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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 11, NOVEMBER 20075395MIMO Transmit Beamforming Under UniformElemental Power ConstraintXiayu Zheng, Student Member, IEEE, Yao Xie, Student Member, IEEE, Jian Li, Fellow, IEEE, andPetre Stoica, Fellow, IEEEAbstract—We consider multi-input multi-output (MIMO)transmit beamforming under the uniform elemental powerconstraint. This is a nonconvex optimization problem, and it isusually difficult to find the optimal transmit beamformer. First,we show that for the multi-input single-output (MISO) case, theoptimal solution has a closed-form expression. Then we proposea cyclic algorithm for the MIMO case which uses the closed-formMISO optimal solution iteratively. The cyclic algorithm has a lowcomputational complexity and is locally convergent under mildconditions. Moreover, we consider finite-rate feedback methodsneeded for transmit beamforming. We propose a simple scalarquantization method, as well as a novel vector quantizationmethod. For the latter method, the codebook is constructed underthe uniform elemental power constraint and the method is referredas VQ-UEP. We analyze VQ-UEP performance for the MISO case.Specifically, we obtain an approximate expression for the averagedegradation of the receive signal-to-noise ratio (SNR) caused byVQ-UEP. Numerical examples are provided to demonstrate theeffectiveness of our proposed transmit beamformer designs andthe finite-rate feedback techniques.Index Terms—Finite-rate feedback, multi-input multi-output(MIMO), multi-input single-output (MISO), quantization, transmit beamforming, uniform elemental power constraint.I. INTRODUCTIONXPLOITING multi-input multi-output (MIMO) spatialdiversity is a spectrally efficient way to combat channelfading in wireless communications. Although the theory andpractice of receive diversity are well understood, transmit diversity has been attracting much attention only recently. Generally,the transmit diversity systems belong to two groups. In thefirst group, the channel state information (CSI) is available atthe receiver, but not at the transmitter. Orthogonal space-timeblock codes (OSTBC) [1], [2] have been introduced to achievethe maximum possible spatial diversity order. In the secondgroup, the CSI is exploited at both the transmitter and thereceiver via MIMO transmit beamforming, which has recentlyEManuscript received October 18, 2006; revised February 19, 2007. The associate editor coordinating the review of this manuscript and approving it forpublication was Dr. Eric Serpedin. This work was supported in part by the Office of Naval Research under Grant No. N00014-07-1-0193 and the NationalScience Foundation by Grants ECS-0621879 and CCF-0634786.X. Zheng and J. Li are with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611 USA (e-mail: li@dsp.ufl.edu).Y. Xie was with the Department of Electrical and Computer Engineering,University of Florida, Gainesville, FL 32611 USA. She is now with the Department of Electrical Engineering, Stanford University, Stanford, CA 94305 USA.P. Stoica is with the Department of Information Technology, Uppsala University, SE-75105 Uppsala, Sweden.Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2007.896058attracted the attention of the researchers and practitioners alike,due to its much better performance compared to OSTBC [3],[4]. Compared to OSTBC, MIMO transmit beamforming canachieve the same spatial diversity order, full data rate, as well asadditional array gains. However, implementing MIMO transmitbeamforming schemes in a practical communication systemrequires additional considerations.First, optimal transmit beamformers obtained by the conventional, i.e., the maximum ratio transmission (MRT) approachmay require different elemental power allocations on the various transmit antennas, which is undesirable from the antennaamplifier design perspective. Especially in an orthogonal frequency division multiplexing (OFDM) system, this power imbalance can result in high peak-to-average power ratio (PAPR),and hencewise reduce the amplifier efficiency significantly [5].These practical problems have been considered in [6]–[8] fornew transmit beamfomer designs, and have also been addressedfor transmitter designs in a downlink multiuser system [9].Second, we need to consider how to acquire the CSI at