“People Hearing Without Listening:” An Introduction To Compressive Sampling

The conventional approach to sampling signals or images follows the celebrated Shannon sampling theorem: the sampling rate must be at least twice the maximum frequency present in the signal (the so-called Nyquist rate). In fact, this principle underlies nearly all signal acquisition protocols used in consumer audio and visual electronics, medical imaging devices, radio receivers, and so on. In the field of data conversion, for example, standard analog-to-digital converter (ADC) technology implements the usual quantized Shannon representation: the signal is uniformly sampled at or above the Nyquist rate. This paper surveys the theory of compressive sampling also known as compressed sensing, or CS, a novel sensing/sampling paradigm that goes against the common wisdom in data acquisition. The CS theory asserts that one can recover certain signals and images from far fewer samples or measurements than traditional methods use. To make this possible, compressive sampling relies on two tenets, namely, sparsity, which pertains to the signals of interest, and incoherence, which pertains to the sensing modality. • Sparsity expresses the idea that the “information rate” of a continuous time signal may be much smaller than suggested by its bandwidth, or that a discrete-time signal depends on a number of degrees of freedom which is comparably much smaller than its (finite) length. More precisely, compressive sampling exploits the fact that many natural signals are sparse or compressible in the sense that they have concise representations when expressed in the proper basis Ψ. • Incoherence extends the duality between time and frequency and expresses the idea that objects having a sparse representation in Ψ must be spread out in the domain in which they are acquired, just as a Dirac or a spike in the time domain is spread out in the frequency domain. Put differently, incoherence says that unlike the signal of interest, the sampling/sensing waveforms have an extremely dense representation in Ψ.