Multi-Mode Precoding for MIMO Wireless Systems David J

Multi-Mode Precoding for MIMO Wireless Systems David J. Love, Member, IEEE, Robert W. Heath, Jr., Member, IEEE

Abstract— Multiple-input multiple-output (MIMO) wireless the mean squared error (MSE) [5], [6], maximizing the receive systems obtain large diversity and capacity gains by employing minimum distance (with receive distance defined as the two- multi-element antenna arrays at both the transmitter and re- norm of the difference between two unique symbol vectors ceiver. The theoretical performance benefits of MIMO systems, however, are irrelevant unless low error rate, spectrally efficient after multiplication by the channel) [7], maximizing the mini- signaling techniques are found. This paper proposes a new mum singular value [22], and maximizing the mutual informa- method for designing high data-rate spatial signals with low error tion assuming Gaussian signaling [4], [17], [20], [26]. These rates. The basic idea is to use transmitter channel information precoding methods provide probability of error improvements to adaptively vary the transmission scheme for a fixed data-rate. compared with unprecoded spatial multiplexing [2], but linear This adaptation is done by varying the number of substreams and the rate of each substream in a precoded spatial multiplexing precoding does not guarantee full diversity order and full rate system. We show that these substreams can be designed to growth. This follows from the fact that linear precoders usually obtain full diversity and full rate gain using feedback from the limit the number of substreams transmitted in order to obtain receiver to transmitter. We model the feedback using a limited improved error rate performance [2]. feedback scenario where only finite sets, or codebooks, of possible The full rate and full diversity problem was first discussed precoding configurations are known to both the transmitter and receiver. Monte Carlo simulations show substantial performance in space-time code design in [27], [28]. These open-loop gains over beamforming and spatial multiplexing. (i.e. no channel knowledge at the transmitter) space-time Index Terms-Diversity methods, MIMO systems, Quantized codes use spatial and temporal redundancy combined with precoding, Rayleigh channels, Spatial multiplexing. sphere decoding at the receiver. Unfortunately, these codes are difficult to decode even using the low-complexity sphere decoder. Given these results, the natural question is whether I.INTRODUCTION full rate and full diversity gain transmission can be obtained Wireless systems employing multi-element antenna arrays in a more practical solution at the expense of feedback from at both the transmitter and receiver, known as multiple- the receiver to transmitter. input multiple-output (MIMO) systems, promise large gains in This problem was also addressed for the special case of capacity and quality compared with single antenna links [1]. antenna subset selection precoding in [29], [30]. These papers Spatial multiplexing is a simple MIMO signaling approach that studied systems where the size of the antenna subset, along achieves large spectral efficiencies with only moderate trans- with the spatial multiplexing constellation, could be varied mitter complexity. Receivers for spatial multiplexing range in order to guarantee full diversity performance for a fixed from the high complexity, low error rate maximum likelihood data-rate. Various selection criteria were proposed for both decoding to the low complexity, moderate error rate linear dual-mode (i.e. selecting between spatial multiplexing and receivers [2], [3]. Unfortunately, rank deficiency of the matrix selection diversity) and multi-mode antenna selection. These channel can cause all spatial multiplexing receivers to suffer methods provided substantial performance improvements com- increases in the probability of error [2]–[4]. pared with traditional spatial multiplexing. Antenna subset Linear transmit precoding, where the transmitted data vector selection, however, is quite limited because the precoding is premultiplied by a precoding matrix that is adapted to some matrices are restricted to columns of the identity matrix. form of channel information, adds resiliency against channel In this paper, we propose a modified version of linear ill-conditioning. Linear precoded spatial multiplexing has been precoding called multi-mode precoding. Multi-mode precoding proposed for transmitter’s with full channel state information varies the number of substreams contained in the precoded (CSI) [5]–[7], channel first-order statistics [8]–[10], channel spatial multiplexing vector assuming that the transmitted data- second order statistics [11]–[15], partial subspace knowledge rate is independent of the number of substreams chosen [16], or limited feedback from the receiver (which includes for transmission. This allows the transmitter to adapt the antenna subset selection) [4], [17]–[26]. Optimization tech- signal using a combination of linear precoding and adaptive niques for choosing the precoding matrix include minimizing modulation. This substream selection has been studied independently in related work [31], [32]. We present methods This material is based in part upon work supported by the Texas Advanced Research (Technology) Program under Grant No. 003658-0614-2001, the Na- for choosing the multi-mode precoder based on minimizing tional Instruments Foundation, the Samsung Advanced Institute Technology, the probability of error and maximizing the mutual informa- and the SBC Foundation. This work appeared in part at the 2004 IEEE Global tion assuming independent and identically distributed (i.i.d.) Telecommunications Conference, Dallas, Texas, November-December 2004. D. J. Love is with the Center for Wireless Systems and Applications, School Gaussian signaling. Because it is often impractical to assume of Electrical and Computer Engineering, Purdue University, West Lafayette, perfect channel information at the transmitter, we present a IN 47907 USA. (email: [email protected]). R. W. Heath, Jr. is with the limited feedback multi-mode precoder using limited feedback Wireless Networking and Communications Group, Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX approaches developed in [23]–[25], [33], [34]. Limited feed- 78712 USA. (email: [email protected].). back multi-mode precoders use precoder codebooks, finite sets 2 of precoder matrices, for each of the supported substream II.SYSTEM OVERVIEW numbers. These codebooks are designed offline and stored The Mt transmit and Mr (with Mt ≤ Mr) receive antenna at both the transmitter and receiver. The chosen multi-mode MIMO wireless system studied in this paper is shown in Fig. precoder is then conveyed from the receiver to transmitter over 1. For each channel use, R bits are demultiplexed into M a limited capacity feedback channel using a small number of different bit streams. Each bit stream is modulated using the th bits. same constellation S, producing a vector sk at the k channel use. This means that each substream carries R/M bits of Multi-mode precoding represents a new approach to information. The spatial multiplexing symbol vector1 s = diversity-multiplexing tradeoff in MIMO wireless systems. k [s s . . . s ]T is assumed to have power constraints so Previous work in diversity-multiplexing tradeoff, see for ex- k,1 k,2 k,M that E [s s∗] = Es I . Note that this means the average of ample [27], [28], [35]–[37], emphasized that every spatio- sk k k M M the total transmitted power at any channel use is independent temporal signaling method has an accompanying diversity- of the number of substreams M. multiplexing gain tradeoff curve. Diversity systems, such as orthogonal space-time block codes or transmit beamforming, v1,k maximize the diversity gain for a fixed rate, while spatial + Spatial Spatial Multiplexing multiplexing maximizes the multiplexing gain. Our system Multiplexer M + s bits Receiver k actually “adapts” the transmission scheme between different s F s k M k yk FM M ,k chosen from r H diversity-multiplexing curves by varying the number of sub- M: # substreams v codebook M streams for transmission. This allows multi-mode precoding Low-rate feedback path Mode to maximize the multiplexing gain if the rate is allowed to selector vary with SNR and maximize the diversity gain if the rate is Fig. 1. Block diagram of a limited feedback precoding MIMO system. held constant. This adaptation is in the spirit of that studied in [36] but is more general because we are not restricted to An Mt × M linear precoding matrix FM maps sk to only two transmission types (such as Alamouti coding and an Mt-dimensional spatial signal that is transmitted on Mt spatial multiplexing as in [36]). Unlike [36], we do not switch transmit antennas. The transmitted signal vector encounters between space-time coding and spatial multiplexing but rather an Mr × Mt matrix channel H before being added with an vary the number of spatial multiplexing substreams and the Mr-dimensional white Gaussian noise vector vk. Assuming precoder. perfect pulse-shaping, sampling, and timing, this formulation Limited feedback multi-mode precoding is also one so- yields an input-output relationship lution to the important problem of covariance optimization y = HFM s + v (1) for transmitters without any form of CSI besides feedback. where the channel use index k has been suppressed because we This problem has previously been studied in [38]–[41]. Unlike are interested in vector-by-vector detection of sk. We assume [38], we are concerned with maximizing the average mutual that H has i.i.d. entries with each distributed according to information rather than obtaining a better quantized estimate CN (0, 1). We employ a block-fading model where the channel of the optimal waterfilling covariance matrix. Our approach is constant over multiple frames before independently taking also uses fixed codebooks known to both the transmitter and a new realization. As well, the noise vector v is assumed to receiver as opposed to the random vector quantization (RVQ) have i.i.d. entries distributed according to CN (0,N0). technique described in [39], [40]. Most importantly, we do We assume that the receiver has perfect knowledge of H and not require that the codebook be redesigned when the SNR FM . The matrix HFM can be thought of as an effective chan- changes as is assumed in [38], [41]. We give a technique nel, and the receiver decodes y using this effective channel and that can generate codebooks off-line. These codebooks can a spatial multiplexing decoder. Spatial multiplexing decoders then be used regardless of the operating SNR. Practically, include ML, sub-optimal ML, linear, and V-BLAST receivers. this is a large savings compared to the algorithms in [38], An ML receiver decodes to an estimated signal vector bs using [41] that require vector quantization algorithms that generate 0 2 thousands of channel realizations and then perform iterative bs = argmin ky − HFM s k2 . (2) 0 M optimizations to be run given an SNR value. s ∈S Sub-optimal ML receivers, such as the sphere decoder [42], This paper is organized as follows. Section II introduces [43], solve the optimization in (2) by performing reduced com- the multi-mode precoded spatial multiplexing system model. plexity ML decoding over a reduced set of candidate vectors. Criteria for choosing the optimal matrix from the codebook are 1We use CN (0, σ2) to denote the complex Gaussian distribution with presented in Section III. Multi-mode precoding using limited independent real and imaginary parts distributed according to N (0, σ2/2), feedback from the receiver to the transmitter is considered in ∗ for conjugate, T for transpose, ∗ for conjugate transpose, + for the matrix pseudo-inverse, IM for the M × M identity matrix, log2 is the base two Section IV. The relationship between limited feedback multi- th logarithm, ln is the natural logarithm, λj {H} to denote the j largest singular mode precoding and covariance quantization is explored in value of a matrix H, tr() for the trace operator that gives the sum of the Section V. Section VI illustrates the performance improve- diagonal elements of a matrix, k·k2 for the matrix two-norm, k·kF to denote ments over previously proposed techniques using Monte Carlo the matrix Frobenius norm, |a| to denote the absolute value of a complex number a, |M| to denote the cardinality of a set M, A\B denotes the simulations of the symbol error rate and mutual information. elements in the set A that are not in B, and Es[·] to denote expectation with Conclusions are presented in Section VII. respect to random variable s. 3

