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Multidimensional chirp algorithms for computing Fourier transforms.

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society

|January 1, 1992

Summary

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This study introduces advanced multidimensional chirp algorithms for efficiently computing Fourier transforms. These algorithms optimize discrete Fourier transform calculations, even when lattices are not reciprocal, using novel factorization and modified transforms.

Area of Science:

Digital Signal Processing
Computational Mathematics
Fourier Analysis

Background:

Multidimensional chirp algorithms are used to compute Fourier transforms of functions with vector variables.
Existing algorithms face challenges when the output lattice is not reciprocal to the input lattice.

Purpose of the Study:

To develop efficient discrete multidimensional chirp algorithms for computing Fourier transforms.
To address limitations of existing algorithms when dealing with non-reciprocal lattices.

Main Methods:

The study extends continuous multidimensional chirp algorithms to discrete versions.
For symmetric matrices, it utilizes multidimensional Bluestein chirp algorithm with pointwise multiplication operations (PMOs) and convolution operations (COs).
For non-symmetric matrices, it factors the Fourier transform into Fresnel transforms, factors the matrix M into symmetric matrices A and B, and modifies Fresnel transforms using A and B, each factored into PMOs and COs.

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Main Results:

Efficient computation of discrete Fourier transforms over arbitrary lattices is achieved.
The algorithms leverage fast convolution methods for efficient implementation.
The modified approach for non-symmetric matrices decomposes the computation into manageable PMOs and COs.

Conclusions:

The developed discrete multidimensional chirp algorithms provide an efficient method for Fourier transform computation.
These algorithms are versatile, handling both symmetric and non-symmetric transformation matrices.
The findings offer significant improvements in computational efficiency for signal processing applications.