Joint space-time auxiliary-vector filtering for DS/CDMA systems with antenna arrays

1406 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 47, NO. 9, SEPTEMBER 1999

Joint Space–Time Auxiliary-Vector Filtering forDS/CDMA Systems with Antenna Arrays

Dimitris A. Pados,Member, IEEE,and Stella N. Batalama,Member, IEEE

Abstract— Direct-sequence/code-division multiple-access(DS/CDMA) communication systems equipped with adaptiveantenna arrays offer the opportunity for jointly effective spatialand temporal (code) multiple-access interference (MAI) andchannel noise suppression. This work focuses on the developmentof fast joint space–time (S–T) adaptive optimization proceduresthat may keep up with the fluctuation rates of multipath fadingchannels. Along these lines, the familiar S–T RAKE processoris equipped with a single orthogonal S–T auxiliary vector (AV)selected under a maximum magnitude cross-correlation criterion.Then, blind joint spatial/temporal MAI and noise suppressionwith one complex S–T degree of freedom can be performed.This approach is readily extended to cover blind processing withmultiple AV’s and any desired number of complex degrees offreedom below the S–T product. A sequential procedure forconditional AV weight optimization is shown to lead to superiorbit-error-rate (BER) performance when rapid system adaptationwith limited input data is sought. Numerical studies for adaptiveantenna array reception of multiuser multipath Rayleigh-fadedDS/CDMA signals illustrate these theoretical developments.The studies show that the induced BER can be improved byorders of magnitude, while at the same time significantly lowercomputational optimization complexity is required in comparisonwith joint S–T minimum-variance distortionless response orequivalent minimum mean-square-error conventional filteringmeans.

Index Terms— Adaptive equalizers, antenna arrays, code-division multiple access, fading channels, interferencesuppression, multipath channels, spread-spectrumcommunication.

I. INTRODUCTION

W HILE signal spreading (DS/CDMA) may lead to signif-icant capacity increases for wireless cellular and PCS

(personal communications services) networks [1], additionallarge capacity gains are available from exploiting the spatiallocation of cellular system users [2]. This is possible when thebase stations are equipped with multiple antenna elements.Usually, an antenna array is a set of identical omni-directional transceivers arranged in a straight line (linear array)

Paper approved by R. Kohno, the Editor for Spread-Spectrum Theory andApplications of the IEEE Communications Society. Manuscript received April14, 1998; revised December 21, 1998. This work was supported by theNational Science Foundation under Grant CCR-9805359 and the U.S. AirForce Office of Scientific Research under Grant F49620-99-1-0035. This paperwas presented in part at the 1998 Conference on Information Sciences andSystems, Princeton University, Princeton, NJ, March 1998, and at the 1998International Conference on Telecommunications, Porto Carras, Greece, June1998.

The authors are with the Department of Electrical Engineering, StateUniversity of New York at Buffalo, Buffalo, NY 14260-2050 USA (e-mail:[email protected]; [email protected]).

Publisher Item Identifier S 0090-6778(99)07459-0.

or in a circle (circular array) [3]. Certainly, other geomet-ric configurations are possible, and nonidentical orthogonallypolarized elements can also be deployed [4]. Theoretically,an antenna array of elements can provide a mean signalpower gain of over spatially and temporally white additiveGaussian noise (linear signal-to-noise ratio (SNR) increase bya factor of [5].

