Approaching MIMO-OFDM Capacity With Zero-Forcing V ... - IEEE Xplore

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 10, OCTOBER 2008

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Approaching MIMO-OFDM Capacity With Zero-Forcing V-BLAST Decoding and Optimized Power, Rate, and Antenna-Mapping Feedback Rui Zhang, Member, IEEE, and John M. Cioffi, Fellow, IEEE

Abstract—This paper studies capacity-approaching transmission schemes for the multiple-input multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) channel, under the assumption that the channel state information (CSI) is completely known at the receiver but only partially available at the transmitter via a limited-rate feedback channel. A vertical Bell Labs layered space-time (V-BLAST)-based transmission structure is considered, where multiple data streams are independently encoded at the transmitter (i.e., horizontal encoding) and successively decoded at the receiver by the zero-forcing-based generalized decision feedback equalizer. A closed-loop V-BLAST extension is presented whereby transmit powers, rates, and antenna mappings for multiple data streams at different OFDM tones are jointly optimized at the receiver and then returned to the transmitter via the feedback channel. Two low-complexity algorithms for optimization of feedback parameters are proposed: one is based on the Lagrange dual-decomposition method and the other is a greedy algorithm. Antenna and tone grouping techniques by exploiting the MIMO-OFDM channel space-frequency correlations are also proposed to reduce the feedback complexity. Simulation results show that by only a moderate amount of feedback, the proposed closed-loop V-BLAST scheme improves substantially the throughput of the conventional open-loop V-BLAST scheme without feedback and, furthermore, approaches closely the MIMO-OFDM channel capacity achievable by the eigenmode transmission that requires the complete CSI at the transmitter. Index Terms—Adaptive coding and modulation (ACM), convex optimization, multiple-input multiple-output (MIMO), orthogonal frequency-division multiplexing (OFDM), partial channel feedback, spatial multiplexing, V-BLAST.

I. INTRODUCTION ULTIPLE-input multiple-output orthogonal frequencydivision multiplexing (MIMO-OFDM) is a promising technology for support of high-rate and broadband transmissions over frequency-selective fading channels with multiple transmit and multiple receive antennas. When the channel state information (CSI) is perfectly known at both the transmitter and receiver, the MIMO-OFDM channel capacity can be achieved

M

Manuscript received August 10, 2007, revised June 26, 2008. First published July 25, 2008; current version published September 19, 2008. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Geert Leus. This paper was presented in part at the IEEE Global Communications Conference, San Francisco, CA, Nov. 27–Dec. 1, 2006. R. Zhang is with the Institute for Infocomm Research, 119613 Singapore (e-mail: [email protected]). J. M. Cioffi is with the Department of Electrical Engineering, Stanford University, Stanford, CA 94305 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2008.928965

by signaling through the channel’s eigenmodes at each OFDM tone along with “water-filling”-based power and rate adaptations [1]. However, the eigenmode transmission relies on the complete CSI at the transmitter. For systems where the transmitter can acquire the CSI only via a feedback channel from the receiver, the amount of feedback overhead can be substantial for MIMO-OFDM channels with a large number of antennas and/or a large number of multipath delays with significant path gains. Consequently, a great deal of research has studied partial CSI feedback schemes for MIMO and MIMO-OFDM channels. Generally speaking, the optimal strategy (e.g., [2]–[4]) for transmit adaptation over block-fading (BF) MIMO channels feedback bits for each under a limited-rate feedback, say, block transmission, is to partition the multidimensional space of random MIMO channels into 2 disjoint regions and associate each region with one unique set of transmit parameters that may include the number of transmitted data streams as well as the transmit power, rate, and beamforming vector assigned to each data stream. Thereby, for each block transmission, the receiver can first determine the region where the instantaneous MIMO channel is located and then feed back the -bit representation of this region to the transmitter for adapting the transmission accordingly. Finding optimal MIMO channel partitions and their corresponding transmit parameters for arbitrary number of transmit and receive antennas is in general still an open problem in literature. Furthermore, such optimization not only relies on the complete knowledge on the fading channel distribution but also usually requires a costly computational complexity. Therefore, study on robust and low-complexity MIMO feedback schemes is still an important area for research. One commonly adopted feedback strategy for MIMO channels is to send back efficient representation of precoding (beamforming) vectors, one for each transmitted data stream (e.g., [5]–[10]). The precoding vector is usually drawn from a finite codebook that is known to both the transmitter and receiver and predetermined based on the feedback rate and some assumed statistical properties of the channel (e.g., isotropic fading distribution, transmit antenna correlations, and so on). Then, at each fading state, the receiver selects from the codebook one precoding vector (or precoding matrix if spatial multiplexing with more than one data stream used) corresponding to the best achievable performance [e.g., bit error rate (BER) minimization or capacity maximization] to return to the transmitter via the feedback channel. The simplest design for precoding vectors is probably the transmit-antenna selection (e.g., [11]–[13]) for which the codebook consists

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of merely indexes of transmit antennas and the feedback information becomes the binary index of the selected antenna for transmission. Recently, reduced-feedback precoding for MIMO-OFDM channels by exploiting channel correlations across adjacent OFDM tones has also been considered in, e.g., [14]–[18]. Although the MIMO precoding technique has been shown in the literature to be able to improve substantially both the MIMO channel capacity and transmission reliability, there is usually a common drawback associated with this approach: when the distribution of the fading channel deviates drastically from the presumed one, the channel-mismatched codebook may cause a notable performance degradation. This paper considers a special form of MIMO feedback that does not contain any precoder information (i.e., the precoding matrix at the transmitter is constantly the identity matrix) but instead contains transmit power and rate adaptations based solely on the instantaneous MIMO channel realization. The proposed feedback scheme does not require any knowledge on the MIMO channel distribution and, hence, may be found useful when the fading environment exhibits heterogeneous statistical properties such that a constant precoding codebook having a limited size is unlikely to be effective for all possible channel realizations. There are two commonly deployed encoding methods for MIMO spatial multiplexing, known as “horizontal” encoding and “vertical” encoding, respectively [19]. For the former, each data stream for spatial multiplexing is first independently encoded and then transmitted by a different antenna, while for the latter, a single encoder is used to spread coded information bits across all transmit antennas. The decoding methods for the horizontal and vertical encoding are also different. The former allows for parallel or successive decoding (by, e.g., the vertical Bell Labs layered space-time (V-BLAST) receiver [20], [21] or the equivalent generalized decision feedback equalizer (GDFE) [22]), while the latter requires maximum likelihood (ML)-based or approximate ML-based joint decoding (e.g., [23]). Successive decoding usually has a lower complexity to implement than joint decoding; however, it has a critical issue to tackle in practice on decoding error propagation. For instance, in the originally proposed open-loop V-BLAST [20], [21], equal power and rate are assigned to all data streams because of the lack of channel knowledge at the transmitter. As a result, the achievable rate of the open-loop V-BLAST is limited by the transmit antenna with the smallest channel capacity because incorrectly decoding one data stream may cause failures in decoding the remaining data streams due to error propagation. Although changing the decoding order of transmit antennas can prevent error propagation to some extent, the achievable rate of the open-loop V-BLAST is still far from the MIMO channel capacity [21]. This paper incorporates per-antenna-based power and rate feedback into the V-BLAST, termed the closed-loop V-BLAST, as an effective solution to improve the achievable rate of the open-loop V-BLAST. First, because of power control applied to each transmit antenna, more reliable transmission for each data stream is achievable and hence error propagation is effectively avoided. Secondly, adaptive rate allocation to different transmit antennas makes the achievable throughput of the closed-loop V-BLAST no longer limited by one transmit