thetransmitter. Recent focus has been on the finite-rate feedbacktechniques for the current conventional transmit beamforming[10]–[15]. These techniques attempt to efficiently feed backthe transmit beamformer (or the CSI) from the receiver to thetransmitter via a finite-rate feedback channel, which is assumedto be delay and error free, but bandwidth-limited. The problemis formulated as a vector quantization (VQ) problem [16],[17] and the goal is to design a common codebook, whichis maintained at both the transmitter and the receiver. Forfrequency-flat independently and identically distributed (i.i.d.)Raleigh fading channels, various codebook design criteriacan be used and the theoretical performance (e.g., outageprobability [12], operational rate-distortion [14], capacity loss[15]) can be analyzed for the multi-input single-output (MISO)case. The feedback schemes can be readily extended to thefrequency-selective fading channel case via OFDM. The relationship among the OFDM subcarriers can also be exploited toreduce the overhead of feedback by vector interpolation [18].We address the aforementioned problems as follows. First,we consider MIMO transmit beamformer design under the uniform elemental power constraint. This is a nonconvex optimization problem, which is usually difficult to solve, and no globallyoptimal solution is guaranteed [6]. Generally, we can relax theoriginal problem to a convex optimization problem via semidefinite relaxation (SDR). The relaxed problem can be solved viapublic domain software [19]. We can then obtain a solution tothe original nonconvex optimization problem from the solutionto the relaxed one by, for example, a heuristic method [20] (referred to as the heuristic SDR solution). Interestingly, we find1053-587X/ 25.00 2007 IEEE
5396out that in the multi-input single-output (MISO) case, the optimal solution has a closed-form expression and is referred toas the closed-form MISO transmit beamformer. (Similar resultshave appeared in [6]–[8] for equal gain transmission (EGT).)We then propose a cyclic algorithm for the MIMO case whichuses the closed-form MISO optimal solution iteratively and thesolution is referred to as the cyclic MIMO transmit beamformer.The cyclic algorithm has a low computational complexity andis shown via numerical examples to converge quickly from agood initial point. The numerical examples also show that theproposed transmit beamforming approach outperforms the conventional one with peak power clipping. Meanwhile, the cyclicsolution has a comparable performance to the heuristic SDRbased design and outperforms the latter when the rank of thechannel matrix increases.Second, we consider finite-rate feedback schemes for theproposed transmit beamformer designs. A simple scalar quantization (SQ) method is proposed; by taking advantage of theproperty of the uniform elemental power constraint, the numberof parameters to be quantized can be reduced to less than onehalf of their conventional counterpart. VQ methods are also discussed. Although the existing codebooks [10]–[12], [14], [15]can be used with some modifications by the MISO closed-formsolution, the performance may not be optimal since they donot take into account the uniform elemental power constraintin the codebook construction. We propose in this paper a VQmethod for transmit beamformer designs whose codebook isconstructed under the Uniform Elemental Power constraint(referred to as VQ-UEP). The generalized Lloyd algorithm[16] is adopted to construct the codebook. When the numberof feedback bits is small, VQ-UEP performs similarly to theconventional VQ (CVQ) method without uniform elementalpower constraint. For the MISO case, we further quantifythe performance of VQ-UEP by obtaining an approximateclosed-form expression for the average degradation of thereceive signal-to-noise ratio (SNR). It is shown that this approximate expression is quite tight and that we can use it as aguideline to determine the number of feedback bits needed inpractice, for a desired average degradation of the receive SNR.The remainder of this paper is organized as follows. Section IIdescribes the conventional MIMO transmit beamforming andits limitations. Section III presents our closed-form MISO andcyclic MIMO transmit beamformer designs under the uniformelemental power constraint. In Section IV, we consider thefinite-rate feedback