OFDM for Underwater Acoustic Communications

Appendix A

Compressive Sensing

Since the term compressive sensing was coined a few years ago [53, 100] this subject has been under intensive investigation [24, 56, 70]. It has found broad application in imaging, data compression, radar, and data acquisition to name a few. Overviews on compressive sensing can be found in, e.g., [56, 70]. Extensive references and resources on compressive sensing are also available online [1].

In a nutshell, compressive sensing is a novel paradigm where a signal that is sparse in a known transform domain can be acquired with much fewer samples than usually required by the dimensions of this domain. The only condition is that the sampling process is incoherent with the transform that achieves the sparse representation and sparse means that most weighting coefficients of the signal representation in the transform domain are zero. While it is obvious that a signal that is sparse in a certain basis can be fully represented by an index specifying the basis vectors corresponding to nonzero weighting coefficients plus the coefficients—determining which coefficients are nonzero would usually involve calculating all coefficients, which requires at least as many samples as there are basis functions. The definition of incoherence usually states that distances between sparse signals are approximately conserved as distances between their respective measurements generated by the sampling process. In this sense the reconstruction problem has per definition a unique ...