2758 ieee transactions on signal processing, vol. …wcan.ee.psu.edu/papers/sy_tsp2005double.pdfonal...

2758 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 7, JULY 2006

MIMO-CDMA Systems: Signature and BeamformerDesign With Various Levels of Feedback

Semih Serbetli, Member, IEEE, and Aylin Yener, Member, IEEE

Abstract—We investigate the signature and beamformer designproblem for the uplink of multiple-input multiple-output code-di-vision multiple-access (MIMO-CDMA) systems with the sumcapacity and the systemwide mean-square error (MSE) as perfor-mance metrics. We first construct iterative algorithms to find thejointly optimum temporal signatures and transmit beamformersunder the assumption of perfect temporal signature and transmitbeamforming feedback. Next, motivated by the need to reducethe amount of feedback, we present a low-complexity sequentialorthogonal temporal signature assignment algorithm for giventransmit beamformers. Our approach is based on minimizing thedifference between the performance of the MIMO-CDMA systemand the described upper bounds at each signature assignmentstep. We next note that the transmit beamformers can be shapeddepending on the channel state information (CSI) available at thetransmitter. We investigate the cases of various levels of availablefeedback resulting in different beamformer structures and presenta joint orthogonal temporal signature assignment and beam-forming algorithm. We observe that as the available feedback levelis increased, the performance of the joint signature and beam-former assignment algorithm approaches the performance upperbounds. In particular, we observe that a substantial performancegain is obtained when the individual channel state information isavailable at the transmitter side.

Index Terms—Mean-square error (MSE), multiple-inputmultiple-output code-division multiple-access (MIMO-CDMA)system, signature sequences, sum capacity, transmit beamforming.

I. INTRODUCTION

CONSIDERING the rapidly increasing demand for highdata rate and reliable wireless communications, high-ca-

pacity multiuser transmission schemes are of great importancefor next generation wireless systems. Code-division multiple-access (CDMA) systems emerge as promising candidates tomeet these challenges. In addition, recent studies indicate thatusing multiple antennas at the transmitter and receiver can dra-matically increase the performance of wireless communicationsystems [1]–[4]. It is well known that CDMA system perfor-mance is multiaccess interference limited and can be improvedby efficient interference management techniques. To that end,

Manuscript received August 18, 2004; revised August 30, 2005. This workwas supported in part by the National Science Foundation under CAREERAward 02-37727. This paper was presented in part at the Conference onInformation Sciences and Systems (CISS’03), Baltimore, MD, March 2003,and CISS’04, Princeton, NJ, March 2004. The associate editor coordinating thereview of this manuscript and approving it for publication was Prof. GregoriVazquez.

S. Serbetli was with the Electrical Engineering Department, The Pennsyl-vania State University, University Park, PA 16802 USA. He is now with PhilipsResearch, Eindhoven, The Netherlands (e-mail: [email protected]).

A. Yener is with the Electrical Engineering Department, The PennsylvaniaState University, University Park, PA 16802 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TSP.2006.874787

multiuser detection and interference avoidance exploit the tem-poral structure of the received signal [5]–[7]. Recent literaturehas suggested that, the use of multiple antennas for CDMA sys-tems provides additional interference suppression by exploitingthe spatial structure of the system [8]–[10].

The uplink performance of a multiuser system can be im-proved significantly by carefully adopting the transmittersof each user. Optimum or near-optimum transmit strategieshave been investigated for multiuser systems with variousperformance metrics up to date. For single-antenna systems, in-formation theoretic sum capacity and user capacity of a CDMAcell, and iterative algorithms that converge to the optimumtransmitters, i.e., signature sequence sets, are given in [6],[11]–[14]. Similarly, for narrowband multiuser multiple-inputmultiple-output (MIMO) systems, transmitters have been opti-mized with respect to sum capacity and the mean-square error(MSE) [15], [16].

In the case of a CDMA system where each user and the basestation are equipped with multiple antennas, termed hereafter aMIMO-CDMA system, users’ transmissions can be coordinatedboth temporally and spatially. Transmitter design, in this case,entails judicious choice of temporal signatures and transmitbeamformers. Most of the transmission schemes previously pro-posed for MIMO-CDMA systems rely on assigning orthogonalsignature sequences to the antennas/symbols without reusingthe signatures for different users, or integrating space–timecodes to the system [17], [18].

The premise of this paper is that, as in the case of single-an-tenna CDMA systems, and narrowband multiuser MIMOsystems, we can utilize the channel state related feedbackat the transmitter side to substantially improve the uplinkperformance of a MIMO-CDMA system, where each user iseventually assigned a unique signature sequence, and a transmitbeamformer. Our objective in this paper is to design the ap-propriate transmit beamformers and signatures considering thesum capacity, and the systemwide mean squared error (MSE)as the performance metrics. We first investigate the unlimitedreliable feedback case and construct iterative algorithms to findthe optimum transmit beamformer and signature set. Motivatedby milder feedback requirements, we constrain the signatureset to an orthogonal set and investigate the joint orthogonalsignature assignment and transmit beamformer design. Incontrast to references [17], [18], in this paper, we consider anoverloaded system where the orthogonal signatures are reusedby several users and such users interfere with each other. Insuch a scenario, efficient grouping of the users that will sharethe same orthogonal signatures, and designing the beamformerwith available feedback for each user, are new design problemsthat we encounter. We present a low-complexity orthogonaltemporal signature assignment algorithm that aims to approach

1053-587X/$20.00 © 2006 IEEE

SERBETLI AND YENER: MIMO-CDMA SYSTEMS 2759

Fig. 1. MIMO-CDMA system model.