schemes, where a simple SQ method andVQ-UEP are proposed. In Section V, we focus on the MISOcase and quantify the average degradation of the receive SNRcaused by VQ-UEP by obtaining an approximate closed-formexpression. Numerical examples are given in Section VI todemonstrate the effectiveness of our designs. We concludethe paper in Section VII. The following notations are adoptedthroughout this paper.Notation: Bold upper and lower case letters denote matricesand vectors, respectively. We useto denote the transposeto denote the conjugate transpose. stands for the abandsolute value of a scalar anddenotes the two-norm of a vector.is the complex set;andare the complex- andmatrices, respectively.is the trace of areal-valuedIEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 11, NOVEMBER 2007is the expectation,is the ensemble averagematrix.denotes the variance.is the vector formed by theanddenotes the floor operation.phase angles of andII. MIMO TRANSMIT BEAMFORMINGMIMO communication systemConsider antransmit andreceive antennas in a quasi-staticwithfrequency flat fading channel. At the transmitter, the comis modulated by the beamformerplex data symbol, and then transmitted intoa MIMO channel. At the receiver, after processing with the, thecombining vectorsampled combined baseband signal is given by(1)whereis the channel matrix with itsthelementdenoting the fading coefficient between the this thetransmit and th receive antennas, andnoise vector with its entries being independent and identicallydistributed (i.i.d.) complex Gaussian random variables withzero-mean and variance . Note that in the presence of interference, i.e., when is colored with a known covariance matrix, we can use pre-whitening at the receiver to get(2)in (1) is now reHence (2) is equivalent to (1) except thatand the whitened noise has unit variance.placed byWithout loss of generality, we focus on (1) hereafter.and the receive combiningThe transmit beamformervectorin (1) are usually chosen to maximize the receiveSNR. Without loss of generality, we assume that, and. Then the receive SNR isexpressed as(3)To maximize the receive SNR, the optimal transmit beamformeris chosen as the eigenvector corresponding to the largest eigenvalue of[14] (referred to as MRT in [6]), which is also thecorresponding to its dominant sinright singular vector ofgular value. The optimal combining vector is given by, which can be shown to be the left singularcorresponding to its dominant singular value (revector offerred to as maximum ratio combining (MRC) in [6]). Thus, the, wheremaximized receive SNR isdenotes the maximum eigenvalue of a matrix. The covariance matrix of the transmitted signal is(4)The average transmitted power for each antenna is(5)wheredenotes the th diagonal element of . (Note thatrepif the constellation of is phase shift keying (PSK),mayresents the instantaneous power.) The average power
ZHENG et al.: MIMO TRANSMIT BEAMFORMING5397amounts to omitting the rank-one constraint yielding the following semidefinite program (SDP) [22]:(8)The dual form of (8) is given by [20](9)Fig. 1. Transmit power distribution across the index of the transmit antennasfor a (4,1) system.vary widely across the transmit antennas, as illustrated in Fig. 1,which shows a typical example of transmit power distributionacross the antennas. The wide power variation poses a severeconstraint on power amplifier designs. In practice, each antennausually uses the same power amplifier, i.e., each antenna has thesame power dynamic range and peak power, which means thatthe conventional MIMO transmit beamforming can suffer fromsevere performance degradations since it makes the power clipping of the transmitted signals inevitable.III. TRANSMIT BEAMFORMER DESIGNS UNDER UNIFORMELEMENTAL POWER CONSTRAINTWe consider below both MIMO and its degenerate MISOtransmit beamformer designs under the uniform elementalpower constraint.A. Problem Formulation and SDRGiven MRC at the receiver, maximizing the receive SNR in(3) under the uniform elemental power constraint is equivalentto:(6)This is a nonconvex optimization problem, which is usually difficult to solve, and no globally optimal solution is guaranteed[20], [21], [6]. The problem in (6) can be reformulated aswithdenoting anwhere-dimensional all one column vector, andis a diagonal matrix with on its diagonal. The problem in (9) is also