Sigma Delta Quantization with Harmonic Frames and Partial Fourier Ensembles
Sigma Delta Quantization with Harmonic Frames and Partial Fourier Ensembles
复制标题
使用谐波框架和部分傅里叶系综的 Sigma Delta 量化
DOI：
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发表时间：
2015
影响因子：
1.2
通讯作者：
Rongrong Wang
中科院分区：
文献类型：
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作者：
Rongrong Wang
Sigma Delta （ΣΔdocumentclass[12pt]{minimal} uspackage {amsmath} uspackage {wasysym} uspackage {amsfonts} uspackage {amssymb} uspackage {amssymb} uspackage {amssfs} uspackage {upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$Sigma Delta $$end{document}）量化，是一种最早出现于20世纪60年代的量化方法，现已广泛应用于相机、手机、雷达等各种数码产品中。该方法通过对输入信号以超奈奎斯特速率采样，对量化噪声具有很强的鲁棒性。压缩感知（CS）是一种节俭的采集方法，它利用客观信号的稀疏性结构来减少无损采集所需的样本数量。通过将这个减少的数视为稀疏信号集的有效维数，可以将相对过采样/次采样率定义为实际采样率与有效维数之间的比率。当通过Sigma Delta量化记录这些“压缩”模拟测量时，自然会出现一个问题：对于带限函数，先前显示的信号重建误差是否会随着普通过采样率的增加而多项式衰减，现在会随着相对过采样率的增加而多项式衰减？回答这个问题是这个方向的主要目标之一。到目前为止，CS中的量化研究仅限于证明高斯和亚高斯传感矩阵的误差收敛结果，因为比特数和/或样本数增长到无穷大。在本文中，我们为更现实的傅立叶传感矩阵提供了第一个结果。其主要思想是在将傅里叶样本输入量化器之前对其进行随机排列。我们发现随机排列可以有效地增加测量的低频功率，从而提高ΣΔdocumentclass[12pt]{minimal} uspackage {amsmath} uspackage {wasysym} uspackage {amsfonts} uspackage {amssymb} uspackage {mathrsfs} uspackage {upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$Sigma Delta $$end{document}量化的质量。
Sigma Delta (ΣΔdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$Sigma Delta $$end{document}) quantization, a quantization method first surfaced in the 1960s, has now been widely adopted in various digital products such as cameras, cell phones, radars, etc. The method features a great robustness with respect to quantization noises through sampling an input signal at a Super-Nyquist rate. Compressed sensing (CS) is a frugal acquisition method that utilizes the sparsity structure of an objective signal to reduce the number of samples required for a lossless acquisition. By deeming this reduced number as an effective dimensionality of the set of sparse signals, one can define a relative oversampling/subsampling rate as the ratio between the actual sampling rate and the effective dimensionality. When recording these “compressed” analog measurements via Sigma Delta quantization, a natural question arises: will the signal reconstruction error previously shown to decay polynomially as the increase of the vanilla oversampling rate for the case of band-limited functions, now be decaying polynomially as that of the relative oversampling rate? Answering this question is one of the main goals in this direction. The study of quantization in CS has so far been limited to proving error convergence results for Gaussian and sub-Gaussian sensing matrices, as the number of bits and/or the number of samples grow to infinity. In this paper, we provide a first result for the more realistic Fourier sensing matrices. The main idea is to randomly permute the Fourier samples before feeding them into the quantizer. We show that the random permutation can effectively increase the low frequency power of the measurements, thus enhance the quality of ΣΔdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$Sigma Delta $$end{document} quantization.