the performance bounds for given transmit beamformers. Sincethe performance of the orthogonal temporal signature assign-ment algorithm is a function of the transmit beamformers ofthe users, we next tackle the problem of designing the transmitbeamformers for different levels of feedback available at thetransmit side of each user, and combine the proposed orthog-onal temporal signature assignment algorithm with perfectlycontrolled transmit beamforming, eigentransmit beamformingand antenna selection. Throughout the paper, we first developsignature and beamformer design algorithms for single-pathsynchronous channels, and then extend our results for multipathand asynchronous channels. Additionally, to provide a perfor-mance benchmark for the proposed algorithms, we developan upper bound for the sum capacity and a lower bound forthe MSE. We observe that, for both metrics, joint design oftransmit beamformers and signatures results in performancethat approaches to the described performance bounds withincreased level of available feedback at the transmit side.

The organization of the paper is as follows. The system modelis given in Section II. The performance metrics and the corre-sponding performance bounds are formulated in Section III. Thesignature and beamformer design algorithm with perfect feed-back is proposed in Section IV. The orthogonal temporal sig-nature assignment algorithm for given transmit beamformers ispresented in Section V. In Section VI, the impact of transmitbeamformers is investigated, and the joint transmit beamformerand temporal signature selection methods are proposed for dif-ferent feedback levels. Numerical results demonstrating per-formance of the algorithms are presented in Section VII. Sec-tion VIII concludes the paper.

II. SYSTEM MODEL AND ASSUMPTIONS

We consider the uplink of a single-cell MIMO-CDMA systemwith users and processing gain . The common receiver isequipped with receive antennas. User has transmitantennas, and unit energy signature waveform . Usertransmits its symbol, , by precoding it with an 1 unit-norm transmit beamforming vector, (Fig. 1), and it passesthrough a multipath channel with an impulse response

(1)

where , and are the complex MIMO channel andthe delay associated with the th path of the th user and thedelay of user , respectively. For simplicity, each user’s channelis assumed to be composed of exactly paths and the delaysare assumed to be chip synchronous i.e., and

. Without loss of generality, the asynchronous multi-path channel can be modeled with paths

for each user. The received signal is chip matched filtered andsampled at the chip rate over the entire observation interval of

chips. MIMO channels and the temporal signa-tures of the users provide effective spatial temporal signatures,

, for the users. Assuming and using the mul-tipath model in [7], the effective spatial temporal signature ofuser , , can be expressed

(2)

with ......

. . .. . .

...

...

(3)

where is the chip matched filtered and sampled form of ,and is the transmit power of user .

Throughout the paper, we assume that the transmit beam-forming vector for each user is given and fixed first. We relaxthis assumption later and investigate the effect of transmit beam-forming on the system. The received vector at the common des-tination is given by

(4)

where is the zero-mean complex Gaussian noise vector with, and denotes the Hermitian of a vector

or matrix. We assume that the channel realizations of each pathand delays of the users are constant and perfectly known at thereceiver side. Despite the fact that (4) is a general model forasynchronous multipath channels, it adds little insight in termsof the performance and complicates the derivations. Thus, in thesequel, we will first assume synchronous users with single-pathchannels, i.e.,

......

......

. . ....

(5)


with , where is the single-path MIMOchannel of user . We will extend the results to multipath andasynchronous channels, at the conclusion of each section.

Our aim in this work is to design transmit beamformer andsignature sets that will optimize the performance metric chosenfor MIMO-CDMA systems with the number of users exceedingthe dimensionality of the temporal signal space, i.e., systemswith users.

III. PERFORMANCE METRICS

We consider two systemwide performance metrics: the sumcapacity of and the total incurred MSE in a MIMO-CDMAcell. Both metrics have been considered for single-antennaCDMA and multiuser MIMO systems [12], [15], [16], [19],and are equally meaningful metrics representing the systemperformance: the former representing the information theoreticachievable sum rate of all users, and the latter involving the em-ployment of linear receivers and is the sum of the MSE incurredby each user in the system. Therefore, we will consider the jointdesign of temporal signatures and transmit beamformers of allusers in the cell with aim of i) maximizing the informationtheoretic sum capacity, or ii) minimizing the total MSE witheach user employing a linear receiver.

A. Sum Capacity

The sum capacity of a multiuser system with effective signa-tures is given by [11]:

(6)

The sum capacity optimization problem (OP1) becomes

s.t. (7)

where is a function of and .

B. Systemwide MSE

The systemwide MSE with effective signaturesand linear receivers is

MSE

(8)

For given effective signatures , the receivers min-imizing the MSE results in the well-known receivers1:

1Throughout the paper, ()y should be substituted for () if an inversionproblem arises for an interference limited system, i.e., � ! 0.

[20]. When thesereceivers are employed, the MSE is [16]

MSE

(9)The MSE optimization problem (OP2) becomes

MSE

s.t. (10)

It is easily seen that, for both OP1 and OP2, the performance isa function of the effective signatures and thesubspace of the effective signatures is defined by the span ofmatrices . The resulting dimension of the users’ signa-ture space is on the order of the number of receivers multipliedby . However, each user’s signature space is constrained byits channel realization. Transmitter optimization in this caseentails optimization of temporal signatures and transmit beam-formers given the channels of the users. The resulting optimumtransmitters are a function of the particular channel realizationsof the users, and do not have a closed form. However, as isexplained below, we can easily construct performance upperbounds. Given the hardship of the optimization problem athand, it is meaningful to try to construct algorithms such thatthe performance of the resulting temporal signatures and thetransmit beamformers approach the performance upper bounds,and hence the actual optimum performance.

