A New Rearrangement Inequality and Its Application to Index Assignment
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This paper investigates the rearrangement problem whose objective is to maximize the bilinear form x^T H y associated with a certain matrix H by explicitly characterizing the optimal permutations of the two vectors x and y. We relate the problem to Hardy's discrete version of the famous Riesz rearrangement inequality and prove a new rearrangement inequality. The new inequality is applied to 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 so as to minimize the channel mean-squared distortion. The so-called zigzag mapping was known to be asymptotically optimal under maximum-likelihood (ML) decoding, but how to construct an optimal index assignment for any given signal-to-noise ratio (SNR) is still open. By relating the index assignment problem to the new rearrangement inequality, we prove that the zigzag mapping is optimal for all SNRs when either ML decoding or minimum mean-square-error (MMSE) decoding is used.
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