A linear receiver applies an M × Mr matrix transformation III.MULTI-MODE PRECODER SELECTION G to y and then independently detects each entry of Gy. If The selection of the mode and precoder matrix will deter- a zero-forcing (ZF) linear receiver is used, G = (HF )+. M mine the performance of the entire system. Because we are Minimum mean squared error (MMSE) decoding uses G = interested in constructing a high-rate signaling scheme with [F∗ H∗HF + (N M/E )I ]−1F∗ H∗. V-BLAST decod- M M 0 s M M low error rates, we will present bounds on the probability ing uses successive cancellation with ordering using ZF or of vector symbol error (i.e. the probability that the receiver MMSE receivers. An overview of the V-BLAST algorithm returns at least one symbol in error). We will also review can be found in [44]. the capacity results for MIMO systems both with and without Note that the total instantaneous transmitted power for this ∗ ∗ transmitter CSI. system is given by s FM FM s. The precoder matrix FM must therefore be constrained in order to limit the transmitted power. 2 We will restrict λ1{FM } ≤ 1 (i.e. the largest squared singular A. Performance Discussion value of FM ) in order to limit the peak-to-average ratio. This The selection criterion used to choose M and F must tie E [s∗F∗ F s] ≤ E M means that s M M s regardless of the modulation directly to the resulting performance of the precoded spatial scheme or the value of M. It was shown in [5], [24] that multiplexing system. We will address selection details based matrices that optimize MSE, capacity, the minimum effective 2 on two different performance measures: probability of error channel singular value (i.e. maximize λmin{HFM }), and total and capacity. effective channel power (i.e. maximize kHF k ) over the set M F 1) Probability of Error: The probability of error analysis of all matrices will be broken down into the three different receiver cate- gories: ML, linear, and V-BLAST. Note that the ML bounds Mt,M 2 L(Mt,M) = {L ∈ C | λ1{L} ≤ 1} (3) can also be employed for the popular sphere decoder [42], [43] which provides ML performance using a low-complexity are all members of the set search. In order to provide a fair tradeoff, we will use the probability of vector symbol error as the probability of error Mt,M ∗ U(Mt,M) = {U ∈ C | U U = IM }. metric. This choice was made to provide a fair comparison between modes that transmit symbol vectors of different For this reason, we will further restrict that FM ∈ U(Mt,M) dimensionality. for any chosen value of M. Intuitively, this means that the Maximum Likelihood Receiver precoding attempts to diagonalize the channel without any ML performance is commonly characterized as a function of form of power pouring among the spatial parallel channels. the receive minimum distance deﬁned as

