The inverse chirp z-transform (ICZT) can be used with chirp contours that perform partial or multiple revolutions on the unit circle, according to engineers at Iowa State University.

Last year, Iowa State engineers Alexander Stoytchev and Vladimir Sukhoy created a closed-form solution for the ICZT, which generalises the inverse fast Fourier transform (IFFT) off the unit circle in the complex plane, and also wrote a fast algorithm for computing it.

Now the pair have published ‘Numerical error analysis of the ICZT algorithm for chirp contours on the unit circle’, a paper showing how their algorithm functions on the unit circle.

“It is numerically accurate for large areas of the parameter space,” according to the paper. “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 generalisation allows the use of non-orthogonal frequency components, thus lifting one of the main restrictions of the IFFT.”

This is an interesting connection because Farey sequences often appear in number theory, said the university. They have shown that the singularities of the ICZT of size n are related to the elements of the Farey sequence of order n-1.

In short:
IFFTs work with equi-spaced sampling points that fully cover the unit circle
ICZT can work with contours that cover only a fraction of the unit circle and contours perform multiple revolutions over the circle – enabling the use of certain (non-orthogonal) frequency components

On the unit circle, the ICZT algorithm needs only 64bit floating-point numbers and works in O(n log n) time, according to the university.

“This algorithm is more general than the IFFT, but maintains the same speed,” explained Stoytchev.

It can pair with the existing CZT (forward chirp) algorithm to do back-to-back signal analysis and signal synthesis. Application is foreseen in signal processing, electronics, medical imaging, radar, sonar and wireless communication.

Numerical error analysis of the ICZT algorithm for chirp contours on the unit circle is published in Scientific reports, and is available free in full.

Diagram courtesy of Alexander Stoytchev