Generalizing the inverse FFT off the unit circle

Vladimir Sukhoy,Alexander Stoytchev

doi:10.1038/s41598-019-50234-9

Vladimir Sukhoy, Alexander Stoytchev

Open Access

https://doi.org/10.1038/s41598-019-50234-9

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Journal: Scientific Reports	Publication Date: Oct 8, 2019
Citations: 13	License type: open-access

Affiliation: Iowa State University

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.

Highlights

The Fourier transform and its inverse appear in many natural phenomena and have numerous applications
O(n2) is a lower bound for the complexity of any inverse chirp z-transform (ICZT) algorithm that works with an n-by-n matrix in memory
ICZT procedure described in the Methods section

Summary

Introduction

The Fourier transform and its inverse appear in many natural phenomena and have numerous applications. The fast Fourier transform (FFT) and the inverse FFT (or IFFT) algorithms compute the discrete versions of these transforms. Both of these algorithms run in O(n log n) time, which makes them practical. An efficient algorithm for computing the forward chirp z-transform was described 50 years ago[1,2,3,4,5] It was derived using an index substitution, which was originally proposed by Bluestein[1,5], to compute the transform using fast convolution. O(n2) is a lower bound for the complexity of any ICZT algorithm that works with an n-by-n matrix in memory

Methods

Results

Conclusion