Dispersion of infinite constellations in fast fading channels

Abstract

In this work we extend the setting of communication without power constraint, proposed by Poltyrev, to fast fading channels with channel state information (CSI) at the receiver. The optimal codewords density, or actually the optimal normalized log density (NLD), is considered. Poltyrev's capacity for this channel is the highest achievable NLD, at possibly large block length, that guarantees a vanishing error probability. For a given finite block length n and a fixed error probability ǫ, there is a gap between the highest achievable NLD and Poltyrev's capacity. As in other channels, this gap asymptotically vanishes as the square root of the channel dispersion V over n, multiplied by the inverse Q-function of the allowed error probability. This dispersion, derived in the paper, equals the dispersion of the power constrained fast fading channel at the high SNR regime. Connections to the error exponent of the peak power constrained fading channel are also discussed. I. INTRODUCTIONWireless communication channels are traditionally modeled as fading channels, where the transmitted signal is multiplied by a fading process and observed with additive white Gaussian noise (AWGN). In a fast fading channel the fading process is composed of fading coefficients, modeled as independent and identically distributed (i.i.d.) random variables. This is a reasonable model for many practical wireless communication systems, such as systems that use a (pseudo) random interleaver between the transmitted digital symbols (e.g. BICM techniques) over, e.g., a frequency selective wireless channel. Here we will assume that a perfect knowledge of the channel state information (the fading coefficients) is available at the receiver.Classical coding problems over the fading channels often include a peak or an average power restriction of the transmitted signal. Without power constraint the capacity of the channel is not limited, since we can choose an infinite number of codewords to be arbitrarily far apart from each other, and hence get an arbitrarily small error probability and infinite rate. Nevertheless, coded modulation methods ignore the power constraint by designing infinite constellations (IC), and then taking only a subset of codewords which are included in some "shaping region" to get a finite constellation (FC) that holds the power constraint. Hence, IC is a very convenient framework for designing codes.Poltyrev studied in [1] the IC performance over the AWGN without power constraint. He defined the density (the average number of codewords per unit volume) and the normalized log density (NLD) of the IC, in analogy to the number of codewords and the communication rate in the power constraint model, respectively. He showed that the highest achievable NLD over the unconstrained AWGN channel, with arbitrarily small error probability, is limited by a maximal NLD, sometimes termed the 'Poltyrev's capacity'. He also derived an exact term for the maximal NLD and error exponent bounds using random coding and sphere packing techniques, f...

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