Performance of Sigma–Delta Quantizations in Finite Frames

Abstract

In this paper, we extend the results that we derived in , to the case of filter banks (FBs) based transmission. We consider first- and second-order sigma-delta (SD) quantization in the context of an oversampled digital Fourier transform (DFT) FBs (DFT-FBs). In this context, we investigate the case of Odd- and Even-stacked DFT FBs. We establish the set of conditions that guarantee that the reconstruction minimum squares error (MSE) behaves as <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/(r</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> denotes the frame redundancy and we derive the corresponding MSE upper-bounds closed-form expressions. The obtained results demonstrate that overoversampled FBs that are subject to the first- and second-order SD can exhibit a reconstruction error behavior according to <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/(r</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). Furthermore, the established results are shown to be true under the quantization model used in [3]-[6], as well as under the widely used additive white quantization noise assumption.