Construction of chaotic sensing matrix for fractional bandlimited signal associated by fractional fourier transform

Abstract

Fractional Fourier transform (FrFT) is a powerful tool for the non-stationary signals because of its additional degree of freedom in the time-frequency plane. Due to the importance of the FrFT in signal processing, most of the bandlimited sampling theorems in traditional frequency domain have been extended to fractional Fourier bandlimited signals based on the relationship between the FrFT and regular integer order Fourier transform (FT). However, the implementations of those existing extensions are not efficient because of the high sampling rate which is related to the maximum fractional Fourier frequency of the signal. Compressed Sensing (CS) is a useful tool to collect information directly which reduces sampling pressure, computational load as well as saving the storage space. The construction of sensing matrix is the basic issue. Most of CS demand that the sensing matrix is constructed by random under-sampling which is uncontrollable and hard to be realized by hardware. This paper proposes a deterministic construction of sensing matrix for the multiband signals in the fractional Fourier domain (FrFD). We give the sparse basis of the signal and derive the sensing matrix based on the analog to information conversion technology. The sensing matrix is constructed by random sign matrix and Toeplitzed matrix. The sub-sampling method is used to obtain the structural signal. Theoretically, the matrix satisfies the incoherent condition and the entire structure of system is practical. We show in this paper that the sampling rate is much lower than the Nyquist rate. The signal reconstruction is studied based on the framework of compressed sensing. The performance of the proposed sampling method is verified by the simulation. The probability of the successful reconstruction and the mean squared error (MSE) are both analyzed. The numerical results suggest that proposed system is effective for a spectrum-blind sparse multiband signal in the FrFD and demonstrate its promising potentials.