The compressed-sampling filter (CSF)

Abstract

The common approaches to sample a signal generally follow the well-known Nyquist-Shannon's theorem: the sampling rate must be at least twice the maximum frequency presented in the signal. A new emerging field, compressed sampling (CS), has made a paradigmatic step to sample a signal with much less measurements than those required by the Nyquist-Shannon's theorem when the unknown signal is sparse or compressible in some frame. We call a compressed-sampling filter (CSF) one for which the function relating the input signal to the output signal is pseudorandom. Motivated by the theory of random convolution proposed by Romberg (for convenience, called the Romberg's theory) and the fact that the signal in complex electromagnetic environment may be spread out due to the rich multi-scattering effect, two CSFs via microwave circuit to enable signal acquisition with sub-Nyquist sampling have been constructed, tested and analyzed. Afterwards, the CSF based on surface acoustic wave (SAW) structure has also been proposed and examined by the numerical simulation. The results have empirically shown that by the proposed architectures the S-sparse n- dimensional signal can be exactly reconstructed with O(S log n) real-valued measurements or O(S log(n/S)) complex-valued measurements with overwhelming probability.