Compressive Sensing Matrix Design for Fast Encoding and Decoding via Sparse FFT

Compressive sensing (CS) is proposed for signal sampling below the Nyquist rate based on the assumption that the signal is sparse in some transformed domain. Most sensing matrices (e.g., Gaussian random matrix) in CS, however, usually suffer from unfriendly hardware implementation, high computation cost, and huge memory storage. In this letter, we propose a deterministic sensing matrix for collecting measurements fed into sparse fast Fourier transform (sFFT) as the decoder. Compared with the conventional paradigm with Gaussian random matrix at encoder and convex programming or greedy method at decoders, sFFT can reconstruct sparse signals with very low computation cost under the comparable number of measurements. But, the limitation is that the signal must be sparse in the frequency domain. We further show how to relax this limitation into any domains with the transformation matrix or dictionary being circulant. Experimental and theoretical results validate that the proposed method achieves fast sensing, fast recovery, and low memory cost.