ieee transactions on information theory, vol. 51, no. …140.113.144.123/publications/performance of...

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 10, OCTOBER 2005 3493

Performance of Linear Reduced-Rank MultistageReceivers for DS-CDMA in Frequency-Selective

Fading ChannelsSau-Hsuan Wu, Member, IEEE, Urbashi Mitra, Senior Member, IEEE, and C.-C. Jay Kuo, Fellow, IEEE

Abstract—The performance of a set of linear reduced-rankmultistage filter banks is studied in the context of multiuserdetection for direct-sequence (DS) code-division multiple-access(CDMA) systems. The set of filter banks under consideration iscomprised of the minimum mean-square error (MMSE), the min-imum output energy (MOE), the best linear unbiased estimator(BLUE), and the maximum-likelihood (ML) detector. Based ona common framework for the multistage implementations of theaforementioned filter banks, the signal-to-interference plus noiseratios (SINRs) and bit-error rates (BERs) of these reduced-rankfilter banks are studied for multipath Rayleigh-fading channels.A generic BER formula is provided for coherent detection andnoncoherent differential detection schemes constructed under thiscommon framework. Analysis shows that all of these performancemeasures are characterized by a kernel matrix mmse whosetrace forms the output SINR of the MMSE filter bank. Throughinvestigating the recursive structure of mmse, the output SINRsare proven to be monotonically increasing with the number ofstages and upper-bounded by a number equal to the paths of thedesired user’s channel. The condition for asymptotically achievingthis upper bound is also provided, which leads to the notion of ef-fective user capacity of linear reduced-rank multiuser detection aswell as serves as a test for the existence of a BER floor for coherentdetection. In addition, the channel mismatch due to differentialdetection is also shown to yield a BER floor for noncoherentdetection. Based on this analysis, a simple yet effective rule forchoosing the number of stages is provided for both coherent andnoncoherent linear multistage multiuser detection.

Index Terms—Best linear unbiased estimator (BLUE), code-di-vision multiple-access (CDMA), maximum likelihood (ML),minimum mean-square error (MMSE), minimum output en-ergy (MOE), multipath fading channels, multistage Wienerfiltering (MSWF), multiuser detection, reduced-rank filtering,time-varying channels.

I. INTRODUCTION

REDUCED-rank linear filtering for interference suppres-sion of direct-sequence (DS) code-division multiple-ac-

cess (CDMA) has recently been studied from several perspec-tives, such as the principal component approach in [2], the par-tial despreading technique in [3], and the multistage Wiener fil-

Manuscript received November 5, 2003; revised April 13, 2005. This workwas supported in part by the Integrated Media Systems Center, a National Sci-ence Foundation Engineering Research Center, under Cooperative AgreementEEC-9529152, and in part by the National Science Foundation under GrantNSF/ANI-0087761.

The authors are with the Department of Electrical Engineering, University ofSouthern California, Los Angeles, CA 90089-2564 USA (e-mail: [email protected]; [email protected]; [email protected]).

Communicated by X. Wang, Associate Editor for Detection and Estimation.Digital Object Identifier 10.1109/TIT.2005.855592

tering (MSWF) methods discussed in [1], [4], [5]. The principleof reduced-rank filtering is to process the desired user’s signal ina subspace that has a dimension as low as possible, while main-taining the signal-to-interference plus noise ratio (SINR) at alevel as high as desired. The purpose of reduced-rank filtering isto improve its adaptive performance under time-varying channelstatistics. By reduced-rank techniques, the number of samplesfor estimating the channel statistics, which are required for thefilter’s implementation, can be significantly reduced, thus im-proving the filter’s adaptability.

For investigating the performance of a reduced-rank filter, itis crucial to characterize the filter’s SINR evolution with re-spect to its rank. Even though the maximum achievable SINR ofa reduced-rank filter is determined by its full-rank steady-statevalue, its effective run-time figure is highly related to the accu-racies of the online estimates of time-varying channel statistics,which, in turn, depend on the rank of the filter. Therefore, un-derstanding the mechanism by which the SINR changes withthe applied ranks is of great interest to the application of a re-duced-rank scheme. To this end, a large-system analysis (see[6]) was conducted in [7] for the steady-state output SINRsof a number of linear reduced-rank filters over additive whiteGaussian noise (AWGN) channels. It is shown in [7] that therank of the MSWF is directly proportional to the number of ap-plied stages, and that the full-rank SINR of the MSWF can beclosely achieved with only few stages of implementations in afully loaded CDMA system.

Despite theoretical results obtained for AWGN channels,practical wideband systems such as DS-CDMA usually operatein time-varying multipath channels. The design considera-tions for reduced-rank multistage filtering in such channelsare quite different from those in AWGN channels. To exploitfrequency diversity offered by multipath channels, variationsof the MSWF have been proposed for multiuser detectionin DS-CDMA, e.g., the reduced-rank decorrelating RAKEreceiver in [8], the multistage receiver in [9] (which is basedon the correlation–substraction architecture presented in [10]),as well as the noncoherent differential detection scheme in[11]. Evolving from the MSWF [12], we developed a multi-stage (MS) structure for the construction of a class of linearreduced-rank filter banks in [1]. This MS structure coversdesign criteria ranging from the minimum mean-square error(MMSE), the minimum output energy (MOE), and the bestlinear unbiased estimation (BLUE) to the maximum-likelihood(ML) detection. These four filter banks (FBs) were shown toshare one common multistage interference suppressor, modulo

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3494 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 10, OCTOBER 2005

Fig. 1. A common framework for MRC and EGC receivers in multipath fadingchannels.

distinctive scaling matrices at the suppressor’s output to meeteach one’s optimization criterion. Based on the proposedmultistage interference suppressor, we further presented acommon framework in [1] for efficient implementations ofcoherent maximal ratio combining (MRC) and noncoherentdifferential equal gain combining (EGC) [13] multiuser re-ceivers. This framework is shown in Fig. 1. For MRC receivers,the estimation scaling module is not needed andis replaced by given channel coefficient . For EGC re-ceivers, matrices and can be any of scaling matricesof MMSE/MOE/BLUE/ML filter banks.

To continue our research in [1], we conduct thorough per-formance analysis on receivers that are constructed using thisframework in current research. The performance of both co-herent MRC and noncoherent EGC schemes is fully investigatedfor frequency-selective Rayleigh-fading channels. By using themoment-generating function (MGF) approach, we show thatsystematic and widely applicable analytical results can be ob-tained without the use of large-system random-sequence anal-ysis as introduced in [6].

One major contribution of this research is to present a unifiedapproach to performance analysis from different perspectives,e.g., from SINRs to bit-error rates (BERs), and from effectiveuser capacity to the existence of BER floors. To assist in theanalysis, an auxiliary performance measure, referred to as theoutput SINR, is defined to characterize the average SINR andBER in multipath fading channels. All these measures are foundto be characterized by a kernel matrix, denoted by ,whose trace forms the output SINR of the MS-MMSE-FB.With the kernel matrix and its dual matrix , whosetrace forms the average SINR of the MS-ML-FB, a genericclosed-form BER formula is derived for detectors constructedunder the common framework in Fig. 1. Moreover, throughthe recursion form of , SINRs and BERs can be fur-ther formulated with respect to the number of applied stages.Then, important properties for SINRs and BERs, such as theconditions for the equivalence of different receivers, can beestablished for all possible stages. As a special yet importantresult, the BERs of reduced-rank receivers in each class ofcombinings are shown to be equal in the presence of flatRayleigh-fading channels.