We now explain how we obtain the performance bounds. Wefirst note that, from a system point of view, the MIMO-CDMAsystem can be viewed as a CDMA system with a processing gain

with structural constraints on the effective signatures. Thereceived power of user is

(11)

For a single-path synchronous MIMO-CDMA systemwith unit-norm temporal signatures (11) simplifies to

. Thus, the performance ofa single-antenna CDMA system with processing gain ,and received powers , forms an upper bound for theperformance of the MIMO-CDMA system. For single-an-tenna, single-cell CDMA systems, the optimum signatures formaximizing sum capacity and minimizing the MSE coincide[14]. For , the performance upper bound issimply the performance of an underloaded CDMA system withprocessing gain , and orthogonal effective signature se-quences are optimum [12]. The resulting sum capacity and theMSE values yield our performance upper bounds. In particular,the sum capacity upper bound, is [12]

(12)


and the systemwide MSE lower bound, MSE is [19]

MSE MSE (13)

Note that the performance metrics in (6) and (9) and perfor-mance bounds in (12) and (13) are for single-path synchronouschannels. Using the model in (2) and (3), the sum capacity andthe MSE expressions of asynchronous multipath channels canbe found by replacing the term in (6), (9) with ,and the bounds are

MSE MSE (14)

C. Effect of Imperfect Channel State Information

Throughout our formulation of the performance metrics, wehave assumed that the receiver has perfect channel state infor-mation (CSI) of all users. In practical scenarios, CSI errors willbe present and should be taken into account in system design.The effect of imperfect CSI on the sum capacity and the MSE ofa MIMO link is studied in references [21]–[23]. [21] shows that,when estimation errors exist, the capacity has a lower bound,and the channel estimation error acts like additional Gaussiannoise. [23] shows that the effect of CSI errors can be reflectedin transceiver design via additional Gaussian noise with totalMSE as the performance metric. Thus, the formulation of theperformance metrics and following development for both sumcapacity and the systemwide MSE performance metrics can beadapted to the case of imperfect CSI by simply considering thechannel estimation errors as a source of additive Gaussian noise,reducing the effective signal-to-noise ratio (SNR) of each user.In the sequel, for clarity of exposition, we will assume that theerrors in CSI is already reflected in our designs as additionalGaussian noise.

In the following sections, we pose the problem of optimizingthe performance in terms of sum capacity and MSE in the pres-ence of various levels of feedback and devise iterative algo-rithms. The performance bounds, and MSE will beused as benchmarks to evaluate the performance of the algo-rithms.

We reiterate that throughout the rest of the paper, the per-formance metrics as well as the algorithms are formulated firstfor the synchronous single-path uplink of a MIMO-CDMA cell.This assumption is valid, for instance, when the channel gain ofone path is comparatively large than the others and the delaysof users are small with respect to the processing gain. The de-velopment is then extended to address the multipath and asyn-chronous channel models using the general channel model in(2)–(3) at the conclusion of each subsection.

IV. SIGNATURE AND BEAMFORMER DESIGN WITH

PERFECT FEEDBACK

Our aim in this section is to design temporal signature andtransmit beamformer sets that achieve performances near theperformance bounds developed in the previous section. We as-sume that we have a feedback channel to the transmitter sidewhere the signatures and beamformers obtained for each usercan be communicated exactly. Here after, we will term this sce-nario the perfect feedback case.

A. Sum Capacity

Recall that the sum capacity of the MIMO-CDMA system is

(15)

Using the properties for squarematrices andfor rectangular matrices and , the sum capacity canbe expressed in terms of user ’s parameters as

(16)

(17)

where is the interferencecovariance matrix of user , represents the terms indepen-dent of user , and is the term de-noting the effect of user on the sum capacity. Recalling that

, we can express as

(18)

Thus, the sum capacity can be expressed in terms of the signa-ture of user , , as

(19)

and in terms of the transmit beamformer of user , , as

(20)

We will use (19) and (20) in the construction of the iterative al-gorithm to find the capacity maximizing signatures and beam-formers.

B. Systemwide MSE

Let us now follow the same approach for the systemwideMSE. Recall that the MSE of the MIMO-CDMA system witheffective signatures and the corresponding optimum linearMMSE receivers is

MSE (21)


where . Using matrix inversion lemma,can be expressed as

(22)

In this form, the MSE can be expressed as a function of param-eters of user , plus the contributions of the remaining users asfollows:

MSE

(23)

where represents terms independent of user , andis the term denoting the effect of user on

the MSE. Once again, using , we can easilyexpress as

(24)

Consequently, the total MSE, in terms of the signature of user, , can be expressed as

MSE (25)

and, in terms of the transmit beamformer of user , , as

MSE (26)

C. Iterative Algorithms

Equations (25) and (26), as well as (19) and (20), suggest aniterative way of improving the performance. Specifically, itera-tive algorithms that maximize the performance at each step canbe devised by optimizing the temporal signature and transmitbeamformer of one user, say user , and , at each step.From the perspective of user , the performance can be im-proved by choosing and to maximize the second termsin (19) and (20), in terms of the sum capacity, and the secondterms in (25) and (26), in terms of the MSE. Sum capacity im-provement is accomplished by choosing to be the eigenvectorwhich corresponds to the maximum eigenvalue ofin (19), and to be the eigenvector which corresponds to themaximum eigenvalue of in (20), i.e., maximizing

in (18). Similarly, in (24) can bemaximized by choosing to be the generalized eigenvectorwhich corresponds to the maximum generalized eigenvalue of

and in (25), and choosingto be the generalized eigenvector which corresponds to the

