Capacity of the Discrete-Time AWGN Channel Under Output Quantization

TL;DR

This paper characterizes the capacity limits of discrete-time AWGN channels with output quantization, proving the optimality of discrete input distributions with limited mass points and demonstrating near-capacity performance with low-bit quantizers at various SNR levels.

Contribution

It proves that optimal input distributions have at most K+1 mass points for K-bit quantization and employs numerical methods to optimize quantizers, showing minimal capacity loss at moderate to high SNR.

Findings

Optimal input distributions have at most K+1 mass points for K-bit quantization.

2-3 bit quantization retains 80-90% of the infinite-precision capacity at SNRs up to 20 dB.

Low-precision quantization is effective even at moderate to high SNR levels.

Abstract

We investigate the limits of communication over the discrete-time Additive White Gaussian Noise (AWGN) channel, when the channel output is quantized using a small number of bits. We first provide a proof of our recent conjecture on the optimality of a discrete input distribution in this scenario. Specifically, we show that for any given output quantizer choice with K quantization bins (i.e., a precision of log2 K bits), the input distribution, under an average power constraint, need not have any more than K + 1 mass points to achieve the channel capacity. The cutting-plane algorithm is employed to compute this capacity and to generate optimum input distributions. Numerical optimization over the choice of the quantizer is then performed (for 2-bit and 3-bit symmetric quantization), and the results we obtain show that the loss due to low-precision output quantization, which is small at low signal-to-noise ratio (SNR) as expected, can be quite acceptable even for moderate to high SNR values. For example, at SNRs up to 20 dB, 2-3 bit quantization achieves 80-90% of the capacity achievable using infinite-precision quantization.