aSDP. Both (8) and (9) can be solved by using a public domainSDP solver [19]. The worst case complexity of solving (9) is[23]. We can obtain the optimal solution to (9), whosedual is also the optimal solution to (8). ThenAssume that the optimal solution to (8) isfor anyunder the uniform eleis one, then wemental power constraint. If the rank ofto (6) as the eigenvector correobtain the optimal solutionsponding to the nonzero eigenvalue of. If the rank ofis greater than one, we can obtain a suboptimal solutionfromvia a rank reduction method. For example, the heuristicas the eigenvector corresponding tomethod in [20] choosesthe dorminant eigenvalue of. The Newton-like algorithmpresented in [24] uses the SDR solution as an initial solutionand then uses the tangent-and-lift procedure to iteratively findthe solution satisfying the rank-one constraint. However, theapproximate heuristic method is preferred, as shown in ourlater discussion, due to its simplicity.Interestingly, we show below that the optimal solution to (6)has a closed-form expression for the MISO case. Moreover,we propose a cyclic algorithm for the MIMO case which usesthe closed-form MISO optimal solution iteratively. The cyclicmethod has a low complexity and numerical examples in Section VI show that it converges quickly given a good initial point.Furthermore, we also show in Section VI that the performanceof the cyclic algorithm is comparable to that of the HeuristicSDR solution and in fact better when the rank of the channelmatrix is large. Hence, the former is preferred over the latter inpractice.B. MISO Optimal Transmit Beamformerbe the row channel vector for the MISO case.LetWe\ consider the maximization problem in (6)(7)where, and the inequalitymeans that the matrix is positive semidefinite. Notethat in (7), the objective function is linear in , the constraintsare also linear in , and theon the diagonal elements ofis convex. However, thepositive semidefinite constraint onrank-one constraint on is nonconvex. The problem in (7) canbe relaxed to a convex optimization problem via SDR, which(10)where the equality holds when, withdenoting the unit-norm columnvector having the angles of , and. Note that the
5398IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 11, NOVEMBER 2007optimal solution is not unique due to the angle ambiguity; yetas the optimal solution to (6) for simplicity.we may take(This result can also be found in [6]–[8] for EGT.)A. Scalar Quantizationunder the uniform elNote that the transmit beamformeremental power constraint can be expressed asC. The Cyclic Algorithm for MIMO Transmit BeamformerDesign.(14)The original maximization problem for (6) is(11)Inspired by the cyclic method (see, e.g., [25]), we solve theproblem in (11) in a cyclic way for the MIMO case. The cyclicalgorithm is summarized as follows.to an initial value (e.g., the left singular1) Step 0: Setvector of corresponding to its largest singular value).that maximizes (11) for2) Step 1: Obtain the beamformerfixed at its most recent value. By takingas the“effective MISO channel,” this problem is equivalent to (6)for the MISO case. The problem is solved in (10) and theoptimal solution is:where the transmit beamformerparametersoftions, we obtainis a function. Via simple manipula-.(15)where. Since, we caninsteadreduce one parameter and quantize.ofDenote(12).that maxi3) Step 2: Determine the combining vectormizes (11) forfixed at its most recent value. The opis the MRC and has the form:timal(13)Iterate Steps 1 and 2 until a given stop criterion is satisfied.An important advantage of the above algorithm is that bothSteps 1 and 2 have simple closed-form optimal solutions. Alsothe cyclic algorithm is convergent under mild conditions [25].We remark here that the cyclic algorithm is flexible andor. A useful onewe can add more constraints onis the uniform elemental power constraint on the receiveantennas (or equal gain combining (EGC) [11], [6]), i.e.,. Then we only have toin Step 2 of each itermodify (13) asation. Given a good initial value (e.g., the one as given in Step0), the cyclic algorithm usually converges in a few iterations inour numerical examples, and the computational complexity ofeach iteration is very low, involving just (12) and (13).IV. FINITE-RATE FEEDBACK FOR TRANSMITBEAMFORMING DESIGNSIn the aforementioned transmit beamformer designs, we haveassumed that the transmitter has perfect knowledge on the CSI.However, in many real systems, having the CSI known exactlyat the transmitter is hardly possible. The channel information isusually provided by the receiver through a bandwidth-limited finite-rate feedback channel, and SQ or VQ methods, which havebeen widely studied for source coding [16], [17], can be used toprovide the feedback information. To focus on our problem, weassume herein that the receiver has perfect CSI, as usually donein the literatures [10]–[12], [14], [15].