antenna with the smallest channel capacity. Note that if there is only one transmit antenna assigned positive transmit rate and power, the closed-loop V-BLAST becomes identical to the transmit-antenna selection. The closed-loop V-BLAST has been considered for MIMO channels [24]–[27] as well as MIMO-OFDM channels [28]. This paper differs from the above prior work in the following aspects. First, this paper considers the V-BLAST receiver that utilizes the zero-forcing (ZF)-based successive decoding (also known as the ZF-GDFE [22]), while the receiver in [27] and [28] employs the minimum mean-squared error (MMSE)-based successive decoding (also known as MMSE-GDFE [22]). Consequently, the corresponding feedback power and rate optimization for these two receiver structures are also different. Secondly, the criterion adopted in [24]–[26] for feedback optimization is the minimized BER with fixed-rate transmission, while in this paper a generalized optimization framework is provided for evaluating the performance limit of the closed-loop V-BLAST. The main contributions of this paper are summarized as follows. This paper formulates joint optimization of feedback parameters in the closed-loop V-BLAST as a convex optimization problem. For the ZF-based V-BLAST receiver, selection of active transmit antennas as well as their decoding order can produce different tradeoffs in the resultant spatial subchannel gains at each OFDM tone. Therefore, these parameters need to be jointly optimized along with feedback transmit powers and rates at the receiver. An exhaustive search for the optimal antenna selection and decoding order (probably different from one OFDM tone to another) at all OFDM tones is shown to be practically infeasible for MIMO-OFDM due to the prohibitive computational complexity. This paper thus presents two low-complexity algorithms for this problem. The first algorithm is based on the Lagrange dual-decomposition method, which breaks the original problem into parallel subproblems, each independently solving the corresponding optimization at one OFDM tone. Consequently, the overall complexity becomes linear in the number of OFDM tones. The second approach is a simple greedy algorithm that provides a suboptimal solution but has a close-to-optimal performance with even lower complexity than the dual decomposition. This paper also considers modifications of the developed algorithms for further reduction of the feedback complexity by exploiting space–frequency correlations in the MIMO-OFDM channel. The remainder of this paper is organized as follows. Section II provides the MIMO-OFDM channel model. Section III illustrates the closed-loop V-BLAST architecture. Section IV formulates the optimization problem for determining feedback parameters and presents various solutions to this problem. Section V provides the simulation results. Section VI concludes this paper. Notation: Scalars are denoted by lower case letters, e.g., . Boldface lower case letters are used for vectors, e.g., , and denotes boldface upper case letters for matrices, e.g., . the trace of a square matrix . For any general matrix , and denote its transpose and conjugate transpose, respectively, and denotes its th element. denotes the idendenotes a diagonal matrix with all the ditity matrix. denotes the agonal elements represented by the vector .

ZHANG AND CIOFFI: APPROACHING MIMO-OFDM CAPACITY

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Euclidean norm of a complex vector . denotes statistical denotes the space of matrices with comexpectation. plex entries. The distribution of a circular symmetric complex Gaussian vector with the mean vector and the covariance ma, and means “distributed as.” trix is denoted by and denote, respectively, the The notations maximum and the minimum between two real numbers and , and . II. CHANNEL MODEL This paper considers an MIMO-OFDM channel with transmit antennas, receiver antennas, and orthogonal OFDM tones. It is assumed that the transmission is on a block basis and the channel is slow-fading. For simplicity, the BF model is considered in this paper, i.e., the channel remains constant during each transmission block but can probably vary from block to block. Let denote the fading state of the channel during each block transmission and assume that is ergodic and stationary. The cyclic prefix in each OFDM symbol is assumed to be sufficiently long to suppress any possible intersymbol interference caused by the multipath propagation. By applying standard OFDM modulation and demodulation at the transmitter and the receiver, respectively, the demodulated signal for one transmission block of interest can be expressed as (1) and are In the above, the received and transmitted signal vector at the th OFDM ; denotes tone, respectively, is the number of OFDM the OFDM symbol index and symbols in each transmission block; is the frequency-domain channel matrix at the th tone and is , where expressed as represents the channel associated with its th column the th transmit antenna, ; is the additive noise at the receiver, and it is assumed that . Assuming that the time-domain channel responses have delayed multipath taps, then denotes the time-domain channel matrix for the th tap, . The frequency-domain channel matrix can then be expressed as (2) for . Assuming that the transmitted signals at different tones, , , are independent vectors with zero means and covariance matrices, (the expectation is taken over ), the average transmit power over all OFDM tones for each transmission block of interest is then given by (3) s and can be changed according to the fading In general, state of each transmission block, though for brevity they are not written here explicitly as functions of . The average

Fig. 1. The closed-loop V-BLAST structure for MIMO-OFDM.

signal-to-noise ratio (SNR) over different fading states at the receiver is then defined as

SNR

(4)

It is further assumed that It then follows from (4) that

,

.

.

III. CLOSED-LOOP V-BLAST FOR MIMO-OFDM This section presents an extension of the open-loop V-BLAST transmission structure in [20] and [21] for MIMO-OFDM, referred to as the closed-loop V-BLAST. The closed-loop V-BLAST differs from the open-loop V-BLAST in that there exists a feedback channel that enables the receiver to send to the transmitter the optimized transmit adaptation based on the instantaneous channel realization, which is assumed to be perfectly known at the receiver. Fig. 1 provides an illustration of the closed-loop V-BLAST architecture, which is described in more detail as follows. • Demultiplexer (Demux): All information bits of each transmission block corresponding to a target rate are first divided into , ,1 data streams, each , carrying a portion of the total bits equal to , where denotes the rate for the th data stream. • Encoder: Each data stream is then independently processed for channel coding and modulation. Modulated symbols of each data stream are then divided into consecutive -tuples, each tuple assigned to one of OFDM symbols. For the th data stream, the signals assigned to the th tone of the th OFDM symbol are denoted by , , . is then amplified according to • Power Amplifier: its assigned data stream and tone position at each OFDM , , symbol. For the th data stream, , and the at every th tone are amplified according to . outputs are denoted by

M min(M ;M ) is required by the

1As will be shown later in this paper, considered ZF-based V-BLAST receiver.

