Design of optimal feedback filters with guaranteed closed-loop stability for oversampled noise-shaping subband quantizers

We consider the oversampled noise-shaping subband quantizer (ONSQ) with the quantization noise being modeled as white noise. The problem at hand is to design an optimal feedback filter. Optimal feedback filters with regard to the current standard model of the ONSQ inhabit several shortcomings: the stability of the feedback loop is not ensured and the noise is considered independent of both the input and the feedback filter. Another previously unknown disadvantage, which we prove in our paper, is that optimal feedback filters in the standard model are never unique. The goal of this paper is to show how these shortcomings can be overcome. The stability issue is addressed by showing that every feedback filter can be "stabilized" in the sense that we can find another feedback filter which stabilizes the feedback loop and achieves a performance equal (or arbitrarily close) to the original filter. However, the other issues are inherent properties of the standard model. Therefore, we also consider a so-called extended model of the ONSQ, which includes the influence of the feedback filter on the noise power and indirectly also the influence of the inputs statistical properties. We show that the optimal feedback filter for this extended model is unique and automatically achieves a stable feedback loop. We also give an algorithm to compute it.