Sub-Nyquist Sampling of Pulse Streams Based on the Real Part of Fourier Coefficients

The Finite Rate of Innovation (FRI) theory can realize the sub-Nyquist sampling of pulse streams signal by a sampling rate much lower than its Nyquist frequency. Most classical FRI reconstruction algorithms operate on the basis of Fourier coefficients, and there is a lot of singular value decomposition of complex matrices, which reduces the efficiency of the algorithm. To solve this problem, an FRI sampling and reconstruction method based on the real part of Fourier coefficients is proposed in this paper. Firstly, the discrete cosine transform is used to obtain the real part of Fourier coefficients information from the low-speed sampling value of the pulse flow signal, and the Toeplitz matrix of the real part is used in the reconstruction algorithm to improve the efficiency of the Singular Value Decomposition (SVD). Secondly, in order to improve the robustness of the classical annihilating filter algorithm, a covariance matrix decomposition algorithm and a null space searching algorithm are proposed from the rotation invariant feature and the null space property of the real covariance matrix. The two methods are based on the discrete cosine transform to estimate characteristic parameters of the pulse stream signal. For the conjugate root problem, a new method of deconjugation based on the alternating direction multiplier is proposed in this paper. The simulation results show that using the real part information of Fourier coefficients can greatly improve the efficiency of the algorithm and ensure the accuracy of parameter estimation when the rate of innovation of the signal is high.