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• Antenna Mapping: Let . is mapped to a corresponding Next, each element of transmit antenna. This antenna mapping is the same for all OFDM symbols but probably different from tone to tone.2 Let the antenna mapping at tone be specified by , the vector denotes the transmit antenna at tone assigned e.g., to carry the first data stream. Define the matrix, , parameterized by , as follows: if otherwise.

(5)

The input and output after the antenna mapping can then be expressed as (6) where is the transmitted signal vector given in (1). not only selects (out of ) active transmit Clearly, data streams but also speciantennas at tone to carry fies the mapping between these selected antennas and their assigned data streams. • OFDM Modulator: Let . is then modulated as the th OFDM symbol, , . radiated by the th transmit antenna, data streams with To summarize, a total number of , respectively, are independently transmit rate encoded and simultaneously transmitted (i.e., horizontal enis zero for any , then , , and no coding). If data are assigned to the th data stream. Hence, the closed-loop V-BLAST dynamically adapts the number of data streams or the degree of spatial multiplexing. Notice that although one transmit antenna may be assigned to carry a data stream at one tone but not assigned any data stream at another tone, in general all transmit antennas may be active. This is in contrast to the transmit antenna selection in the case of a flat-fading MIMO channel for which some transmit antennas are completely switched off if they are not assigned any data stream. In general, there may be three sets of parameters for the receiver to feed back in the closed-loop V-BLAST: , i) rate assignments for different data streams: ; ii) power assignments for each data stream at different , , ; OFDM tones: iii) transmit antennas assigned to each data stream at different , . tones: For the V-BLAST, each independently encoded data stream is analogous to a virtual user in an equivalent Gaussian multiple-access channel (MAC). Therefore, each data stream of the V-BLAST can be decoded by employing the well-known multiuser detection techniques [29], which in general can be either linear or nonlinear, and either ZF-based or MMSE-based. 2This is because active transmit antennas and their corresponding decoding orders optimized at the receiver, as will be shown later in this paper, may also be variable at different OFDM tones in order to exploit the channel frequency diversity. As a result, transmit antenna mapping needs to ensure that the signals at different OFDM tones from the j th data stream are all decoded j th in the order by the successive decoding at the receiver.

Depending on the receiver structure, the optimization of feedback parameters in the closed-loop V-BLAST can also be different. In [27] and [28], the closed-loop V-BLAST with MMSE-based successive decoding has been studied, while in this paper the ZF-based successive decoding is considered. The MMSE-based V-BLAST receiver is the optimal multiuser detector that achieves the capacity region of the Gaussian MAC [30], and is thus also optimal for the closed-loop V-BLAST. However, as shown in [28], the optimization of feedback parameters for the MMSE-based successive decoding is associated with a prohibitive computational complexity, especially when the number of transmit and/or receive antennas is large. Furthermore, it is hard to incorporate practical (non-Gaussian) modulation and coding constraints into the feedback optimization for the MMSE-based receiver [28]. In this paper, the ZF-based V-BLAST receiver is considered to overcome the above difficulties. The ZF-based V-BLAST receiver with successive decoding is parameterized by a set of projection vectors for each data , stream at different OFDM tones, denoted by , . Notice that is dependent on , which specifies the assigned transmit antennas for data streams at tone . For some given , the QR decomposition at tone (for brevity, the fading state of the channel matrix is dropped) after the column-wise selection and permutation can be expressed as based on (7) where satisfies , is a posdiagonal matrix, and is a monic itive lower triangular matrix. The projection vector for the th data stream at tone , , can then be obtained as the th column . The ZF-based V-BLAST receiver first applies the proof jection vectors to the received signals to obtain (8) (9) where and (6), (7), and (9), and by using the fact that can be simplified as

. From , (10)

Assuming that the decoding order for data streams is from one to , from (10) it is observed that the th data stream 1, is interfered merely by the decoded data streams one to which correspond to the lower and off-diagonal elements and thus can be subtracted from . It is also of observed that the interferences from not-yet-decoded data 1 to have been completely removed by the ZF streams operation of the projection vector. Notice that because has the size of , the maximum value of needs to be such that the first decoded data stream can successfully remove the interference from the other data streams using the ZF-based projection. Assuming that the interference subtraction is perfect, the equivalent channel for the th data stream can be characterized by channel


gains, denoted by

, at each tone , where

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is obtained

from . The receiver then jointly decodes the signals for the th data stream , . Note that the overall decoding delay due to successive decoding is approximately OFDM symbol periods, which is usually tolerable in practice if is a small number. Given the set of antenna mappings and power assign, the achievable rate of each data stream , ments , averaged over all OFDM tones, can be obtained as

TABLE I COMPARISON OF COMPUTATIONAL COMPLEXITY

Minimize

Subject to (12)

(11) Note that the factor 1/2 in front of the achievable rate expression is measured in bits/real dimension (or b/s/Hz). is because In the above, denotes the “gap”[31], which is assumed to be equal for all data streams and accounts for the rate loss from the actual channel capacity owing to nonideal (non-Gaussian) signaling. The gap is usually determined by the required BER and the employed modulation and coding scheme (MCS). For example, for the uncoded M-ary quadrature amplitude modulation with the BER of 10 , has the value of 8.8 dB. With a capacity-achieving MCS, can become close to one (0 dB). It was shown in [32] that it is possible to employ a universal Gaussian codebook with different powers assigned to codeword symbols based on their experienced fading gains to achieve the capacity of a scalar fading channel. Practical capacity-approaching MCS for a scalar fading channel may be, e.g., turbo or low-density parity-check code combined with bit-interleaved coded modulation. This result implies that in the closed-loop V-BLAST, by employing a scalar capacity-achieving code for each data stream along with optimal power assignments at different OFDM tones, each data stream can transmit reliably at a maximum rate given (0 dB). Furthermore, it is sufficient to feed in (11) with instead back the overall rate per data stream , of its exact rate components at different tones. IV. OPTIMAL FEEDBACK PARAMETERS This section first provides an optimization framework for determining the optimal feedback parameters for the closed-loop V-BLAST. Then, two low-complexity algorithms for this problem are presented. Lastly, this section presents modifications of the proposed algorithms for further feedback complexity reduction. A. Optimization Problem In this paper, transmit optimization is assumed to achieve the goal of minimizing the average transmit power for each transmission block to support a target average transmit rate . Similar optimization techniques can be developed for maximizing the transmit rate of each block under some given transmit power constraint. The following optimization problem is thus considered for each transmission block:

The optimization variables in the above problem are the antenna mapping vectors and the power assignments . After they are determined, the associated rate assignments for each can be obtained accordingly from (11). For data stream a given set of , are uniquely determined from the in QR decomposition and, hence, the optimal values for the above problem can be easily found by using the standard water-filling algorithm [33]. The main difficulty for solving the . problem at hand lies in the search for the optimal values of Note that the problem in (12) might not be convex because the left-hand-side (LHS) function of its constraint is a maximum of a set of concave functions, and is thus not necessarily concave [34]. Therefore, it is unclear whether this problem can be solved by using standard convex optimization techniques. Next, three candidate algorithms are considered for this problem, and their computational complexities are summarized in Table I. 1) Exhaustive Search: A direct approach to solve the problem at hand is to search over all possible antenna mappings at all OFDM tones and find the optimal one with the smallest , the complexity power consumption . For a given set of is mainly due to two parts: i) QR decomposition to obtain and ii) water-filling algorithm to compute channel gains . The first part requires the complexity optimal values of for each tone3 and hence in total. of The second part has the complexity of . At each tone, possible ways of assigning each of data there are streams a different transmit antenna. Hence, the number of , which distinct antenna mappings over all tones is is . The total computational complexity for the exhaustive search can thus be shown to be . This complexity is in the exponential of both and , and hence becomes infeasible even for a small number of tones and transmit antennas.4 2) Lagrange Dual-Decomposition Method: A more efficient algorithm to find the optimal antenna mapping than the exhaustive search is based on the Lagrange dual-decomposition method [36]. The first step of this method is to introduce the 3QR decomposition can be implemented by either the Gram–Schmidt algorithm or the Householder transform [35]. 4When computing the QR decomposition, a further reduction of complexity is possible by utilizing the fact that some different antenna mappings may have similar QR decompositions. Nevertheless, even if the complexity for computing QR decompositions is ignored, it can be easily verified that a total complexity of ( ) is still necessary for the exhaustive search.

OM

MN

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positive dual variable associated with the rate constraint in (12) and to write the Lagrangian [34] of the original (primal) problem as

(19) is the positive dual variable associated with the conwhere , for , respectively. From (18) and straint can be obtained as the following (19), the optimal values for “water-filling” solutions: (20)

(13) By substituting them into (17), it follows that Next, the Lagrange dual function [34] is defined as (14) , and the opThe dual problem is then defined as . From timal value of the dual problem is denoted by (13) and (14), is a concave function because it is a pointwise minimum of a family of affine functions of [34]. Hence, the optimal value of , denoted by , which maximizes the dual , can be found by using convex optimization techfunction niques. The corresponding optimal value of the dual problem then serves as a lower bound for the optimal value of the orig[34] inal problem, denoted by (15) and is nonzero for In general, the duality gap between a nonconvex optimization problem. However, for the problem at hand, it is shown in Appendix I that the duality gap is incan be obtained by first minimizing the deed zero. Hence, and then maxiLagrangian to obtain the dual function over all positive values of . mizing Consider first the problem for obtaining with a given . can From (13) and (14), it is interesting to observe that also be written as (16)

(21) From (21), can be obtained by the minimization over all that maximizes possible . The remaining task is to find over all , which can be done by a bisection search [34] for . In summary, the following algorithm can be used to solve the problem at hand. Algorithm 4.1: • Initialize

,

.

• Repeat 1. Set

;

2. From (21), find the optimal

, for

; ,

3. Set

,

; 4. If

, set ; otherwise set

.

• Until where is a small positive constant that controls the algorithm accuracy.

where

(17) can be obtained through solving independent Hence, , . The above practice subproblems, each for is usually referred to as the dual decomposition.5 For tone with , a given antenna mapping , the minimization in (17) over , is a convex optimization problem. Hence, by the Karush–Kuhn–Tucker (KKT) conditions [34], the optimal at tone need to satisfy the following equations values of simultaneously: (18) 5Other applications of the Lagrange dual-decomposition method for resource allocation in communication systems can be found in, e.g., [28], [37]–[39].

In the above algorithm, is an upper bound for . One possible method to obtain is provided in Appendix II. The computational complexity of the dual-decomposition method can be obtained as follows. The number of iterarequires a complexity of tions of the bisection search for . In each iteration, like the exhaustive search, the . Hence, the complexity can be shown to be , which is a sigtotal complexity is ) reduction compared to the exhaustive nificant (a factor of search. Nevertheless, the complexity is still in the exponential of . 3) Greedy Algorithm: The greedy algorithm iteratively finds a suboptimal transmit antenna mapping at each OFDM tone. In the first iteration, the algorithm applied to tone computes the for each transmit annorm of the channel gain vector , and then selects the one with the largest tenna, norm to carry the th data stream that is decoded th in the order at the receiver. In the second iteration, each remaining


projects itself into the null space of the selected channel gain vector obtained in the first iteration, and the transmit antenna with the largest vector norm after the projection is assigned to th data stream. This process iterates until all carry the data streams at tone are assigned with different transmit antennas. The following algorithm summarizes the greedy algorithm for finding the antenna mapping at tone . Algorithm 4.2: and

• Initialize

.

• Repeat 1.

;

2.

;

3.

;

4. For

,

, ;

5. • Until

. .

The computational complexity of the greedy algorithm is mainly due to the channel QR decompositions and can be shown , which becomes linear in both and to be and is thus a significant reduction compared to for the dual decomposition. With the antenna mappings found by the greedy algorithm, the corresponding power and rate assignments can be easily found by using the standard water-filling algorithm. The greedy algorithm is different from the rule proposed in [21], [40], and [41] for finding decoding orders of transmit antennas in the open-loop V-BLAST. In the above prior work, it is assumed that there is no feedback power and rate for each transmit antenna, and, hence, equal power and rate are assigned to each data stream radiated by a corresponding transmit antenna. In this case, the rule for assigning decoding orders for transmit antennas is quite intuitive, i.e., always selecting the transmit antenna that has the largest channel gain (after projecting into the null space of channel vectors of those not-yet-decoded) to be decoded first. This rule is to minimize the error prorogations into subsequent decoding stages. In contrast, the closed-loop V-BLAST has feedback power control to protect each data stream, and hence the criterion for selecting transmit antennas and their corresponding decoding orders changes to maximize the sum-rate of all data streams (or equivalently, minimize the total transmit power given the sum-rate). As a result, the decoding orders given by the greedy algorithm are simply reverted, i.e., always selecting the transmit antenna that has the largest channel gain among the remaining unassigned antennas to be decoded last. B. Feedback Reduction The feedback parameters and their associated complexities in the closed-loop V-BLAST are summarized as follows: i) rate assignments : ;