In addition to analytical SINR and BER formulas, an impor-tant issue is the determination of the number of applied stagesfor a reduced-rank multiuser receiver. This, in general, has todo with the rank of the filter, the dimension of multiple-ac-cess interference (MAI), as well as the degree of channel mis-match resulting from differential EGC. By investigating the lim-

iting properties of output SINRs and BERs in the high-SNRregime, we show that the number of stages decides the effec-tive user capacity of the DS-CDMA system. The effective usercapacity is characterized by the degrees of freedom, , of-fered by multistage filters, and the dimension of interference,

, due to system loading and the multipath channel effect. If, BER floors will not occur for MRC schemes; oth-

erwise, SINRs will saturate at high SNR and lead to BER floorsdespite increasing signal power. In addition to the saturation ef-fect of output SINRs, performance bounds for EGC schemesare also governed by the channel mismatch resulting from dif-ferential combining. BER floors of EGC schemes are shown tobe functions of both the test condition and the au-tocorrelation value of a time-selective Rayleigh-fading channel.Hence, due to the SINR saturation and/or channel mismatch,we may observe significant BER degradation even if the filter’sfull-rank output SINR can be nearly achieved by reduced-rankfiltering.

We begin our study by providing a detailed description of thesystem model in Section II for asynchronous DS-CDMA overmultipath fading channels. Some known results on reduced-rankmultistage filter banks are briefly reviewed in Section III. Thesteady-state output SINRs and their recursion forms are studiedin Section IV, followed by the BER analysis for both MRCand EGC schemes in Section V. Numerical simulations are pro-vided in Section VII to verify the accuracy of theoretical studies.Based on analytic formulas, the effective user capacity of re-duced-rank filtering and the channel mismatch effect are ana-lyzed in Section VI. Finally, concluding remarks are drawn inSection VIII.

II. SYSTEM MODEL

The following convention is adopted in this paper. A boldfacecapital letter denotes a matrix of dimension by , and aboldface lower case letter denotes a vector of dimension

. The Hermitian of a matrix or a vector is denoted byor , and the transpose of a matrix or a vector is denoted by

or . For convenience of presentation, the time index ofa variable is suppressed if its explicit expression is not essential.

A standard asynchronous DS-CDMA model is considered inthis study. The baseband representation of the transmitted signalfor the th user can be written as

(1)

where is the symbol duration, and , , and are thedata bit at time , energy per bit, and relative delay with refer-ence to the base station, respectively, for the th user. The trans-mitted symbols are identically and independently dis-tributed, taking on values with equal probabilities. Dataare binary phase-shift keying (BPSK) and differential BPSK(DBPSK) modulated for MRC and EGC schemes, respectively.The spreading waveform is given by

WU et al.: PERFORMANCE OF LINEAR REDUCED-RANK MULTISTAGE RECEIVERS 3495

Fig. 2. An example to illustrate the delay pattern of a three-path DS-CDMA channel of user k.

where is the signature sequence for the th

user with a period of . The function is a normalized chippulse shaping function of duration , with beingthe spreading gain.

The th user’s signal propagates through a multipathfading channel with complex impulse response

where is the Dirac delta function, is the number ofpaths of user , is the time delay associated with the th tapof the tapped-delay-line channel model, and is the fadingprocess corresponding to the tap. The value of is assumedto be constant during one symbol interval and changes fromsymbol to symbol. Thus, can be modeled by a discrete-time fading process , where is a time-invariantnonnegative channel gain and is a complex zero-meanwide-sense-stationary Gaussian process satisfying

(2)

where stands for the Kronecker delta function. The autocor-relation between two consecutive channel states is given by

(3)

The received signal due to the th user is given by

(4)

where . The overall received signal is, where is the number of users and is

complex zero-mean white Gaussian noise.To reveal the relationship between the hierarchical structure

of multistage filtering and its effects on various performance

measures for CDMA systems, we consider one-shot detection.Under this constraint, important properties such as SINRs andBERs can be more easily characterized and clearly presentedin recursive expressions of the applied filtering stages. Never-theless, the analytical functions of SINRs and BERs can still beobtained for delay spreads larger than based on the approachprovided in this paper. However, the resultant expressionsare far more complex and may not provide useful insightsinto the structure of multistage interference suppression forCDMA systems. Therefore, the maximum delay spread is setto for all users. Thus, for thefirst path of every user, we have its delay . The delayof other paths satifies , .

Without loss of generality, user 1 is chosen to be the desireduser. It is assumed that the delay time for each path of the desireduser is known and the receiver’s clock is synchronized with thereception of the first path of the desired user, i.e., and

, . We define

where denotes the largest integer less or equal to the argu-ment, ,

and

where denotes the remainder of divided by . An exampleof a three-path delay pattern for user is illustrated in Fig. 2.The light-shaded regions in this figure belong to symbols thatarrive earlier inside the sampling interval of . In contrast,the dark-shaded regions belong to symbols that arrive later inthe same transmission paths of the same sampling interval.

The received signal is passed through a filter matched to thechip pulse shaping function and then sampledat the chip rate. We define

and


Both vectors are of dimension . Due to asynchronousarrival times, the contribution of the signature sequence fromthe earlier arrival symbol in path of Fig. 2 can be expressed as

and the contribution from the later arrival symbol in path isequal to

Thus, the discrete-time received signal vector obtained by col-lecting consecutive samples of the matched filter output isgiven by (e.g., [13])

(5)

where . The filtered noise vector is com-plex Gaussian distributed with covariance matrix . Note that

. For convenience of analysis, the discrete-time re-ceived signal is rewritten as

(6)

where , ,, and . The matrix

is referred to as the steering matrix for the desired user’ssignal and vector is the aggregate of intersymbol inter-ference (ISI), MAI, and the AWGN noise relative to the desireduser’s signal, i.e.,

(7)

For the rest of this paper, we set .

III. LINEAR MULTISTAGE REDUCED-RANK FILTERING

The essence of reduced-rank filtering lies in projecting thereceived signal vector onto a lower dimensional subspace soas to reduce the number of filter coefficients to be estimated.For reduced-rank filtering using the multistage structures devel-oped in [1], [4], [12], the key idea is based on the successiveorthogonal projection of the received signal onto certain sub-spaces. Specifically, let be a matrix of dimensionand , with its column vectors forming a basis for an

-dimensional subspace. Consider a transformation of, with a matrix

(8)

where is obtained via QR factorization of, and satisfies , denoted as, provided that steering matrix is of full column

rank. The reduced-rank MMSE/MOE/BLUE/ML filter banksoperating on the transformed received signal are givenby (see [1])

(9)

(10)

(11)

(12)

where , , and

, with and . Itwas shown in [1] that these four filter banks share a commonmultistage implementation modulo distinctive output scalingmatrices. The generic form of their soft outputs is given by

(13)

where and , andis the autocorrelation matrix of and

the cross-correlation matrix between and .

For a scaling matrix , we have for theMMSE filter bank, which is denoted by . For other filterbanks, their scaling matrices are defined by ,

, and , where

(14)

(15)

and where is the autocorrelation matrix of ,and is the error correlation matrix of the MMSEfilter as shown in Fig. 3. Furthermore, let

and

where

and , , are formed via QR factorizationof . By successively projecting the transformed re-ceived signal onto the column space of , the matrix

can be decomposed into a multistage ex-pression as shown in Fig. 3, where the output scaling matrixis for the MS-MMSE filter bank, for theMS-MOE filter bank, for the MS-BLUE filterbank,and for the MS-ML filter bank. The objective of fil-tering is for the estimation of .


Fig. 3. The common structure of the D-stage MMSE/MOE/BLUE/ML filter banks in multipath fading channels.

