An Introduction to Compressed Sensing

Conventional signal sampling is that the sampling rate must be at least twice the maximum frequency presented in the signal. In the field of signal sampling or compressed, Shannon theorem plays an implicit role. However, the signal is uniformly sampled at or above the Nyquist rate. Thus, this sampling is not a effective approach. Fortunately, Candes and Tao, Donoho have introduced a new sampling theory: Compressed sensing, which was beyond Nyquist sampling theory. CS theory asserts that one can recover certain signals and images from far fewer samples or measurments than traditional methods use. To make this possible, CS relies on two priciples: signal sparsity and incoherence of measurment matrix.

On channel estimation problem, we can introduce CS theory. In general, there always exist a large delay spread and only has a few dominant taps in high data rate wireless communication. that is to say, this multipath channel has sparsity. In the process of channel estimation, training matrix can be designed Toeplitz structre which satisfies Restrict Isometry property (RIP).

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Good overview.

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