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TABLE II FEEDBACK COMPLEXITY OF CLOSED-LOOP V-BLAST

ii) antenna mappings : ; : . iii) power assignments This section presents two techniques to further reduce the amount of feedback for antenna mappings and power assignments, respectively. Table II summarizes the feedback complexity for the closed-loop V-BLAST before and after applying these techniques. 1) Tone Grouping: The technique of tone-grouping or toneclustering reduces the amount of feedback for antenna mapby dividing OFDM tones into disjoint groups, pings (each group consists of adjacent tones, and for convenience is assumed to be divisible by ) and then assigning all tones in each group the same set of antenna mappings , . The feedback complexity for antenna mapafter applying this technique. pings becomes The tone grouping utilizes the fact that the MIMO channels of adjacent OFDM tones are usually highly correlated and, hence, their optimal antenna mappings are also very likely to be similar. As a result, these adjacent tones can be grouped together and assigned with an identical antenna mapping without a notable performance degradation. Some cases of the tone grouping with different values of are listed as follows. : The receiver feeds back the complete antenna • mappings for all tones. : The receiver feeds back the antenna mappings • for disjoint bands, where is the number of delayed is approximately equal to the multipaths in (2) and channel coherence bandwidth. : The receiver does not feed back any antenna map• pings, and a fixed set of antenna mappings is used for all tones. Without loss of generality, this paper assumes that , , in this case. With the tone grouping, the optimal feedback parameters can also be obtained by solving the problem in (12) with an additional constraint if

(22)

The above constraint can be easily incorporated into the algorithms previously developed. Algorithm 4.1 remains mostly unchanged except that the dual-decomposition method needs to be applied to disjoint grouped frequency bands. Algorithm 4.2 needs to modify the criterion for determining for the th . In this paper, it is simply assumed that group, the sum of channel gains of each transmit antenna from all tones for transmit antenna in the in each group (e.g., th group) is the criterion for assigning antenna mapping in the greedy algorithm. 2) Antenna Grouping: The goal of antenna grouping is to reduce the amount of power feedback in the closed-loop

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cation to each group can be obtained by solving the following modified problem of (12):

Minimize

Subject to

(24)

The solutions to the above problem can be obtained by modifying the standard water-filling algorithm. The derivation is omitted here for brevity, and the optimal power values can be , , where is the unique expressed as root of the following equation: (25) Fig. 2. Space-frequency channel gains by the V-BLAST receiver with ZF-based successive decoding for a Rayleigh-fading MIMO-OFDM channel , , and . with

M = M = M = 4 N = 256

L=4

for , respectively, and is the “water level” with which the rate constraint becomes equality in (24). V. SIMULATION RESULTS

V-BLAST. Suppose that the set of antenna mappings at all tones is already resolved by using either Algorithm 4.1 or Algorithm 4.2 or their modifications after applying the tone grouping. The antenna grouping collects the transmit antennas at all OFDM tones corresponding to the same assigned data stream (or equivalently, the same decoding order) into a group and assigns them an identical power. More specifically, the following new constraints can be added to the problem in (12):

(23) As a result, the total feedback complexity for powers is reduced to after the antenna grouping is applied, from which can be a significant reduction for MIMO-OFDM with a large number of tones. In addition to reducing the feedback complexity, the antenna grouping is also able to achieve closely the performance with a complete power feedback. This can be explained by observing from all tones of each the channel gains group . Fig. 2 plots these channel gains for a Rayleigh-fading MIMO-OFDM channel with , , . It is assumed that the antenna mappings at and different tones are obtained by the greedy algorithm (Algorithm 4.2). It is observed that the channel gains of each group can have very different fading statistics compared to those of the other groups. The channel gains corresponding to a higher group index (i.e., decoded later) have a larger mean and smaller variations (i.e., less fading) compared to those from lower group indexes. Based on these observations, it follows that it might not be necessary to allocate different powers to transmit antennas at all tones because most of the performance gain over the equal-power allocation can be obtained by only varying the powers assigned to different groups. The optimal power allo-

This section presents simulation results to evaluate the performance of the closed-loop V-BLAST and compares it to that of the conventional open-loop V-BLAST and other existing MIMO feedback schemes in the literature. The MIMO-OFDM tones and channel is assumed to have equal-energy multipath delays. The Rayleigh-fading channel model is assumed for simplicity, and thus the time-domain , , are independent random channel matrices matrices, each having entries independently distributed as . Two antenna configurations are considered: i) and ii) . Monte Carlo simulations are used to average the fading effects over 5000 independently generated MIMO-OFDM channels. If not stated otherwise, it is assumed that the capacity-achieving code is used and thus the gap is one (0 dB). A constant-rate transmission is assumed for each fading block, and the required average transmit power over all randomly generated MIMO-OFDM channels, which can be equivalently represented by the average SNR defined in (4), is used as the performance measure. Note that the maximum constant-rate that is achievable over all the fading states given an average power constraint is known as the channel delay-limited capacity [42], which is achievable by the eigenmode transmission along with water-filling-based power and rate adaptations [43]. Figs. 3 and 4 compare the achievable rates of the closed-loop V-BLAST with the full transmit power, rate, and antenna-mapping feedback and the open-loop V-BLAST without feedback, for the 4 4 and 2 4 antenna configuration, respectively. Two algorithms for finding antenna mapping in the closed-loop V-BLAST, namely, the Lagrange dual-decomposition method (Algorithm 4.1) and the greedy algorithm (Algorithm 4.2), are also compared. For both antenna configurations, it is observed that the greedy algorithm sustains a rate almost equal to that obtained by the Lagrange dual-decomposition method for all SNR


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M =M =4

Fig. 5. Comparison of the achievable rates of the closed-loop V-BLAST under different amounts of feedback for transmit antenna mappings for MIMO-OFDM and . denotes the number of grouped bands channels with for the tone grouping.

Fig. 3. Comparison of the achievable rates of the closed-loop and open-loop . V-BLAST for MIMO-OFDM channels with

M =4 M =2

Fig. 4. Comparison of the achievable rates of the closed-loop and open-loop and . V-BLAST for MIMO-OFDM channels with

values. This result demonstrates the usefulness of the greedy algorithm in achieving the close-to-optimal performance given its very low computational complexity. Two receiver structures for the open-loop V-BLAST are considered: the MMSE-based and ZF-based successive decoding, respectively.6 It is observed that the throughput improvement by the closed-loop V-BLAST over the open-loop V-BLAST is quite substantial for both antenna configurations. This is because the closed-loop V-BLAST has better decoding reliability by power control, as well as more flexible transmit rate assignments and antenna mappings. 6For this open-loop V-BLAST, equal power and rate are assigned to all OFDM tones of four transmit antennas in the 4 4 case and of two randomly selected transmit antennas in the 2 4 case. For both antenna configurations, decoding order of transmit antennas at the receiver are chosen to minimize the decoding error propagation like in [40] and [41]. For each randomly generated MIMO-OFDM channel, the sum-power for all transmit antennas is chosen to be the minimum for the assigned equal rate to be achievable for all data streams.

