Sphere Packings, Lattices and Groups

Transmit beamforming and receive combining are simple methods for exploiting the significant diversity that is available in multiple-input multiple-output (MIMO) wireless systems. Unfortunately, optimal performance requires either complete channel knowledge or knowledge of the optimal beamforming vector; both are hard to realize. In this article, a quantized maximum signal-to-noise ratio (SNR) beamforming technique is proposed where the receiver only sends the label of the best beamforming vector in a predetermined codebook to the transmitter. By using the distribution of the optimal beamforming vector in independent and identically distributed Rayleigh fading matrix channels, the codebook design problem is solved and related to the problem of Grassmannian line packing. The proposed design criterion is flexible enough to allow for side constraints on the codebook vectors. Bounds on the codebook size are derived to guarantee full diversity order. Results on the density of Grassmannian line packings are derived and used to develop bounds on the codebook size given a capacity or SNR loss. Monte Carlo simulations are presented that compare the probability of error for different quantization strategies.

Grassmannian beamforming for multiple-input multiple-output wireless systems

Most of the recent advances in the design of high-speed wireless systems are based on information-theoretic principles that demonstrate how to efficiently transmit long data packets. However, the upcoming wireless systems, notably the fifth-generation (5G) system, will need to support novel traffic types that use short packets. For example, short packets represent the most common form of traffic generated by sensors and other devices involved in machine-to-machine (M2M) communications. Furthermore, there are emerging applications in which small packets are expected to carry critical information that should be received with low latency and ultrahigh reliability. Current wireless systems are not designed to support short-packet transmissions. For example, the design of current systems relies on the assumption that the metadata (control information) is of negligible size compared to the actual information payload. Hence, transmitting metadata using heuristic methods does not affect the overall system performance. However, when the packets are short, metadata may be of the same size as the payload, and the conventional methods to transmit it may be highly suboptimal. In this paper, we review recent advances in information theory, which provide the theoretical principles that govern the transmission of short packets. We then apply these principles to three exemplary scenarios (the two-way channel, the downlink broadcast channel, and the uplink random access channel), thereby illustrating how the transmission of control information can be optimized when the packets are short. The insights brought by these examples suggest that new principles are needed for the design of wireless protocols supporting short packets. These principles will have a direct impact on the system design.

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Toward Massive, Ultrareliable, and Low-Latency Wireless Communication With Short Packets

In this paper, we consider a multiple-input-multiple-output (MIMO) fading broadcast channel and compute achievable ergodic rates when channel state information (CSI) is acquired at the receivers via downlink training and it is provided to the transmitter by channel state feedback. Unquantized (analog) and quantized (digital) channel state feedback schemes are analyzed and compared under various assumptions. Digital feedback is shown to be potentially superior when the feedback channel uses per channel state coefficient is larger than 1. Also, we show that by proper design of the digital feedback link, errors in the feedback have a minor effect even if simple uncoded modulation is used on the feedback channel. We discuss first the case of an unfaded additive white Gaussian noise (AWGN) feedback channel with orthogonal access and then the case of fading MIMO multiple access (MIMO-MAC). We show that by exploiting the MIMO-MAC nature of the uplink channel, a much better scaling of the feedback channel resource with the number of base station (BS) antennas can be achieved. Finally, for the case of delayed feedback, we show that in the realistic case where the fading process has (normalized) maximum Doppler frequency shift 0 ? F < 1/2, a fraction 1 - 2F of the optimal multiplexing gain is achievable. The general conclusion of this work is that very significant downlink throughput is achievable with simple and efficient channel state feedback, provided that the feedback link is properly designed.

/pdf/multiuser-mimo-achievable-rates-with-downlink-training-and-3wsje5js4s.pdf

Multiuser MIMO Achievable Rates With Downlink Training and Channel State Feedback

Elements of Information Theory (2nd ed.). Thomas M. Cover and Joy A. Thomas

A recent result of Zheng and Tse states that over a quasi-static channel, there exists a fundamental tradeoff, referred to as the diversity-multiplexing gain (D-MG) tradeoff, between the spatial multiplexing gain and the diversity gain that can be simultaneously achieved by a space-time (ST) code. This tradeoff is precisely known in the case of independent and identically distributed (i.i.d.) Rayleigh fading, for Tgesnt+nr-1 where T is the number of time slots over which coding takes place and nt,nr are the number of transmit and receive antennas, respectively. For T nt case, we present two general techniques for building D-MG-optimal rectangular ST codes from their square counterparts. A byproduct of our results establishes that the D-MG tradeoff for all Tgesnt is the same as that previously known to hold for Tgesnt+n r-1

Explicit Space&#8211;Time Codes Achieving the Diversity&#8211;Multiplexing Gain Tradeoff

