Abstract: Recent studies in Compressive Sensing (CS) have shown that sparse signals can be recovered from a small number of random measurements, which raises the question of whether random measurements can provide an efficient representation of sparse signals in an information-theoretic sense. To examine the influence of quantization errors, the average distortion introduced by quantizing compressive sensing measurements was studied using rate distortion theory. Both uniform quantization and non-uniform quantization were considered. The asymptotic rate-distortion functions were obtained when the signal was recovered from quantized measurements using different reconstruction algorithms. Both theoretical and experimental results shows that encoding a sparse signal through simple scalar quantization of random measurements incurs a significant penalty relative to direct or adaptive encoding of the sparse signal, but compressive sensing is able to exploit the sparsity to reduce the distortion, so quantized compressive sampling is suitable to be used to encode the low sparse signal.

Key words: Compressive Sensing (CS), quantization, rate distortion function, lossy compression, signal reconstruction

摘要： 压缩感知理论表明稀疏信号能由少量的随机测量值恢复，从信息理论的角度来看，随机测量值能否有效表示稀疏信号仍是一个值得探讨的问题。针对压缩感知测量值的量化，将率失真理论作为工具研究压缩测量值的量化带来的平均失真度，包括均匀量化和非均匀量化两种情况，并进一步得到由量化测量值重构信号的率失真性能极限。理论分析和实验结果表明，相对于信号的自适应编码随机观测过程会引起较大的失真，但是压缩感知能利用信号的稀疏度来减小量化后的重构失真，这说明量化压缩感知适用于低稀疏度的信号。

关键词: 压缩感知, 量化, 率失真函数, 有损压缩, 信号重构