The key difference between multi-mode precoding and pre- 2 0 00 2 dmin,rec = min kHFM (s − s )k2 . (4) viously proposed linear precoders is that M is adapted using s0,s00∈SM :s06=s00 current channel conditions by allowing M to vary between Using the receive minimum distance, the probability of vector 1 and Mt. We refer to the value of M as the mode of the symbol error can be bounded using the nearest neighbor union precoder. Usually, only a subset of the Mt possible modes can be chosen. For example, a necessary condition that motivates bound (NNUB) as using only a subset of {1,...,M } is that R/M is an integer s  t d2 for only a few of the modes between 1 and M . As well, it Es min,rec t Pe(H, Es/N0) ≤ MNe(M,R)Q   (5) might be practical from an implementation complexity point- MN0 2 of-view to support only a small subset of modes if Mt is large. We will denote the set of supported mode values as M. where Ne(M,R) is a nearest neighbor scale factor for the For example, if R = 8 bits and Mt = 4 then only modes in constellation S. Thus to minimize this bound, the receive the set M = {1, 2, 4} can be supported. Another example is minimum distance must be maximized. The minimum receive dual-mode precoding where M = {1,Mt}. distance, however, can be bounded as We assume that a selection function g : CMr ×Mt → M µ ¶ 2 2 2 picks the “best” mode according to some criterion, and we set dmin,rec ≥ min kHFM uk2 dmin(M,R) u∈U(M,1) M = g(H). After M is chosen, the precoder FM is selected 2 2 = λ {HFM }d (M,R) (6) from a set FM ⊆ U(Mt,M) using a selection function fM : M min Mr ×Mt C → FM . Therefore, FM = fM (H). This means that 0 00 2 where dmin(M,R) = mins06=s00 |s − s | denotes the min- there are |M| different precoder selection functions. imum distance for the constellation S used when R/M bits We assume that the transmitter has no prior knowledge of are modulated per substream. the matrix H and that FM is designed using data sent from Linear Receiver the receiver over a limited capacity feedback channel. This It was shown in [22] that the effective SNR of the kth assumption of zero prior channel knowledge approximates a substream after linear processing is given by frequency division duplexing (FDD) system where the forward E and reverse frequency bands are separated by a frequency SNR(ZF )(F ) = s (7) k M ∗ ∗ −1 bandwidth much larger than the channel coherence bandwidth. MN0[FM H HFM ]k,k 4

for ZF decoding and The notation CUT is used because this is commonly called the uninformed transmitter (UT) capacity (i.e. no transmitter (MMSE) Es SNR (FM ) = −1 CSI) [2]. Note that this is not really a “capacity” expression k ∗ ∗ MN0 −1 MN0[F H HFM + IM ] M Es k,k in the sense of distribution maximization because we assume (8) a ﬁxed distribution [45]. We will, however, refer to (13) as for MMSE decoding where A−1 is entry (k, k) of A−1. The k,k the capacity of the effective channel HF when there is no minimum substream SNR, given by M form of CSI at the transmitter in order to follow existing

SNRmin(FM ) = min SNRk(FM ), (9) terminology in the MIMO literature. When transmitter and 1≤k≤M receiver both have perfect knowledge of H and FM , the is an important parameter that will be used to characterize capacity is found by waterﬁlling [1]–[3]. performance. For ZF and MMSE decoding, SNRmin(FM ) can be bounded by [22] B. Selection Criteria E SNR (F ) ≥ λ2 {HF } s . (10) We will present probability of error and capacity selection. min M M M MN 0 The probability of error selection is based on the previous Therefore, the minimum singular value of the effective channel work in [22], [29], [30] while the capacity selection is similar is often an important parameter in linear precoded MIMO to work presented in [17], [22]. systems. Probability of Error Selection We are more interested, however, in tight bounds on the Assuming a probability of error selection criterion, the optimal probability of vector symbol error. Given a matrix channel selection criterion would obviously be to choose the mode and H, the conditional probability of vector symbol error can be precoder that provide the lowest probability of vector symbol bounded by the NNUB as error. Selection using this criterion, however, is unrealistic Ãr ! 2 because closed-form expressions for the probability of vector dmin(M,R) Pe(H, Es/N0) ≤ MNe(M,R)Q SNRmin symbol error conditioned on a channel realization are not 2 available to the authors’ knowledge. The NNUB can be suc-

