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

Vladimir Sukhoy,Alexander Stoytchev

doi:10.1038/s41598-020-60878-7

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.

Highlights

The Inverse Chirp Z-Transform (ICZT) is a generalization of the Inverse Fast Fourier Transform (IFFT), which is one of the most popular and useful algorithms[1,2]
This paper studies the properties of the inverse chirp z-transform (ICZT) for the special case when the magnitudes of A and W are equal to 1, which restricts the contour to lie on the unit circle
This enables applications in which the CZT is paired with the ICZT to how the FFT is often paired with the inverse fast Fourier transform (IFFT)

Summary

Introduction

The Inverse Chirp Z-Transform (ICZT) is a generalization of the Inverse Fast Fourier Transform (IFFT), which is one of the most popular and useful algorithms[1,2]. Unlike the IFFT, the ICZT can work with a contour that performs a partial revolution, a full revolution, or more than one revolution The effect of this generalization is that the frequency components specified by the sampling points are no longer restricted to be harmonically related or orthogonal. Many applications require both signal analysis and signal synthesis These tasks have been performed with the FFT and IFFT algorithms that were published in 19653. The ICZT algorithm 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 to how the FFT is often paired with the IFFT. The behavior of the error in those cases is more complicated and is related to Farey sequences

Methods

Results

Discussion

Conclusion