A recent result of Zheng and Tse states that over a quasi-static channel, there exists a fundamental tradeoff, referred to as the diversity-multiplexing gain (D-MG) tradeoff, between the spatial multiplexing gain and the diversity gain that can be simultaneously achieved by a space-time (ST) block code. This tradeoff is precisely known in the case of i.i.d. Rayleigh-fading, for T>= n_t+n_r-1 where T is the number of time slots over which coding takes place and n_t,n_r are the number of transmit and receive antennas respectively. For T < n_t+n_r-1, only upper and lower bounds on the D-MG tradeoff are available. 
In this paper, we present a complete solution to the problem of explicitly constructing D-MG optimal ST codes, i.e., codes that achieve the D-MG tradeoff for any number of receive antennas. We do this by showing that for the square minimum-delay case when T=n_t=n, cyclic-division-algebra (CDA) based ST codes having the non-vanishing determinant property are D-MG optimal. While constructions of such codes were previously known for restricted values of n, we provide here a construction for such codes that is valid for all n. 
For the rectangular, T > n_t case, we present two general techniques for building D-MG-optimal rectangular ST codes from their square counterparts. A byproduct of our results establishes that the D-MG tradeoff for all T>= n_t is the same as that previously known to hold for T >= n_t + n_r -1.

Explicit Space-Time Codes Achieving The Diversity-Multiplexing Gain Tradeoff

The problem of constructing space-time (ST) block codes over a fixed, desired signal constellation is considered. In this situation, there is a tradeoff between the transmission rate as measured in constellation symbols per channel use and the transmit diversity gain achieved by the code. The transmit diversity is a measure of the rate of polynomial decay of pairwise error probability of the code with increase in the signal-to-noise ratio (SNR). In the setting of a quasi-static channel model, let n/sub t/ denote the number of transmit antennas and T the block interval. For any n/sub t/ /spl les/ T, a unified construction of (n/sub t/ /spl times/ T) ST codes is provided here, for a class of signal constellations that includes the familiar pulse-amplitude (PAM), quadrature-amplitude (QAM), and 2/sup K/-ary phase-shift-keying (PSK) modulations as special cases. The construction is optimal as measured by the rate-diversity tradeoff and can achieve any given integer point on the rate-diversity tradeoff curve. An estimate of the coding gain realized is given. Other results presented here include i) an extension of the optimal unified construction to the multiple fading block case, ii) a version of the optimal unified construction in which the underlying binary block codes are replaced by trellis codes, iii) the providing of a linear dispersion form for the underlying binary block codes, iv) a Gray-mapped version of the unified construction, and v) a generalization of construction of the -ary case corresponding to constellations of size /sup K/. Items ii) and iii) are aimed at simplifying the decoding of this class of ST codes.

A unified construction of space-time codes with optimal rate-diversity tradeoff

We show that for any (Q/spl times/M) space-time code S having a fixed, finite signal constellation, there is a tradeoff between the transmission rate R and the transmit diversity gain /spl nu/ achieved by the code. The tradeoff is characterized by R/spl les/Q-/spl nu/+1, where Q is the number of transmit antennas. When either binary phase-shift keying (BPSK) or quaternary phase-shift keying (QPSK) is used as the signal constellation, a systematic construction is presented to achieve the maximum possible rate for every possible value of transmit diversity gain.

Rate-diversity tradeoff of space-time codes with fixed alphabet and optimal constructions for PSK modulation

This article presents a new asymptotically exact lower bound on pairwise error probability of a space-time code as well as an example code that outperforms the comparable orthogonal-design-based space-time (ODST) code. Also contained in the article are an exact expression for pairwise error probability (PEP), signal design guidelines, and some observations relating to the reception of ODST codes.

Hsiao-feng Lu

Papers

Explicit Space–Time Codes Achieving the Diversity–Multiplexing Gain Tradeoff

Explicit Space-Time Codes Achieving The Diversity-Multiplexing Gain Tradeoff

A unified construction of space-time codes with optimal rate-diversity tradeoff

Rate-diversity tradeoff of space-time codes with fixed alphabet and optimal constructions for PSK modulation

Remarks on space-time codes including a new lower bound and an improved code

Hsiao-feng Lu

Papers

Explicit Space&#8211;Time Codes Achieving the Diversity&#8211;Multiplexing Gain Tradeoff

Explicit Space-Time Codes Achieving The Diversity-Multiplexing Gain Tradeoff

A unified construction of space-time codes with optimal rate-diversity tradeoff

Rate-diversity tradeoff of space-time codes with fixed alphabet and optimal constructions for PSK modulation

Remarks on space-time codes including a new lower bound and an improved code

Explicit Space–Time Codes Achieving the Diversity–Multiplexing Gain Tradeoff