For the analysis of the MSWF, a more convenient way ofgenerating the subspace , spanned by basis vectors for the

-stage filter banks, has been given in [1], [4],1 where blockingmatrices are redefined as , and the matrixof basis vectors is given by

(16)

The matched filter bank at each stage is constructedusing the formula

and

(17)

Matrix decomposition is obtained by applying theGram–Schmidt orthogonalization process to the right-handside of (17) so that . The autocorrelation matrixof the received signal in multipath Rayleigh-fading channelsis given by

(18)

where , ,and .

To simplify the notation, is rewritten in the form of

(19)

where the subscript indicating the matrix rank is dropped.In the sequel, all operations are conducted with this more gen-eral reduced-rank transformed matrix . To indicate the totalnumber of stages and to avoid confusion with that is theoutput autocorrelation matrix of the matched filter in Fig. 3,

and are used to denote and (see (14) and (15))for the -stage implementations of filter banks. They can bewritten as

(20)

1The subspace derivation for the single-path case in [4] was generalized tothe multipath case in [1].

(21)

which are obtained by replacing with and within (14) and (15), respectively.

IV. OUTPUT SINRS OF MULTISTAGE FILTER BANKS

With the common multistage interference suppressor, the im-plementational differences between MMSE/MOE/BLUE/MLfilter banks become small and only lie in scaling the matrix

to the suppressor’s output. However, this does not meantheir performance is the same. In this section, we will study theinfluence of the scaling matrix on the performance measuresof the average SINR and the output SINR, and characterize theeffect of multistage filtering on these SINRs. The output SINRsof four reduced-rank filter banks will be studied in the contextof coherent MRC detection for short-code DS-CDMA in multi-path Rayleigh-fading channels. We will show in Section V thatthese output SINRs also play important roles in characterizingthe BER performance of the corresponding reduced-rank detec-tors, either for coherent MRC or noncoherent differential EGC.

A. Average Versus Output SINRs for Linear Filtering

We start by deriving the filter bank that maximizes the SINRvalue. This will be examined from two angles, i.e., the averageSINR and the output SINR.

Conditioned on channel parameters, the filter bank that max-imizes the instantaneous SINR is given by [14]

(22)

which is obtained using the Cauchy inequality. This filterbank is actually the full-rank ML filter bank

[1]. The corresponding maximal SINR isequal to , conditioned on the currentfading coefficient vector. Taking the expectation with respect to

of a Rayleigh-fading channel yields the following averageSINR:

(23)


The instantaneous SINRs of the MMSE, MOE, and BLUE re-ceivers can also be evaluated by substituting their correspondingfilters into (22). However, deriving the closed-form formulasof the average SINRs for multipath fading channels is compli-cated, and does not provide much insights into their achievableperformance. To get around this problem, we define in the fol-lowing an alternative performance measure, called the outputSINR, for evaluating the average SINRs of different filter banksover multipath fading channels.

We consider the MRC filter bank in form , where isnot a function of fading coefficient vector , and define theoutput SINR to be the ratio of the mean squared filter output tothe output’s variance; namely,

SINR (24)

Evaluation of the output SINR is more straightforward thanthe average SINR.2 Moreover, as shown in the sequel, theoutput SINR possesses important properties in characterizingthe performance of linear reduced-rank filtering, and is crucialto the BER analysis in multipath fading channels. Throughoutthis paper, this SINR measure is referred to as the output SINRand will be extensively used for the performance evaluation ofthe MMSE/MOE/BLUE filter banks in contrast to the measuregiven in (23), which is solely used for the ML filter bank andreferred to as the average SINR.

Next, we determine the filter that maximizes the output SINRas given by (24). Ignoring the desired user’s self-induced ISI, 3

the autocorrelation matrix of conditioned on is given by

(25)Thus, the variance of MRC filter’s output becomes

(26)

To deal with the fourth moment term, we use the followinglemma [15]. If , then we have the followingequality for Hermitian matrices and :

(27)Consider filter banks that satisfy the constraint:4 isHermitian, and , and . Thus, for

2Note that the output SINR is not equivalent to the average SINR, since6= E .

3It was mentioned in Section II that only one-shot detection is considered tocharacterize the performance impact of multistage filtering on interference sup-pression. Under this constraint, the ISI is relatively small in comparison withthe overwhelming MAI. Thus, we ignore ISI so as to give a more succinct ex-pression for analysis.

4All filter banks considered within the scope of this paper satisfy this con-straint. It is more straightforward to derive the maximizing filter with this con-straint. However, the MMSE filter can still be proven to maximize the outputSINR without imposing this constraint.

Rayleigh-fading channels, we have . Substituting theseterms back into (27) plus the fact that

we have

Based on the above results, a filter bank that maximizes theoutput SINR satisfies

(28)

Taking the derivative with respect to and setting to zero yields

(29)

The solution is exactly the full-rank MMSE filter bank, and the corresponding output SINR is equal to

.

B. Output SINRs of Reduced-Rank Multistage Filter Banks

We have given SINR-maximizing filter banks that are opti-mized for two different SINR definitions above. However, tobetter understand the distinction and similarity among the fouraforementioned filter banks, we shall first derive the outputSINR for each one of them and then study the relationshipbetween their SINRs. To facilitate the analysis, we ignore thedesired user’s self-induced ISI as done before. The steady-stateoutput SINR of each multistage receiver is given for the multi-stage Rayleigh-fading channel in the following proposition.

Proposition 1: The SINR values of MRC MMSE/MOE/BLUE/ML receivers are given by

SINR

(30)

SINR

(31)

SINR

(32)

SINR

(33)


where , ,, and

[cf. (21)].Proof: The proof is given in Appendix A.

Note that these SINRs are all characterized by and. Moveover, as shown in (23), the average SINR, in the

sense of (22), of the full-rank ML receiver is simply equal to. By (104), the average SINR of the reduced-

rank ML receiver is equal to

SINR

(34)

Thus, and are called kernals of the MMSE and theML filter banks, respectively.

In addition to the above closed-form formulas of SINRs.From (101)–(104) and (109)–(112), the conditional covariancematrices of these filter banks can also be obtained as

(35)

(36)

(37)

(38)

where the decision statistic of MRC is given by. Employing the positive semidefinite property of

covariance functions, we obtain the following properties.

Lemma 1: Let .Then, we have the following bounds.

A. The diagonal matrix of eigenvalues is bounded by

(39)

B. The output SINR of the MMSE filter bank is bounded by

SINR (40)

C. and.

D. The lower bound of

(41)

Proof:

A. From (37), we have so that .Similarly, from (35), we have . Combiningthese two constraints, we get .

B. By (30), the output SINR of the MMSE filter bank issimply equal to . Thus, we have

SINR .C. It is clear from (21) that

(42)

Thus, and share the same eigenstructure, i.e.,matrix . Thus, we have

(43)

(44)

(45)

D. It follows directly from (38) and the fact that the varianceis positive semidefinite.

From the preceding expressions, we know that SINR isupper-bounded by . Recall from (29) that the full-rank MMSEfilter bank maximizes the output SINR defined by (24). Thus,constitutes the upper bound for the output SINRs of MRC filterbanks. This poses a question whether the output SINR can serveas a performance measure as the average SINR. We justify itsvalidity by showing that the eigenvalues of and areincreasing with the desired user’s receiving power if the self-induced ISI is ignored.

Proposition 2: Consider the case where the self-induced ISIcan be ignored, i.e., . Let and be two real positiveconstants satisfying , and let , ,be the eigenvalues of , , with amplitudematrix , . Then, , .