2

2

M =4 M =2 D

Fig. 5 compares the achievable rates of the closed-loop V-BLAST with different amounts of feedback for transmit-antenna mappings by applying the tone grouping. Four cases with different numbers of grouped bands are compared with , , , decreasing amount of feedback: and . The 2 4 antenna configuration is considered. (fixed antenna mapping), the For all cases except greedy algorithm is used to find antenna mappings for different grouped bands. In this simulation, the full power and rate feedback is assumed. It is observed that fixed antenna mapping at all tones suffers from a severe rate loss compared to the other three cases with antenna-mapping feedback. This is because in , antenna selection diversity at different the case of tones is crucial to the achievable rate and, hence, the feedback of antenna mappings (this feedback also selects two active transmit antennas from four available ones) at different tones boosts up the achievable rate significantly. It is also observed (i.e., the number of grouped bands is equal to that that of independent multipaths of the MIMO-OFDM channel) is a good choice for balancing the feedback complexity and the achievable rate. Fig. 6 shows the achievable rates of the closed-loop V-BLAST under different amounts of power feedback for the 4 4 antenna configuration. Three schemes with their corresponding power feedback complexities are considered: ; i) full power feedback: ii) reduced power feedback after the antenna grouping is ap; plied: iii) equal-power feedback for which the same feedback power is assigned to all transmit antennas at all tones: . The full rate and antenna-mapping feedback are assumed, and the greedy algorithm is used to find the antenna mapping at each tone. It is observed that for all SNR values, the antenna grouping achieves almost the identical rate by the full power feedback. At asymptotically high SNR, it is observed that the achievable rates

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M=

Fig. 6. Comparison of the achievable rates of the closed-loop V-BLAST under different amounts of power feedback for MIMO-OFDM channels with .

M =4

M =

Fig. 7. Comparison of the achievable rates of the closed-loop V-BLAST and other closed-loop feedback schemes for MIMO-OFDM channels with .

M =4

of all three power feedback schemes converge to be identical, and the gain by adapting power assignments becomes minimal. In contrast to rate and antenna-mapping feedback, it is observed that the throughput gain by power feedback is much less substantial. Nevertheless, power feedback (at least the equal-power feedback) is still necessary in the closed-loop V-BLAST because it ensures the reliability of successive decoding. Figs. 7 and 8 compare the achievable rates by the proposed closed-loop V-BLAST and three other feedback schemes for the 4 4 and 2 4 antenna configuration, respectively. These feedback schemes for comparison include the eigenmode transmission with the complete CSI feedback, a vertical-encoding scheme with ZF-based linear receivers (e.g., [44]) and optimized power feedback,7 and the per-tone-based transmit-an7For this scheme, the ZF-based linear receiver decomposes the MIMO subchannels. Then, water-filling channel at each tone into based power allocations for sub-channels at all OFDM tones are computed. Note that for this scheme with the 2 4 antenna configuration, two randomly selected transmit antennas out of four available ones are used to carry two transmitted data streams.

min(M ;M ) 2

M =4

Fig. 8. Comparison of the achievable rates of the closed-loop V-BLAST and other closed-loop feedback schemes for MIMO-OFDM channels with . and

M =2

tenna selection with feedback of selected antennas and optimized power assignments.8 For the closed-loop V-BLAST, the greedy algorithm is used for determining transmit antenna , and the equal-power feedback mappings with and the full rate feedback are assumed. Note that the eigenmode transmission achieves the channel delay-limited capacity, which is also the upper bound for the achievable rate by any MIMO-precoding feedback scheme with a finite codebook size. It is observed that the rate gap between the closed-loop V-BLAST and the channel delay-limited capacity is very in Fig. 7 but increases when small for in Fig. 8. This is so because the closed-loop V-BLAST with per-antenna-based feedback is incapable of capturing the full transmit beamforming gain by the eigenmode transmission or MIMO-precoding feedback, which becomes more dominant at high SNR. On the other hand, it is observed that the throughput gain achievable by the closed-loop V-BLAST (with reduced feedback) over the closed-loop ZF-based linear receiver (with full power feedback) is still substantial for both antenna configurations. This is mainly because of the gain by successive decoding over linear decoding. Similarly, substantial throughput gains are also observed for the closed-loop V-BLAST over the per-tone-based transmit-antenna selection feedback, especially at high SNR when spatial multiplexing gain achievable by the closed-loop V-BLAST becomes more dominant over diversity gain by transmit-antenna selection. For all previous simulation results, it is assumed that the feedback delay is negligible compared to the channel coherence time and, hence, the channel is unchanged during each feedback period. Next, the performance of the closed-loop V-BLAST is evaluated under the channel estimation error due to the feedback delay. For simplicity, it is assumed that the actual time-domain , at each fading state are channel matrices 8For this scheme, a greedy algorithm similar like Algorithm 4.2 is used to select one transmit antenna at each tone (probably different from tone to tone). Then, the power allocations for selected antennas over different tones are optimized by the water-filling algorithm.


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M =

Fig. 9. Additional transmit power margin to compensate for the channel variation in the closed-loop V-BLAST for MIMO-OFDM channels with .

M =4

equal to , where is the channel matrix used at the receiver for assigning feedback parameters like in is the error matrix that is identically previous simulations, but independently distributed compared to , and the param, controls the amount of channel error due to eter , is equivalent to no channel the feedback delay. Notice that for the complete channel mismatch estimation error and between the transmitter and the receiver sides. For simplicity, , the equal-power feedthe greedy algorithm with back, and the full rate feedback are assumed for the closed-loop V-BLAST. The 4 4 case is considered. Fig. 9 plots the additional transmit power, termed power margin, assigned equally to all data streams at the transmitter to ensure a block error probability9 of 10 with a fixed transmit rate of 6 b/s/Hz versus the channel variation parameter . Note that the transmit power margin is used for compensating the channel estimation error at the receiver. It is observed that the closed-loop V-BLAST is very robust to channel estimation . errors, e.g., only 1.7 dB power margin is required for At last, Fig. 10 shows the achievable rate of the closed-loop V-BLAST when practical non-Gaussian MCS is applied. It is assumed that each data stream uses the same adaptive MCS that has an SNR gap and a discrete bit-loading granularity . The greedy algorithm with , the per-antenna power feedback, and the full rate feedback are assumed for the closed-loop V-BLAST, and the 4 4 case is considered. Because the antenna mapping at different tones has been resolved by the greedy algorithm, the V-BLAST receiver decomposes the MIMO channel at each OFDM tone into parallel scalar channels, and therefore the optimal discrete bit-loading algorithm [45] can be applied. It is observed that the closed-loop only supports a discrete set of V-BLAST with nonzero transmit rates, and increasing from 0.5 to 1 bit results in a further transmit power loss of 0–0.5 dB under the same gap dB. 9Since successive decoding is used at the receiver, a decoding error of the whole block is declared if any of the data streams is not decoded correctly.

