Title: Optimal Constellations for the Low SNR Noncoherent MIMO Block Rayleigh Fading Channel
ArXiv ID: 0706.3710
Date: 2008-09-22
Authors: ** - Rao, S. (주 저자) - Hassibi, B. (논문 초안 날짜: 2018년 11월 9일) — **
📝 Abstract
Reliable communication over the discrete-input/continuous-output noncoherent multiple-input multiple-output (MIMO) Rayleigh block fading channel is considered when the signal-to-noise ratio (SNR) per degree of freedom is low. Two key problems are posed and solved to obtain the optimum discrete input. In both problems, the average and peak power per space-time slot of the input constellation are constrained. In the first one, the peak power to average power ratio (PPAPR) of the input constellation is held fixed, while in the second problem, the peak power is fixed independently of the average power. In the first PPAPR-constrained problem, the mutual information, which grows as O(SNR^2), is maximized up to second order in SNR. In the second peak-constrained problem, where the mutual information behaves as O(SNR), the structure of constellations that are optimal up to first order, or equivalently, that minimize energy/bit, are explicitly characterized. Furthermore, among constellations that are first-order optimal, those that maximize the mutual information up to second order, or equivalently, the wideband slope, are characterized. In both PPAPR-constrained and peak-constrained problems, the optimal constellations are obtained in closed-form as solutions to non-convex optimizations, and interestingly, they are found to be identical. Due to its special structure, the common solution is referred to as Space Time Orthogonal Rank one Modulation, or STORM. In both problems, it is seen that STORM provides a sharp characterization of the behavior of noncoherent MIMO capacity.
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formulation. Systems that assume coherent transmission by arguing that the channel can be acquired at the receiver by the use of pilot-symbol assisted transmission to perform explicit channel estimation are inherently suboptimal in general while not taking into account the resources, namely energy and degrees of freedom, needed for pilot transmissions, as they should.
The study of noncoherent fading channels at low SNR is motivated by their application in wideband (WB) and ultra-wideband (UWB) channels. In such scenarios, the signal power is spread over a large bandwidth, rendering the SNR per degree of freedom low. Transmissions over wideband fading channels experience both time and frequency selectivity. However, within a short window of time or frequency, the channel fading coefficients are known to be highly correlated. One widespread approach therefore to deal with frequency-selectivity is to divide the original wideband channel into several parallel narrowband channels such that each narrowband channel experiences flat fading or a single tap coefficient. To deal with time-selectivity, a common approach is to model each narrowband channel through block fading. In the block fading model, the channel coefficients are assumed fixed for a duration in time following which they assume independent and identically distributed realizations (here adequate interleaving across time and frequency windows is implicitly assumed). In this work, we model the wideband channel as a block faded narrowband channel in the low SNR regime. This simplifying channel modeling assumption helps captures the essence of the orignal wideband channel, and is widely adopted in the analysis of MIMO fading channels.
The study of noncoherent SISO fading channels at low SNR dates back to the 1960’s. Two equivalent notions of optimality in the literature that are indicators of energy efficiency in the low SNR regime are (1) the input being first order optimal with respect to Shannon capacity or (2) the input achieving the minimum energy per bit or E b N 0 min required for reliable communication. A classical result by Shannon [1] is that in the limit of infinite bandwidth or vanishing SNR, the minimum energy/bit required for reliable communications over an AWGN channel is -1.59dB. Early work by Kennedy [2], Jacobs [3] (also see Gallager [4] and the references therein) studied wideband SISO Rayleigh fading channels with an average power constrained input and showed that in the limit of infinite bandwidth or vanishing SNR, the required minimum energy/bit is again -1.59dB, the same as that of an AWGN channel. A remarkable observation then was that the minimum energy/bit required is the same whether or not the receiver has knowledge of the channel fading coeffecients. Telatar and Tse [5], and Verdu [6] show that the minimum energy/bit is -1.59 dB even for fairly general multipath SISO fading channels and general MIMO fading channels, respectively. A common approach adopted to obtain E b N 0 min for fading channels is to consider the achievable rate of a certain scheme (often M-ary Frequency Shift Keying or MFSK), which is transmitted at arbitrarily low duty cycles (cf. [2,4,5]). The required result is then obtained by either showing that the energy/bit of the scheme at vanishing SNR matches that of the AWGN channel, or by deriving an upper bound on capacity that is tight with respect to the achievable lower bound. However, this approach fixes the input a priori, and therefore no determination can be made as to the necessary conditions for a constellation to achieve the minimum energy/bit. The characterization of the class of signals (more generally, input distributions) that are both necessary and sufficient to achieve the minimum energy/bit had been an important and long standing open problem.
Signals such as arbitrarily low duty-cycled FSK tend to have prohibitively large peak-to-average-power ratios (PAPR) and are consequently difficult to implement in practice. Such signals are therefore referred to as “peaky” signals in the literature. Using certain types of fourth moments of the input as measures of peakiness, Medard and Gallager [7], and Subramanian and Hajek [8] showed that signaling that is not peaky in either time or frequency dimensions cannot achieve the minimum energy/bit as SNR → 0. Verdu [6] formalized this notion further for fairly general noncoherent MIMO fading channels and established that flash signaling, where the input distribution converges to a zero mass and a non-zero mass that is transmitted with vanishing probability as SNR → 0, is both necessary and sufficient to achieve the minimum energy/bit. While noncoherent communications is sufficient to transmit at the AWGN minimum energy/bit of -1.59dB, the work in [6] resolves another major difficulty. It introduces and explains the crucial role of wideband slope (S 0 ) at large but finite bandwidths. The wideband slope is a measure of how fast the energy/bit of the optima
