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

Area of Science:

• Digital Signal Processing
• Complex Analysis
• Numerical Analysis

Background:

• The inverse fast Fourier transform (IFFT) is a fundamental algorithm in signal processing, typically operating on the unit circle in the complex plane.
• Generalizations of the IFFT, such as the inverse chirp z-transform (ICZT), allow computations off the unit circle.
• Existing methods often restrict the use of orthogonal frequency components.

Purpose of the Study:

• To investigate the application of the inverse chirp z-transform (ICZT) using chirp contours on the unit circle.
• To evaluate the numerical accuracy and computational complexity of the ICZT for these specific contours.
• To explore the potential of the ICZT to handle non-orthogonal frequency components.

Main Methods:

• Algorithmic evaluation of the inverse chirp z-transform (ICZT) for chirp contours on the unit circle.
• Analysis of computational complexity, showing it is equivalent to the inverse fast Fourier transform (IFFT) at O(n log n).
• Numerical error analysis, correlating accuracy with the polar angle between contour points and Farey sequences.

Main Results:

• The ICZT is shown to be numerically accurate for a significant parameter space when applied to unit circle chirp contours.
• The computational complexity remains efficient at O(n log n), matching the IFFT.
• The error profile is demonstrably linked to the Farey sequence of order n-1, providing a theoretical basis for accuracy.
• The generalized ICZT successfully accommodates non-orthogonal frequency components.

Conclusions:

• The inverse chirp z-transform (ICZT) offers a numerically stable and computationally efficient generalization of the IFFT for unit circle contours.
• This advancement overcomes the IFFT's limitation regarding non-orthogonal frequency components, broadening its applicability.
• The findings provide a theoretical framework for understanding and predicting the numerical accuracy of the ICZT in this context.