The Structure Of One Kind Of Wavelets And Their Applications

Wavelet analysis is a relatively recent development in applied mathematics, which is one of the two separate branches of Fourier transform. Another one is fractional Fourier transform.Wavelet's primary point is that it can sufficiently reveal some property of questions by transformation. So, wavelet translation has successful application in many fields. Harmonic wavelet is one of the wavelet types. Because it can be constructed from simple functions and a band–limited wavelet, it has much superior property.The fractional Fourier transform emerged as an extension of the ordinary Fourier transform, which is one of the most common tools in signal processing. In fact, the multiplicity is caused by two reasons:one is that there are different eigenfunctions for Fourier transform; the other is that there are a lot of techniques to fractionalize the eigenvalues of Fourier transform in the procedure of constructing the fractional Fourier transform. The standard Chirp-type fractional Fourier transform is one type of the fractional Fourier transforms.Shannon Sampling Theorem allows us to completely recover a continuous-time band-limited signal from its digital samples. Sub-band coding approach can be used for this goal by less digital samples. It's the reason why we research the sub-band coding approach. The mathematical foundation of the sub-band coding approach is the wavelet packet of the wavelet analysis.The work of this paper is based on the Shannon wavelet's sub-band coding approach to research the following two questions: One is that whether the sub-band coding approach can be used for the standard Chirp-type fractional Fourier transform. If can, how can it be? The other question is that as to the harmonic wavelet which kind of the wavelets can be the sub-band coding approach to be used.