(16)where, withand denoting the number ofquantization levels and feedback index of , respectively, andwhereis the number of feedback bits for . After obtainingthe transmit beamformer from (10) or the cyclic algorithm into the “closest”Section III.C, we quantize the parameters. Hence for(via round off) grid pointsthis scalar quantization scheme, we need to send the index setfrom the receiver to the transmitter, whichbits. The receive combing vector isrequires.The choice ofis known as a counting problem [26],which hascombinations. The optimal setis the one that maximizes. However, this exhaustive search istoo complicated for practical applications. One simple subopapproximately equal. Specifically,timal approach is to makeandlet. Then we can letbits for the firstparametersandbits for the remainingparameters.We remark here that for the conventional MIMO transmitbeamformer without uniform elemental power constraint, theSQ requires about twice as many parameters. In this case, thetransmit beamformer is expressed as.(17)
ZHENG et al.: MIMO TRANSMIT BEAMFORMING5399whereis the th amplitude andis the th phase of the transmit beamformer vector, respectively,parameters.and hence there are totally, the receiver first chooses thebookoptimal transmit beamformer as:(20)B. Ad-Hoc Vector QuantizationVector quantization can be adopted to further reduce thefeedback overhead. In this case, both the transmitter and thereceiver have to maintain a common codebook with a finitenumber of codewords. The codebook can be constructed basedon several criteria. One approach is to directly apply the existing codebooks (e.g., [10]–[12], [14], [15]) constructed forthe conventional transmit beamformer designs obtained withoutthe uniform elemental power constraint. Among them, the criteria (e.g., [10], [14], [15]) that can be implemented by thegeneralized Lloyd algorithm can always lead to a monotonically convergent codebook. The generalized Lloyd algorithmis based on two conditions: the nearest neighborhood condition(NNC) and the centroid condition (CC) [16], [14], [15]. NNCis to find the optimal partition region for a fixed codeword,while CC updates the optimal codeword for a fixed partitionregion. The monotonically convergent property is guaranteeddue to obtaining an optimal solution for each condition. Maximizing the average receive SNR is a widely used criterion todesign the codebook [10], [12], [14] and will also be adoptedhere for codebook construction. Some modifications are stillneeded as below when the uniform elemental power constraintis imposed.Let a codebook constructed for the conventional transmit, wherebeamforming beis the number of codewords in the codebook , and is thenumber of feedback bits. The receiver first chooses the optimalcodeword in the codebook as:(18)where the operatorreturns a global maximizer. Thenfrom the receiver to thewe need to feed back the index ofbits. The transmit beamformertransmitter, which requiressatisfying the uniform elemental power constraint is obtainedas:(19).and the receive combining vector isHowever, the codebook may not be optimal for our proposedtransmit beamformer designs, since it is ad-hocly constructedwithout the uniform elemental power constraint (referred to asthe ad-hoc vector quantization (AVQ) method).C. Vector Quantization Under Uniform Elemental PowerConstraintLike AVQ, herein we also maximize the average receive SNR,while the codebook is constructed under the uniform elementalpower constraint (referred to as “VQ-UEP”). For a given code-and the corresponding vector quantizer is denoted as. Then we need to feedback the index offrom the rebits, and the receiveceiver to the transmitter withcombining vector is.Now the design problem becomes finding the codebook,which can be constructed off-line as follows. First, we generatefrom a sufficiently largea