Abstract
The coprime discrete Fourier transform (DFT) filter banks provide an effective scheme of spectral sensing for wide-sense stationary (WSS) signals in case that the sampling rate is far lower than the Nyquist sampling rate. And the resolution of the coprime DFT filter banks in the Fourier domain (FD) is 2π/MN, where M and N are coprime. In this work, a digital fractional Fourier transform- (DFrFT-) based coprime filter banks spectrum sensing method is suggested. Our proposed method has the same sampling principle as the coprime DFT filter banks but has some advantages compared to the coprime DFT filter banks. Firstly, the fractional power spectrum of the chirp-stationary signals which are nonstationary in the FD can be sensed effectively by the coprime DFrFT filter banks because of the linear time-invariant (LTI) property of the proposed system in discrete-time Fourier domain (DTFD), while the coprime DFT filter banks can only sense the power spectrum of the WSS signals. Secondly, the coprime DFrFT filter banks improve the resolution from 2π/MN to 2π sin α/MN by combining the fractional filter banks theory with the coprime theory. Simulation results confirm the obtained analytical results.
Highlights
Power spectrum plays an important role in signal processing such as array processing [1,2,3,4], spectral estimation [5,6,7], signal detection and estimation [8,9,10,11,12], and so on
We describe how to construct a sub-Nyquist system by using two low-speed coprime sampling analog-todigital converters (ADCs) and two digital fractional Fourier transform (DFrFT) filter banks and derive the mathematical expression of the output, which shows that the proposed system can effectively sense the fractional power spectrum of nonstationary signals without loss of any information
We prove that the coprime pair of DFrFT filter banks is linear time invariant (LTI) in discrete-time Fourier domain (DTFD) based on the low-pass filter theory in discrete-time fractional Fourier domain (DTFrFD), as the corresponding chirp modulated forms of the nonstationary signals are stationary, the decimate operation can be performed in the cross-correlation analysis between the outputs of the coprime pair of DFrFT filter banks, and the conventional power spectrum of the stationary signals can be acquired based on the uniqueness of the passband. ird, in terms of the polyphase representation of the filters in DTSFrFD, the decimator at the output of the filters can be moved to the left of the polyphase subfilters, resulting in an efficient coprime DFrFT filter banks
Summary
Power spectrum plays an important role in signal processing such as array processing [1,2,3,4], spectral estimation [5,6,7], signal detection and estimation [8,9,10,11,12], and so on. Is paper aims to develop a coprime digital fractional Fourier transform (DFrFT) filter banks theory for sensing the fractional power spectrum of the chirp stationary signals, which are nonstationary in the usual sense or in the FD [22, 33]. We prove that the coprime pair of DFrFT filter banks is linear time invariant (LTI) in DTFD based on the low-pass filter theory in DTFrFD, as the corresponding chirp modulated forms of the nonstationary signals are stationary, the decimate operation can be performed in the cross-correlation analysis between the outputs of the coprime pair of DFrFT filter banks, and the conventional power spectrum of the stationary signals can be acquired based on the uniqueness of the passband (see section 3.4 for details).
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