Signals, Systems, and Signal Processing

This chapter gives a brief overview of sampling based on sparsity. The idea is that a signal which is not bandlimited can sometimes be reconstructed from a sampled version if we have a priori knowledge that the signal is sparse in a certain basis. These results are very different from the results of Shannon and Nyquist, and are sometimes referred to as sub-Nyquist sampling theories. They can be regarded as generalizations of traditional sampling theory, which was based on the bandlimited property. Examples include sampling of finite-duration signals whose DFTs are sparse. Sparse reconstruction methods are closely related to the theory of compressive sensing, which is also briefly introduced. These are major topics that have emerged in the last two decades, so the chapter provides important references for further reading.