M =M =4

Fig. 10. The achievable rate of the closed-loop V-BLAST under practical adap. tive MCS for MIMO-OFDM channels with

VI. CONCLUDING REMARKS This paper presents a practical partial-channel-feedback scheme to support capacity-approaching spatial multiplexing for the frequency-selective fading MIMO-OFDM channel. The proposed scheme is a closed-loop extension of the well-known V-BLAST transmission scheme. Though the conventional open-loop V-BLAST is severely compromised in practice owing to its poor diversity performance and error propagation, the proposed closed-loop V-BLAST overcomes these difficulties by adaptively assigning transmit powers, rates, and antenna mappings at all OFDM tones. It is shown by simulation results that even with a moderate amount of feedback by applying antenna and tone grouping, the closed-loop V-BLAST is still able to approach closely the MIMO-OFDM channel capacity achievable by the eigenmode transmission. The main challenge for determining feedback parameters for the closed-loop V-BLAST is optimization of transmit antenna mappings together with transmit powers and rates. This paper presents low-complexity algorithms for this problem and reveals some new insights on finding good transmit antenna mappings for the closed-loop V-BLAST. This paper studies the feedback power and rate optimization for each independent fading block. With consecutive block transmission and the sufficiently large channel coherence time, there is likely strong correlation between the MIMO-OFDM channels during consecutive block transmission. As a result, if a constant-rate transmission per block is required, the optimal power and rate assignments for consecutive blocks cannot change abruptly. This observation has an important consequence, i.e., only feedback of the increments or decrements of powers and rates (also known as “energy swapping” and “bit swapping,” respectively, in [31]) between the current transmission block and its preceding one is sufficient. This fact can be used to reduce further the amount of feedback for each transmission block. By doing this, the time-correlated CSI at the receiver is conveyed to the transmitter by incrementally encoding the feedback transmit powers, rates, and antenna mappings.

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APPENDIX I PROOF OF ZERO DUALITY GAP This appendix proves that the duality gap between the primal and dual problem in Section IV-A-2) is indeed zero. First, beis concave, from (16) and (21), it follows that the cause satisfies , i.e., optimal dual variable (26) Let denote the optimal values for in (26), By substituting the optimal powers corresponding to

. ,

, into the LHS of (12) and using (26), it can be verified that the achievable sum-rate is indeed equal to . As a result, from (13), it follows that . Because can be no less than , the obtained as well. Hence,

must be optimal for the primal problem is established.

APPENDIX II UPPER BOUND OF An upper bound for the optimal dual variable in Algorithm 4.1 can be obtained as follows. Let and be any set of antenna mappings and power assignments, respectively, satisfying (27) The following equalities/inequalities can be shown: (28) (29) (30) (31) where (29) is from the zero duality gap, (30) is due to are not the Lagrangian minimizers in the fact that , and (31) is by substituting (27) into (13). Let (14) for . From (31), it follows that . REFERENCES [1] G. Raleigh and J. M. Cioffi, “Spatial-temporal coding for wireless communications,” IEEE Trans. Commun., vol. 46, pp. 353–366, 1998. [2] A. Narula, M. J. Lopez, M. D. Trott, and G. W. Wornell, “Efficient use of side information in multiple-antenna data transmission over fading channels,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1423–1436, Oct. 1998. [3] V. Lau, Y. J. Liu, and T. A. Chen, “On the design of MIMO blockfading channels with feedback-link capacity constraint,” IEEE Trans. Commun., vol. 52, pp. 62–70, Jan. 2004. [4] D. J. Love and R. W. Heath, Jr, “Multimode precoding for MIMO wireless systems,” IEEE Trans. Signal Process., vol. 53, pp. 3674–3687, Oct. 2005. [5] D. J. Love, R. W. Heath, Jr, and T. Strohmer, “Grassmannian beamforming for multiple-input multiple-output wireless systems,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2735–2747, Oct. 2003.

[6] K. K. Mukkavilli, A. Sabharwal, E. Erkip, and B. Aazhang, “On beamforming with finite rate feedback in multiple antenna systems,” IEEE Trans. Inf. Theory, vol. 49, pp. 2562–2579, Oct. 2003. [7] G. Jöngren and M. Skoglund, “Quantized feedback information in orthogonal space-time block coding,” IEEE Trans. Inf. Theory, vol. 50, pp. 2473–2486, Oct. 2004. [8] L. Collin, O. Berder, P. Rostaing, and G. Burel, “Optimal minimum distance-based precoder for MIMO spatial multiplexing systems,” IEEE Trans. Signal Process., vol. 52, pp. 617–627, Mar. 2004. [9] D. J. Love and R. W. Heath, “Limited feedback unitary precoding for spatial multiplexing systems,” IEEE Trans. Inf. Theory, vol. 51, pp. 2967–2976, Aug. 2005. [10] J. C. Roh and B. D. Rao, “Transmit beamforming in multiple-antenna systems with finite rate feedback: A VQ-based approach,” IEEE Trans. Inf. Theory, vol. 52, pp. 1101–1112, Mar. 2006. [11] R. W. Heath, Jr, S. Sandhu, and A. Paulraj, “Antenna selection for spatial multiplexing systems with linear receivers,” IEEE Commun. Lett., vol. 5, pp. 142–144, Apr. 2001. [12] A. F. Molisch, M. Z. Win, and J. H. Winters, “Capacity of MIMO systems with antenna selection,” in Proc. IEEE Int. Commun. Conf. (ICC), 2001, vol. 2, pp. 570–574. [13] R. S. Blum and J. H. Winters, “On optimum MIMO with antenna selection,” in Proc. IEEE Int. Commun. Conf. (ICC), 2002, vol. 1, pp. 386–390. [14] J. Choi and R. W. Heath, Jr, “Interpolation based transmit beamforming for MIMO-OFDM with limited feedback,” IEEE Trans. Signal Process., vol. 53, pp. 4125–4135, Nov. 2005. [15] J. Choi, B. Mondal, and R. W. Heath, Jr, “Interpolation based unitary precoding for spatial multiplexing MIMO-OFDM with limited feedback,” IEEE Trans. Signal Process., vol. 54, pp. 4730–4740, Dec. 2006. [16] S. Zhou, B. Li, and P. Willett, “Recursive and trellis-based feedback reduction for MIMO-OFDM with rate-limited feedback,” IEEE Trans. Wireless Commun., vol. 5, pp. 3400–3405, Dec. 2006. [17] H. Zhang, Y. Li, V. Stoplman, and N. V. Waes, “A reduced CSI feedback approach for precoded MIMO-OFDM systems,” IEEE Trans. Wireless Commun., vol. 6, pp. 55–58, Jan. 2007. [18] T. Pande, D. J. Love, and J. V. Krogmeier, “Reduced feedback MIMOOFDM precoding and antenna selection,” IEEE Trans. Signal Process., vol. 55, pp. 2284–2239, May 2007. [19] A. Paulraj, R. Nabar, and D. Gore, Introduction to Space-Time Wireless Communications. Cambridge, U.K.: Cambridge Univ. Press, 2003. [20] P. W. Wolnainsky, G. J. Foschini, G. D. Golden, and R. A. Valenzuela, “V-BLAST: An architecture for achieving very high data rates over the rich-scattering wireless channel,” in Proc. ISSSE, Pisa, Italy, 1998. [21] G. J. Foshini, G. Golden, R. Valenzuela, and P. Wolniansky, “Simplified processing for high spectral efficiency wireless communication employing multi-element arrays,” IEEE J. Sel. Areas Commun., vol. 17, pp. 1841–1852, Nov. 1999. [22] G. Ginis and J. M. Cioffi, “On the relation between BLAST and the GDFE,” IEEE Commun. Lett., vol. 5, no. 9, pp. 364–366, Sep. 2001. [23] B. M. Hochwald and S. T. Brink, “Achieving near-capacity on a multiple-antenna channel,” IEEE Trans. Commun., vol. 51, pp. 389–399, Mar. 2003. [24] S. H. Nam, O. S. Shin, and K. B. Lee, “Transmit power allocation for a modified V-BLAST system,” IEEE Trans. Commun., vol. 52, pp. 1074–1079, Jul. 2004. [25] N. Wang and S. D. Blostein, “Minimum BER transmit optimization for two-input multiple-output spatial multiplexing,” Proc. IEEE GLOBECOM, vol. 6, Dec. 2005. [26] H. Zhuang, L. Dai, S. Zhou, and Y. Yao, “Low complexity per-antenna rate and power control approach for closed-loop V-BLAST,” IEEE Trans. Commun., vol. 51, pp. 1783–1787, Nov. 2003. [27] S. T. Chung, A. Lozano, H. C. Huang, A. Sutivong, and J. M. Cioffi, “Approaching the MIMO capacity with a low-rate feedback channel in V-BLAST,” EURASIP J. Appl. Signal Process., no. 5, pp. 762–771, 2004. [28] R. Zhang, Y. C. Liang, R. Narasimhan, and J. M. Cioffi, “Approaching MIMO-OFDM capacity with per-antenna power and rate feedback,” IEEE J. Sel. Areas Commun., vol. 25, pp. 1284–1297, Sep. 2007. [29] S. Verdu, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 2003. [30] M. K. G. VaranasiT, “Optimum decision-feedback multiuser equalization with successive decoding achieves the total capacity of the Gaussian multiple-access channel,” in Proc. Asilomar Conf. Signals, Syst., Comput., 1998, pp. 1405–1409. [31] J. M. Cioffi, “Digital communications,” Stanford Univ., Stanford, CA, 2007, unpublished course notes.