training setnumber of channel realizations. Next, starting from an initialcodebook (e.g., a codebook obtained from the conventionaltransmit beamformer designs or one obtained via the splittingmethod [16]), we iteratively update the codebook accordingto the following two criteria until no further improvement isobserved., assign a training ele1) NNC: for given codewordsto the th regionment(21)wherecodeword .2) CC: for a given partitionsatisfywordsis the partition set for the th, the updated optimum code-(22). Letbe Hermitian square root oflation results inforand. A simple reformu-(23))This problem is identical to (6) ( is replaced byand can be efficiently solved by the cyclic algorithm proposed in Section III.C.V. AVERAGE DEGRADATION OF THE RECEIVE SNRFor frequency flat i.i.d. MISO Rayleigh fading channels,various analysis approaches have been proposed to quantifythe vector quantization effect (outage probability [12], operational rate-distortion [14], capacity loss [15], etc.). Theseanalyses provide theoretical insights into the vector quantization methods and can serve as a guideline for determiningthe optimum number of feedback bits needed for the conventional transmit beamforming. We quantify below the effect ofVQ-UEP with finite-bit feedback on our closed-form MISO
5400IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 11, NOVEMBER 2007transmit beamformer design. Letloss of generality, we assumedegradation of the receive SNR is defined as:. Without. The averagecodewordbeta distribution, the random variable[15], with the pdf:has a(30)(24)whereis the partition set (or Voronoi cell) for the th codewordis the probabilitythat a channel realization belongs to the partition , andandthe last equality is due to the independence between[14], [26]. Obviously, we havethe normalized vector.A. Maximum Average Receive SNRUsingNow we consider the conditional density. Generally, each Voronoi cell [10], [12], [15], [16] obtained fromthe generalized Lloyd algorithm has a very complicated shapeand it is difficult to obtain an exact closed-form expression for. We adopt herein the approximate method used in[12], [15] to analyze the problem at our hand.is reasonably large, we can approximate the probaWhenbilityas. The Voronoicells can be considered as identical to each other. We then apas a spherical segment on theproximate each Voronoi cellsurface of a unit hypersphere:(31)whereis the maximum average value ofachieved by perfect feedback in our MISO transmitis the minimumbeamformer design, and the parameterin each Voronoi cell. We need to solve thevalue offollowing equation related to to obtain :in (10), we get:(32)Using the pdf in (30), we get(25)where the last equality is due to the i.i.d. property of.in (25) has the probability density function (pdf) asThefollows [27]:(33)Thus, for the Voronoi cellpdf of as, we approximate the conditional(26)The mean and variance of(34)are, respectivelywhere(27)otherwise(28)Combining (27) and (28) into (25), we getis the indication function.From the conditional pdf(35)in (34), we obtain(29)B. Approximate Value ofNote that the vector is considered as uniformly distributed[10], [12], [14], [15]. For a fixedon the unit hypersphere(36)
ZHENG et al.: MIMO TRANSMIT BEAMFORMING5401C. Quantifying the Average Degradation of the Receive SNRNow we quantify the average degradation of the receive SNR.in (24) using the approximate conditional pdfFrom (36), we observe that the average receive SNR is(37)Combining (29) and (37) into (25), we obtain the followingproposition.Proposition 5.1: For i.i.d. MISO Raleigh fading channels,the average degradation of the receive SNR, for an -antennatransmit beamforming system with an-size VQ-UEPcodebook, can be approximated as(38)The average degradation of the receive SNR in (38) can beproven to be monotonically decreasing with respect to nonnegative real number (see Appendix). Given a degradation amount, this proposition provides a guideline to determine the necessary number of feedback bits. That is, we can always findthe optimum integer number of feedback bits (via, e.g., theof theNewton’s method) with the average degradationreceive SNR being less than or equal to. Similarly, the average receive SNR in (37) can be shown to be monotonicallyincreasing with respect to , and we can determine the needednumber of feedback bits with the average receive SNR beingless or equal to a desired .Although our analysis shares some similar features to those in[7] and [8], our results are more accurate (see Section VI). In [7]and [8], the authors found the pdf ofvia making more approximations. Under high-resolution approximations, the average degradation of the receive SNR givenin [7], [8] has the form:(39)Both (38) and (39) are compared with numerically determined average receive SNR loss at the end of the next sectionand (38) is shown to be more accurate than (39).VI. NUMERICAL EXAMPLESWe present below several numerical examples to demonstratethe performance of the proposed MISO and MIMO transmitFig. 