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SAR vibrometry using fractional Fourier transform processing

A novel signal-processing approach is reported for vibrometry in synthetic aperture radar (SAR) imaging systems. The approach exploits the conventional deramp process; however, in place of Fourier-transform processing we utilize the fractional Fourier transform (FRFT) as a processing tool. The FRFT is geared toward non-stationary signals and chirped sinusoids particularly. A simplified mathematical expression is developed to describe the reflectivity of the aimed patch of ground containing vibrating targets as a function of space and time. Under the approximation that the velocities of vibrating point targets are constant during each probing chirped pulse, it is shown that the returned echo after the deramp process is a superposition of sinusoids that are chirped according to the Doppler effects induced by the vibrating point targets. By applying the multiangle centered discrete fractional Fourier transform (MA-CDFRFT) to the demodulated echoes, the instantaneous velocities of the vibrating point targets are estimated from the two coordinates of each peak in the MA-CDFRFT's frequency-angle plane. By repeating this process where a sequence of successive pulses are used to interrogate the vibrating targets, the velocities of the targets are estimated in each pulse, thereby generating a piecewise-linear estimate of the history of the vibration velocity in time. Theoretical performance evaluation of the proposed technique is carried out using real SAR-system parameters and simulated vibrating targets. The interplay amongst minimum detectable velocity, maximum detectable vibration frequency, pulse duration and chirp rate is determined analytically.