TABLE ISIGNATURE AND BEAMFORMER DESIGN WITH PERFECT FEEDBACK

maximum generalized eigenvalue of andin (26). First, each user is assigned a random sig-

nature sequence and a beamformer. Next, the two iterative algo-rithms constructed to improve the sum capacity and the MSE, it-erate over the users, optimizing the signature and then the beam-former (or vice versa) of each user while all other parametersare fixed, improving the performance at each step and yieldinga monotonic sequence of performance values. This implies thatthe algorithms, which produce monotonic sequences that arebounded by the corresponding performance bound, i.e., (12) or(13), are convergent [24]. A formal proof that the algorithmsconverge to the global optimum point eludes us due to the non-convexity of the problem, however, our numerical result willshow that the performance of the algorithms consistently comesvery close to the performance bounds given in Section III.

We term this class of algorithms, whose outline we present inTable I, signature and beamformer design with perfect feedback(PF). Note that, in Table I, we can obtain the particular algo-rithm according to or MSE, by replacing with or

respectively. Note also that the update mechanisms do notdepend which particular definitions of and , i.e., (3) or(5), are used. Thus, the algorithm can be implemented for bothsingle-path synchronous channels as well as multipath and asyn-chronous channels. The algorithms proposed in Table I can berun online or offline. Online implementation of the algorithmsrequires the feedback of the temporal signature and beamformerinformation to the users at each step whereas the offline imple-mentation requires the feedback of the resulting optimum tem-poral signature and beamformer of each user to the user.

V. ORTHOGONAL TEMPORAL SIGNATURE ASSIGNMENT

In the previous section, we proposed iterative algorithms todesign temporal signatures and transmit beamformers that im-prove the performance of the MIMO-CDMA system at eachstep. It is important to note that this approach eventually re-quires perfect feedback of the resulting signatures and transmitbeamformers to the transmitter side. The resulting signaturesand transmit beamformers are a function of the channel param-eters. This implies that the algorithm would have to be run eachtime the channel changes. In practice, such amount of feedbackmay be overwhelming, rendering the implementation imprac-tical. Hence, it may be necessary to look for ways to reduce the


feedback requirement in exchange for a modest sacrifice in per-formance.

An effective way to significantly reduce the amount of feed-back to the transmitter side, which we will follow for the re-mainder of this paper, is to constrain the temporal signature setto be a set of orthogonal signatures. Because we haveusers in the system, we will assume the largest such signatureset, i.e., orthogonal signatures. In contrast to [17] and [18],the temporal signatures will be reused between the users up to

times since . Users transmitting with thesame signature will be distinguished via the spatial signatures.In this case, since each orthogonal signature can be labeled with

bits, the feedback required, i.e., the label of each signa-ture, is bits long. For the development of our signatureassignment algorithm, we will assume that the orthogonality be-tween the signatures is preserved.

When orthogonal temporal signatures are used, the optimiza-tion problems formulated in Section III reduce to a multiuserpartitioning problem, i.e., finding subsets of users that willbe assigned to each temporal signature. [25] shows that thisproblem is equivalent to finding a maximum independent set ina vertex weighted graph where each vertex represents a subsetof the users. Specifically, in [25], finding the optimum partitionwith the highest average vertex weight is studied, where theweight of each vertex represents the sum capacity achieved bythe corresponding subset of users. When the MSE is consideredas the performance metric, the equivalence of the orthogonalsignature assignment problem, and finding the maximum inde-pendent set in vertex weighted graph is still valid, with vertexweights defined as the MSEs achieved by the correspondingsubset of users. In [26], weighted versions of finding themaximal independent set is shown to be NP-complete, andseveral approximation methods with polynomial complexityare presented in [27], [28]. Thus, for both performance met-rics, the sum capacity and the MSE, the orthogonal temporalsignature assignment problem is NP-complete. Fortunately,a low-complexity orthogonal temporal signature assignmentalgorithm that performs close to the performance bounds canbe derived as explained next.

Recall that the optimum unconstrained effective signaturesfor are those that are mutually orthog-onal. However, the predefined spatial signatures of the usersmay prevent the effective signatures to be orthogonal to eachother. Thus, we conclude that, a good heuristic is to try to assignthe orthogonal temporal signatures to those users whose effec-tive signatures, ’s, are as close to being orthogonal aspossible. Observe that assigning more than users to the sametemporal signature is likely to cause high correlation among theusers. Besides the fact that each temporal signature should notbe assigned to more than users, intuitively, it is logical to as-sign no more than Number of users users to eachtemporal signature for the sake of fairness.

The above observations suggest that an step temporal sig-nature assignment algorithm that tries to select the spatially lesscorrelated users to ’share’ each temporal signature is a good can-didate for near-optimum performance. Specifically, at each step,the number of users that will be assigned for each temporal sig-

nature is Number of users which is guaranteed not to ex-ceed .

We now describe the proposed temporal signature assignmentalgorithm. The development assumes a fixed beamformer foreach user. We will address the choice of beamformers in Sec-tion VI. Note that, at each orthogonal temporal signature as-signment step, a performance penalty is incurred with respectto the performance bounds. Thus, intuitively, an effective tem-poral signature assignment strategy is to assign the user suchthat the least penalty is incurred. That is to say, we will aim forthe smallest difference between the performance of the MIMO-CDMA system and the prescribed performance bound.