≤ MNe(M,R) · cessfully employed in place of this bound for asymptotically s  tight selection. Using the NNUB results in Section III-A, the E d2 (M,R)  2 s min  following selection criterion is obtained. Q λM {HFM } . MN0 2 Probability of Error Selection Criterion: Choose M and (11) FM such that 2 (SV ) where SNRmin is computed for the given linear receiver. (SV ) λm{Hfm (H)} 2 g (H) = argmax dmin(m, R) (14) V-BLAST Receiver m∈M m Similarly to linear receivers, the V-BLAST receiver probability (SV ) 2 0 fM (H) = argmax λM {HF }. (15) 0 of error can be bounded using the SNR of the weakest F ∈FM substream after cancellation and before detection. Combining the NNUB with the results in [32], this gives The function (15) corresponds to finding the optimal precoding matrix from a limited feedback codebook FM con- P (H, E /N ) ≤ MN (M,R) · ditioned on a specific mode number M. This optimization, e s 0  e  s thus, corresponds to a fixed mode limited feedback codebook E d2 (M,R)  2 s min  matrix selection. The mode number is determined in (14) using Q λM {HFM } . MN0 2 the optimal codebook precoder matrix for each mode. Thus (12) the receiver would send both the chosen mode number (i.e. g(SV )(H)) and the optimal precoder given that mode (i.e. Once again, the minimum singular value of the effective (SV ) fM (H)) back to the transmitter. channel HFM plays an important role in system performance. This criterion is computed by first searching for the pre- Relation to Unitary Precoding (SV ) coder (denoted by fm (H)) in each mode’s codebook It must be noted that each receiver uses the same performance (i.e. Fm) that maximizes the effective channel minimum bound that relates back to the minimum singular value of squared singular value. The optimal mode is then deter- λ2 {HF }. the precoded channel matrix M M Under the assump- mined by computing the receive minimum distance bound tion of maximum singular value constrained precoders (i.e. (SV ) λ2{Hf (H)}d2 (m, R)/m for each mode in M and λ {F } ≤ 1), unitary precoding is optimal with respect to m min 1 M returning the mode with the largest bound. These optimizations minimizing the probability of error bound. correspond to multiple brute force searches. 2) Capacity: The mutual information of the channel HF M Capacity Selection assuming uncorrelated Gaussian signaling on each substream, While capacity selection is not optimal from a probability of denoted CUT , is known to be µ ¶ error point-of-view, it can provide insight into the attainable Es ∗ ∗ spectral efficiencies given the multi-mode precoding system CUT (FM ) = log2 det IM + FM H HFM . (13) MN0 model when an ML or V-BLAST receiver is used. Because 5 the uninformed transmitter capacity is evaluated in closed- where the codebook is explicitly a function of the SNR. form given a matrix channel, the following criterion can be Our approach will be based on maximizing the asymptotic succinctly stated. performance measures of diversity gain and multiplexing gain. Capacity Selection Criterion: Choose M and FM such that Diversity gain is a fundamental performance parameter (Cap) (Cap) in MIMO communications that has been given a variety g (H) = argmax CUT (fm (H)) m∈M of definitions. For probability of error selection, we will (Cap) 0 define diversity as the negative of the probability of error fM (H) = argmax CUT (F ). (16) 0 curve’s asymptotic slope. Following [26], [46], [47], we define F ∈FM diversity gain for capacity selection as the negative of the prob- IV. LIMITED FEEDBACK MULTI-MODE PRECODING ability of outage curve’s asymptotic slope. Mathematically, a MIMO wireless system is said to have diversity gain d [35] if We now consider the implementation of multi-mode precod-  log(Pe(Es/N0)) ing when the transmitter has no form of channel knowledge − limE /N →∞ , if probability of  s 0 log(Es/N0) besides feedback. This design makes multi-mode precoding  error selection; practical even in systems that do not meet the assumption of d =  log(Pout(R(Es/N0))) − limEs/N0→∞ , if capacity full transmitter CSI.  log(Es/N0)  selection. A. Codebook Model The diversity gain is always bounded above by the product of The design of an adaptive modulator in a system without the number of transmit and receive antennas, MtMr. Diversity transmitter CSI is daunting because we must find a simple gain, or diversity order, is one of the fundamental parameters method that allows the transmitter to adapt to current channel for MIMO systems. Therefore, it will be essential that we conditions. We will overcome the lack of transmitter CSI by maximize the diversity gain using as few bits of feedback as using a low-rate feedback channel that can carry a limited possible. number of information bits, denoted by B, from the receiver As in [35], let ˙= denote exponential equality. This means d to the transmitter. that f(Es/N0) ˙=(Es/N0) if In this limited feedback scenario, the precoder FM is chosen log f(Es/N0) from a finite set, or codebook, of N different M × M lim = d. (17) M t Es/N0→∞ log(Es/N0) precoder matrices F = {F , F ,..., F }. Thus, M M,1 M,2 M,NM The following lemma addresses the conditions sufficient to we assume that there is a codebook for each supported mode obtain full diversity order. value. Because there are a total of X Lemma 1.A If N1 ≥ Mt and the rate is fixed, multi-mode Ntotal = Nm precoding provides full diversity order. m∈M Proof: Each selection criterion will be treated separately. codeword matrices, a total of Probability of Error Selection & Ã !' Combining the probability of error results in Section III-A X reveals that the probability of vector symbol error with an B = dlog (N )e = log N 2 total 2 m ML, linear, or V-BLAST receiver can be bounded as m∈M P (E /N ) = E [P (H, E /N )] bits is required for feedback. Feedback can thus be kept to a e s 0 H " e s 0 reasonable amount by varying the size of NM for each mode. There are two main problems associated with this codebook- ≤ EH MNe(M,R) · based limited feedback system. First, we must determine how s  to distribute the Ntotal codewords among the modes in M. E d2 (M,R)  2 s min  The second problem is how to design F given M and N . Q λM {HFM } . M M MN0 2 We present solutions for both of these problems in Sections IV-B and IV-C. (18) Bounding this by the transmit diversity case gives B. Diversity-Multiplexing Codeword Distribution P (E /N ) e s 0   The feedback amount B is often specified offline by general s E d2 (1,R) system design constraints. For example, only B bits of control   2 s min  ≤ EH Ne(1,R)Q max λ1{Hf} . overhead might be available in the reverselink frames. For this f∈F1 N0 2 reason, we will assume that B is a fixed system parameter. · B Thus, we wish to understand how to distribute Ntot = 2 ≤ EH Ne(1,R) · codeword matrices among the |M| modes.   sµ ¶ In most standardized wireless scenarios, the system can E λ2 {W}d2 (1,R)  2 s Mt min  support a wide range of different SNR values. Because we Q max |hk,l|2 k,l N0 2MrMt do not want to redesign the codebooks each time the SNR changes, we will take a different approach than [38], [41] (19) 6

where W = [f1 f2 ··· fMt ] is constructed by taking Mt channel realization by vectors from F1. The upper-bound in (19) rolls off with full diversity if the vectors in F span CMt . This condition was Pe(Es/N0) 1 s  shown in [48] to be satisfied when N ≥ M and F is a 1 t 1 E d2 (M ,R) non-trivial codebook design. ≤ M N (M ,R)Q  λ2 {H} s min t  t e t min N 2M Capacity Selection 0 t We need to show that ≤ M N (M ,R) · (25)  t e t  µ ¶ v u 2 Pout(R) = P r max max CUT (F) < R (20) u 1 Es dmin(Mt,R) M F∈FM Q t min h i  . 1≤k≤Mt 3 ∗ −1 N0 2 Mt (H H) rolls off with full diversity. Note that k,k (26) µ µ ¶ ¶ Es 2 Lets assume that R = r log(Es/N0) with r < Mt. Fol- Pout(R) ≤ P r max log2 1 + kHfk2 ≤ R . (21) f∈F1 N0 lowing the technique used in [35], we will obtain an upper- bound using a QAM signaling assumption. This means that we Rearranging gives can choose the constellation such that the minimum distance −r/(2Mt) µ ¶ satisfies dmin(Mt,R) ≥ α(Es/N0) where α is a real Es 2 ¡ R ¢ Pout(R) ≤ P r max kHfk2 ≤ 2 − 1 . (22) constant. Using this assumption, the probability of error can f∈F1 N 0 be bounded as This is bounded further as Pe(Es/N0) µ 2 ¶ λ {W} E ¡ ¢ ≤ MtNe(Mt,R) · (27) Mt s 2 R    Pout(R) ≤ P r 2 2 kHkF ≤ 2 − 1 v M M N0 u µ ¶ t r u α2 E 1−r/Mt −MtMr  u s  ˙=(Es/N0) (23) EH Q t min h i  1≤k≤Mt 3 ∗ −1 N0 2Mt (H H) k,k with W defined as before. This is simply the outage proba- −(Mr −Mt+1)(1−r/Mt) bility of orthogonal space-time block codes with an SNR shift ˙=(Es/N0) (28) λ2 {W} Mt of 2 2 and was shown to possess full diversity order in Mt Mr where (28) follows from the results in [49]. Thus multi-mode [35]. precoding with probability of error selection can achieve a The achievability of full diversity gain is a substantial multiplexing gain of Mt. benefit. Spatial multiplexing has limited diversity order per- Capacity Selection formance (ex. achieves diversity order Mr for overly complex For the capacity case, we will use the outage probability maximum likelihood decoding), so a large diversity increase assuming that R = r log(Es/N0) with r < Mt. such as this adds considerable error rate improvements. P (r log(E /N )) Just as diversity order is often used to characterize proba- out µ s µ0 µ ¶¶ ¶ bility of error performance, multiplexing gain can be used to Es ∗ ≤ P r log det IMt + H H ≤ r log(Es/N0) characterize spectral efficiency. Let R(Es/N0) = r log(Es/N0) MtN0 denote the supported data rate as a function of SNR. A MIMO −Mr (Mt−r) ˙=(Es/N0) . (29) wireless system is said to have a multiplexing gain of c [35] if Thus, we have shown that capacity selection yields a multi- R(Es/N0) plexing gain of M . c = lim , (24) t Es/N0→∞ log(Es/N0) In order to satisfy the conditions in Lemma 1, we will

