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

Abstract

Abstract. We design alternative dual frames for linearly reconstructing signals from Sigma-Delta (Σ∆) quantized finite frame coefficients. In the setting of sampling expansions for bandlimited functions, it is known that a stable rth order Sigma-Delta quantizer produces approximations where the approximation error is at most of order 1/λ r , and λ > 1 is the oversampling ratio. We show that the counterpart of this result is not true for several families of redundant finite frames for R d when the canonical dual frame is used in linear reconstruction. As a remedy, we construct alternative dual frame sequences which enable an rth order Sigma-Delta quantizer to achieve approximation error of order 1/N r for certain sequences of frames where N is the frame size. We also present several numerical examples regarding the constructions.

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

“…However, in [18] it is shown that even for higher order schemes one cannot "robustly" expect more than 1/N 2 error with the canonical dual for natural classes of frames, such as the harmonic frames. That is, in the case of harmonic frames, some oversampling rates achieve the correct approximation error but in large part it is not achieved.…”

“…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

“…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.…”

“…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. It was proved in [21] that for Σ∆ schemes of order 3 or higher one cannot obtain these sought-for approximation rates, at least in the case of harmonic frames, if one uses canonical dual frames in the reconstruction. As a remedy, two main approaches have been developed.…”

“…This approach is less computationally intensive as there is no need to compute a dual frame, but it only works for rather specific analysis frames. A different approach used in [21], and in this paper, allows a more flexible choice of analysis frame and chooses a non-canonical dual frame that is specially designed for the particular quantization scheme.…”