Proof: It was shown in [1] that

(46)

If the self-induced ISI as given in (7) can be ignored, is nolonger a function of . By (21), is independent of . Theinequality leads to

Thus, we obtain

(47)

which implies . Furthermore, by (43), we conclude. This completes the proof.

C. Equivalence of Output SINRs

Proposition 1 shows that output SINRs are all related to thekernels of SINR and SINR , which are

and

respectively. In fact, these two kernels are also related to eachother by (45). Inspired by the common multistage structure ofthese filter banks, we are interested in whether there exists amultistage structure for these two kernels and what are the con-ditions that make output SINRs of different filter banks behavesimilarly. We investigate the second question in this section,which leads to a sufficient condition on the equality of outputSINRs as given in the following proposition. The multistagestructure of SINRs will be discussed in detail in Section IV-D.


In Section V, we will show that these two kernel matrices alsocharacterize the BER performance of the filter banks.

Proposition 3: For steady-state output SINRs of -stageMRC multiuser receivers, equalities of SINRSINR , where MOE,BLUE,ML , hold under the fol-lowing conditions.

A. SINR SINR if

and

B. SINR SINR , if

C. SINR SINR , if and only if , ,namely

D. SINR SINR , , if

and

Proof: The proof is given in Appendix B.

For the conditions specified in item D above, it is challengingto achieve given the form of in (20). Moreover,since , this implies thatcolumn vectors of are mutually orthogonal. Recall thatis a function of the spreading sequence and path delays of thedesired user. The equality is in general difficult to achieve inpractice except for the case of . Thus, we are led to thefollowing corollary.

Corollary 1: For multistage MRC MMSE/MOE/BLUE/MLreceivers in DS-CDMA systems with flat Rayleigh-fading chan-nels, we have

SINR SINR SINR SINR (48)

Proof: In flat-fading channels, , , and are scalars. By (40), . Thus,

According to Proposition 3, the equality holds for any spreadingsequence of the desired user.

This result plays a crucial role in relating output SINRs toBERs. As shown later in Section VII, this corollary coincideswith the fact that the BER is the same for all multistageMMSE/MOE/BLUE/ML receivers in flat-fading channels.Results for more general multipath channels will be given inSection V.

D. Recursive Structures of SINRs

Given the closed-form output SINRs of multistage filterbanks, there arises an interesting yet practical question; namely,whether the output SINRs increase with an increasing numberof stages. If so, by what mechanism? This question can beinvestigated from two angles with different SINR definitions.For the ML filter bank, it is more straightforward to workon the average SINR as defined in (34), since it is equal to

the trace of . The answer lies in the structure of kernel. For the other three filter banks, we can study their

output SINRs as given in (30)–(32). The output SINRs of theMMSE/MOE/BLUE filter banks are only functions of kernel

if and are fixed. Hence, studying the properties ofSINRs can be done by examining the recursion structuresof kernel matrices and . This boils down to theanalysis of and with respect to the number of stages. Toapproach this goal, we first explore the structure of .

For fixed and , is a constant matrix. The evolutionof output SINRs solely depends on the evolving structure of

. It was shown in [12] that forand for the MSWF in AWGN channels. Thismeans that has a tri-diagonal structure. This propertyalso holds for filter banks studied in this paper for multipathRayleigh-fading channels, since we have

(49)

for . Similarly, for

[cf. (17)]. For more details, we refer to [1, Proposition 1]. Inaddition, we have

(50)

so that is a block tri-diagonal matrix of the form

. . ....

. . .. . .

.... . .

. . .

(51)

Let denote , for . Byapplying the block tri-diagonal property to (20) and recursivelyemploying the matrix inversion lemma [16] for every resultantmatrix , we get

(52)

Recall the following property of the MSWF [1], [12]. Con-sider a filter bank pruned at stage . Then, the re-moved part from stage to stage forms a reduced-rankMMSE filter bank that minimizes Thecorresponding error correlation matrix

is equal to , with in the laststage and SINR , [cf. (30), where


is replaced by to indicate the total number of stages].Therefore, the output SINR of , denoted by SINR , isequal to , .5 Furthermore, similarto the structure of (52) for , we have

(53)

That is, possesses a recursive structure of the form

(54)

with . This structure can be clearly seen for flatRayleigh-fading channels.

For the scalar case of flat fading, we use and to denoteand , respectively. The output SINR of matched filter

, denoted by SINR , at each stage is equal to . Theoutput SINR of MMSE filter constructed with stages tobecomes

SINRSINR

(55)

with SINR at the last stage. An interesting propertyrevealed by (55) is that the SINR of the MMSE filter differsfrom that of the matched filter in their denominators. SubtractingSINR from the denominator of SINR gives SINR .As a result, the output SINR of the multistage MMSE receivercan be expressed of a continued-fraction form given by

SINR

...

(56)

where and are both scalars in this case.This form also applies to multistage MOE/BLUE/ML filterbanks since their output SINRs are equal in flat Rayleigh-fadingchannels by Corollary 1.

It is worthwhile to point out that a continued-fraction rep-resentation for the SINR value of the MSWF was derived in[7] for AWGN channels based on the large-system random-se-quence analysis. Although both SINRs can be expressed in con-tinued-fraction form, the problem setup, parameters definitions,and recursion forms involved in these two equations are dif-ferent. Equation (56) is for DS-CDMA over flat-fading chan-nels. The spreading gain is finite and spreading matrix isfixed for the analysis conducted in this paper. In contrast, [7, eqs.(44) and (45)] are for a large DS-CDMA system over AWGNchannels, with and . Moreover, variables

and in [7, eq. (45)] are SINRs of the -stageand -stage MMSE filters, respectively. However, SINRin (55) is not the output SINR of an -stage MMSE filter. It is theoutput SINR of the filter , embedded in the recursion structurefrom stage down to the bottom stage of a -stage MMSEfilter.

It was shown in (56) that the output SINR is increased byeach additional stage. However, that only holds for flat-fading

5For the convenience of notation, we made AAA explicit in expressing thecross-correlation E(yyyddd ) = HHH UUU AAA at the first stage, where ddd � b .For stages i = 2 to D, E(yyy ddd ) = HHH UUU .

channels. For multipath fading channels, we need to show thatthe eigenvalues of increase with the number of stages. To dothat, we first rewrite for better notation kernel matrices of the

-stage and -stage MMSE filter banks, respectively, by

(57)

(58)

Then, we characterize the relation between eigenvalues ofand of by exploiting the recursive structure of

, which is given in the following theorem.

Theorem 1: The eigenvalues of of the multistageMMSE filter bank are monotonically increasing with thenumber of stages. i.e.,

Proof: The proof is given in Appendix C.

Since the output SINR of the reduced-rank MMSE filter bankis simply equal to , a direct consequence of Theorem1 is that SINR increases with the number of stages.

For the output SINRs of the MOE and the BLUE filter banks,their numerators are constants while their denominators arefunctions of . For the BLUE filter bank, the denominator

is and, as a result, SINR increases with thenumber of stages. For the MOE filter bank, the denominator is

. Similar to what has been done in the proof ofTheorem 1, one can show that the eigenvalues ofdecrease as the eigenvalues of decrease. Therefore,SINR also increases with the number of stages.

For the ML filter bank, we should investigate the multistagestructure of since SINR . We can rewrite

(59)

(60)

which are kernels of the -stage and -stage ML filterbanks, respectively. The property of is described in fol-lowing theorem.

