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

Space-Time Block Coding for Wireless Communications

In this chapter we describe the six families of so-called ‘classical’ simple groups. These are the linear, unitary and symplectic groups, and the three families of orthogonal groups. All may be obtained from suitable matrix groups G by taking G′/Z(G′). Our main aims again are to define these groups, prove they are simple, and describe their automorphisms, subgroups and covering groups.

The classical groups

Understand the fundamentals of wireless and MIMO communication with this accessible and comprehensive text. Viewing the subject through an information theory lens, but also drawing on other perspectives, it provides a sound treatment of the key concepts underpinning contemporary wireless communication and MIMO, all the way to massive MIMO. Authoritative and insightful, it includes over 330 worked examples and 450 homework problems, with solutions and MATLAB code and data available online. Altogether, this is an excellent resource for instructors and graduate students, as well as an outstanding reference for researchers and practicing engineers.

Foundations of MIMO Communication

Motivated by the current interest in ultra-reliable, low-latency, machine-type communication systems, we investigate the tradeoff between reliability, throughput, and latency in the transmission of information over multiple-antenna Rayleigh block-fading channels. Specifically, we obtain finite-blocklength, finite-SNR upper and lower bounds on the maximum coding rate achievable over such channels for a given constraint on the packet error probability. Numerical evidence suggests that our bounds delimit tightly the maximum coding rate already for short blocklengths (packets of about 100 symbols). Furthermore, our bounds reveal the existence of a tradeoff between the rate gain obtainable by spreading each codeword over all available time-frequency-spatial degrees of freedom, and the rate loss caused by the need of estimating the fading coefficients over these degrees of freedom. In particular, our bounds allow us to determine the optimal number of transmit antennas and the optimal number of time-frequency diversity branches that maximize the rate. Finally, we show that infinite-blocklength performance metrics such as the ergodic capacity and the outage capacity yield inaccurate throughput estimates.

Short-Packet Communications Over Multiple-Antenna Rayleigh-Fading Channels

To evaluate the unitary integrals, such as the well-known Harish-Chandra-Itzykson-Zuber integral, character expansions were developed by Balantekin, where the matrix integrand is a group member; i.e., a square matrix with a nonzero determinant. Recently, this method has been exploited to derive the joint eigenvalue distributions of the Wishart matrices; i.e., HH* where H is the complex Gaussian random channel matrix of a multiple-input multiple-output (MIMO) system. The joint eigenvalue distributions are used to calculate the moment generating function of the mutual information (ergodic capacity) of a MIMO channel. In this paper, we show that the previous integration framework presented in the literature is not correct, and results in incorrect joint eigenvalue distributions for the Ricean and full-correlated Rayleigh MIMO channels. We develop a new framework to apply the character expansions for integrations over the unitary group, involving general rectangular complex matrices in the integrand. We derive the correct distribution functions and use them to obtain the capacity of the Ricean and correlated Rayleigh MIMO systems in a unified and straightforward approach. The integration technique proposed in this paper is general enough to be used for other unitary integrals in engineering, mathematics, and physics.

On the Application of Character Expansions for MIMO Capacity Analysis

The high computational complexity of maximum likelihood (ML) decoding can impact many applications such as code division multiple access (CDMA) and multiple-input multiple-output (MIMO) systems. The sphere decoder (SD) as an efficient ML decoder has therefore received significant attention in the wireless research community. This letter presents a new statistical method to reduce the complexity of the Schnorr and Euchner sphere decoder (SESD). The method uses a set of bounds, which are computed using the conditional probability based on the minimum metric of the current solution. A lookup tabic for the bounds can be computed offline. The proposed method is effective for any number of antennas with complexity savings about 50% or more over the conventional SD approach.

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A Statistical Pruning Strategy for Schnorr-Euchner Sphere Decoding

Multiple-input multiple-output (MIMO) systems achieve significant diversity and array gains by using transmit beamforming. When complete channel state information (CSI) is not available at the transmitter, a common set of beamformers (codebook) is used by both the transmitter and the receiver. For each channel realization, the best beamformer is selected at the receiver and its index is sent back to the transmitter via a limited feedback channel. In this paper, a codebook design method using the genetic algorithm is proposed, which reduces the design complexity and achieves large minimum-distance codebooks. Exploiting the specific structure of these beamformers, an order and bound algorithm is proposed to reduce the beamformer selection complexity at the receiver side. The exact bit error rate (BER) of the optimal beamforming in finite-series expression is used to facilitate the BER analysis of limited feedback beamforming. By employing a geometrical approach, an approximate BER of limited feedback beamforming is derived when the codebook size is relatively large (high resolution analysis). The simulation results show that the approximate BER is comparatively tight even for small size codebooks.

On the design, selection algorithm and performance analysis of limited feedback transmit beamforming

The character expansion method was introduced by Balantekin [Phys. Rev. D 62, 085017 (2000)] for integration over the unitary group and, in particular, for calculating the well-known Harish–Chandra–Itzykson–Zuber integral where the coefficient matrices in the integrand are square matrices with nonzero determinants. However, in some applications such as the capacity analysis of multiple-input multiple-output channels in wireless communications and information theory, or applying the color-flavor transformation to lattice quantum chromodynamics in physics, or the theory of random matrices in mathematics, the integration over the unitary group is required where general rectangular complex matrices appear in the integrand. In this paper, we use the character expansion of groups to generalize two integrals over the unitary group that have general rectangular complex matrices in the integrand. Although we consider only two integrals, we believe that the integration framework presented here can be used for other...

Generalization of some integrals over unitary matrices by character expansion of groups

Despite primary space-time coding where the channel state information (CSI) is available at the receiver only, the capacity and performance of multiple-input multiple-output (MIMO) systems can be increased significantly when a complete or partial CSI is available at the transmitter. Recently, limited feedback methods including antenna subset selection and unitary preceding have been proposed for orthogonal space-time codes where a partial knowledge of the channel is available at the transmitter via an error-free, zero-delay feedback channel. In this paper, we propose a general structure matrix rather than a unitary one for precoding. By maximizing the signal-to-noise ratio (SNR) per received symbol, we find the optimal precoder for general space-time codes with rate⩽1 symbol per channel use. The performance of the optimal scheme is analytically evaluated. Next, we extend the result for limited feedback systems. Simulation results show that the proposed precoder outperforms the previous work.

A. Ghaderipoor

Papers

On the Application of Character Expansions for MIMO Capacity Analysis

A Statistical Pruning Strategy for Schnorr-Euchner Sphere Decoding

On the design, selection algorithm and performance analysis of limited feedback transmit beamforming

Generalization of some integrals over unitary matrices by character expansion of groups

Optimal Precoder for Rate⩽1 Space-Time Block Codes