In contrast to fixed sectorized antenna arrays (also knownas switched-beam systems) and phased-arrays where only thesignal phase is adjustable, -element adaptive antenna arraysare supposed to be gain- and phase-adjustable and able toperform online beamforming with full degrees offreedom [2], [4], [6]. Incorporation of adaptive antenna arraysin direct-sequence/code-division multiple-access (DS/CDMA)cellular systems, which is the topic of this paper, is presentlygaining significant interest. Examples of recent work includesignature matched-filtered code processing combined with alightweight principal component analysis approach for spaceprocessing (in [7] and [8] and with capacity studies in [9])and static (that is nonadaptive) space–time (S–T) processingin the form of a bank of decorrelators, one per antenna element[10]. Adaptive processing proposals involve disjoint spaceminimum mean-square error (MMSE) [minimum-variance dis-tortionless response (MVDR)] optimum and RAKE tempo-ral processing, or disjoint MMSE (MVDR) optimum spatialand MMSE (MVDR) optimum temporal, or jointly opti-mum MMSE (MVDR) S–T processing [11] with post-filteringViterbi decoding in [12], [13]. Other recent contributionsinclude joint S–T RAKE processing [14], low-complexityS–T RAKE receivers [15], and joint S–T adaptive MVDRprocessing [16]–[18] for systems with aperiodic spreadingcodes.

In this work we wish to maintain the principles ofjointS–T adaptive processing for jointly effective S–T detection ofthe information bits of the user(s) of interest in the presenceof DS/CDMA multiple-access interference (MAI), multipathfading phenomena, and additive ambient multichannel noise.We recognize that attempts for disjoint space and time opti-mization may fall far behind the potential of joint processingas shown in [19] in the context of array radar detection.

On the other hand, for a DS/CDMA system withantenna elements, system processing gainand resolvablemultipaths (usually is between 2 and 4 including the directpath, if any [4]), jointly optimum processing under minimummean-square criteria requires processing in theS–T product space. As an example, ifand filter optimization needs to be carried out

0090–6778/99$10.00 1999 IEEE

PADOS AND BATALAMA: DS/CDMA SYSTEMS WITH ANTENNA ARRAYS 1407

in the complex vector space. Adaptive sample-matrix-inversion (SMI) implementations of the MMSE/MVDR filtersolution are known to require data samples several times theS–T product to approach the performancecharacteristics of their ideal counterparts [20], [21]. In fact,theoretically, system optimization with less thandata samples is not even possible due to the singularity ofthe underlying sampled autocovariance matrix. With CDMAchip rates at 1.25 MHz [4], and typical fading ratemeasurements between 4.5 Hz for mobiles on foot and 70 Hzfor vehicle mobiles (carriers are assumed at 900 MHz) [2],the fading channel may fluctuate significantly—as quickly asevery 280 data symbols. In this context, adaptive SMI filteroptimization in the vector space becomes an unrealisticgoal.

This paper focuses on fast adaptive joint S–T optimizationthrough small data sets that can “catch up” with multipath fad-ing communications channels. In this direction, first we extendthe auxiliary-vector (AV) framework—originally introduced in[22] for time-domain processing of DS/CDMA signals—tocover joint S–T processing in complex data vector spaces. Inthis context, the familiar joint S–T RAKE filter is accompaniedby a single orthogonal auxiliary vector selected under amaximum magnitude cross-correlation criterion for blind S–TMAI and noise suppression withone degree of freedom.Further generalization leads to adaptive generation of multipleauxiliary vectors, which are orthogonal to each other and to theS–T RAKE filter, for blind S–T MAI and noise suppressionwith any desirable number of degrees of freedom belowthe S–T product. Interestingly enough, this generalizationparallels well-known “blocking-matrix” processing techniquesthat have been used successfully for radar and antenna arrayapplications in the past [23]–[26] and recently for time-domain-only CDMA signal processing [27]. However, to avoidSMI altogether even for reduced-dimension

blocking-matrix processing, here we generate a data-dependent sequence ofauxiliary vectors through successiveconditional optimization. Performancewise, we maintain thatin small-sample support situations, conditional optimizationoutperforms -dimension blocking-matrix processing withoutever inverting the underlying covariance matrix.

In summary, the developments in this work cover adaptivejoint S–T processing procedures with any desirable numberof degrees of freedom, from one up to the full S–T product.The choice, of course, depends on the size of the assumedavailable data set, which in turn depends on the channel fadingrate and other pertinent considerations. Given the prevailingDS/CDMA system parameters and mobile channel fadingrates, we have to argue in favor of very few degrees of freedomwith conditional optimization, as we discuss in the sequel.

