Sub-Nyquist Sampling and Recovery of Pulse Streams With the Real Parts of Fourier Coefficients

We consider the problem of sampling pulse streams with known shapes. The recent finite rate of innovation (FRI) framework has shown that such signals can be sampled with perfect reconstruction at their rate of innovation, which is usually much lower than the Nyquist rate. Although FRI sampling of pulse streams was treated in various works, either the work was unstable for high rate of innovation, or the sampling stage was complex and redundant. In this paper, we propose an FRI sampling and recovery method for pulse streams, which is based on the real parts of the Fourier coefficients. The proposed method is simple and efficient, and leads to stable recovery even when the rate of innovation is very high. This is achieved through modulating the input signal in each channel with a properly chosen cosine signal, followed by filtering with a low-pass filter. Since the modulating process will lead to the signal spectrum aliasing, we propose a spectrum de-aliasing algorithm to solve this problem, resulting in the real parts of a band of Fourier coefficients from each two channels. Combining with the multi-channel sampling structure, we propose a more efficient way to obtain arbitrary frequency bands from the aliased spectrum, which improves the utility of the signal spectrum. By using a sparsity-based recovery algorithm, the time delays and amplitudes of the pulse streams can be recovered from the obtained real parts of the Fourier coefficients. Finally, simulation results have shown that the proposed scheme is flexible and exhibits better noise robustness than previous approaches.