The Convolution Theorem Related To The Fractional Wavelet Transform

As the convolution theory is the basic theory in the linear time invariant (LTI) system,for every continuous time’s output in LTI system, it can be obtain by the convolutionthrough the system impulse response of the input signal. The convolution theorem is widelyused in communication system, the signal processing field and so on. Its study has achievedfruitful results.The wavelet transform and the fractional Fourier transform have a wide range ofapplications in the signal processing research. However, the wavelet transform is limited inthe time-frequency plane, although the Fourier transform overcome such limitation, and inthe fractional domain it can provide signal representations, but it can not obtain localstructures of the signal. While the fractional wavelet transform not only rectify thelimitation of the wavelet transform and the fractional Fourier transform, but also inheritsthe advantages of multiresolution analysis of the wavelet transform and the capability ofsignal representations in the fractional domain which is similar to the fractional Fouriertransform. Due to the variability of the parameters more flexible, so it is increasingly beingused in all fields.The paper contains the following knowledge: Firstly introduces the theories offractional wavelet transform. In order to get a more thorough grasp of the definition andproperties of the fractional wavelet transform, this paper put the definition and properties ofthe wavelet transform and the fractional Fourier transform in the front place, which leads tothe theories of fractional wavelet transform. As an emerging of transformation, thefractional wavelet transform has a widely applications in the image denoising, ecg signaldenoising and image encryption. Then introduce the relevant theories of convolution.Starting from the classical Fourier transform convolution theory, this paper then introducesthe convolution theory of the wavelet transform. As the correlation theory has the similarcharacter to the convolution theory, and it is also an important role in the signal processingresearch, so the correlation theory of the wavelet transform is given in this paper. At the lastof the paper, comprehensively study the contents of the above, which leads to theconvolution theorem and the correlation theorem of the fractional wavelet transform.