In this note we will show that the so called Sobolev dual is the minimizer over all linear reconstructions using dual frames for stable <i>r<sup>th</sup></i> order ΣΔ 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.
The Rangan-Goyal (RG) algorithm is a recursive method for constructing an estimate <i>x</i><sub>N</sub> ∈ <b>R</b><sup>d</sup> of a signal <i>x</i> ∈ <b>R</b><sup>d</sup>,
given <i>N</i> ≥ <i>d</i> frame coefficient measurements of <i>x</i> that have been corrupted by uniform noise. Rangan and Goyal
proved that the RG-algorithm is constrained by the Bayesian lower bound: lim inf<sub>N→∞</sub><i>N</i><sup>2</sup> <b>E</b>||<i>x</i> − <i>x</i><sub><i>N</i></sub>||<sup>2</sup> > 0. As
a positive counterpart to this, they also proved that for every <i>p</i> < 1 and <i>x</i> ∈ <b>R</b><sup>d</sup>, the RG-algorithm satisfies
lim<sub>N→∞</sub> <i>N<sup>p</sup></i>||<i>x</i> − <i>x<sub>N</sub></i>|| = 0 almost surely. One consequence of the existing results is that one "almost" has mean
square error <b>E</b>||<i>x</i> − <i>x<sub>N</sub></i>||<sup>2</sup> of order 1/<i>N</i><sup>2</sup> for random choices of frames. It is proven here that the RG-algorithm
achieves mean square error of the optimal order 1/<i>N</i><sup>2</sup>, and the applicability of such error estimates is also
extended to deterministic frames where ordering issues play an important role. Approximation error estimates
for consistent reconstruction are also proven.
Sigma-Delta (ΣΔ) schemes are shown to be an effective approach for quantizing finite frame expansions. Basic error estimates show that first order ΣΔ schemes can achieve quantization error of order 1/<i>N</i>, where <i>N</i> is the frame size. Under certain technical assumptions, improved quantization error estimates of order (log<i>N</i>)/<i>N</i><sup>1.25</sup> are obtained. For the second order ΣΔ scheme with linear quantization rule, error estimates of order 1/<i>N</i><sup>2</sup> can be achieved in certain circumstances. Such estimates rely critically on being able to construct sufficiently small invariant sets for the scheme. New experimental results indicate a connection between the orbits of state variables in ΣΔ schemes and the structure of constant input invariant sets.
Conference Committee Involvement (1)
26 August 2007 | San Diego, California, United States