Multiple transmit antennas in a downlink channel can provide tremendous capacity (i.e., multiplexing) gains, even when receivers have only single antennas. However, receiver and transmitter channel state information is generally required. In this correspondence, a system where each receiver has perfect channel knowledge, but the transmitter only receives quantized information regarding the channel instantiation is analyzed. The well-known zero-forcing transmission technique is considered, and simple expressions for the throughput degradation due to finite-rate feedback are derived. A key finding is that the feedback rate per mobile must be increased linearly with the signal-to-noise ratio (SNR) (in decibels) in order to achieve the full multiplexing gain. This is in sharp contrast to point-to-point multiple-input multiple-output (MIMO) systems, in which it is not necessary to increase the feedback rate as a function of the SNR

MIMO Broadcast Channels With Finite-Rate Feedback

It is now well known that employing channel adaptive signaling in wireless communication systems can yield large improvements in almost any performance metric. Unfortunately, many kinds of channel adaptive techniques have been deemed impractical in the past because of the problem of obtaining channel knowledge at the transmitter. The transmitter in many systems (such as those using frequency division duplexing) can not leverage techniques such as training to obtain channel state information. Over the last few years, research has repeatedly shown that allowing the receiver to send a small number of information bits about the channel conditions to the transmitter can allow near optimal channel adaptation. These practical systems, which are commonly referred to as limited or finite-rate feedback systems, supply benefits nearly identical to unrealizable perfect transmitter channel knowledge systems when they are judiciously designed. In this tutorial, we provide a broad look at the field of limited feedback wireless communications. We review work in systems using various combinations of single antenna, multiple antenna, narrowband, broadband, single-user, and multiuser technology. We also provide a synopsis of the role of limited feedback in the standardization of next generation wireless systems.

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An overview of limited feedback in wireless communication systems

Multi-user MIMO (MU-MIMO) networks reveal the unique opportunities arising from a joint optimization of antenna combining techniques with resource allocation protocols. Furthermore, it brings robustness with respect to multipath richness, allowing for compact antenna spacing at the BS and, crucially, yielding the diversity and multiplexing gains without the need for multiple antenna user terminals. To realize these gains, however, the BS should be informed with the user's channel coefficients, which may limit practical application to TDD or low-mobility settings. To circumvent this problem and reduce feedback load, combining MU-MIMO with opportunistic scheduling seems a promising direction. The success for this type of scheduler is strongly traffic and QoS-dependent, however.

Shifting the MIMO Paradigm

Multiple-input multiple-output (MIMO) wireless systems use antenna arrays at both the transmitter and receiver to provide communication links with substantial diversity and capacity. Spatial multiplexing is a common space-time modulation technique for MIMO communication systems where independent information streams are sent over different transmit antennas. Unfortunately, spatial multiplexing is sensitive to ill-conditioning of the channel matrix. Precoding can improve the resilience of spatial multiplexing at the expense of full channel knowledge at the transmitter-which is often not realistic. This correspondence proposes a quantized precoding system where the optimal precoder is chosen from a finite codebook known to both receiver and transmitter. The index of the optimal precoder is conveyed from the receiver to the transmitter over a low-delay feedback link. Criteria are presented for selecting the optimal precoding matrix based on the error rate and mutual information for different receiver designs. Codebook design criteria are proposed for each selection criterion by minimizing a bound on the average distortion assuming a Rayleigh-fading matrix channel. The design criteria are shown to be equivalent to packing subspaces in the Grassmann manifold using the projection two-norm and Fubini-Study distances. Simulation results show that the proposed system outperforms antenna subset selection and performs close to optimal unitary precoding with a minimal amount of feedback.

Limited feedback unitary precoding for spatial multiplexing systems

In this paper, we present space shift keying (SSK) as a new modulation scheme, which is based on spatial modulation (SM) concepts. Fading is exploited for multiple-input multiple-output(MIMO) channels to provide better performance over conventional amplitude/phase modulation (APM) techniques. In SSK, it is the antenna index used during transmission that relays information, rather than the transmitted symbols themselves. This absence of symbol information eliminates the transceiver elements necessary for APM transmission and detection (such as coherent detectors). As well, the simplicity involved in modulation reduces the detection complexity compared to that of SM, while achieving almost identical performance gains. Throughout the paper, we illustrate SSK's strength by studying its interaction with the fading channel. We obtain tight upper bounds on bit error probability, and discuss SSK's performance under some non-ideal channel conditions (estimation error and spatial correlation). Analytical and simulation results show performance gains over APM systems (3 dB at a bit error rate of 10-5), making SSK an interesting candidate for future wireless applications. We then extend SSK concepts to incorporate channel coding, where in particular, we consider a bit interleaved coded modulation (BICM) system using iterative decoding for both convolutional and turbo codes. Capacity results are derived, and improvements over APM are illustrated (up to 1 bits/s/Hz), with performance gains of up to 5 dB.

