1A Bound on the MSE of Oversampled Dithered Quantization with Feedback

Abstract—We analyze the behaviour of the mean squared error (MSE) achievable by oversampled, uniform scalar quantization using feedback, pre- and post-filters of un-restricted order, when encoding wide-sense stationary discrete-time random sources having (possibly) unbounded support. Our results are based upon the use of subtractively dithered uniform scalar quantizers. We consider the number of quantization levels, N, to be given and fixed, which lends itself to fixed-rate encoding, and focus on the cases in which N is insufficient to avoid overload. In order to guarantee the stability of the closed-loop, we consider the use of a clipper before the scalar quantizer. Our results are valid for zero-mean sources having independent innovations whose moments satisfy some mild requirements, which are met by infinite-support distributions such as Gaussian and Laplacian. We show that, for fixed N, the MSE can be made to decay with the oversampling ratio λ as O(e−c0λ1/3) when λ tends to infinity, where c0, [0.5(N−1)]2/3. We note that the latter bound is asymptotic in λ but not in N, and that it includes clipping errors. Index Terms—Oversampling, quantization, Σ ∆ converters. I