Capacity of the discrete-time AWGN channel under output quantization

Abstract

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 log 2 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.

Multi-Gigabit communication: the ADC bottleneck¹

“…We begin by recalling some results we obtained last year while investigating these issues for the classical bandlimited Additive White Gaussian Noise (AWGN) channel [3], [4]. Apart from its fundamental significance, the AWGN channel model also forms an excellent approximation for the near line-of-sight, 60 GHz "point-and-shoot" links, where the use of directional antennas (possibly fixed beam) at each end can cut down drastically on multipath.…”

“…In [3], [4], we performed a Shannon-theoretic capacity analysis for this channel, which included an optimization over the choice of the input modulation, as well as over the choice of the quantization thresholds. The results are encouraging: the capacity loss relative to unquantized transmission is of the order of 10-15% at moderate signal-to-noise ratio (SNR), and uniform pulse amplitude modulation (PAM) with quantizer boundaries chosen to be the maximum likelihood (ML) decision regions (i.e., chosen midway between the constellation points) is close to optimal.…”

Signal processing for multiGigabit communication

“…We first recall and interpret some results obtained last year [2], [3]. Consider a line-of-sight radio link employing linear modulation, with ideal carrier synchronization (no frequency or phase offset between receiver local oscillator and incoming carrier wave) and ideal timing synchronization (Nyquist sampling).…”

“…The key results from [2], [3], where we investigate the capacity of the quantized AWGN channel (1) under an average power constraint on the input, can be summarized as follows: 1) Capacity for the quantized AWGN channel (1) is achievable with a discrete input distribution, with the number of mass points at most K + 1, where K is the number of quantization bins. In fact, in all our numerical computations of capacity, at most K mass points suffice.…”

Modulation Diversity in Fading Channels with a Quantized Receiver

“…In case of fading channels, high error floors in the bit error performance have been reported, and it seems difficult to avoid this behavior [2] [3]. On the other hand, channel capacity results show that even with 2-bit quantizers, the capacity of a quantized output channel is not far from that of a channel with unquantized output [4] [5]. Therefore, there appears to be a gap between the theoretical limits of communication with quantized receivers, and the current state of art.…”