On quantization of finite frame expansions: sigma-delta schemes of arbitrary order

Abstract

In this note we will show that the so called Sobolev dual is the minimizer over all linear reconstructions using dual frames for stable r th order Σ∆ quantization schemes under the so called White Noise Hypothesis (WNH) design criteria. We compute some Sobolev duals for common frames and apply them to audio clips to test their performance against canonical duals and another alternate dual corresponding to the well known Blackman filter.

Distributed Noise-Shaping Quantization: I. Beta Duals of Finite Frames and Near-Optimal Quantization of Random Measurements

“…With these tools, it is possible to obtain fairly strong bounds on the analysis distortion of sub-Gaussian frames, but not quite as strong as those obtainable for the Gaussian case. The main problem stems from the presence of the c l term in (22) which, as explained in [27,28], is characteristic of the sub-Gaussian case. As such, this term prevents us from stating an exact analog of Theorem 1.1.…”

“…Quantization of finite frames by sigma-delta modulation goes back to [3,2]. Alternative duals for frames were introduced in the context of sigma-delta quantization in [22,21,4], and more recently have been effectively used for Gaussian random frames in [16] and more generally for sub-Gaussian random frames in [20]. For these frames, an r-th order sigma-delta quantization scheme coupled with an associated Sobolev dual can achieve the bound D a (E, L) L −1 (cr) 2r (m/k) −r with high probability for some constant c. Optimizing this bound for r yields D a (E, L) L −1 e −c √ m/k for another constant c which is sub-optimal in terms of its dependence on both the alphabet size L and the oversampling factor m/k and L. In contrast, our result achieves near-optimal dependence on both parameters: Not only is the dependence on m/k exponential, but the rate can also be made arbitrarily close to its maximum value log 2 L. To the best of our knowledge, this is the first result of this kind.…”