The rest of this paper is organized as follows. The all-important received signal model is presented in Section II.The core concept of AV filtering is developed in Section III.Extensions in the form of blocking-matrix processing and se-quences of conditionally optimized weighted auxiliary vectorsare pursued in Section IV. Numerical results are presentedin Section V, and a few concluding remarks are drawn inSection VI.

II. SIGNAL MODEL

Receiver design strategies and signal processing algorithmsare dictated by signal and channel modeling assumptions. Asa brief overview, in this work we consider mobile-to-basetransmissions of DS/CDMA users simultaneous in timeand frequency with processing gain The transmissions takeplace over multipath fading additive white Gaussian noise(AWGN) channels. The fading is modeled by multiplicativecomplex Gaussian random variables (Rayleigh-distributed am-plitude and uniformly distributed phase). The multipath fadingchannels are modeled by tapped-delay lines with independentfading per path. The received signal is collected by a narrow-band antenna array of elements. For illustration purposes,we consider uniform linear arrays. Identical fading is assumedto be experienced by all antenna elements for each path ofeach user signal (no antenna diversity). Details and notationare given below.

The contribution of the th user, to thetransmittedsignal is denoted by

(1)

where is the th transmitted data (information)bit, denotes energy, is the carrier phase with carrierfrequency and is the assumed normalized usersignature waveform given by

(2)

where is the th bit of the spreading sequenceof the th user, is the chip waveform, is the chipperiod, and if in (1) denotes the information bit durationthen is the spreading processing gain.

After multipath fading channel “processing,” the total signaldue to all users received at the input of a narrow-band uniformlinear array of elements is given by

(3)

where is the total number of resolvable multipaths (withoutloss of generality, the number of resolvable multipaths isassumed to be the same for all users) and

are independent zero-mean complexGaussian random variables that model the fading phenomenaand are assumed to remain constant over several bit intervals.We recall that, in fact, measurements have shown that mobileson foot operating at MHz induce a typical fadingrate of about 4.5 Hz, while fast-moving vehicles may causefading rates as high as 70 Hz [2]. Moreover,in (3) denotesthe relative transmission delay of userwith respect to user0 with and with bandlimited to , thetap-delay line channel model has taps spaced at chip intervals

The array response vector for the th path ofthe th user signal is defined by

(4)


where identifies the angle of arrival of theth multipathsignal from the th user, is the carrier wavelength, andisthe element spacing of the antenna (usually Finally,

in (3) denotes an -dimension complex Gaussian noiseprocess that is assumed white both in time and space.

After carrier demodulation

(5)

where withbeing the total carrier phase absorbed into the channel

coefficient. Assuming synchronization at the reference antennaelement with the signal of the user of interest,for example, user 0, direct sampling of at the chip rate

(or chip-matched filtering and sampling at the chip rate)over the multipath extended chip period preparesthe data for one-shot detection of theth information bit ofinterest We visualize the S–T data in the form of a

matrix

(6)

To avoid cumbersome two–dimensional filtering opera-tions and notation, we decide at this time to “vectorize”

by sequencing all matrix columns in the formof a vector

(7)

From now on, denotes the joint S–T data in thecomplex vector domain.

The cornerstone for any form of joint S–T filtering is theS–T RAKE filter that we define compactly for user 0 as thecross correlation between the received S–T dataand thedesired information bit

(8)

where the statistical expectation operation is taken withrespect to only. Equation (8), in the form of sampleaveraging, can also serve as a supervised estimator of the S–TRAKE filter for slowly fading channels

(9)

where is a sequence of S–T received data vectors.We recognize that if sample averaging extends over multiplemultipath channel realizations , then convergeseither to the direct path S–T signature of user 0 when Riceanfading is assumed or to the0 vector when no direct path exists,i.e., Rayleigh fading is assumed. In either case, in order tocapture the prevailing full S–T effective signature of user 0, (9)needs to be continuously reinitialized at a rate that is consistentwith the fading rate or a corresponding forgetting factor needsto be embedded in the sample average procedure [28]. We arecareful to point out that estimation of the S–T RAKE filter

through supervised or blind estimation procedures as in [8],[9], [14], [15], or [29] is not the primary subject of this work.What we are concerned with in this paper is to minimize theinduced bit-error rate (BER) of the overall system with a givenS–T RAKE vector and a few data samplesCellular system designers may then take the available lowerBER to increase the number of active users for a given BERquality threshold, or to reduce the transmit power of the mobilehandset, or to increase the range of the base station transceiver(cell size).

