Fractional Fourier transform: Time-frequency representation and structural instantaneous frequency identification

Abstract

The fractional Fourier transform involving a rotation angle as an additional variable is capable to transform a signal to any fractional Fourier domain in between time and frequency domain. With insight into the important role of rotation angle and time–frequency projection of fractional Fourier transform, the time–frequency representation of fractional Fourier transform is in this paper theoretically derived in terms of the window. It is demonstrated that the fractional Fourier transform in time–frequency domain is an algorithm of classical Fourier transform combined the chirp window that is adaptive dilation and translation. The window parameter properties are investigated in details and compared with those of wavelet window. To realize the structural instantaneous frequency identification, the analytical ridge point and corresponding ridge line based on the time–frequency representation of fractional Fourier transform is defined and presented. The fractional Fourier transform based instantaneous frequency identification algorithm is established accordingly by extracting ridge line. The effectiveness of proposed method is verified by a numerical multi-component non-stationary signal, a three-story frame on the shaking table excited by non-stationary inputs with different signal-to-noise ratios and a cable experiment with the time-varying tensioned forces. The identified instantaneous frequency results are compared with those obtained from short-time Fourier transform and wavelet transform. Both numerical and experimental verifications have demonstrated that the proposed fractional Fourier transform based method is capable of identification of instantaneous frequencies involving non-stationary signals with a sound robustness to noise. The fractional Fourier transform can be an alternative tool to process the non-stationary signals in time–frequency domain.