2. Performance comparison of various transmit beamformer designs withperfect CSI at the transmitter. (a) (4,1) MISO case. (b) (4,2) MIMO case. Notethat the (4,2) UEP TxBm and (4,2) Heuristic SDR curves almost coincide witheach other in (b).beamformer designs under the uniform elemental power constraint. We assume a frequency flat Rayleigh channel model. In thewithsimulations, we use QPSK for the transmitted symbols.First, we consider the bit-error-rate (BER) performance ofour proposed MISO and MIMO transmit beamformer with perfect CSI available at the transmitter. For comparison purposes,we also implement several other designs. The “Con TxBm” denotes the conventional transmit beamforming design withoutthe uniform elemental power constraint. The “TxBm with Clipping” stands for the conventional design with peak power clipping, which means that for every transmit antenna, ifwill be clipped by. The “Heuristic SDR” refers to the Heuristic SDRsolution described in Section III.A. We denote “UEP TxBm”as the closed-form MISO and the cyclic MIMO transmit beamformer designs under uniform elemental power constraint.Fig. 2 shows the bit-error-rate (BER) performance comparison of various transmit beamforming designs for both the (4,1)
5402IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 11, NOVEMBER 2007Fig. 3. Performance comparison of various transmit beamformer designs forthe (8,8) MIMO case.MISO and (4,2) MIMO systems. The “Con TxBm” achievesthe best performance since it is not under the uniform elemental power constraint. Under the uniform elemental powerconstraint, the “UEP TxBm” schemes have much better perfor, formance than the “TxBm with Clipping.” Atexample, the improvement is about 1.5 dB for the (4,2) MIMOsystem. In the MIMO system, we note that our “UEP TxBm”achieves almost the same performance as the “Heuristic SDR.”Interestingly, if we increase both the transmit and receiveantennas to 8, as shown in Fig. 3, our “UEP TxBm” outperforms the “Heuristic SDR.” The performance degradation of“Heuristic SDR” is caused by reducing the high rank optimalsolution to (8) to a rank-one solution heuristically. We note herethat our “UEP TxBm” is also much simpler than the “HeuristicSDR” (see the discussions in Section III).We examine next the effects of the two quantization methods(SQ and VQ) on the overall system performance. We usedescribed inherein the suboptimal combination ofSection IV-A for SQ due to its simplicity (although the optimalone can provide a better performance). We show in Figs. 4–7the BER performance of various quantization schemes for ourproposed and conventional transmit beamformer designs, with. We notevarious numbers of feedback bitsthat VQ-UEP outperforms the AVQ for all cases. When the), VQ-UEP cannumber of feedback bits is small (e.g.,provide a similar performance as that of CVQ, even though thelatter is not under the uniform elemental power constraint! TheVQ-UEP performance approaches
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 11, NOVEMBER 2007 5395 MIMO Transmit Beamforming Under Uniform Elemental Power Constraint Xiayu Zheng, Student Member, IEEE, Yao Xie, Student Member, IEEE, Jian Li, Fellow, IEEE, and Petre Stoica, Fellow, IEEE Abstract—We consider multi-input multi-output (MIMO) transmit beamforming under the uniform
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 8, AUGUST 2007 4241 Autonomous Spectrum Balancing for Digital Subscriber Lines Raphael Cendrillon, Member, IEEE, Jianwei Huang, Member, IEEE, Mung Chiang, Member, IEEE, and Marc Moonen, Fellow, IEEE Abstract—The main performance bottleneck of modern digital subscriber line (DSL) networks is the crosstalk among different
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