III. B LIND AUXILIARY -VECTOR JOINT

SPACE-TIME (S–T) PROCESSING

With signal model and notation already in place, we beginthis section with a direct statement of the problem that weinvestigate. With respect to a specific DS/CDMA user ofinterest (user 0), we assume that we have available the jointS–T RAKE filter as defined in the previous section and afinite set of joint S-T data No knowledgeof the information bits of the user of interest in

is assumed and the MAI S-T population is consideredunknown (unknown interfering user signatures, locations, andmultipaths). We wish to determine the best detection strategyfor a DS/CDMA system with antenna elements, spreadinggain and resolvable multipaths per user, given the S-TRAKE vector and a total number of S-T data vectors.

The first well-known approach that we may consider is thestraight S-T RAKE-based [11], [30] detector that does notutilize the available data set

(10)

where denotes the information bit decision for the user ofinterest, identifies the sign operation (zero-thresholdhard limiter), extracts the real part of a complexnumber, and is the Hermitian of We note that jointS-T RAKE processing is equivalent to cascade space and timeRAKE processing, for cases in which cascade realizations arepreferred for implementation purposes. In any case, the outputvariance of the S-T RAKE filtered data is

(11)

whereAnother well-known approach is joint S-T MMSE optimum

filtering (proposed in [11] for joint S-T processing and in[31]–[34] for time-only processing of CDMA signals). It isstraightforward to obtain the Wiener filter solution [28] forour problem if we view the sequence of information bits ofthe user of interest as the “desired” filter output when the filterinput is the S-T data vectors in (7). The solution is

(12)

where is an arbitrary positive scalar, is the joint S-TRAKE filter previously defined, and If wechoose then we obtain the well-knownequivalentMVDR filter (which was used in [35] for time


processing of plain asynchronous channels and in [36], [37]for time processing of multipath channels)

(13)

for which and

(14)

Direct comparison of (12) or (13) with the S-T RAKE filtershows that S-T RAKE processing is mean square (MS)

optimal only if the joint S-T disturbance (multichannel MAIand noise) is white Gaussian. Certainly, this is not the casefor the general multiuser multipath CDMA signal model in(5). To tap, however, the merits of optimum MS processingand to form the filter, exact knowledge of the S-Tdata autocovariance matrix is required. Since is unknown,we may use the available data set to form a sample-averageestimate and then proceed with SMI

[20]. In this case, and bitdecisions are made as follows:

(15)

It is known that batch SMI procedures outperform su-pervised least mean square (LMS) implementations of theMMSE filter [20] or blind-constraint LMS implementationsof the MVDR solution in terms of small-sample convergencecharacteristics [6]. However, at least

data samples are required for to be invertible withprobability one, and in fact, data sizes many times the S–Tproduct are necessary for to approachthe performance characteristics of the ideal filterreasonably well. Unfortunately, for typical valuesand data transmission rates, filter adaptation over the requirednumber of symbol intervals can not keep up with typicalchannel fluctuation (fading) rates, as explained earlier inSection I.