require that N1 ≥ Mt and NMt = 1 when Ntot ≥ Mt + 1. and this rate is the supremum of all rates that can be trans- Following the uninformed transmitter results in [1], [50], when mitted with a non-zero diversity order. Multiplexing gain is B < log2(Mt + 1) we will first set NMt = 1 and allocate B a fundamental property and, in our system, will be bounded the remaining 2 − 1 matrices to N1. Thus we can state the from above by min(Mt,Mr) = Mt. following allocation criterion. The next lemma addresses multiplexing gain. Probability of Error and Capacity Allocation Criterion for B Lemma 1.B If NMt > 0 and the rate is allows to grow with B ≤ log2(Mt + 1): Set NMt = 1 and N1 = 2 − 1. SNR, multi-mode precoding provides full multiplexing gain. When B > log2(Mt + 1), the first step in assigning Proof: We know that the multiplexing gain is upper- codewords is the determination of a distortion function. The bounded by min(Mt,Mr) = Mt. Thus we need to find a distortion function must be specific to the selection function suitable lower-bound on multiplexing gain for each selection used in order to maximize performance. We will design the scenario. distortion function by attempting to force the quantized multi- Probability of Error Selection mode precoder to perform identically to an unquantized (or We can bound the probability of error conditioned on the perfect CSI) multi-mode precoder. 7

∗ ∗ T To determine the allocation of matrix codewords among the VM H HVM = ΣM ΣM , and modal codebooks, we will use rate-distortion theory. Each selection criteria will motivate a different definition of distortion. ¯ ³ ´¯2 ° °2 Using the defined distortion, the distortion-rate function is the ¯ ∗ ¯ 1 ° ∗ ∗° 1 − max ¯det VM F ¯ ≤ min °VM VM − FF ° . minimum obtainable distortion for a given feedback rate. We F∈FM 2 F∈FM F will upper-bound the distortion-rate function for each mode in order to determine an allocation scheme. Let the channel’s singular value decomposition be given by Let M = {m1, m2, ..., m|M|} denote the set of possible modes. We will allocate the codewords to minimize the total ∗ H = VLΣVR (30) distortion-rate function assuming a uniform mode distribution which is given by where VL is an Mr × Mr unitary matrix, VR is an Mt × Mt unitary matrix, and Σ is a diagonal matrix with λi{H} at |M| position (i, i). We will define the probability of error design ¡ ¢ X 1 distortion to be the average loss of the mode selection function. D log Nm ,..., log Nm = D(mi, log Nm ) 2 1 2 |M| |M| 2 i This can be expressed as i=1 (34) (P e) using the appropriate distortion. By plugging (31) into (34) D (M, log2 NM ) · 2 and removing the common scale factors among all modes, the dmin(M,R) = min EH · probability of error selection is equivalent to minimizing the FM :|FM |=NM M ¯ ¯ # allocation cost function ¯ ¯2 ¯ 2 2 ¯ ¯λM {HVM } − max λM {HF}¯ F∈FM X|M| 2 ¡ ¢ d (mi,R) d2 (M,R) £ ¤ (P e) min −2/Mt(Mt+1) min 2 A Nm1 ,...,Nm|M| = Nmi . ≤ min EH kHkF · mi 2M F :|F |=N i=1 M M M · ¸ (35) ° °2 ° ∗ ∗° EH min °VM VM − FF ° Similarly, the capacity selection allocation should minimize F∈FM F the cost function d2 (M,R) ≤ min M M 3N −2/Mt(Mt+1) (31) M r t M ¡ ¢ X|M| where V is the matrix taken from the first M columns of (Cap) −2/Mt(Mt+1) M A Nm1 ,...,Nm|M| = Nmi . (36) VR and (31) follows from the fact that a Gaussian random i=1 variable gives the worst possible distortion-rate function (i.e. the largest minimum achievable distortion for a given rate and given variance) with a mean squared error [51]. In Both (35) and (36) must be minimized subject to particular the bound results from dividing up the NM matrices (or equivalently log NM bits) among the Mt(Mt + 1) real 2 ∗ X|M| parameters in the Mt ×Mt complex matrix VM VM for scalar Nmi = Ntot, (37) quantization. The capacity selection will use the conditional i=1 distortion-rate function given by

