Derivation of Subband Coding Gain: The Most General Case

Consider the subband coder of figure 1, consisting of k = 0, 1, ..., M − 1 channels. Each channel has an analysis filter H k (z), a synthesis filter F k (z), down/up-sampling ratio m k , and a b k bit quantizer. The input is x[n], after passing through the analysis filters it becomes x k [n], and the subband signals are x k [n]. The subband coder is maximally decimated, M −1 k=0 m k = 1. (1) Assumption 1: Assume the input x[n] to be wide sense stationary (WSS) [1, page 806]. Assumption 2: Assume the input x[n] to be zero mean. It follows that all subsequent signals (including quantization noise) become zero mean. (This assumption does not cause any difficulty in applying subband coding to nonzero mean signals such as image, since that system is equivalent to a zero mean system obtained by subtracting the mean from the input.)