Discrete chirp-Fourier transform and its application to chirp rate estimation

The discrete Fourier transform (DFT) has found tremendous applications in a lmost all fields, mainly because it can be used to match the multiple frequ encies of a stationary signal with multiple harmonics. In many applications , wideband and nonstationary signals, however, often occur. One of the typi cal examples of such signals is chirp-type signals that are usually encount ered in radar signal processing, such as synthetic aperture radar (SAR) and inverse SAR imaging. Due to the motion of a target, the radar return signa ls are usually chirps, and their chirp rates include the information about the target, such as the location and the velocity. In this paper, we study discrete chirp-Fourier transform (DCFT), which is a nalogous to the DFT. Besides the multiple frequency matching similar to the DFT, the DCFT can be used to match the multiple chirp rates in a chirp-typ e signal with multiple chirp components. We show that when the signal lengt h N is prime, the magnitudes of all the sidelobes of the DCFT of a quadrati c chirp signal are 1, whereas the magnitude of the mainlobe of the DCFT is rootN. With this result, an upper bound for the number of the detectable ch irp components using the DCFT is provided in terms of signal length and sig nal and noise powers. We also show that the N-point DCFT performs optimally when N is a prime.