(Cap) D (M, log NM ) N = N ≥ M , and N = N = 1. Thus, this ·¯ µ2 µ ¶¶ m1 1 t m|M| Mt ¯ distortion-rate function based on a uniform mode distribution ¯ Es ∗ ∗ = EH ¯log2 det IM + VM H HVM MN0 will allow the codeword allocation to be done independently µ µ ¶¶¯ # of the SNR. ¯2 Es ∗ ∗ ¯ − max log2 det IM + F H HF ¯ It is easily seen that the following allocation will minimize F∈FM MN0 · µ ¶¸ (36). 1 Es T ≤ E det I + Σ Σ · Capacity Allocation Criterion for B > log2(Mt + 1): If 2(ln(2))2 H M MN M M 0 B ≤ log2(Mt(|M| − 1) + 1), set NMt = 1,N1 = Mt, and · ° ° ¸ 2B −M −1 ∗ 2 t ° ∗° Nmi = |M|−2 for 1 < i < |M|. If B > log2(Mt(|M| − EH min °VM V − FF ° (32) M 2B −1 F∈FM F 1) + 1), set N = 1 and N = for 1 ≤ i < |M|. "µ ¶ # Mt mi |M|−1 M 2 E Mt In the cases where this yields non-integer allocations, the ≤ t E 1 + s λ2{H} N −2/Mt(Mt+1) 2 H 1 M allocation can be adjusted by giving any extra matrices to the (ln(2)) N0 lower order modes. (33) Eq. (35) can be minimized by a numerical search or convex ˜ ˜ where ΣM is the matrix constructed from the first M columns optimization techniques. Let Nm1 = N1 − Mt and Nmi = of Σ. Note that (32) follows by using |ln(x) − ln(y)| ≤ |x−y|, Nmi for i > 1. Using this notation, we can reformulate (35) 8 as Probability of Error ³ ´ Following Section IV-B, we will define the distortion as ˜(P e) ˜ ˜ A Nm1 ,..., Nm|M|−2 · µ ¶¸ d2 (M,R) ³ ´−2/M (M +1) min 2 2 t t EH λmin{HFopt} − max λmin{HFi} 2 ˜ F ∈F = dmin(1,R) Nm1 + Mt M i (41) |M|−X 2 2 dmin(mi,R) −2/M (M +1) where Fopt is the precoder in U(Mt,M) that maximizes + N˜ t t 2 mi λ {HF }. This can be bounded as mi min opt i=2 · µ ¶¸ 2 d2 (M,R) dmin(m|M|−1,R) min 2 2 + · EH λmin{HFopt} − max λmin{HFi} M Fi∈F m|M|−1 2  −2/M (M +1) d (M,R) £ ¤ |M|−2 t t min 2 X ≤ EH λM {H} ·  ˜  M·µ ¶¸ Ntot − Mt − 1 − Nml (38) 2 ∗ l=2 EH 1 − max λmin{VRFi} Fi∈F where we have omitted the N|M| term. The values of 2 dmin(M,R) £ 2 ¤ N˜ ,..., N˜ that minimize A˜ can be easily determined ≤ EH λM {H} · m1 m|M|−1 M P|M|−2 Ã ! with N˜ = N − 1 − M + N˜ . These 2 m|M|−1 tot t l=2 ml δproj values can be used to determine N ,...,N with ∆proj (δproj ) + (1 − ∆proj(δproj )) m1 m|M|−1 4 N = 1. Therefore, the following criterion can be used m|M| (42) for codeword allocation. Probability of Error Allocation Criterion for B > where log2(Mt + 1): Minimize (35) such that N1 ≥ Mt,NMt = 1, δproj = min dproj (F1, F2) (43) P|M| F1,F2∈FM :F16=F2 and i=1 Nmi = Ntot. This minimization can be done using a numerical search or by using convex optimization techniques and ∆proj(δproj ) is N times the normalized volume of a on (38). metric ball in G(Mt,M) of radius δproj /2. It should be noted that the assumption of a uniform mode Using metric ball arguments similar to [33] and the asymp- distribution is a rough approximation. In general, environmen- totic metric ball volumes derived in [52], the bound in (42) tal effects such as spatial correlation will play a large role in can be approximately minimized by thinking of the set FM the notion of an “optimal” mode. As well, the probability of as a set of subspaces rather than as a set of matrices. The mode selection is inherently dependent on the rate at which the bound can thus be minimized by maximizing the minimum system transmits. Based on simulations, the uniform allocation projection two-norm subspace distance δproj . tends to be overly conservative in that it biases codeword Probability of Error Design Criterion: Design FM such that distribution to the lower mode numbers. δproj = min dproj (F1, F2) (44) F1,F2∈FM :F16=F2 C. Codebook Criterion Given the Number of Substreams is maximized. Now that we have determined an algorithm that gives an Capacity allocation of the possible codebook matrices among the modes, The capacity distortion is defined as "¯ ¯ # it is now imperative to present the design of FM for each ¯ ¯2 ¯ ¯ mode. EH ¯CUT (Fopt) − max CUT (Fi)¯ (45) Fi∈F For a given mode, the codebook FM is a set of matrices in U(Mt,M). Each matrix in FM defines an M-dimensional where Fopt is the precoder in U(Mt,M) that maximizes CUT . subspace in CMt . The set of all M-dimensional subspaces in The distortion cost function can be bounded as CMt is the Grassmann manifold G(M ,M). A finite set of ¯ # t ¯2 subspaces in the Grassmann manifold can be thought of as ¯ EH [|CUT (Fopt) − max CUT (Fi)¯ a subspace code [52]. A large variety of different subspaces Fi∈F · µ ¶¸ distances can be used for subspace coding [53]. These include 2 Es T the projection two-norm distance given by ≤ (1/ ln(2)) EH det IM + Σ Σ · MN0 µ ·¯ ³ ´¯ ¸¶ ∗ ∗ ∗ 2 dproj(F1, F2) = kF1F1 − F2F2k2 (39) ¯ ¯ 1 − max EH ¯det VRFi ¯ (46) Fi∈F and the Fubini-Study distance given by · µ ¶¸ E T ≤ (1/ ln(2))2E det I + s Σ Σ · ∗ H M MN dFS(F1, F2) = arccos |det (F1F2)| . (40) ¡ 0 ¢ 1 − cos2 (δ /2) ∆ (δ ) (47) The codebooks will be designed by choosing and then FS FS FS minimizing a distortion function. We will break the presen- where δFS and ∆(δFS) are defined using the Fubini-Study tation into the two different selection cases following the distance. Using the metric ball volume approximations from development in [54]. [52] and differentiating the resulting bound tells us that we 9

want to maximize δFS in order to approximately minimize In addition, multi-mode precoding uses a set of the form in the distortion. (50) that only requires a multiplicative scale factor when the Capacity Design Criterion: Design FM such that SNR changes. In contrast, [41] uses the Lloyd algorithm. The Lloyd algorithm, described in [64], uses a set of test channels δFS = min dFS(F1, F2) (48) F1,F2∈FM :F16=F2 (usually numbering in the thousands to hundred of thousands) and then repeats the two steps of generating a codebook is maximized. for each Voronoi region (i.e. the subset of the test channels that map to the same covariance matrix) and redefining the Discussion Voronoi regions for the generated codebook. The algorithm Note that the column vectors in F correspond to beamforming 1 converges rather quickly [64] to a locally optimal solution vectors [33], [34]. The design of limited feedback beamform- but still suffers because the random matrix being quantized is ing was explored in [33], [34], [55]–[58]. In particular, it highly dependent on the SNR. was shown in [33], [34] that the set of vectors should be designed by thinking of the vectors as representing lines in VI.SIMULATIONS CMt . The lines can then be optimally spaced by maximizing the minimum angular separation between any two lines. This Limited feedback multi-mode precoding was simulated to is seen by noting that when M = 1 both δproj and δFS are exhibit the available decrease in probability of error and ∗ the increase in capacity. The capacity results are compared maximized by minimizing |f1 f2| for any two distinct vectors f1 with the results in [38], [41]. We also consider both full and f2 in F1. The set FMt is trivially designed because we will channel knowledge [2], [3], [65] and limited feedback [33], require that FMt = {IMt }. This precoder matrix corresponds to sending a standard spatial multiplexing vector. [34], [58] beamforming. Probability of error simulations used For M < Mt, codebooks can be designed using the matrix the probability of error selection criterion, while capacity codebook design algorithms for non-coherent constellations in simulations used the capacity selection criterion. [59]. The only modification needed is to use the correct sub- Experiment 1: This experiment addresses the probability of space distance when performing the optimization. In addition, vector symbol error of 4 × 4 dual-mode precoding with a ZF algebraic design techniques can be used for certain values of receiver. The results are shown in Fig. 2. The rate is fixed at M and NM [60], [61]. Numerical design techniques have also R = 8 bits per channel use with QAM constellations. Because been studied in [62], [63]. the system is dual-mode, the set of supported modes is M = {1, 4}. Four bits of feedback is assumed to be available. V. RELATION TO COVARIANCE QUANTIZATION This means that F1 contains 15 vectors and F4 = {I4}. Limited feedback beamforming using four bits (see [33], [34]) The capacity analysis for MIMO systems with transmitter and spatial multiplexing are simulated for comparison. Multi- CSI relies on optimizing the transmit covariance matrix. A mode precoding provides approximately a 0.6 dB performance MIMO system has a general input-output relationship improvement over limited feedback beamforming. These gains y = Hx + v are modest because of the restriction to dual-mode precoding. with H and v defined as in (1). The mutual information is Spatial Multiplexing maximized by optimizing the covariance matrix Beamforming MM Precode ∗ Q = Ex [xx ] . (49)

