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Symmetric convolution of asymmetric multidimensional sequences using discrete trigonometric transforms.

T M Foltz¹, B M Welsh

¹Air Command and Staff College, Maxwell AFB, AL 36112-6426, USA.

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

|February 13, 2008

Summary

This study derives the symmetric convolution-multiplication property for discrete trigonometric transforms using the discrete Fourier transform. This method extends to multidimensional asymmetric sequences and filters, like those modeling atmospheric turbulence.

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Area of Science:

Signal Processing
Applied Mathematics
Digital Image Processing

Background:

The discrete Fourier transform (DFT) is fundamental in signal processing, known for diagonalizing circulant matrices.
Convolution and multiplication properties are crucial for analyzing and implementing filters and transforms.
Extending these properties to multidimensional asymmetric sequences presents computational challenges.

Purpose of the Study:

To provide an alternative derivation of the symmetric convolution-multiplication property for discrete trigonometric transforms.
To extend this property to multidimensional asymmetric sequences.
To demonstrate a practical application in designing a two-dimensional finite impulse response filter.

Main Methods:

Utilizing the diagonalization property of the discrete Fourier transform on circulant matrices.
Extending the derived property to multiple dimensions via block circulant matrices.
Applying the convolution-multiplication property through the product of trigonometric transforms followed by an inverse transform.

Main Results:

An alternate derivation of the symmetric convolution-multiplication property for discrete trigonometric transforms.
Generalization of this property to multidimensional asymmetric sequences.
A framework for efficient computation of multidimensional symmetric convolution.

Conclusions:

The discrete Fourier transform provides an elegant method for deriving and extending convolution-multiplication properties.
The generalized property enables efficient implementation of multidimensional filters, such as nonlinear phase FIR filters for atmospheric turbulence modeling.
This approach offers a unified perspective on transform properties in signal processing.