Second-order sigma-delta (\(\Sigma \Delta\)) quantization of finite frame expansions
The authors consider the quantization problem for finite unit-norm tight frame expansions in \({\mathbb R}^d\). They analyze the performance of a second-order sigma-delta scheme and show that it outperforms the standard quantization techniques. Let \(\{e_n\}_{n=1}^N \subset {\mathbb R}^d\) be a finite unit-norm tight frame of \({\mathbb R}^d\), i.e., that \(\| e_n\| =1\) and that there exists \(A>0\) such that for all \(x \in {\mathbb R}^d\) \[ \sum_{n=1}^N | \langle x,\,e_n\rangle| ^2 = A\,\| x\| ^2. \] Here \(\| \cdot \| \) is the Euclidean norm. Then a given signal \(x\in {\mathbb R}^d\) is decomposed into a frame expansion \(x={d \over N}\,\sum_{n=1}^N x_n\,e_n\). In a quantization step, the coefficients \(x_n\), which are real or complex numbers, are replaced by discrete representations \(q_n\) using a second-order sigma-delta quantization scheme. The quantization error \(\| x - \tilde x\| \) with \({\tilde x} = {d \over N}\, \sum_{n=1}^N q_n\,e_n\) is estimated. It is shown that for special finite frames the quantization error is of order \(N^{-2}\). But in general, the second-order scheme only gives a quantization error of order \(N^{-1}\).
scientific article; zbMATH DE number 5013515