[32] G. Caire and S. Shamai, “On the capacity of some channels with channel state information,” IEEE Trans. Inf. Theory, vol. 45, pp. 2007–2019, Sep. 1999. [33] T. Cover and J. Thomas, Elements of Information Theory. New York: Wiley, 1991. [34] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. [35] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1985. [36] S. Boyd, “EE392o Lecture Notes,” Stanford University, Stanford, CA, 2003. [37] L. Xiao, M. Johansson, and S. P. Boyd, “Simultaneous routing and resource allocation via dual decomposition,” IEEE Trans. Commun., vol. 52, pp. 1136–1144, Jul. 2004. [38] R. Cendrillon, W. Yu, M. Moonen, J. Verlinden, and T. Bostoen, “Optimal multiuser spectrum balancing for digital subscriber lines,” IEEE Trans. Commun., vol. 54, pp. 922–933, May 2006. [39] M. Mohseni, R. Zhang, and J. M. Cioffi, “Optimized transmission of fading multiple-access and broadcast channels with multiple antennas,” IEEE J. Sel. Areas Commun., vol. 24, pp. 1627–1639, Aug. 2006. [40] D. Wubben, R. Bohnke, J. Rinas, V. Kuhn, and K. D. Kammeyer, “Efficient algorithm for decoding layered space-time codes,” Electron Lett., vol. 37, no. 22, pp. 1348–1350, Oct. 2001. [41] B. Hassibi, “An efficient square-root algorithm for BLAST,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), Jun. 2000, vol. 2, pp. 737–740. [42] S. Hanly and D. Tse, “Multi-access fading channels—Part II: Delaylimited capacities,” IEEE Trans. Inf. Theory, vol. 44, pp. 2816–2831, Nov. 1998. [43] E. Biglieri, G. Caire, and G. Taricco, “Limiting performance of blockfading channels with multiple antennas,” IEEE Trans. Inf. Theory, vol. 47, pp. 1273–1289, May 2000. [44] M. R. Mckay and I. B. Collings, “Capacity and performance of MIMOBICM with zero-forcing receivers,” IEEE Trans. Commun., vol. 53, pp. 74–83, Jan. 2005. [45] J. Campello, “Practical bit loading for DMT,” in Proc. IEEE ICC, 1999, vol. 2, pp. 801–805.

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Rui Zhang (S’00–M’07) received the B.S. and M.S. degrees in electrical and computer engineering from National University of Singapore, Singapore, in 2000 and 2001, respectively, and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, in 2007. Since 2007, he has been a Research Fellow with the Institute for Infocomm Research, Singapore. His recent research interests include cognitive radio networks, cooperative communication systems, and multiuser MIMO transmission systems.

John M. Cioffi (S’77–M’78–SM’90–F’96) received the B.S. degree from the University of Illinois, Urbana-Champaign, in 1978 and the Ph.D. degree from Stanford University, Stanford, CA, in 1984, both in electrical engineering. He was with Bell Laboratories from 1978 to 1984 and IBM Research from 1984 to 1986. He has been a Professor of electrical engineering with Stanford University since 1986. He founded Amati Communications Corporation in 1991 (purchased by Texas Instruments in 1997) and was Officer/Director from 1991 to 1997. He currently is on the Board of Directors of ASSIA (Chair), Afond, Teranetics, and ClariPhy. He is on the Advisory Board of Portview Ventures, Wavion, MySource, and Amicus. His specific interests are in the area of high-performance digital transmission. He has published more than 250 papers and has received more than 80 patents. Dr. Cioffi is a member of the National Academy of Engineering. He received the Hitachi America Professorship in Electrical Engineering from Stanford (2002), the IEEE Kobayashi Medal (2001), the IEEE Millennium Medal (2000), the IEE J. J. Tomson Medal (2000), the 1999 University of Illinois Outstanding Alumnus Award, the 1991 IEEE Communications Magazine Best Paper Award, the 1995 ANSI T1 Outstanding Achievement Award, the National Science Foundation Presidential Investigator Award (1987–1992), the ISSLS 2004 Outstanding Paper Award, and the Marconi Fellow Award (2006).