Performance of Sigma-Delta quantizers in the context of oversampled filter banks

Abstract

We consider the computation of the MSE for the first-order SigmaDelta (SD) quantizers in the context of an oversampled Filter Bank (FB). We use the same SD quantizer model than the one used by Benedetto et al. [1], we establish that the reconstruction minimum squares error (MSE) behaves as 1 r 2 where r denotes the frame redundancy. This result is shown to be true both under the quantization model used in [1] as well as under the widely used additive white quantization noise assumption. INTRODUCTIONSigma-Delta (SD) quantizers technique gained recently an increased interest with the fast advances done in the VLSI technologies. In a frame based framework, SD techniques can also be used to quantize finite frame expansions for R M [1]. Quantized overcomplete expansions are useful in different applications such as A/D conversion of band-limited signals [2]. The accuracy that can be reached using such techniques depends on two factors: the efficiency of the quantization scheme and the reconstruction algorithm. Extensive research work has been investigated to design reconstruction algorithms that are optimal or near optimal in terms of asymptotic (large redundancy values r) accuracy. It has been proved that in some particular cases: if the reconstruction is consistent then the Mean Square Error (MSE) of the reconstruction error can be upper-bounded by O( 1 r 2 ) [3]. However, for the not consistent linear reconstruction algorithms, it has been shown experimentally that the MSE of the reconstruction error is order of magnitudeO( 1 r ). Recently, Benedetto et al. [1], introduced an alternative approach with reference to the first-order SD quantizers. He mainly proved that the optimal upper-bound of the reconstruction error is obtained in the case of the harmonic tight and normalized frame and its behavior follows O( 1 r 2 ). All the existing work dealing with the computation of the reconstruction accuracy, considers the case of finite frame expansions in R M . In [4], we have proved that the finite-dimensional frame, namely Cyclic Geometrically Uniform (CGU) frame (a set of frames that include the harmonic one) in R M or C M , leads to an MSE upperbounded by O( 1 r 2 ) when we use the same SD quantizer model and assumptions as [1]. In this paper, we investigate the more general case of infinitedimensional signals and oversampled filter banks which represent a convenient way to implement important classes of infinitedimensional frames. We address several problems in the context of the first-order SD quantizers in the context of an oversampled Filter Bank (FB). Our specific contributions vis-a-vis previous work on quantization of finite frames reported in [1] include:• using techniques similar to those in [1], we show that in the case of the first-order SD quantizers in the context of an oversampled Filter Bank (FB), the MSE behaves as 1 r 2 with r denoting the frame redundancy,• the results in [1] use a deterministic model to describe the quantizers in the feedback loop. We demonstrate that the 1 r 2 -behavior of the recon...

Use of first‐ and second‐order sigma–delta quantisers in context of odd‐stacked cosine modulated filter banks

“…It has been proved that in some particular cases, if the reconstruction is consistent then the mean square error (MSE) of the reconstruction error can be upper bounded by O (1/ r 2 ) [2], and for the inconsistent linear reconstruction algorithms it has been shown experimentally that the MSE of the reconstruction error has an order of magnitude O (1/ r ). Note that such results were also validated in the case of digital Fourier transform filter banks (FBs) in [3].…”