Theorem 2: The eigenvalues of of the multistage MLfilter bank are monotonically increasing with the number ofstages. i.e.,

Proof: With the Matrix Inversion Lemma, (44) can berewritten as

(61)

which leads to

(62)

This implies that each pair of eigenvalues, and ,, forms a parabolic function with asymptotic lines at

and , respectively, as shown in Fig. 4. Due tothe physical constraints and [cf. (39) and (41),respectively], only the upper branch is valid. This means that


Fig. 4. Parabolic curves of eigenvalues � and � , i = 1 . . .L.

is monotonically increasing with . The relation betweenincrements and can be given by their Laurent seriesexpansion as

(63)

Since increases with the number of stages by Theorem 1,so does .

Similarly, since SINR , we conclude that theSINR increases with the number of stages using Theorem 2.From Theorem 2, we see that it suffices to use either orin representing SINRs, since one is fully characterized by theother. However, it is sometimes more convenient to use inanalysis since goes to infinity as approaches . In contrast,we have .

V. BERS OF LINEAR MULTISTAGE MULTIUSER RECEIVERS

In the preceding section, we analyzed the properties ofoutput SINRs for multistage MRC filter banks under the as-sumption that the channel state information is available. In theabsence of channel state information (especially in fast fadingenvironments), noncoherent schemes such as EGC schemesare often used in practice. Due to the common structure ofthe MMSE/MOE/BLUE/ML filter banks, a variety of EGCschemes can be derived without additional implementationcomplexity; namely, using two temporally adjacent outputsfrom one or two filter banks. The use of two adjacent out-

puts from the same or two distinct filter banks is called thehomogenous or the heterogenous EGC, respectively. Basedon the derived SINR results, we continue to investigate therelationship between the rank of the filter and its associatedBER in this section. In particular, we will focus on BERs ofall MRC and homogenous EGC schemes along with someheterogenous EGC schemes of special interest.

A. Review of Existing Results

Our analysis is along the lines of the BER analysis for ho-mogeneous EGC MMSE/MOE multiuser receivers presented in[13]. This approach can be easily extended to the BER analysisfor both MRC and EGC schemes. It is briefly reviewed as fol-lows. Let us define the matrix

and (64)

where is a combining vector with the same dimension of .Note that for the MRC filter banks andfor the EGC filter banks. Vectors and may comefrom different filters. The decision statistic can be expressedas

(65)

In the presence of Gaussian random noise, the probability den-sity function of is Gaussian quadratic. Its MGF, denoted by

, is given by

(66)


TABLE IM(s) FOR MAXIMAL RATIO COMBINING FILTER BANKS

where . By assigning for BPSK (orfor DBPSK), the BER is given by [13]

(67)

where the residues, denoted by are taken with respect to theright half-plane (RHP) poles of . Using this formula, theBER analysis for MRC and EGC schemes can be conducted inthe same framework. We begin with the analysis for MRC filterbanks and then turn to the more complicated case of EGC filterbanks. Finally, a generic BER formula is developed for thesetwo classes of combinings.

B. BER for Maximal Ratio Combining (MRC)

With (67), it suffices to derive MGF, , for each filterbank. Since

and

we have

(68)

Substituting into (66) yields

(69)

This expression can be further simplified with the followinglemma (see [16]):

(70)which yields

(71)

As stated earlier in Section IV-A, is Hermitian.The signal correlation matrix and the output correlation ma-trix can be defined for filter bank as

(72)

(73)

where . Let . Equation (71) canbe rewritten as

(74)

Furthermore, for MS-MMSE/BLUE/ML filter banks, it can beshown that . Thus, we have

(75)

This equation allows us to separate the RHP and the left half-plane (LHP) poles of directly. The first determinant onthe right-hand side determines the RHP poles and the secondone determines the LHP poles.

For the MS-MOE filter bank, we do not have a proof to showthat (75) is true for general multipath fading channels. However,it holds for flat-fading channels and, by simulations, we haveheuristically verified the applicability of this expression in mul-tipath fading channels. Thus, we have the following conjecture:(75) provides a general formula of for MRC schemes.

Despite the generic BER formula, our main objective is toinvestigate the relation between BERs and the number of ap-plied stages. By (75), the analysis can be simplified to the ex-amination of the structures of and

for each type of reduced-rank filter banks. Followingthe procedure in [1] for the derivation of reduced-rank filterbanks, expressions of and of each type can also beobtained. Since the derivation contains mainly routine proce-dures, we leave details to Appendix D and summarize results of

for each type of filter banks in Table I.From (43) and Table I, it is clear that all terms are func-

tions of or, equivalently, functionsof , explicitly or implicitly, whose trace is the output SINR ofthe MMSE filter bank. As shown in (39) and Theorem 1, theeigenvalues of increase with the numberof stages and are upper-bounded by . This implies that theRHP poles for each MGF in Table I will move closer to theorigin. Intuitively, the residues of will become smaller,thus bringing down the BER. More detailed analysis will begiven in Section VII along with the BER comparison betweenMRC and EGC schemes.

C. BER for Differential EGC

Two types of multistage schemes will be analyzed for dif-ferential EGC. The first one is the homogeneous EGC, wherea filter’s output is multiplied by the conjugate transpose ofits preceding one, and the corresponding decision statisticis . The other one is the hetero-geneous EGC, where a filter’s output is combined with thepreceding output of a different filter, and its decision statistic


TABLE IIM(s) FOR HOMOGENEOUS EGC SCHEMES

is . For homogeneous EGCMMSE/MOE/BLUE filter banks, can be expressed ofthe following form (for details, please see Appendix E):

(76)

where

and (77)

It is perhaps a little bit surprising to see that expressions in (75)and (76) share the same structural form. Their main differenceslie in the presence of the autocorrelation value [cf. (3)], of twoadjacent channel states in (76), which accounts for the energy

loss due to inexact channel combining, and that and inthe MRC formula (75) are replaced by their squares in the EGCformula (76). These differences result from the fact that thereare two filters’ outputs contributing to the decision statistic ofthe EGC scheme, and that serves as an estimate ofchannel vector . In contrast, there is only one filteroutput involved in the decision statistic of the MRC scheme and

is given in this case. Therefore, only the squared roots ofand appear in for MRC schemes. Moreover, since

perfect channel information is available for MRC schemes, pa-rameter , which is equal to unity, has no influence on the finalexpression of . The expression of for each typeof the homogeneous EGC filter bank is listed in Table II. Again,the forms of are all explicit functions of kernel matrix

. Thus, discussion given in the pre-ceding section on the convergence of BERs for MRC schemesapplies to homogeneous EGC schemes as well.

For heterogeneous EGC, there are two combining schemesof special interest, i.e., the BLUE-MMSE and the BLUE-MLcombinings. The BLUE filter is a minimum variance unbiased(MVU) estimator for in the presence of Gaussian noise,while the MMSE and ML filter banks maximize SINRs in thesenses of (23) and (22), respectively. It is shown in Appendix Fthat for these two combinings is merely a generaliza-

tion of (76) with and . Thus,we have

(78)

where , , , and. The corresponding are given by

-

(79)

-

(80)

where

and

Comparing results for MRC and homogeneous EGCschemes, it is clear that (75) and (76) are only special cases of(78) for general heterogeneous EGC schemes. For MRC, thereis only one filter’s output involved in each decision statistic.

Thus, only and are present in (75). For homogeneousEGC, two outputs of the same filter contribute to each decisionstatistic so that and are present in (76). Although the ap-plicability of (78) to all possible heterogeneous EGC schemesis not fully verified,6 this formula provides a fairly genericBER expression for the two types of combining schemes of ourinterest.

