate S-DFD filters in the presence of multi path. All advantage of this approach, relative to the training-based approach, is that fewer data may be needed to obtain accurate estimates of the lW1) filters. A disadvantage is that other-cell interference is treated as background noise and is therefore not suppressed. In general, we observe that any of the adaptive techniques discussed earlier for the linear receiver can also be applied to a DFD, where any necessary side in fo rmation must be provided for all users to be demodulated.

Asynchronous CDMA with Multipath

A requirement of the lWD is that the observation window and sampling times must be the same for all demodulated users. This is in contrast with linear receivers in which the timing can be adjusted separately for each user. Still, asynchronous CDMA can be accommodated by expanding the window of observation to include multiple received symbols. Of course, the received vector corrcspending to each desired user must be contained within this interval. If the received signals arc chip-asynchronous, then the different timing offsets across users becomes an issue. However, in principle, this can be solved with fractior ial chi p sampling com bincd with some excess bandwidth (i.e., see 172.1 and [73]).

In analogy with the linear MMSE receiver, the MMSE DFD filters for asynchronous CDMA arc IIR, even in the absence of multipath, Consequently, we must represent the filters as FC z) and B C z), each of wh ich Ius ,1 matrix impulse response. For the general case with rnulripath, the MMSE solution for F(z) and n(z) can be inferred from the results in rS9]. Namely, the matrix transfer functions arc dctcrmi ned by a spectral factorization of the equivalent multi-input/multi-output channel response.

For adaptive estimation it is convenient to approximate F(z) and B(z) as FIR filters. In the absence of multipath, near MMSE performance is generally attainable i f'the filters span three symbol intervals. Th is rernai ns true when mulriparh is present, provided that the delay spread is small relative to the symbol interval. Of course,

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if the multipath SP,1l1S multiple symbols, then more memory is required in the J)BD to equalize the multi-channel intcrsvmhol intcrfcrencc. The latter situation may arise for high data rate services, which usc a relatively SlD,l11 processing gain.

A simple approach to estimating tilt; BrR DH) filters adaprivclv is described in [70].1 f the filters span three symbol intervals, then the received vector r[i] is formed by stacking r[i - II , I: [iJ, and r[i+ 1]. The sequence of stacked vectors {rlin is then tile input to ,1I1 embedded DFD, which has the 3N>< j( fccdforward matrix F. The output of 11 is stacked in a similar way, and is used 1"0_(0111- pute the 31C X J( feedback matrix B. Of course, increasing the dimension of the filters in this way means that many more cocfficicnrs must be estimated.

Coded Performance

a: w co

Number of Training Bits

.... ro. Bit error rate versus number of training iterations for adaptive multiuser DFDs. The processing gain N = 8, there are four asynchronous users, and

f .. , / No = 6 dB. Tile channel for each user consists of three Rayleigh fading poths (power profife [0 dB, -2 dB, -4 dB]).

The results in Figs. 6 ,HId 7 apply to a coded system and assumes an S-DrD with succcssivc decoding 152]. There is relatively little work so far on the performance of MMSb DFDs with actual codes, although some results arc presented in 162], [ 701- Those results indicate that the S-DFD can otter a substantial gain in performance relative to a linear receiver provided that the users vary substantially in powel'. For example, in third-generation systems, received pow!.: r is proportional to rate. This causes a significant power imbalance which an S-DFD can exploit.

Conclusions

Significant progress has been made during rhc past decade ill ovcrcoming obsraclcs which have prevented so far the introduction of multiuser detection in commercial CDMA systems. The llSC ofshort codes, in particular, enables adaptive solutions that require little side information. We have indicated how these solutions can be extended to exploit mulripath divers it}' even without training or explicit channel estimates. Convergence and rrackiug srill remain issues ill the presence of bursty interference and moderate to fast fade rates. Reduced-rank methods may be useful in these situations, and this observation is stimulating work along these lines. Finally, we have discussed mulriuscr decision-feedback techniques, which can also be made adaptive with short codes and can offer significant benefits relative to linear receivers.

Of course, signitlcant challenges still remain, and multiuser detection continues to be an acrive area of research. The topics discussed in this article represent a subset of ropics within multiuser detecrion which arc currently being studied by many investigators.In particular, we have not discussed many practical issues with rcccivcr design and MD. Additional issues, such as the role

MAY 2000

MD can play in the support of integrated services with different information rate and quality of service requirements, arc just t:l11erging.

Acknowledgment

The work of Michael Honig was supported by NSF NCR-9628365 and ARO DAAD19-99-1-0288. The work of Michail Tsatsanis was supported by NSF NCR-9706658, NSf/CAREER CCR-9733048 NSF/Wireless CCR-9979295 and NJCSTjWireless.

Michael L. Honig (Bellow) received the B.S. degree in electrical engineering from Stanford University in 1977 and the M.S. and Ph.D. degrees in electrical engineering from the University of California, Berkeley, in 1978 and 1981, respectively. He subsequently joined Bell Laboratories in Holmdel, NJ, where he worked on local area networks and voiceband data transmission. In 1983 he joined the Systems Principles Research Division at Bcllcorc, where he worked OIl digital subscriber lines and wireless communications. He was a vi~iting lecturer at Princeton University during the Pall of 1993. Since the Fall of 1994 he has been with Northwestern University where he is a Professor ill the Electrical and Computer Engineering Department. He is an Editor for the IEEE Transactions on it!fiWHUttion 'I 'hemy and has served as an Editor fix the lIillH Transactions on Communications and as a Guest Editor tor the hUrl/penn Transactions on '1 'elecommunications and Wi1'Cim Personal Commurticati()]U, He has also served on tile D igi ta I Signal Processing Technical Committee tor the IEEE Signal Processing Society, He is currently serving as a member of the Board of Go vcrnors lor the I EEE Information Theory Society.

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MichailK Tsatsanis received his degree in c1ectricalcnginccring from the National Technical University of Athens, Greece, ill 19R7, <Inti the M.Sc. and Ph.D degrees in electrical engineering from the University of Virginia in 1990 and 1993, respectively. He is currently an Associate Professor ill the Electrical and Computer Engim:ering Department at Stevens Institute of Technology, NJ. His general research interests lie in the areas of statistical signal and array processing, with applications to communi" cations and networking. His current interests focus on signal processing techniques 6Jr wireless communicarions including blind equalization, multiuser detection, fading channel cstim arion and tracking, and signal processing methods for networking problems, He is a member of the IEEE technical committee 011 SPCOM. He has served as a member of the organizing committee for the 1996 IEEE Signal Processing Workshop on SSAP and as the technical co-chair of the organizing committee tor the 1999 IEEE Workshop 011 Signal Processing AdWllecs in Wireless Communications. He received the 1998 NSF CA REEK award and the 1999 $1' Society Y ollng Author Best Paper Award. He is an Associate Editor for the Ililili Transactions Ott Signfl! l'roccssitJg and IEEE Communications Letters.

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