Numerical error analysis of the ICZT algorithm for chirp contours on the unit circle

Abstract

This paper shows that the inverse chirp z-transform (ICZT), which generalizes the inverse fast Fourier transform (IFFT) off the unit circle in the complex plane, can also be used with chirp contours that perform partial or multiple revolutions on the unit circle. This is done as a special case of the ICZT, which in algorithmic form has the same computational complexity as the IFFT, i.e., O(n log n). Here we evaluate the ICZT algorithm for chirp contours on the unit circle and show that it is numerically accurate for large areas of the parameter space. The numerical error in this case depends on the polar angle between two adjacent contour points. More specifically, the error profile for a transform of size n is determined by the elements of the Farey sequence of order n − 1. Furthermore, this generalization allows the use of non-orthogonal frequency components, thus lifting one of the main restrictions of the IFFT.