This discussion brings us to the core concept of this paper.First, without loss of generality, let us assume that the S-TRAKE filter is normalized, that is Then,we consider an arbitrary fixed “auxiliary” vector that isorthonormal with respect to , i.e.,

and (16)

We propose S-T adaptive processing witha single joint S-Tdegree of freedom in the form of where [22]

(17)

Decision making is carried out as seen in Fig. 1

(18)

With an arbitrary but fixed auxiliary vector that satisfiesthe constraints of (16), the filter in (17) can be optimizedwith respect to the single complex scalarThe MS-optimum

Fig. 1. Joint S–T auxiliary-vector detection.

value of can be identified from two different—yet equiv-alent—points of view. Since is by con-struction distortionless in the direction , forany we may seek the value of that minimizesthe output variance (energy) This way wesuppress in the MS-sense the overall disturbance contributionat the filter output (multipath-processed MAI and multichannelambient noise) without any cancellation (suppression) of thesignal of the user of interest. An equivalent interpretationmotivated by Fig. 1 is to look for the value ofthat minimizesthe MS difference between the S-T main-beam (S-T RAKE)processed data and the orthogonal AV processed data

The solution to the firstminimum-output variance problem can be obtained by directdifferentiation of the variance expression with respect to thecomplex scalar The solution to the second mean square-error (MSE) problem can be obtained by direct application ofthe optimum linear MS estimation theorem [28]. The solutionis, of course, identical in both cases, and it is identified by thefollowing proposition.

Proposition 1: The value of the complex scalar that

minimizes the output variance of the filteror minimizes the MSE between the main-beam processed data

and the AV processed data is

(19)

Proposition 1 identifies the optimum value of for anyfixed auxiliary vector We now turn our attention to theselection of an auxiliary vector subject to an appropriatelychosen criterion. We understand that there exists a vectorin the subspace orthogonal to for which and theoptimum filter coincide. This solution, however, is notdesirable since it reverts to processing with fulldegrees of freedom and requires inversion of the pertinent S-T data autocovariance matrix. The selection criterion for theauxiliary vector that we propose in this work is motivatedby the MSE interpretation of the filter in Fig. 1, and it isthe maximization of the magnitude of the cross correlationbetween the processed data and the AV processeddata subject to the constraints in (16)

subject to and (20)


Fig. 2. “Blocking” matrix processing withP S–T auxiliary vectors.

In terms of a physical intuitive interpretation of this selectioncriterion, we might say that the orthonormal to auxiliaryvector that maximizes can capture the most—inthe maximum-magnitude cross-correlation sense—of the dis-turbance contribution present at the S-T RAKE filter output.“Captured” disturbance contributions are then subtracted fromthe S-T RAKE output as explained in the context of Propo-sition 1 and shown in Fig. 1. We note that both the criterionfunction and the orthonormality constraints are phaseinvariant. In other words, if maximizes under thegiven constraints, so does for any phase Therefore,without loss of generality, we may identify the unique auxiliaryvector that is a solution to the constraint optimization problemin (20) and places on the nonnegative real line

The following proposition identifies thisoptimum auxiliary vector according to our criterion. The proofis given in the Appendix.

Proposition 2: The auxiliary vector

(21)

maximizes the magnitude of the cross correlation between themain-beam processed data (w.l.o.g. and theAV processed data subject to the constraints

andProposition 2 completes the design of the joint S-T

auxiliary-vector detector. As a brief summary, the sequenceof calculations is as follows: is given by (21), thenis calculated by (19), the filter structure takes the form of(17), and decisions are made according to (18). The S-Tdata autocovariance matrix that appears in the expressionfor the ideal and may be sample average estimated,

, as usual.Straightforward algebraic manipulations show that the out-

put variance of the AV filter as a function of the auxiliaryvector is

(22)

It is interesting to observe that, as it turns out, the AV selectioncriterion that we proposed corresponds to the maximization ofthe numerator of the second term in the expressionsubject to the constraints in (16). Maximization of the secondterm as a whole, that is including the denominator, with respect

to with and unconstrained norm leads tooptimization with full degrees of freedomthat takes us back to the optimum MVDR solutionFinally, direct comparison of the variance expressions in (11),(14), and (22) shows that

(23)

with equalities throughout if is diagonal (white S-T dis-turbance). To the extent that linear-filtered (with

taps) multichannel multipathed MAI can be closelymodeled by Gaussian noise [38], (23) offers a direct rankingof the respectiveideal BER’s (that is forperfectlyknown S-T autocovariance matrices and inverses andperfectlyknownS-T RAKE vectors).

