Conditions for optimality of scalar feedback quantization

This paper presents novel results on scalar feedback quantization (SFQ) with uniform quantizers. We focus on general SFQ configurations where reconstruction is via a linear combination of frame vectors. Using a deterministic approach, we derive two necessary and sufficient conditions for SFQ to be optimal, i.e., to produce, for every input, a quantized sequence that is a global minimizer of the 2-norm of the reconstruction error. The first optimality condition is related to the design of the feedback quantizer, and can always be achieved. The second condition depends only on the reconstruction vectors, and is given explicitly in terms of the Gram matrix of the reconstruction frame. As a by-product, we also show that the the first condition alone characterizes scalar feedback quantizers that yield the smallest MSE, when one models quantization noise as uncorrelated, identically distributed random variables.