Second-order Sigma–Delta (ΣΔ) quantization of finite frame expansions

Abstract

The second order Sigma-Delta (Σ∆) scheme with linear quantization rule is analyzed for quantizing finite unit-norm tight frame expansions for R d. Approximation error estimates are derived, and it is shown that for certain choices of frames the quantization error is of order 1/N 2 , where N is the frame size. However, in contrast to the setting of bandlimited functions there are many situations where the second order scheme only gives approximation error of order 1/N. For example, this is the case when quantizing harmonic frames of odd length in even dimensions. An important component of the error analysis involves extending existing stability results to yield smaller invariant sets for the linear second order Σ∆ scheme.

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

“…There appears to be two lines for the canonical dual in the second order scheme. This can be attributed to the fact that, for even and odd oversampling the so called boundary terms are quite different (see [2,3]). …”

“…Higher order schemes in the finite frame setting have also been previously studied [3] and continue to be examined [5,7,18]. The higher order Σ∆ schemes for finite frames have provided a few surprises in that some of the early convergence rates for the group of frames being studies were somewhat unexpected given results from the band limited case.…”

Sobolev Duals in Frame Theory and Sigma-Delta Quantization

“…We focus on the class of frames associated with piecewise smooth paths in R d and adopt the terminology of [5]. Frames with this property arise naturally in quantization problems, e.g., [2, 1,4,5]. We say that a function f : [0, 1] → R is piecewise-C 1 if it is C 1 except at finitely many points in [0, 1], and the left and right limits of f and f exist at all of these points.…”

“…A main focus of the subsequent work in [1,7,4,5,21] is to achieve better approximation rates (as the redundancy of the frame increases) for higher order Σ∆ schemes. In particular, one desires an approximation error of order O(N −r ) with an rth order Σ∆ scheme, where N is the number of frame vectors.…”

Alternative dual frames for digital-to-analog conversion in sigma–delta quantization

“…If, on the other hand, E is redundant, i.e., N > d, then the associated Bessel map is an injection from R d into R N , and there are infinitely many linear perfect reconstruction maps. In this case, a typical choice is the canonical reconstruction map R e E , i.e., the linear reconstruction map (3.3) obtained by using the canonical dual frame E of E. The performance of a quantizer is often assessed according to the distortion associated with the canonical reconstruction map, e.g., d MSE (Q, R e E ) in the case of PCM quantizers, see [16,15], and d ∞ (Q, R e E ) in the case of Σ∆ quantizers, see [4,5,8]. In Section 4 we show that among all linear reconstruction maps, R e E minimizes the MSE approximation error for PCM schemes under Bennett's white noise assumption on the distribution of the quantization error.…”