The orthogonal temporal signature set is denoted by .When is assigned to user , its effective signature becomes

. Clearly, user interferes only with the users thatemploy the same the temporal signature .

A. Sum Capacity

When orthogonal temporal signatures are used, the sum ca-pacity can be represented as

(27)

where is the set of users employing temporal signature ,and is the corresponding sum capacity. Assume that useris assigned . Define the set of users assigned to excludinguser as . The contribution of user to thesum capacity is simply

(28)

The capacity of user with its unconstrained effective signature,i.e., the single-user capacity is

(29)

Observe that, from user ’s perspective, the assignment of userto will result in a difference of between the

sum capacity upper bound and the achieved sum capacity. Thisdifference can be expressed as

(30)

Thus, the user with the highest

(31)


will result in the smallest difference from the upper bound fromthat user’s perspective.

B. Systemwide MSE

A similar approach can be developed for the MSE. When or-thogonal temporal signatures are used, the MSE can be repre-sented as

MSE

MSE

(32)

where MSE and are the total MSE and the cardinalityof the user set , respectively. The contribution of user onthe MSE is simply

MSE MSE

(33)

The MSE of the user with unconstrained effective signatureorthogonal to all other users is

MSE (34)

From user ’s perspective, the assignment of user to willresult in a difference of MSE between the MSE. Thus,the user with the highest

MSE(35)

will result in the smallest difference from the lower bound fromthat user’s perspective.

Consistent with the preceding discussion, we propose thefollowing iterative algorithm. At each step, we choose the userthat has the highest or to assign to accordingto the performance metric chosen. This algorithm obviouslyfavors the earlier temporal signatures, since these temporalsignatures will have more users available to select from thatwill minimize the difference metric. However, by limiting thenumber of users using each orthogonal temporal signatureby Number of available users the earlier temporalsignatures may have 1 user more than the later ones and theunfairness is somewhat diminished. An arbitrary user can be

TABLE IIORTHOGONAL TEMPORAL SIGNATURE ASSIGNMENT

chosen from the available users as the first user to start the algo-rithm. The algorithm proposed is summarized in Table II, with

or substituted for for the desired performancemetric, i.e., sum capacity or MSE. We note that the proposedalgorithm assigns users to each signature, rather than assigningsignatures to users. We run the algorithm sequentially over theusers to be able to achieve efficient packing of spatially lowcorrelated users to the same signature.

The orthogonal signature assignment algorithm derived sofar is for single-path synchronous channels. For multipath andasynchronous channels, the asynchronous arrivals of usersthrough multiple paths introduce additional interference. Forthis scenario, the interference from the users assigned to otherorthogonal signatures should also be considered as well whenassigning the temporal signatures to users. Using the sameapproach for orthogonal signature assignment algorithm andmultipath and asynchronous model in (2)–(3), assigning userto signature results in a gap of

(36)for the sum capacity, and

MSE

(37)

for the MSE. Once again, , and is the set ofusers that are assigned a temporal signature except user . Thus,


corresponding decision metrics for multipath and asynchronouschannels become

(38)

(39)

Using the modified decision metrics (38) and (39) in Table II,the orthogonal signature assignment algorithm can be used formultipath and asynchronous MIMO-CDMA systems.

VI. BEAMFORMER DESIGN WITH ORTHOGONAL

TEMPORAL SIGNATURES

The previous section considered the case where transmitbeamforming vectors are fixed. The performance of theMIMO-CDMA system is clearly a function of the choice ofthe transmit beamformers. In turn, the choice of the transmitbeamformers is dependent on the available feedback level atthe transmitter side. In this section, we consider different levelsof feedback at the transmitter side, and determine the corre-sponding transmit beamformers to be employed. Specifically,the perfect transmit beamforming feedback case, the individualCSI feedback case where each user has only its own CSIand can beamform through its eigenmodes, and the antennaselection feedback case where only the index of the transmitterantenna is fed back, are investigated next.

The orthogonal temporal signature assignment algorithm inTable II easily accommodates the beamformer design in accor-dance with the feedback requirement, by an appropriate defini-tion of the effective signature, as explained in the following.

A. Perfect Transmit Beamformer Feedback

In this section, we assume that the transmit beamformers canbe designed without any feedback constraints. The feedback re-lated to the temporal signature assignment is still limited to oneof the labels of the orthogonal signatures. Note that, this is incontrast to the scenario in Section IV, where both signature andbeamformer feedback were assumed to be unconstrained. Themotivation for this section is to obtain a performance benchmarkfor the beamformer design with limited feedback.

For the set of effective spatial signatures,, the contribution of the transmit beamforming

vector of user on the sum capacity, and the MSE can beexpressed as

(40)

When transmit beamformers can be optimized, the performancebounds for the sum capacity and MSE can be reformulated as

MSE (41)

This results in the following metrics, and

(42)

(43)

respectively. We observe that the transmit beamforming vectorthat maximizes is the eigenvector which corresponds to themaximum eigenvalue of ,

and the transmit beamforming vector that maximizes isthe generalized eigenvector which corresponds to the maximumgeneralized eigenvalue of

and .


