Wavelet Algorithm and Overview

**Chapter 1**

**Abstract**

*Karlton Wirsing*

tion about the signal.

**1. Introduction**

**3**

processing, vanishing moment

Time Frequency Analysis of

Wavelet and Fourier Transform

Signal processing has long been dominated by the Fourier transform. However, there is an alternate transform that has gained popularity recently and that is the wavelet transform. The wavelet transform has a long history starting in 1910 when Alfred Haar created it as an alternative to the Fourier transform. In 1940 Norman Ricker created the first continuous wavelet and proposed the term wavelet. Work in the field has proceeded in fits and starts across many different disciplines, until the 1990's when the discrete wavelet transform was developed by Ingrid Daubechies. While the Fourier transform creates a representation of the signal in the frequency domain, the wavelet transform creates a representation of the signal in both the time and frequency domain, thereby allowing efficient access of localized informa-

**Keywords:** time-frequency analysis, Fourier transform, wavelet transform, signal

The Fourier transform has been the basis of digital signal processing since the development of the fast Fourier transform in 1965 by Cooley and Tukey in [1]. Its use for analysis goes back much farther with the development of the Fourier transform by Jean Baptiste Joseph Fourier in 1807 as a solution to thermodynamic equations. By using the Fourier transform, we can take any signal and obtain the amplitude of the sinusoids needed to recreate it. Then we can use this information to obtain the power spectrum of the signal, or we modify the amplitudes and take

A fundamental limitation of the Fourier transform is that the all properties of a signal are global in scope. Information about local features of the signal, such as changes in frequency, becomes a global property of the signal in the frequency domain. There have been various methods proposed to address this limitation; the

Gabor [2] created the windowed Fourier transform in 1946. It applies a window function of a short duration to the signal and the Fourier transform is applied to the resulting data. This method is frequently used; however, there are two limitations with this method. The first is that, since the filtering window is constant, it creates problems if the feature is larger or shorter than the window. The second is that the time resolution is the same for high frequencies as it is for low frequencies. Since as frequency increases, so does the rate of change of the signal, higher frequency

the inverse Fourier transform of the signal, which then filters the signal.

main two are the windowed Fourier transform and wavelets.

#### **Chapter 1**