In the following section, we extend Propositions 1 and 2 tocover joint S-T processing with multiple auxiliary vectors. Wefocus on sequential conditional auxiliary-vector optimization.

IV. PROCESSING WITHMULTIPLE AUXILIARY VECTORS

We consider a set ofauxiliary vectors that are orthonormal withrespect to each other and to the normalized S-T RAKE vector

We may organize the auxiliary vectors in the form ofan matrix whereconstitutes the th column of

(24)

With this setup, the orthonormality conditions translate to

and (25)

We call a “blocking” matrix because it blocks signalcomponents in the S-T direction of interest Next, weconsider S-T processing of DS/CDMA signals with jointS-T degrees of freedom in the form of a vector , asshown in Fig. 2. In this context, the overall linear filter canbe written compactly as

(26)

and decisions are made according to

(27)


Blocking matrix processing techniques have been of consid-erable interest in antenna and radar array processing applica-tions in the past [23]. In fact, some recent research effortsfocus on the selection of the blocking processor based oneigenvector metrics [39], [40] or data-independent discrete-Fourier transforms or Walsh-Hadamard transforms [27]. In thispresent work, instead of eigen analysis or data independentprocessing, we wish to extend the design criterion introducedin the previous section for processing with a single auxiliaryvector to cover blocking-matrix processing with

auxiliary vectors.For an arbitrary but fixed blocking matrix that satisfies

the constraints in (25), the linear filter can be optimizedwith respect to the -dimension complex vector Thereare two equivalent interpretations for the MS-optimum valueof the weight vector Since the overall blocking-matrixfilter in (26) is by construction distortionless in thedirection of interest we may search for theweight vector that minimizes the output variance (energy)

This would suppress all signal components thatare not in the S-T direction of interest. Equivalently, wemay instead identify the vector that minimizes the MSdifference between the main-beamprocessed data and the blocking-matrix processed data

The solution to the first problem can be obtained bydirect differentiation of the variance expressionwith respect to the complex vector The solution to thesecond problem falls in the context of optimum linear MSestimation [28]. The solutions, of course, coincide, and theoptimum weight vector is identified by Proposition 3 below.This result is a generalization of Proposition 1 of the previoussection.

Proposition 3: The -dimension,complex weight vector that minimizes the output variance

of the filter or minimizes the MSE betweenthe main-beam processed data and the blocking-matrixprocessed data is

(28)

With this result, the output variance of theideal filteras a function of the blocking matrix can be seen to be

(29)

Blocking matrix processing with joint S-T degrees offreedom requires inversion of the blocked data auto-covariance matrix as seen in (28). To avoid matrixinversion operations altogether, we may consider recursiveconditionaloptimization of the weighted auxiliary vectors.

The anticipated benefits are: 1) significantly lower computa-tional optimization complexity than unconditionally optimumblocking-matrix processing; and 2) superior BER performancefor small-sample support optimization. The latter is anticipateddue to the curse of dimensionality in data-limited environ-ments, where sequential, one-by-one, conditional optimizationof the AV weights is, in general, preferable to joint vectoroptimization by the numerically unstable operation in (28).Parallel to the blocking-matrix filter definition in (26), letus introduce a new filter consisting of

weighted auxiliary vectors that areorthonormal with respect to each other and to the normalizedS-T RAKE vector

(30)

and are optimized exactly as in (21) and (19) inSection III. For convenience, we repeat the formulas below

(31)

(32)

Next, we upgrade the “main-beam” to and seekan auxiliary vector that maximizes the cross-correlationmagnitude , subject to the orthonormalityconstraints and Arguingas in Proposition 2, we find that is as shown in (33) at thebottom of the page. The conditionally MS-optimum value of

for given and is

(34)