−1 Covariance quantization, proposed in [38], [41], chooses Q 10 from a codebook Q = {Q1, Q2,..., QN }. Assuming that s in (1) consists of independent entries distributed according to ∗ CN (0, Es/M), the covariance matrix will be (Es/M)FM FM . Thus multi-mode precoding quantizes the set of covariance

−2 matrices assuming a rank constraint. Let 10

∗ Prob. of Symbol Vector Error QM = {(Es/M)FF | ∀ F ∈ FM }. (50) This allows multi-mode precoding to be reformulated as covariance quantization with a codebook [ −3 Q = Qm. 10 6 7 8 9 10 11 12 13 14 m∈M E /N b 0 Multi-mode precoding is a rank constrained covariance Fig. 2. Probability of vector symbol error performance for limited feedback quantization. While the codebook matrices in [38], [41] at- dual-mode precoding, limited feedback beamforming, and spatial multiplex- tempt to quantize a waterfilling solution, chooses a covariance ing. rank and then allocates equal power among each mode. This avoids the power allocation problems associated with Experiment 2: In contrast to the first experiment, this waterfilling. experiment considers a 4 × 4 MIMO system transmitting 10

R = 8 bits per channel use with M = {1, 2, 4}. Codebooks four bits of feedback. Interestingly, multi-mode precoding were designed using B = 5 bits of feedback. This resulted outperforms multi-mode antenna selection by 0.4dB using in N1 = 7,N2 = 24, and N4 = 1. Constellations were the same amount of feedback. Unlike multi-mode antenna restricted to be QAM and the receiver was a ZF decoder. selection, multi-mode precoding can use more feedback to Fig. 3 presents the simulation results. Spatial multiplexing, add additional array gain as the eight bit curve demonstrates. unquantized beamforming (i.e. perfect CSI at the transmitter Perfect CSI beamforming is also shown to demonstrate the using maximum ratio transmission/maximum ratio combining multi-mode precoding gain. [2], [65]), and unquantized MMSE precoding are shown for 0 10 comparison. MMSE precoding is implemented by transmit- Beamforming MM Ant Select (4 bit) ting two 16-QAM symbol streams using linear transmit and MM Precode (4 bit) MM Precode (8 bit) receive processing (i.e. precoding and a linear receiver). The MM Precode (∞ bit) MMSE precoding was implemented as in [5] with the sum power constraint and the trace cost function. This means that −1 the unquantized MMSE simulation is using power allocation 10 among the modes. Note that all of the selection criteria provide approximately the same probability of vector symbol error performance. Five bit multi-mode precoding provides approximately a 5dB gain over full CSI beamforming. There −2 10 is more than an 8.5dB gain over spatial multiplexing at an Prob. of Symbol Vector Error error rate of 10−1. Interestingly, MMSE precoding with full transmit channel knowledge, a less restrictive power constraint, and a superior receiver gives only a 1.2dB gain over limited feedback multi-mode precoding. −3 10 −2 0 2 4 6 8 10 12 14 16 0 E /N 10 Spatial Multiplexing b 0 Beamforming (Perfect CSI) MM Precode MMSE Fig. 4. Probability of vector symbol error performance comparison for limited feedback and perfect CSI multi-mode precoding, multi-mode antenna selection, and beamforming.

−1 10 Experiment 4: As mentioned in Section III, multi-mode precoding can be employed with ML, V-BLAST, and linear receivers. Fig. 5 compares the performance of limited feedback multi-mode precoding with ML, V-BLAST, and ZF decoding. We simulated a 4 × 4 MIMO system transmitting R = 8 bits −2 Prob. of Symbol Vector Error 10 per channel use with QAM signaling and M = {1, 2, 4}. The feedback bit total was set to B = 5 with a codeword allocation of N1 = 7,N2 = 24, and N4 = 1. All receivers perform very closely with ZF within 0.5dB of ML decoding.

−3 V-BLAST decoding performs approximately 0.1dB away from 10 −2 0 2 4 6 8 10 12 14 16 ML decoding. E /N b 0 Experiment 5: The capacity gains available with the capacity Fig. 3. Probability of vector symbol error performance for limited feedback selection criterion are illustrated in Fig. 6 for a 2 × 2 MIMO multi-mode precoding, beamforming, and spatial multiplexing. system with B = 2, 3, and 4 averaged over a spatially uncorrelated Rayleigh fading channel. The codewords were B Experiment 3: The third experiment, shown in Fig. 4, allocated as N1 = 2 − 1 and N2 = 1. The plot shows compares the performance of limited feedback multi-mode the ratio of the computed mutual information with the ca- precoding with various amounts of feedback. Again, we con- pacity of a transmitter with perfect CSI using waterfilling sidered a 4 × 4 MIMO system transmitting R = 8 bits per [1]. The capacity of an uninformed transmitter (UT) and the channel use with M = {1, 2, 4}. We used a ZF receiver and limited feedback covariance optimization mutual information QAM constellations. Feedback amounts of B = 4,B = 8, results published in [38] are shown for comparison. Note that and ∞ bits were simulated. The B = 4 bit codeword allocation limited feedback multi-mode precoding outperforms limited used N1 = 4,N2 = 11, and N4 = 1. The B = 8 feedback covariance optimization for both two and three bits bit allocation was N1 = 54,N2 = 201, and N4 = 1. of feedback. This result is striking because, unlike covariance Note that four bits of feedback performs within 1dB of the optimization, multi-mode precoding does not require any form infinite feedback scenario. Adding four more bits of feedback of waterfilling. Thus our scheme, on average, always transmits adds approximately a 0.6dB gain. For comparison, multi-mode with the same power on each symbol substream. The high-rate antenna selection [29] using the probability of error selection feedback performance difference between limited feedback criterion is presented. Multi-mode antenna selection requires covariance optimization and multi-mode precoding can be 11

0 10 MM Precode (ZF) beneﬁt of only requiring a few bits of feedback. MM Precode (VBLAST) MM Precode (ML) 1.1 UT MM Precode (4 bit) MM Precode (5 bit) MM Precode (∞ bit) 1 −1 10

0.9

−2 0.8 10 Prob. of Symbol Vector Error Normalized Capacity 0.7

−3 0.6 10 −3 −2 −1 0 1 2 3 4 5 6 7 8 E /N b 0

0.5 −10 −5 0 5 10 15 20 Fig. 5. Probability of vector symbol error performance comparison of E /N different receivers with multi-mode precoding. s 0 Fig. 7. Capacity comparison of multi-mode precoding with an inﬁnite amount of feedback, multi-mode precoding with four and ﬁve feedback bits, and an most likely attributed to this power-pouring. uninformed transmitter.