VI. NUMERICAL RESULTS

The analytic performance of various MRC and EGC detectorsis verified and compared in this section by computer simulation.We consider a DS-CDMA system with a multipath Rayleigh-fading channel. Each user’s signal in the system propagatesthrough a three-path fading channel (i.e., ). The channelcoefficients on each path change from symbol to symbol withthe normalized Doppler shift . There are 10users in the system, and the signature sequence for each user israndomly chosen from a set of Gold codes of length .For simplicity, all users are assumed to have the same power, andthe power is equally distributed among the paths of a user. AsISI is ignored in our analysis, we consider a system with limitedISI to validate the analysis conducted in the previous sections.Therefore, the path delay for each user is randomly gener-ated with a uniform distribution over under the boundeddelay spread constraint

(81)

Finally, BPSK and DBPSK modulations are employed, respec-tively, for the evaluation of the BER performance for coherentMRC and noncoherent EGC schemes.

Fig. 5 shows the convergence behavior of the output SINRswith respect to the number of stages for multistage MRC filterbanks at 20 dB. Recall that the rank of these filters isequal to . The stage number may vary from to ,with corresponding to the full-rank implementation.The upper plot of Fig. 5 gives the output SINRs defined by (24).The curves of analytic results are calculated using (30)–(33).

6The simplification procedure seems to vary case by case.


Fig. 5. The convergence behavior of the steady-state output SINRs with respect to the stage number, where the SINR values are defined by (24) in the upper plot,and by (34) for the MRC ML filter bank in the lower plot.

The lower plot of Fig. 5 shows the convergence of the averageSINR for the MRC ML filter bank. The analytic curve is ob-tained using (34). Both SINR measures converge rapidly to thatof full-rank ones.

Fig. 6 presents the BER performance for MRC filter banks.The BERs of reduced-rank schemes with and that offull-rank schemes are simulated and compared with analytic re-sults given in Table I. Note that the ML receiver outperforms theother three filter banks, since the ML filter bank maximizes theconditional SINR given by (22). Analytic results are very closeto simulated ones since only self-induced ISI is ignored in theBER analysis for MRC schemes.

Fig. 7 depicts the BER performance for the reduced-rankand the full-rank schemes, using the EGC MMSE-MMSEand MOE-MOE combining methods proposed in [13] and theBLUE-ML combining proposed in [1]. Analytic results areobtained from (156) and Table II. Among the three combiningmethods, the BLUE-ML filter achieves the best performancesince the BLUE filter bank is the ML channel estimator offor the linear Gaussian model and the ML filter bank is thecorresponding ML symbol detector for the same model. Theanalytic results are somewhat more optimistic in comparisonwith the simulated results since ISI is ignored in the BERanalysis for differential EGC schemes.

In Fig. 8, BERs of EGC schemes, where the BLUE filter bankis used as the channel estimator, is compared with that of MRCschemes, where perfect channel information is assumed avail-able. The number of implementation stages is still set to six.

The performance of two SINR maximizing filter banks, i.e.,MMSE and ML, is shown along with that of BLUE-BLUE com-bining. By considering results in Figs. 7 and 8 jointly, we seethat the BLUE-ML combining gives the best performance overall EGC schemes without introducing extra implementationalcomplexity.

Note that, the filters are apparently not able to fully suppressMAI and ISI. Both MRC and EGC schemes suffer from BERfloors in the high-SNR regime as shown in Figs. 6–8. In addi-tion, the eigenvalues of are also observed to saturate at a valueless than . Thus, we see as shown in Fig. 5. Thisraises a question about a filter’s capacity for interference sup-pression, especially for a reduced-rank filter. In Section VII, therelation between the filter’s rank and its limiting capacity for in-terference suppression will be studied in details, which leads tothe notion of effective user capacity.

VII. EFFECTIVE USER CAPACITIES OF LINEAR MULTIUSER

DETECTORS

The BER floor phenomenon brings up an importation ques-tion about the performance limit of linear multiuser detectors,which specifies the best performance a linear filter can achieve.The study involves quantifying a filter’s capacity for interfer-ence cancellation. There are a number of factors to be taken intoconsiderations, e.g., the number of users, the number of channelpaths, the system dimension , and the channel estimation er-rors. From the BER formulae provided in Section V, is


Fig. 6. The BER performance for MRC MMSE/MOE/BLUE/ML filter banks with D = 6.

Fig. 7. The BER comparison of EGC MMSE-MMSE, MOE-MOE, and BLUE-ML filter banks, where reduced-rank receivers are implemented with six stages.


Fig. 8. The performance comparison of BERs for MRC ML, MRC MMSE, EGC BLUE-ML, and EGC BLUE-MMSE withK = 10, L = 3, andD = 6.

a function of for MRC and EGC either explicitly or implic-itly. Simulation results also show that output SINRs are less thantheir theoretical limit; namely, . These imply thatthe BER floor might be another form of the SINR saturation inmultiuser detection. Thus, we attempt to identify the reason be-hind the SINR saturation and determine conditions required toachieve the theoretical limit. In other words, we make connec-tions between the SINR saturation and the BER floor phenom-enon, and seek factors that cause the BER floor.

A. Achievability of the Upper Bound of Output SINRs

It was shown in (39) that . Here, we further studythe condition for achieving the upper bound of , which is givenin the following proposition.

Proposition 4: if and only if .Proof: We first examine the necessary condition. Recall

the expression of in (46)

(82)

If , then .This implies that

(83)

By the positive semidefinite property of the autocorrelation ma-trix, we get

(84)

Next, we show the sufficient condition. If, then

(85)which leads to . Therefore, we have

.

This proposition implies that the upper bound of can beachieved if and only if the residual interference , whichresults from MAI, ISI, and the additive noise, in the signalsubspace can be fully suppressed with filter . However,having seems practically infeasible since

(86)

for and MAI ISI MAI ISIbeing positive semidefinite. We thus look for alternative objec-tives. They include “What should we do to suppress as muchinterference as possible?” “What are the rules in choosing the


ranks of filters and how are they related to the limiting achiev-able performance?” We address these questions for MRC andEGC schemes, respectively, in Sections VII-B and C.

B. Effective User Capacity

The last term in (86) is of full dimension. There is no waywe can approach the condition for except for .This is typically a condition set for analysis in the high-SNRregime. Hence, we proceed our analysis with the high SNR as-sumption. Under this constraint, , and thus,

MAI ISI MAI ISI

Then, the ”closeness” of to provides an indication for re-ceiver’s capability to suppress MAI and ISI. The larger thevalue, the larger user number a system can support. This leadsto the concept of effective user capacity for linear receivers.

To quantify effective user capacity, we first observe fromFig. 3 that the first stage of the multistage filter is actually amatched filter used to extract the desired user’s signal. Theremaining stages are essentially for interference suppression.Thus, we define as a measure to assess afilter’s potential in interference suppression. The effectivenessof interference suppression also depends on the dimensionthe interference can possibly span. Therefore, a measure forevaluating the dimension of interference is defined tobe the rank of , i.e., the rank of the autocorrelationmatrix of MAI plus ISI [cf. (7)], given by

(87)

Note that the Gaussian noise is excluded from this definition.The mutual effects of these two measures on a filter’s capabilityfor interference suppression is characterized in the followingproposition, which provides a condition in fully suppressing in-terference at high SNR.7

Proposition 5: If and , there exists ansuch that .

Proof: Let . Then, we have, and (46) can be written as

as (88)

7Note that SNR means the ratio of the desired user’s symbol power to thevariance of AWGN, i.e., E =N , which should not be confused with the SINRat filter’s output.

Let , where and, be the singular value decomposition of .

Then, we have

(89)

To fully suppress the interference, we need

For the -stage implementation, is of dimension, and is of and

. Therefore, is of and is of. For each column of , there are equations for

variables. Therefore, if , there exist solutionsto . Then

Consequently, this leads to when. This completes the proof.

