Chebyshev Vandermonde-like Measurement Matrix Based Compressive Spectrum Sensing

Compressive sensing is a rapidly growing research area that is being applied in many fields, including mathematics, aerospace, electrical engineering, computer science, and biomedical engineering. This concept proclaims that sparse signals can be recovered from a set of few measurements. It involves two processes, encoding and decoding. The encoding process deals with the question how one should design the linear measurement process and the decoding deals with what algorithm can successfully recover the original sparse signal. To date, both processes are still open problems, particularly how to construct explicit measurement matrices that lead to a successful recovery process. Therefore, in this paper, we review the main properties that can be used to design suitable measurement matrices and propose an explicit construction of measurement matrix based on the Chebyshev polynomial and Vandermonde matrix. The performance of this matrix is evaluated, and its efficiency is compared to those of the existing matrices using several evaluation metrics, namely the recovery error, processing and recovery time, and phase transition diagrams. This matrix is also evaluated in the context of wideband spectrum sensing using metrics such as the probabilities of detection and false alarm. The results show that the proposed matrix outperforms the performances of Gaussian and Toeplitz based models.