Basis function selection for compressed sensing and sparse representations of pulsed radar echoes

Compressed sensing theory supposes that a sparse signal can be sampled at a rate much lower than the Nyquist–Shannon rate and reconstructed with high probability. Such lower sampling rate commonly requires finding a set of the optimal basis functions to sparsely represent the signal first. This paper provides a simple and effective working process to select the basis functions for a family of pulsed radar echoes. The selection process is performed in two steps. First,the waveform matching based on the the known array excitation is carried out to select a mother function from a wavelet dictionary. Second, the spectrum matching principle is used to produce a small set of basis functions from the selected mother function. The proposed method is numerically validated by a pulsed radar system equipped with two different dipole arrays. The results demonstrate that the new method is quite effective. With the selected basis functions, all echoes can be under-sampled at a rate lower than 5%$ 5\% $ of the conventional Nyquist–Shannon rate and reconstructed with the root mean-squared error of less than 2%$ 2\% $.