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Space shift keying modulation for MIMO channels

Feedback in a communications system can enable the transmitter to exploit channel conditions and avoid interference. In the case of a multiple-input multiple-output channel, feedback can be used to specify a precoding matrix at the transmitter, which activates the strongest channel modes. In situations where the feedback is severely limited, important issues are how to quantize the information needed at the transmitter and how much improvement in associated performance can be obtained as a function of the amount of feedback available. We give an overview of some recent work in this area. Methods are presented for constructing a set of possible precoding matrices, from which a particular choice can be relayed to the transmitter. Performance results show that even a few bits of feedback can provide performance close to that with full channel knowledge at the transmitter.

What is the value of limited feedback for MIMO channels

Given a multiple-input multiple-output (MIMO) channel, feedback from the receiver can be used to specify a transmit precoding matrix, which selectively activates the strongest channel modes. Here we analyze the performance of random vector quantization (RVQ), in which the precoding matrix is selected from a random codebook containing independent, isotropically distributed entries. We assume that channel elements are independent and identically distributed (i.i.d.) and known to the receiver, which relays the optimal (rate-maximizing) precoder codebook index to the transmitter using B bits. We first derive the large system capacity of beamforming (rank-one precoding matrix) as a function of B, where large system refers to the limit as B and the number of transmit and receive antennas all go to infinity with fixed ratios. RVQ for beamforming is asymptotically optimal, i.e., no other quantization scheme can achieve a larger asymptotic rate. We subsequently consider a precoding matrix with arbitrary rank, and approximate the asymptotic RVQ performance with optimal and linear receivers (matched filter and minimum mean squared error (MMSE)). Numerical examples show that these approximations accurately predict the performance of finite-size systems of interest. Given a target spectral efficiency, numerical examples show that the amount of feedback required by the linear MMSE receiver is only slightly more than that required by the optimal receiver, whereas the matched filter can require significantly more feedback.

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Capacity of a Multiple-Antenna Fading Channel With a Quantized Precoding Matrix

We study the capacity of a single-user channel with multiple antennas and limited feedback. The receiver has perfect channel knowledge, and can relay B bits, which specify a beamforming vector, to the transmitter. We show that a random vector quantization scheme is asymptotically optimal, and give a simple expression for the associated capacity.

Asymptotic capacity of beamforming with limited feedback

We consider a single-user, point-to-point communication system with M transmit and N receive antennas with independent flat Rayleigh fading between antenna pairs. The mutual information of the multi-input/multi-output (MlMO) channel is maximized when the transmitted symbol vector is a Gaussian random vector with covariance matrix Q. The optimal Q depends on how much channel state information is available at the transmitter. Namely, in the absence of any channel state information, the optimal Q is full-rank and isotropic, whereas with perfect channel knowledge, the optimal Q has columns which are the eigenvectors of the channel, and has rank at most min {M, N}. We assume that the receiver can feed back B bits to the transmitter (per codeword). The feedback bits are used to choose the columns of Q from a random set of i.i.d. vectors. We compute the mutual information as a function of both B and the rank of Q. Our results are asymptotic in the number of antennas, and show how much feedback is needed to achieve a rate, which is close to the capacity with perfect channel knowledge at the transmitter.

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Asymptotic performance of MIMO wireless channels with limited feedback

We study the performance of joint signature-receiver optimization for direct-sequence code-division multiple access (DS-CDMA) with limited feedback. The receiver for a particular user selects the signature from a signature codebook, and relays the corresponding B index bits to the transmitter over a noiseless channel. We study the performance of a random vector quantization (RVQ) scheme in which the codebook entries are independent and isotropically distributed. Assuming the interfering signatures are independent, and have independent and identically distributed (i.i.d.) elements, we evaluate the received signal-to-interference plus noise ratio (SINR) in the large system limit as the number of users, processing gain, and feedback bits B all tend to infinity with fixed ratios. This SINR is evaluated for both the matched filter and linear minimum mean-squared error (MMSE) receivers. Furthermore, we show that this large system SINR is the maximum that can be achieved over any sequence of codebooks. Numerical results show that with the MMSE receiver, one feedback bit per signature coefficient achieves close to single-user performance. We also consider a less complex and suboptimal reduced-rank signature optimization scheme in which the user's signature is constrained to lie in a lower dimensional subspace. The optimal subspace coefficients are scalar-quantized and relayed to the transmitter. The large system performance of the quantized reduced-rank scheme can be approximated, and numerical results show that it performs in the vicinity of the RVQ bound. Finally, we extend our analysis to the scenario in which a subset of users optimize their signatures in the presence of random interference.

Wiroonsak Santipach

Papers

What is the value of limited feedback for MIMO channels

Capacity of a Multiple-Antenna Fading Channel With a Quantized Precoding Matrix

Asymptotic capacity of beamforming with limited feedback

Asymptotic performance of MIMO wireless channels with limited feedback

Signature optimization for CDMA with limited feedback