Generalizing the inverse FFT off the unit circle

Abstract

This paper describes the first algorithm for computing the inverse chirp z-transform (ICZT) in O(n log n) time. This matches the computational complexity of the chirp z-transform (CZT) algorithm that was discovered 50 years ago. Despite multiple previous attempts, an efficient ICZT algorithm remained elusive until now. Because the ICZT can be viewed as a generalization of the inverse fast Fourier transform (IFFT) off the unit circle in the complex plane, it has numerous practical applications in a wide variety of disciplines. This generalization enables exponentially growing or exponentially decaying frequency components, which cannot be done with the IFFT. The ICZT algorithm was derived using the properties of structured matrices and its numerical accuracy was evaluated using automated tests. A modification of the CZT algorithm, which improves its numerical stability for a subset of the parameter space, is also described and evaluated.

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

“…The next experiment studied the behavior of the error function for contours with 2048 points. Our previous experience with the ICZT indicated that the numerical accuracy decreases for contours with large number of points 10 . This is consistent with previous observations that Vandermonde systems can be ill-conditioned and should be solved with double precision or higher 29 .…”

“…The error formulas for a more general case with logarithmic spiral contours that span a 360° arc off the unit circle are given in ref. 10 .…”

“…The fourth term, T, is the sum of three simpler terms 10 . In this special case, however, they are all equal to N log 1 2 because the transform parameters A and W have a magnitude of 1, i.e., |A| = |W| = 1.…”

“…Algorithm 1 computes this expression in O(n log n) time by exploiting the structure of the matrices. See reference 10 for more details, proofs, and the pseudo-code for ToeplitzMultiplyE and all of its dependencies.…”

“…The inverse algorithm, however, remained elusive for the next 50 years. The ICZT algorithm also runs in O(n log n) time 10 , where n is the size of the transform. This enables applications in which the CZT is paired with the ICZT similarly to how the FFT is often paired with the IFFT.…”