For multipath asynchronous channels, the contribution of theusers, the performance bounds, and thus, the decision metrics

and , are functions of both the transmit beamformervectors and the temporal signatures. The performance boundsare

MSE (44)

and in (3) is obtained using signature . Thus, we have

(45)

(46)

For multipath asynchronous channels, the transmit beam-forming vector that maximizes is the eigenvector whichcorresponds to the maximum eigenvalue of

, and the transmit beamformingvector that maximizes is the generalized eigen-vector which corresponds to the maximum generalizedeigenvalue of and

.The orthogonal temporal signature assignment algorithm

with perfect transmit beamforming feedback (PTBF) shouldcompare the performance of the users with their best transmitbeamforming vectors at each step for both single and multipathchannels. The resulting signature assignment and beamformerdesign algorithm is a two-step algorithm. The best transmitbeamformers are found at the first step, and the user with thehighest , i.e., or , is assigned a temporal signaturenext. The orthogonal temporal signature assignment algorithmwith perfect transmit beamforming feedback is the algorithmpresented in Table II with the selection criterion defined as

(47)

After the assignment of the temporal signatures, the users maychoose to update their transmit beamformers improving the per-formance further at each step. In the numerical results, we have

chosen to perform a single iteration in order to provide a faircomparison with the limited feedback designs given in the nexttwo sections, i.e., eigenmode and antenna selection.

B. Individual CSI Feedback

In the case of limited CSI, a feasible scenario is that eachuser knows its own channel; for example in a time division du-plex system. The transmit beamforming vector of each user canbe chosen among the set of eigenvectors of the channel ma-trices, , i.e., , and the only required feed-back between the user and the receiver is which eigenmodesshould be used. Consider the case where one eigenmode is se-lected. When user selects eigenmode ,where is the th eigenvector of . It is clear thatthe received power of the user , ,is maximized when the eigenvector corresponding to the max-imum eigenvalue is used. In this case, the single-user perfor-mance bounds, and MSE for the eigenmode selection re-mains the same as in perfect transmit beamforming case, i.e.,(41).

Observe from (42) and (43) that, the metrics are both depen-dent on the received power and the effective spatial signaturesof the users that are already assigned. Assuming that we willassign users with spatially low correlations to the same orthog-onal temporal signature, intuitively, an effective method is tochoose the eigenmode that will maximize the received powerof the user, i.e., the eigenmode which corresponds to the max-imum eigenvalue. In this case, the transmit beamformer and theeffective spatial signature of user become

(48)

The orthogonal temporal signature assignment algorithm withmaximum received power eigenmode selection (MES) is the al-gorithm presented in Table II with the spatial signatures definedas the spatial signatures with the eigenmodes which correspondto the maximum eigenvalues, i.e., as in (48).

MES requires no feedback to the transmitter side related tobeamforming when time division duplex mode is employed. Itis expected to perform well especially if one eigenvalue is sig-nificantly larger than the others, and the MIMO channels ofthe users are independent of each other. On the other hand, ifthe channels of the users are highly correlated, choosing theeigenmode that maximizes the received power may not signif-icantly outperform the remaining eigenmodes. In that case, wemay choose to employ an orthogonal temporal signature assign-ment algorithm that considers each eigenmode, and chooses theeigenmode of the user that has the best performance in the senseof minimizing the performance difference from a single user’sperspective. We term this algorithm the orthogonal temporalsignature assignment algorithm with generalized eigenmode se-lection (GES).

The algorithm with generalized eigenmode selection is atwo-step algorithm where the best eigenmodes are found atthe first step, and the user with the highest or isassigned a temporal signature next. We have (49) and (50),


shown at the bottom of the page, where is the th eigen-value of . The effective spatial signature of user is

when th eigenmode is selected.The extension of MES and GES for multipath and asyn-

chronous channels requires the users to have their individualCSI for each path, i.e., for user . Usingthis information together with the candidate orthogonal tem-poral signature, , each user has their individual and mayselect one of the eigenvectors of , i.e., , astheir transmit beamforming vector. The MES algorithm formultipath asynchronous channels thus chooses the transmitbeamformer as

(51)

MES for multipath asynchronous channels is the algorithmpresented in Table II with the transmit beamformers of usersdefined as in (51). Similarly, GES for multipath asynchronouschannels should compare each eigenmode and choose the besteigenmode in the sense of minimizing the gap between theachieved performance and performance bound. The decisionmetrics for multipath asynchronous channels become (52) and(53), shown at the bottom of the page. GES for multipath

asynchronous channels is the algorithm presented in Table IIwith these decision metrics.

C. Antenna Selection Feedback

An alternative approach for limited feedback is antenna se-lection, where the only required feedback is which antenna(s)should be used [29], [30]. Consider the case where one trans-mitter antenna is selected. In this case, the spatial signatureof user is that of the selected transmitter antenna, , i.e.,

, where is the th column vector of user’s channel matrix, and the received power of user is

. Following the same approach as in the eigenmodeselection, intuitively, an effective antenna selection method is tochoose the transmitter antenna that will maximize the receivedpower of the user, i.e., the transmitter antenna with the highestnorm . In this case, the spatial signature of the user is

(54)

The orthogonal temporal signature assignment algorithmwith maximum received power antenna selection (MAS) isthe algorithm presented in Table II with the effective spatial

(49)

(50)

(52)

(53)


signatures defined as the spatial signatures of the transmitterantennas with the highest norm.

This approach is expected to perform well if one transmitterantenna has a spatial signature whose norm is significantlylarger than the others, and the MIMO channels of the usersare independent of each other. If this is not the case, a similarapproach to the generalized eigenmode selection that com-pares the performances of all transmitter antennas in terms ofminimizing the performance difference from a single user’sperspective, can be devised. We term this algorithm the orthog-onal temporal signature assignment algorithm with generalizedantenna selection (GAS). Note that the performance boundsthat are defined by the maximum received power of the usersbecome

MSE (55)

for the sum capacity and the MSE, respectively, and they aredifferent than the performance bounds in (41). Using the sameapproach in GES and the performance bounds (55), we have

(56) and (57), shown at the bottom of the page. The resultingsignature assignment and antenna selection algorithm is a two-step algorithm where the best transmit antennas are found at thefirst step, and the user with the highest or is assigneda temporal signature next.

