A Convolution Property for Sinusoidal Unitary Transforms

Yasuo YOSHIDA  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E77-A   No.5   pp.856-863
Publication Date: 1994/05/25
Online ISSN: 
DOI: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section LETTER (Special Section on Signal Processing and System Theory)
Category: 
Keyword: 
DCT,  DST,  convolution-product,  eigenvector,  boundary condition,  

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Summary: 
This paper shows that a convolution property holds for sixteen members of a sinusoidal unitary transform family (DCTs and DSTs), on condition that an impulse response is an even function. Instead of the periodicity of an input signal assumed in the DFT case, DCTs require the input signal to be even symmetric outside boundaries and DSTs require it to be odd symmetric. The property is obtained by solving the eigenvalue problem of the matrices representing the convolution. The content of the property is that the DCT (or the DST) element of the output signal is the product of the DCT (or the DST) element of the input signal and the DFT element of the impulse response. The result for the well-known DCT is useful for a strongly-correlated signal and two examples demonstrate it.