Convolutional Compressed Sensing Using Decimated Sidelnikov Sequences

In many applications of compressed sensing, the data acquisition involves convolution by a filter followed by subsampling. In this letter, we propose to construct a filter with real-valued coefficients by taking the discrete Fourier transform of a decimated binary Sidelnikov sequence. With a random subsampler, we prove that stable recovery can be guaranteed if a signal is sparse in the canonical or the FFT basis. Besides, simulation results also show that if a deterministic subsampler is used, the proposed system can offer similar reconstruction performance as that of a random Gaussian operator for a wide range of signal length.