For multipath asynchronous channels, the orthogonal tem-poral signature assignment algorithm with antenna selectionshould choose one of the transmitter antennas resulting theeffective spatial temporal signature of user , , to be a columnof , i.e., where .Thus, MAS algorithm for multipath asynchronous channelschooses the best transmitter antenna as

(58)

and the decision metrics in GAS algorithm for multipath asyn-chronous channels become (59) and (60), shown at the bottomof the page.

In this section, we proposed several transmit beamformer de-sign algorithms, namely, PFTB, MES, GES, MAS and GAS,combined with orthogonal temporal signature assignment con-sidering the level of available feedback. The feedback require-ments of the algorithms are summarized in Table III. In the nu-

(56)

(57)

(59)

(60)


TABLE IIICOMPARISON OF FEEDBACK REQUIREMENTS

Fig. 2. K = 12 user 2� 2 and 4� 4 MIMO-CDMA system.

merical results shown next, we will investigate the performanceof the proposed methods.

VII. NUMERICAL RESULTS AND DISCUSSION

In this section, we present numerical results related to the per-formance of the signature and beamformer design algorithms.We compare the performance of the transmit strategies for dif-ferent levels of feedback to investigate the benefit gained byexploiting the channel state information. For the sake of a faircomparison, we use the same performance upper bound for allscenarios. In particular, we use the upper bound when each userhas the beamformer which is the eigenvector that correspondsto the maximum eigenvalue of its channel. We note that theperformance of this beamformer provides an upper bound forall scenarios. The simulations are performed for 12 user,2 2 and 4 4 MIMO-CDMA systems with a processing gainof . The channels are realizations of a flat fading channelmodel where all links are assumed to be independent and iden-tically distributed complex Gaussian random variables. The re-ceived SNR of each user is 7 dB. CDF curves for sum capacityand MSE obtained by simulating 10000 channel realizations arepresented.

First, we consider 2 2 and 4 4 MIMO-CDMA systemswith given channel realizations and investigate the performanceof the iterative signature and beamformer design algorithm (PF)described in Table I. Figs. 2 and 3 show the evolution of the al-gorithms for the sum capacity and the MSE respectively. It isobserved that the iterative algorithms proposed converge fast,and the first updates of the temporal signatures and beamformers

Fig. 3. K = 12 user 2� 2 and 4� 4 MIMO-CDMA system.

provide substantial improvement in the performance metric con-sidered, a trend we have observed throughout our simulations,with random starting points. Thus, one may choose to update thetemporal signatures and beamformers only once for each userrestricting the feedback requirement for each user to minimum.We have also observed that different random starting points con-verged to the same sum capacity/MSE though they may requiredifferent number of iterations to converge. We observe that theproposed iterative algorithms perform very close to the perfor-mance bounds. As expected, as the number of transmit/receiveantennas is increased, better performance can be achieved sinceeach added transmit/receive antenna provides additional spatialdiversity.

We compare the performance of the signature and transmitbeamformer design strategies for 12 user 2 2 and 4 4MIMO-CDMA system with different levels of feedback to in-vestigate the benefit of exploiting the CSI in Figs. 4–9 in termsof the sum capacity and MSE. We also provide the performanceof MIMO-CDMA systems, in which temporal and spatial trans-mitter optimization are done separately, to present the gain weobtain by jointly exploiting the spatial and temporal characteris-tics of the system: i) a system in which users employ the eigen-vectors which correspond to the maximum eigenvalues of theirchannels as their beamformers, and are assigned GeneralizedWelch Bound Equality (WBE) signature sequences (General-ized WBE with ES), and ii) a system in which users transmitthrough the antenna that will maximize their received power,and are assigned Generalized Welch Bound Equality (WBE)signature sequences (Generalized WBE with AS). In Fig. 4,


Fig. 4. K = 12 user MIMO-CDMA system with 2� 2 MIMO model andN = 8. Comparison of the sum capacity CDF curves of the upper bound, theMIMO-CDMA systems with different levels of feedback and Generalized WBEsequences.

Fig. 5. K = 12 user MIMO-CDMA system with 2� 2 MIMO modeland N = 8. Comparison of the MSE CDF curves of the lower bound, theMIMO-CDMA systems with different levels of feedback and GeneralizedWBE sequences.

we present the sum capacity performances of all algorithmswe proposed as well as the systems with WBE signatures for2 2 MIMO-CDMA system. We observe that considering bothspatial and temporal domains jointly, i.e., utilizing orthogonalsignatures with appropriate beamformers performs better thanthe performance of the systems with WBE signatures and an-tenna selection or eigenbeamforming. The results in Fig. 5 showthe same performance improvement over generalized WBE se-quences for the MSE performance metric. We observe that theperformance is improved as the level of feedback increases, andthe signature and transmit beamformer design algorithm withiterative updates (PF) performs the best. Notice that the gener-alized beamformer design approach that considers the perfor-mance of all transmitter antennas/eigenmodes with the orthog-onal temporal signature assignment (GAS/GES) outperformsthe maximum received power antennas/eigenmodes selection

Fig. 6. K = 12 user MIMO-CDMA system with 2� 2 MIMO model andN = 8. Comparison of the sum capacity CDF curves of the upper bound andthe MIMO-CDMA systems with different levels of feedback.