To complete the recursive construction of the filter in (30),we assume that are defined for

Then, with “main-beam” the auxiliaryvector that maximizes the cross-correlation magnitude

subject to the orthonormalityconstraints is , as shown in (35), at the bottom of thenext page. The conditionally MS-optimum coefficientbecomes

(36)

When the blocking matrix in (24) consists of theauxiliary vectors defined recursively by (35) and (36), thevariance expressions in (11), (14), (22), and (29) are related

(33)


as follows:1

(37)

In (37), equality holds when is diagonal (no MAI andperfectly white S-T disturbance). When

WhenThe latter statement reinforces the

known result that MS-optimum filtering in thevector space can be achieved with filter optimization in a

vector subspace. If we assume that linearlyfiltered multipath S-T MAI is approximately Gaussiandistributed [38], then (37) identifies the ranking in termsof BER performance of the respectiveideal filters (perfectknowledge of the S-T autocovariance matrix and the S-TRAKE vector is assumed). In small-sample support situations,a sequence of conditionally optimized auxiliary vectors isexpected to lead to superior BER performance.

In the following section, we compare the plain S-T RAKEfilter the full-scale MVDR filter in (13), the singleAV filter in (17), the blocking-matrix filter in (26),and the conditionally optimized sequence of auxiliary vectorsin the form of the filter in (30). The comparisons arefocused on the induced BER as a function of the requiredsample support.

V. NUMERICAL AND SIMULATION STUDIES

We consider the DS/CDMA signal model in (5) for a systemwith antenna elements and processing gain Weassume the presence of active users with fixed signatureassignments and the normalized synchronous signature crosscorrelations between the user of interest and the five interferersare chosen between 0.2 and 0.3. Each user signal experiences

independent Rayleigh fading paths with equal averagereceived energy per path and independent angles of arrival uni-formly distributed in Antenna diversity effectsare not pursued. The array interelement spacing is taken to beone-half the wavelength, and identical fading is assumed tobe experienced by all antenna elements for each path of eachuser signal.

In Figs. 3–7, we examine the BER performance of all filtersdiscussed in this work as a function of the total SNR of

1Instead of an algebraic proof of (37), we may note that all filters areconstrained to have unit response in the direction of interestVVV 0: Then,wwwMVDR is the minimum output variance filter within the whole S-T productvector space,wwwBBB is the minimum output variance filter within the subspacespanned byVVV 0 and the linear combinations of the columns ofBBB,wwwc lies in thesame subspace aswwwBBB but is conditionally optimized and, hence, suboptimumwith respect towwwBBB , wwwAV is the single auxiliary-vector version ofwwwc, andwwwRAKE is thewwwc filter with no auxiliary-vector components at all.

Fig. 3. BER versus total SNR for the S–T MVDR filter with varying samplesupport.

the user of interest (the sum of the received SNR’s overall paths) and the sample support. Following the notation in(5), the total received SNR for user isdefined by where is the varianceof the sampled AWGN, which was assumed to be identicalfor every spatial channel (antenna element). All BER’s for allfilters areanalytically evaluated, that is for every given S-Tmultipath channel realization and every given filter realization,the induced BER is calculated analytically as the expectedprobability of error over all possible user–bit combinations.The results that we present are averages over 100 indepen-dently drawn multipath Rayleigh fading S-T channels and 10independent filter realizations per S-T channel.

In Fig. 3, we study the BER characteristics of the MVDRfilter as defined by (13). The total SNR of the interferers isfixed at 7, 8, 9.5, 10.5, and 12 dB, respectively. Fig. 3 plots theBER of the user of interest as a function of its total SNR overthe 0–12-dB range for various data set sizes. The BER’s ofthe ideal S-T RAKE and the ideal MVDR filter are includedas reference points. While the ideal MVDR filter promisesan improvement of many orders of magnitude over the idealS-T RAKE filter, it is rather disappointing to observe thateven for large data sets of size , thedata-estimated MVDR filter is barely superior to the ideal S-TRAKE filter.

