Optimal Index Assignment for Scalar Quantizers and M-PSK via a Discrete Convolution-Rearrangement Inequality

This paper investigates the problem of finding an optimal nonbinary index assignment from
$M$
quantization levels of a maximum entropy scalar quantizer to
$M$
-PSK symbols transmitted over a symmetric memoryless channel with additive noise following decreasing probability density function (such as the AWGN channel) so as to minimize the channel mean-squared distortion. The so-called zigzag mapping under maximum-likelihood (ML) decoding was known to be asymptotically optimal, but the problem of determining the optimal index assignment for any given signal-to-noise ratio (SNR) is still open. Based on a generalized version of the Hardy-Littlewood convolution-rearrangement inequality, we prove that the zigzag mapping under ML decoding is optimal for all SNRs. It is further proved that the same optimality results also hold under minimum mean-square-error (MMSE) decoding. Numerical results are presented to verify our optimality results and to demonstrate the performance gain of the optimal
$M$
-ary index assignment over the state-of-the-art binary counterpart for the case of 8-PSK over the AWGN channel.