Fig. 7. K = 12 user MIMO-CDMA system with 2� 2 MIMO model andN = 8. Comparison of the MSE CDF curves of the lower bound and theMIMO-CDMA systems with different levels of feedback.

approach (MAS/MES). However, the largest relative gain is dueto the feedback related to individual CSI, i.e., the eigenmode se-lection methods, combined with the orthogonal temporal signa-ture assignment algorithm. We observe in Figs. 4 and 5 that or-thogonal temporal signature assignment with antenna selection,which does not consider the best (single or multiuser) eigen-modes to transmit, performs poorly with respect to the other al-gorithms we proposed. In Figs. 6 and 7, we observe that theperformance of GES, GAS, PTBF and PF are very close tothe described performance upper bounds, and hence the op-timum performance in terms sum capacity and MSE, respec-tively. In Figs. 8 and 9, we also provide similar performancecomparison for a 4 4 MIMO-CDMA system. Observe that fora given outage probability, the difference between the achievedsum capacity and the upper bound is no more than 0.5 bits/s/Hzfor 2 2 MIMO-CDMA system, and 0.25 bits/s/Hz for 4 4MIMO-CDMA system for PTBF, MES and GES, and much less


Fig. 8. K = 12 user MIMO-CDMA system with 4� 4 MIMO model andN = 8. Comparison of the sum capacity CDF curves of the upper bound andthe MIMO-CDMA systems with different levels of feedback.

Fig. 9. K = 12 user MIMO-CDMA system with 4� 4 MIMO model andN = 8. Comparison of the MSE CDF curves of the lower bound and theMIMO-CDMA systems with different levels of feedback.

for PF. Similarly, the difference between the achieved MSE andthe lower bound is no more than 0.2 for 2 2 MIMO-CDMAsystem and 0.05 for 4 4 MIMO-CDMA system for PF, PTBF,MES and GES. As expected, the gap between the performanceof proposed algorithms and the performance bounds decreaseas the dimension of the MIMO system is increased. In Figs. 10and 11, the performance of the algorithm for a multipath andasynchronous system where each user has 3 independent fadingpaths and a maximum delay of 2 chips is presented. We observedthat the proposed algorithms perform close to the bounds formultipath asynchronous channels, as well.

VIII. CONCLUDING REMARKS

In this paper, we proposed signature and beamformer designalgorithms for MIMO-CDMA systems with various levels offeedback. First, we investigated the problem assuming perfectfeedback and derived iterative algorithms that enhance the

Fig. 10. K = 12 user multipath and asynchronous MIMO-CDMA systemwith 4� 4 MIMO model and N = 8. Comparison of the CDF curves of thesum capacity upper bound and the achieved sum capacity with different levelsof feedback.

Fig. 11. K = 12 user multipath and asynchronous MIMO-CDMA systemwith 4� 4 MIMO model and N = 8. Comparison of the CDF curves of theMSE lower bound and the achieved MSE with different levels of feedback.

system performance, i.e., the sum capacity, or the systemwideMSE. Next, motivated by limited feedback requirements,we investigated the orthogonal temporal signature assign-ment problem for given transmit beamformers and proposeda near-optimum signature assignment algorithm with lowcomplexity. Since the performance of the system dependson the transmit beamformers, we have considered transmitbeamformer selection next, and combined temporal signatureassignment with several beamformers resulting from differentlevels of feedback at the transmitter side. We have observedthat as the feedback level at the transmitter is increased, theperformance of the proposed algorithms approaches to theperformance bounds, and consequently to the performanceof the jointly optimum signature and beamformer set. Acomplexity-performance trade-off is observed: more transmitside feedback prompts better performance at the expense of


increased transmitter complexity. Notably, we observed thatthe individual CSI feedback facilitates a substantial gain withmodest complexity requirements.

In this paper, we considered a system where the channelsof the users are given, and the algorithms we propose rely onerror-free, low-delay feedback channels. The effect of feedbackaccuracy remains to be investigated.

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Semih Serbetli (S’02–M’05) received the B.S.degree in electrical and electronics engineering fromBogaziçi University, Istanbul, Turkey, in 2000, andthe Ph.D. degree in electrical engineering from thePennsylvania State University in 2005. His Ph.D.studies focused on transceiver design problems formultiuser multiple antenna systems.

He is currently with Philips Research Laboratories,Eindhoven, The Netherlands. His research interestsinclude transceiver optimization for wireless commu-nication systems with an emphasis on multiple-an-

tenna (MIMO) systems, cooperative communications, and multicarrier commu-nications systems.

Aylin Yener (S’91–M’00) received the B.S. degreesin electrical and electronics engineering and inphysics from Bogaziçi University, Istanbul, Turkey,in 1991 and the M.S. and Ph.D. degrees in electricaland computer engineering from Rutgers University,Piscataway, NJ, in 1994 and 2000, respectively.

During her Ph.D. studies, she was with WirelessInformation Network Laboratory (WINLAB) in theDepartment of Electrical and Computer Engineeringat Rutgers University. Between fall 2000 and fall2001, she was with the Electrical Engineering and

Computer Science Department at Lehigh University, PA, where she was a P.C.Rossin Assistant Professor. Currently, she is an Associate Professor with theElectrical Engineering Department at Pennsylvania State University, UniversityPark. Her research interests include performance enhancement of multiusersystems, wireless communication theory, and wireless networking in general.

Dr. Yener is a recipient of the NSF CAREER award in 2003. She is an Asso-ciate Editor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS.

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