Fig. 4 replicates the studies of Fig. 3 for the blocking-matrix filter in (26) with auxiliary vectors selectedaccording to recursion (35), (36). Some improvement withrespect to small-sample support BER’s can be seen.

(35)


Fig. 4. BER versus total SNR under varying sample support for a S–Tblocking-matrix filter with five auxiliary vectors andunconditionally optimumAV weight vector��:

Fig. 5. BER versus total SNR for the single auxiliary-vector filter withvarying sample support.

Fig. 5 presents the same studies for the single auxiliary-vector filter in (17). Significant BER improvement isachieved for data set sizes equal to or a few times the S–Tproduct

In Fig. 6, we maintain the same setup and study the BERbehavior of a conditionally optimized sequence ofauxiliary vectors in the form of the filter in (30). Theseresults indicate superior performance by orders of magnitudecompared with the previously examined filters for samplesupport between and

Finally, Fig. 7 assumes fixed total SNR for the user ofinterest at 12 dB and merges the results of Figs. 3–6 in theform of BER plots versus required number of data samples.The superiority of the conditionally optimized sequence ofauxiliary vectors (filter with ten auxiliary vectors) isapparent over the whole to

data-support range. In fact, over the

Fig. 6. BER versus total SNR for a sequence of tenconditionallyoptimizedauxiliary vectors with varying sample support.

Fig. 7. BER versus number of data samples for the filters examined inFigs. 3–6 (total SNR of user of interest fixed at 12 dB).

to data-support range ofprimary interest, even the simple lightweightsingleauxiliary-vector filter outperforms, by a couple of orders of magnitude ormore, its computationally intensive blocking-matrix optimizedand full-scale MVDR counterparts.

VI. CONCLUSIONS

Typical fading rate measurements for mobile communica-tion channels with carriers in the vicinity of 900 MHz showthatadaptivereceiver designs for data communication rates onthe order of 19.5 symbols per second may not rely on morethan a couple of hundred data symbols for system adaptation.Linear MS-optimum S–T processing for DS/CDMA antennaarray systems requires optimization in the joint S–T productvector space. The dimension of the joint S–T vector spacetypically exceeds the available data-support size.

In search of adaptive optimization procedures that exhibitlow-computational requirements and superior performance in


(42)

small-sample support environments, we introduced the con-cept of maximum cross-correlation S-T auxiliary-vector (AV)filtering. Then, the extension for adaptive processing with a se-quence of conditionally optimized weighted auxiliary vectorsis readily available. We saw that simultaneous vector opti-mization of the AV weights in the context of blocking-matrixprocessing offers a moderate improvement over standard full-scale MVDR processing. In contrast, the inductive conditionalAV optimization procedure leads to greatly improved BERperformance characteristics when system adaptation relies onlimited data. In this context, conditional AV processing appearsa viable approach for adaptive S-T MAI and noise suppressionin common multipath fading mobile communication channels.

APPENDIX

Proof of Proposition 2: Without loss of generality, we as-sume that (real and nonnegative). We form theLagrarian function

(38)

and we set its conjugate gradient equal to the null vector

(39)

The enforcement of the constraints onin (39) results in asystem of equations to be solved for and Indeed, weobtain

and

(40)

Substitution of and back in (39) leads to the desiredauxiliary vector that satisfies the constraints in (16) andmaximizes the cross-correlation magnitude in (20)

(41)

This result combined with Proposition 1 shows that the overallfilter is actually given by , as shown in(42) at the top of the page, where for simplicity, we assumed

We also note that the above optimization procedure leads tothe same filter regardless of the particular value used forthe constraint on the norm of

ACKNOWLEDGMENT

The authors would like to thank the three anonymousreviewers for their valuable comments and suggestions on theoriginal manuscript. The assistance of G. N. Karystinos inobtaining the numerical and simulation results of Section V isgratefully acknowledged.

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Documents

Joint space-time auxiliary-vector filtering for DS/CDMA systems with antenna arrays