1 Experiment 7: This experiment, shown in Fig. 8, compares limited feedback multi-mode precoding with the feedback 0.95 technique proposed in [41] on a 4 × 4 MIMO system. Again, the mutual informations are normalized by the waterﬁlling 0.9 capacity. Multi-mode precoding was restricted the modes M = {1, 2, 4}, and both schemes used three bits of feedback. 0.85 Multi-mode precoding performs within approximately 1.5dB

0.8 of the algorithm in [41] for all SNRs. Note that the algorithm in [41] used a feedback codebook that was redesigned for each

Normalized Capacity 0.75 SNR while the multi-mode codebooks are ﬁxed for all SNRs.

1 0.7 UT UT Lau et al (3 bit) MM Precode (2bit) 0.95 MMode (3 bit) MM Precode (3bit) 0.65 MM Precode (4bit) Blum (2bit) 0.9 Blum (3bit) Blum (4bit) 0.6 −10 −5 0 5 10 15 20 0.85 E /N s 0 0.8 Fig. 6. Capacity comparison of multi-mode precoding, limited feedback covariance optimization [38], and the uninformed transmitter. 0.75

0.7 Experiment 6: The sixth experiment compares infinite res- Normalized Capacity olution multi-mode precoding, limited feedback multi-mode 0.65 precoding, and the UT capacity for a 3×3 MIMO system. The 0.6 capacities have been normalized by the optimal waterfilling capacity at each SNR. The results are shown in Fig. 7. We 0.55 considered supported modes of M = {1, 2, 3} with four 0.5 and five feedback bits. We designed the limited feedback −10 −5 0 5 10 15 20 E /N s 0 codebook using techniques from Section§ IV. The¨ codeword B ¥matrices were¦ allocated with |F1| = (2 − 1)/2 , |F2| = Fig. 8. Capacity comparison of multi-mode precoding and the covariance B (2 − 1)/2 , and |F3| = 1. The infinite resolution multi- feedback technique designed in [41]. mode precoder obtains within 98.5% of the system capacity when the channel is known to both the transmitter and receiver. Experiment 8: The final experiment examines the diversity The limited feedback case obtains more than 84% of the vs. multiplexing performance using the outage probability. perfect transmitter CSI system capacity with four feedback A 4 × 4 multi-mode precoding system with N1 = 4 and bits and 87.8% with five feedback bits. This comes with the N4 = 1 was simulated. Various rate growths were considered 12

including a ﬁxed rate (R = 30),R = 3 log2(Es/N0),R = of feedback and a moderate number of transmit antennas, the 3.5 log2(Es/N0),R = 3.75 log2(Es/N0), and R = computational complexity will quickly scale to unacceptable 3.9 log2(Es/N0). The results are shown in Fig. 9. The R = 30 levels. A possible sub-optimal solution might be to analytically outage probability curve shows the full diversity rate perfor- determine the mode and/or precoder matrix for an unquantized mance of multi-mode precoding. The other curves demonstrate system and then restrict the brute force optimization to candi- that the growth can get arbitrarily close to R = 4 log2(Es/N0) date multi-mode conﬁgurations around the unquantized case. with a non-zero asymptotic slope of the outage curve. An interesting area for future research is the distribution of the codebook matrices among the different modes. In our 0 10 derivation, we assumed that the modes were equiprobable. This assumption will be unrealistic in most environments. It may be possible to compute an approximate closed-form

−1 solution of the number of matrices to be allocated to each 10 mode given i.) the number of feedback bits, ii.) channel correlation, and iii.) rate. The ideas behind the relationship between multi-mode pre-

−2 10 coding and covariance quantization [38], [41] are also of interest. Future research is needed to obtain a more deﬁnitive

Outage Probability capacity analysis when the transmitter is only constrained to vary the rank of the covariance matrix. We conjecture that

−3 10 the losses relative to waterfilling with a trace constraint (as discussed in [1]) will always be minimal. R=30 For practical implementation, the effect of delay and errors R=3log(SNR) R=3.5log(SNR) in the feedback channel on system performance is something −4 R=3.75log(SNR) 10 R=3.9log(SNR) 20 30 40 50 60 70 80 90 100 that should be studied. These effects will lead to degradation E /N s 0 in the obtainable bit error rate and spectral efficiency. This analysis would provide interesting knowledge into the benefits Fig. 9. Outage probability analysis for multi-mode precoding with various transmission rates. of multi-mode precoding in practical wireless systems.

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David J. Love (S’98-M’05) received the B.S. (with highest honors), M.S.E., and Ph.D. degrees in electrical engineering in 2000, 2002, and 2004, respectively, from The University of Texas at Austin. PLACE During the summers of 2000 and 2002, he was a PHOTO summer researcher at the Texas Instruments DSPS HERE R&D Center in Dallas, Texas. At Texas Instruments, he performed research on physical layer design for next generation wireless systems employing multiple antennas. Since August 2004, he has been with the School of Electrical and Computer Engineering at Purdue University as an Assistant Professor. His research interests are in the design and analysis of communication systems (speciﬁcally multiple antenna, ultra-wide band, and cognitive wireless systems), coding theory, and information theory. Dr. Love is a member of SIAM, Tau Beta Pi, and Eta Kappa Nu. In 2003, he was the recipient of the IEEE Vehicular Technology Society Daniel Noble Fellowship.

Robert W. Heath, Jr. (S’96-M’01) received the B.S. and M.S. degrees from the University of Virginia, Charlottesville, VA, in 1996 and 1997 respectively, and the Ph.D. from Stanford University, Stanford, PLACE CA, in 2002, all in electrical engineering. PHOTO From 1998 to 1999, he was a Senior Member of HERE the Technical Staff at Iospan Wireless Inc, San Jose, CA where he played a key role in the design and implementation of the physical and link layers of the ﬁrst commercial MIMO-OFDM communication system. From 1999 to 2001 he served as a Senior Consultant for Iospan Wireless Inc. In January 2002, he joined the Department of Electrical and Computer Engineering at The University of Texas at Austin where he serves as an Assistant Professor as part of the Wireless Networking and Communications Group. His research lab, the Wireless Systems Innova- tions Laboratory, focuses on the theory, design, and practical implementation of wireless communication systems. Dr. Heath’s current research interests include interference management in wireless networks, sequence design, and coding, modulation, equalization, and resource allocation for MIMO communication systems.