Proposition 5 reveals that MAI and ISI can be fully sup-pressed in the high-SNR regime if . Moreover,under this condition, it follows from Propositions 2 and 4 that

will approach asymptotically, as the desired user’s powerincreases. This implies that BERs will approach zero in thehigh-SNR regime if . This argument is confirmedin Fig. 9. In this simulation study, is set to 60 dB suchthat the noise power is negligible, and we have for eachuser so that . The dimension of interference

is shown in the lower plot with respect to the number ofusers in the system. Cross-referencing the upper and lower plots,we see that BERs jump up dramatically as soon as the condition

is violated.We further illustrate this point in Fig. 10 with simulations by

employing a different number of resolvable transmission paths,, for users. We conclude from these two figures that, as long as

a reduced-rank filter meets the requirement of , in-terference can be effectively suppressed in the high-SNR regimeregardless of the rank of the underlying filter.

However, for situations where the strength of Gaussian noiseis comparable to that of MAI and ISI, or when a system is highlyloaded such that , will become saturated quicklyeven if the number of stages increases. This phenomenon wasillustrated in the output SINR curves of Fig. 5. The saturation of

results in BER degradation and BER floors as shown in Fig. 6.We conclude this section by simulation results in Fig. 11, with

, , and . As long as ,which corresponds to the case of and thefull-rank case , the BER floor can be avoided forMRC so that the performance is close to the single-user bound.This gives us a rule of thumb in choosing the number of stages.If the system is not fully loaded, i.e., , we shouldchoose small enough to just pass the requirement of

. Otherwise, we can use the analytical output SINRformulas provided in Proposition 1 to choose that gives anear-full-rank SINR output.

Proposition 5 alone is not sufficient to characterize the BERperformance for EGC receivers. They still suffer from theBER floor as shown in Fig. 11 even though the condition of


Fig. 9. BERs of the reduced-rank MRC MMSE filter bank, whereD = 1; 3; 6; 9; 12 and the full-rank from left to right.

is met. To understand this phenomenon, wewill analyze the BER performance for EGC schemes in Sec-tion VII-C.

C. The Effect of Channel Mismatch on EGC

For brevity, we use the EGC MMSE receiver as an exampleto analyze the performance of EGC schemes in the high-SNRregime. The result can be extended to other EGC schemes sincethey all share the same multistage structure.

First, we perform simplifications to focus on the analysis ob-jective better. We ignore ISI from all users, which is the secondterm on the right-hand side of (18), and rewrite the simplifiedautocorrelation matrix as

(90)

where

and (91)

The interference covariance matrix with respect to the de-sired user is given by . Then, we have

(92)Equation (76) can thus be rewritten as

(93)

Since and for each filter bank are already available(see Appendix D), can be obtained by subtracting from

. For the EGC MMSE filter bank, we haveand , where and

. Thus, . Substitutingthese terms into (93) yields

(94)The first determinant on the right-hand side of (39) deter-

mines the RHP poles, since . To calculate the residues withrespect to RHP poles, we substitute , ,into the second determinant of (94). For simplicity, we considerthe case where all eigenvalues are distinct. By inserting (94) into(67) and calculating the residues, we get

BER -

(95)

Now, let us consider the limiting performance of the EGCMMSE receiver when (or ). In general, forthe multipath case, we need to evaluate the BER by having every

, . To highlight the key effect, we considerthe flat-fading channel case. Thus, and


Fig. 10. BERs of the full-rank MRC MMSE filter bank with N = 31, where L = 1; 2; and 3 from left to right.

. The upper bound of can be easilyobtained by taking its derivative with respect to and setting tozero. Also, by (43), we have

. Thus, we get

(96)

In the high-SNR regime, . The BER of the EGC-MMSEreceiver can be derived as

BER -

(97)

This shows that the BER is controlled by two factors. The firstterm is due to the channel mismatch between two adjacentsymbol intervals, and the second term is related to .Thus, even if in the high-SNR regime such that

and , the BER floor effect still exists fordifferential EGC because of the autocorrelation value of thefading process.

Actually, with a similar argument, BERs of all EGC schemes,heterogeneous or not, can be shown to be equivalent in a flatRayleigh-fading channel. Note also that (97) has the same formas the BER formula for the DBPSK demodulation of nonspreadsystems [17]. The difference is that is replaced by

for the DBPSK demodulation, where and arevariances of the signal and noise, respectively. In our currentcontext, and are variances of the filtered signal and thefiltered interference, respectively.

The remaining question is the degree that the performanceof EGC schemes deviates from that of MRC schemes. For com-parison, we examine the MRC MMSE detector in flat Rayleigh-fading channels. From Table I, of the MRC MMSE re-ceiver is given by

(98)

In flat-fading channels, the resultant BER reduces to

BER (99)

Similar to EGC schemes, BERs of all MRC schemes are thesame in flat-fading channels. For comparison, we can rewritethe BER of EGC schemes as

BER (100)

As shown in (99), the BER of MRC schemes is only a func-tion of the eigenvalue . When , the eigenvaluecan continue to approach its upper bound of the unity asbecomes larger. As a result, can keep decreasingwithout reaching a BER floor. This result is illustrated in Fig. 12where BERs of both EGC BLUE-MMSE and MRC-MMSEschemes are presented. In this simulation, system parametersare chosen as , , , and


Fig. 11. BERs of the reduced-rank EGC BLUE-ML filter bank versus MRC-ML filter bank, where L = 2, K = 10, N = 31. The top graph is for the EGCBLUE-ML filter bank and the lower graph is for the MRC-ML filter bank.

. For full-rank schemes,we have . As we can see, the BERs of EGCschemes hit a lower bound atdespite the fact that the BER curve of MRC keeps deceasinglinearly in the log scale. On the other hand, for reduced-rankschemes, . The eigenvalue saturates ata value due to residual interference. The SINRsaturation leads to the BER floor atfor MRC schemes as well as the BER floor at

for EGC schemes. This explains why BERs of MRCand EGC schemes behave like two horizontal lines traveling ata fixed distance from each other in the high-SNR regime. Thesame arguments also explain similar phenomena observed inFig. 8 with multipath fading channels, where and

.

VIII. CONCLUSION

The SINRs and BERs of multistage MMSE/MOE/BLUE/MLfilter banks were analyzed for linear reduced-rank multiuser de-tection. A generic BER formula was provided for reduced-rankmultistage receivers using either coherent MRC or noncoherentEGC. Two new performance measures, kernel matricesand , were found to characterize the SINRs and BERs ofmultistage receivers in multipath Rayleigh-fading channels. It

was shown that the output SINRs or BERs of different typesof receivers are equivalent in each class of combinings in flatRayleigh-fading channels.

The output SINRs of all receivers were proven to be monoton-ically increasing with the number of applied stages and upper-bounded by a value equal to the number of resolvable paths ofthe desired user’s transmission channel. The achievability of theupper bound was also established and linked to the BER floorphenomenon in MRC receivers. This finding leads to the notionof effective user capacity. A test condition was provided for theexistence of BER floors. As long as the conditionis satisfied, the BER floor can be avoided.

Being different from MRC schemes, the BER floor of EGCreceivers is also governed by the channel correlation value ,due to the channel mismatch occurring in differential com-bining. As a result, the BER floor will exist in fast-fadingenvironments even if the condition of is met.The floor level depends on the value of of the Rayleigh-fadingchannel process. The lower the is, the higher the BER flooris. For channels with rich scattering or higher system loadingssuch that the condition is violated, extensivesimulation results reveal that output SINRs become saturatedin just a few stages. Thus, an implementation with a largelyreduced rank is able to achieve the full-rank performance. It isthis feature that makes the multistage structure more appealingthan other implementations for reduced-rank filtering.


