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Performance of Sigma–Delta Quantizations in Finite Frames

机译：有限帧中Sigma-Delta量化的性能

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摘要

In this paper, we consider sigma-delta (SD) quantization of geometrically uniform (GU) finite frames. In the first part, we prove that under some conditions, the variant I and II permutation modulation (PM) codes, first introduced by Slepian (1968, 1965), can belong to the class of GU frames. Then, we focus essentially on a subclass of a GU frame, namely, cyclic geometrically uniform (CGU) frame, family of frames containing finite harmonic frames (both in BBC M and BBR M). For first- and second-order SD quantizers, 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 deterministic quantization model used in Benedetto (2004), Benedetto (2006), and Yilmaz (2001) as well as under the widely used additive white quantization noise assumption. For the widely used L th-order noise shaping filter G(z)=(1-z -1)L, we show that the MSE behaves as [ 1/( r 2)] irrespectively of the filter order L. More importantly, we prove also that in the case of tight and normalized CGU frame, when the frame length is too large compared to the filter order and under some conditions on the quantizer, the reconstruction MSE can decay as fast as O([ 1/( r 2 L+1)]). Finally, it is shown that degrees of freedom in CGU frames, when compared to harmonic frames, result in a smaller MSE, albeit MSE prop [ 1/( r 2)] in both cases.
机译：在本文中，我们考虑了几何均匀（GU）有限帧的sigma-delta（SD）量化。在第一部分中，我们证明在某些条件下，由Slepian（1968，1965）首次引入的变体I和II置换调制（PM）码可以属于GU帧的类别。然后，我们主要关注GU框架的子类，即循环几何均匀（CGU）框架，这是包含有限谐波框架的框架族（在BBC M和BBR M中都是如此）。对于一阶和二阶SD量化器，我们确定重建最小平方误差（MSE）表现为[1 /（r 2）]，其中r表示帧冗余。在Benedetto（2004），Benedetto（2006）和Yilmaz（2001）中使用的确定性量化模型下以及在广泛使用的加性白色量化噪声假设下，该结果均显示为真实。对于广泛使用的L阶噪声整形滤波器G（z）=（1-z -1）L，我们证明MSE的行为与[1 /（r 2）]无关，与滤波器阶次L无关。更重要的是，我们还证明，在紧密规范化的CGU帧的情况下，当帧长与滤波器阶数相比过大且在量化器上的某些条件下，重构MSE的衰变速度可高达O（[1 /（r 2 L + 1）]）。最后，表明与谐波帧相比，CGU帧中的自由度导致较小的MSE，尽管在两种情况下MSE prop [1 /（r 2）]。

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