Analytical and Numerical Aspects of Wavelets

Almost every physical phenomenon can be described via a waveform –a function of time, space or some other variables, in particular, sound waves. The Fourier transform gives us a unique and powerful way of viewing these waveforms. Nowadays, wavelet transformation is one of the most popular candidates of the time-frequency-transformations. There are three types of wavelet transforms, namely: continuous, discrete and fast wavelet transforms. In this work we will study Fourier transform together with its properties and present the connections between Fourier transform and wavelet transform. Moreover, we will show how the Wavelet-Galerkin method can be used to solve ordinary differential equations and partial differential equations. For the applications of wavelet transform we will consider two applications; first signal decomposition and reconstruction: in this section we use two filters to decompose a signal using the wavelet decomposition algorithm and then we use similar process to rebuild the original signal using the wavelet reconstruction algorithm. A second application is the audio fingerprint. Assume we have an audio. We read this audio and then convert it into signals. These signals are then divided into a number of frames. Next, we decompose each frame of this audio signal into five layer wavelets. Finally we use the wavelet coefficients to compute the variance, zero crossing, energy and centroid.