Fig. 12. Performance comparison of BERs for EGC , BLUE-MMSE, and MRC MMSE filter banks over a flat Rayleigh-fading channel, where reduced-rankschemes with D = 11 and full-rank schemes are shown.

APPENDIX APROOF OF PROPOSITION 1

Given channel vector , the decision statistic for MRC isgiven by . From (13), (20), and (21), theexpectations of the decision statistics conditioned on forfour reduced-rank filter banks are given by

(101)

(102)

(103)

(104)

where , ,and . The above results follow from the factsthat

(105)

and

(106)

Note that , , and have been used in place of ,, and , respectively.

Similarly, the common term in (13) can be rewritten into theform of

(107)

Using (107) and the definition of in (20), we can computethe conditional autocorrelation matrix of the estimation error via

(108)

where we use the fact that

[cf. (25)]. By multiplying the conditional autocorrelation matrixwith the scaling matrices defined for (13), we obtain conditionalautocorrelation matrices for four filter banks as

(109)

(110)


(111)

(112)

The unconditional covariance matrices can be obtained using. By applying (27) in com-

puting the fourth moments of for (109)–(112), we obtainvariances of the four filter banks as

(113)

(114)

(115)

(116)

Dividing the mean-squared value of each of four filter bank out-puts averaged over by the corresponding covariance yieldsthe output SINR.

APPENDIX BPROOF OF PROPOSITION 3

To characterize the performance difference, we introduce thefollowing lemma to reduce complexities.

Lemma 2: Given a matrix of dimension and twopositive diagonal matrices , we have

(117)Proof: Let and

where or , with . With the Cauchyinequality, we obtain

(118)

The equality holds if , .

We can now apply this lemma to the proof of Proposition 3.Proof: The upper bound for the MMSE-based filter bank

was already established in (40). Now, we investigate the perfor-mance differences between the reduced-rank MMSE filter bankand the reduced-rank MOE/BLUE/ML filter banks.

A. By performing the eigenvalue decomposition for, , we get

(119)Let . Using Lemma 2, we obtain

(120)

Dividing both sides by resultsin

(121)Thus, we have

SINR SINR (122)

The equality holds if , . Then,i.e., , since . To satisfy the upper bound

B. By performing the eigenvalue decomposition for thekernel matrix of SINR , we have

. Then

By Lemma 2, we get

(123)

Dividing both sides by , we obtain the fol-lowing inequality:

(124)which leads to

SINR SINR (125)

The equality holds if , . Then, ,i.e., . To satisfy the upper bound,

.C. From (44), the output SINRs of the MMSE and MOE filter

banks can be expressed in terms of as

SINR (126)

SINR (127)


Let

Subtracting SINR from SINR , the numerator be-comes

(128)

By induction, (129) can be shown (see the bottom of thepage). This follows since . The equality holds ifand only if , . By (44),

D. From B and C, if with , then

SINR SINR SINR

From A, if , , then

SINR SINR

Let , i.e.,

we have

SINR SINR SINR SINR

Combining the above conditions, the equality holds if, and , .

APPENDIX CPROOF OF THEOREM 1

Proof: To prove the theorem, we need to explore the recur-sive structure of in (54). For simplicity, we use to de-note that is positive definite (p.d.), and useto denote that, for ordered eigenvalues of Hermitian matricesand , , . Since is an error cor-relation matrix .8 It is assumed that the filter’s stage isincreased from to .

First, we need to show

Let . By the matrix inversion lemma, we have

8We assume that noise variance N > 0.

(130)

Let be the congruence transformation of . Sinceand is nonsingular, by Sylvester’s law of inertia [18],. Similarly, since is nonsingular, .

Continuing the same procedure, we have every term on the right-hand side being positive definite.

Recall the minimax principle [18]; namely, if matrix, then for any Hermitian matrix , we have .

Staring from the second term of (130) and recursively applyingthe minimax principle, we obtain .Thus,

Similarly, with and

we get

(131)

Let denote of the -stage MMSE filter bank. Then, wehave

by adding the th stage.Second, we would like to show that

, given that

Recall the recursive structure of in (54). Let

and

By the matrix inversion lemma,

(132)

SINR SINR (129)


Given that

we have by the minimax principle. Since ,following the same procedure for (130), we obtain andtherefore, .

Now, starting from the bottom stage, given that

(133)

we have

(134)

which, in turn, results in

(135)Eventually, we have . Notethat and are alternative notations for and ,respectively. Thus, we conclude that

APPENDIX DDERIVATION OF TABLE I

It is clear from (9)–(12) that the simplification of andis related to that of , , and . Consider reduced-rankimplementations of stages. By the matrix inversion lemma[16], we have

(136)

and, by construction, we have

(137)

Applying the above equations to the simplification of

and for each type offilter banks given by (9)–(12) leads to the following results,shown in the table at the bottom of the page.

Substituting these terms into (75) for each filter bank givesthe results of in Table I.

APPENDIX EDERIVATION OF (76)

Define . Itcan be expressed as

(138)

Since and ,we have

(139)

Substituting into (64) gives

(140)

Ignoring the ISI effect in , which is the second term in (138),leads to . Since is Hermitian, the matrix in (140)is also Hermitian. Let and in(70), it is straightforward to show that

(141)

Applying this formula to (140) yields

(142)

where and . Substi-tuting and with results obtained in Appendix D for allreceivers results in as shown in Table II.

APPENDIX FDERIVATIONS OF EGC BLUE-MMSE AND BLUE-ML

FILTER BANKS

For heterogeneous combining schemes

By ignoring ISI, we have

(143)

Applying the above equation to EGC MMSE-BLUE andML-BLUE schemes allows us to perform the BER analysis for

MMSE

MOE

BLUE

ML


each combining scheme. Before getting into the detailed deriva-

tion, we define and ,. From Appendix D, we know that these terms are

both Hermitian. Thus, we redefine and

for the heterogeneous combining since thecontributions now come from two different filters. We divideour discussion into the following two cases.

CASE I: BER for the reduced-rank EGC BLUE-MMSE filterbank

Let . With the same pro-cedure given in Appendix D, it is easy to show that

(144)

(145)

(146)

Thus, we obtain

(147)

Let . Applying the lemma of (70), we get

(148)

Since and , we have

(149)

where .

CASE II: BER for the reduced-rank EGC BLUE-ML filterbank

Let . Similarly, we obtain

(150)

(151)

(152)

where . Therefore, wehave

(153)

Thus, and . Then

(154)

Let and . Again, using (70)followed by some mathematical manipulations, we obtain

(155)

Note that and share common eigenvectors, so do and. Therefore, , since all matrices in-

volved share common eigenvectors [18]. With this property, wehave

(156)

REFERENCES

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[13] S. L. Miller, M. L. Honig, and L. B. Milstein, “Performance analysisof MMSE receivers for DS-CDMA in frequency-selective fading chan-nels,” IEEE Trans. Commun., vol. 48, no. 11, pp. 1919–1929, Nov. 2000.

[14] S. Haykin, Adaptive Filter Theory, 3rd ed. Englewood Cliffs, NJ: Pren-tice-Hall, 1996.

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[16] S. M. Kay, Fundamentals of Statistical Signal Processing: EstimationTheory. Englewood Cliffs, NJ: Prentice-Hall, 1993.

[17] J. G. Proakis, Digital Communications, 3rd ed. New York: McGraw-Hill, 1995.

[18] G. Strang, Linear Algebra and Its Application, 3rd ed. Orlando, FL:Harcourt, Brace, Jovanovich, 1988.

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