USING OF DISCRETE ORTHOGONAL TRANSFORMS FOR CONVOLUTION

Anna Ušáková - Jana Kotuliaková - Michal Zajac

   The convolution is a mathematical way of combining two functions to form a third one. It is the single most important technique in digital signal processing because it relates two signals of interest: input signal, output signal and the impulse response of the transmitting system. The convolution is not used only to compute the output of the system, when the impulse response is known, but also in the image signal processing, eg, for the interpolation of 2-D signals, in speech processing, \eg, homomorphic modelling of the speech, noise reduction in the speech, but also in image processing, etc. The evaluation of the convolution in the time domain is very complicated, especially for the large lengths of the input signal or impulse response. So we try to find another way how to compute convolution. Complicated computation of the convolution in the time domain is transformed into trivial multiplication in the domain of some orthogonal transforms (eg, Fourier transform), for another OT's domain this evaluation is more complicated. In this paper we present a convolution formula for some selected discrete orthogonal transforms (DOT).

Keywords: convolution, orthogonal transform, discrete cosine transform, discrete Hartley transform, discrete sine transform, discrete Fourier transform

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