从广义采样、小波到压缩感知

From Generalized Sampling and Wavelet to Compressed Sensing

采样是对模拟信号进行数字化处理的关键环节。近年来,信号带宽及信息传输速率的快速增长致使采用传统采样机理的信号处理方法面临巨大挑战,小波变换与压缩感知等新型信号处理技术应运而生。在这种情况下,有必要从理论上重新审视经典的Shannon-Nyquist采样定理,研究具有普适性的信号表达与采样重构理论。本文从信号空间投影与函数表达的角度分析了信号表达的本质,介绍了Shannon传统采样与重构理论,以及由Papoulis提出的经Unser等所推广的广义采样与重构理论。从数学上重点探讨了小波变换(Wavelet transform)和压缩感知(Compressed sensing,CS)等现代信号处理及变换方法与广义采样的一致性。同时,通过线性调频(Linear frequency modulation,LFM)信号的实例仿真,说明采样与重构的关系以及在各个方法之间的异同性。

Sampling is the key procedure for the digital processing of analog signal. In recent years, signal processing methods employing traditional sampling mechanism have faced tremendous challenges due to the rapid growth of signal bandwidth and information transmission rate, and some new signal processing technology such as the wavelet transform and the compressed sensing emerged at the right moment. On this occasion, it is necessary to reexamine the classical Shannon-Nyquist sampling theorem in theory, and study universal expression, sampling and reconstitution theory of signal. The nature of signal expression is analyzed from the point of view of signal projection and function representation. Firstly, the Shannon traditional sampling and reconstruction theory, and the generalized sampling and reconstruction theory proposed by Papoulis and extended by Unser are introduced. Then, the consistency between modern sig- nal processing and transforming methods （wavelet transform, compressed sensing） and generalized sam- pling theory is investigated mathematically. Meanwhile, the chirp signal is taken as a simulation example to illustrate the relationship between signal sampling and reconstruction, as well as the similarities and differences in each method.