njit-etd2003-081 - New Jersey Institute of Technology
njit-etd2003-081 - New Jersey Institute of Technology njit-etd2003-081 - New Jersey Institute of Technology
65 3.7 Wavelet Transforms Analysis of signals using appropriate basis functions is one of the fundamental problems in the signal processing field. Fourier proposed the complex sinusoids as the basis functions for signal decomposition [41]. The Fourier transform of a finite energy continuous time signal f (t) , (i.e. f (t) E L2 ) is defined as: The strength of the standard Fourier analysis is that it allows the decomposition of a signal into its individual frequency components and establishes the relative intensity of each frequency component. Because of the infinite durations of these basis functions, any time-local information (e.g. an abrupt change in the signal) is spread over the whole frequency spectrum. Therefore, this transform cannot reflect any time-localized characteristic of f (t) into the frequency domain. It only provides the frequency behavior of f (t) in the interval -00< t
66 [-Ω,Ω ], then its STFT will be localized in the region [-T,T]x [– Q, n] of the timefrequency plane. Of course, the uncertainty principle prevents the possibility of having arbitrary high resolution in both time and frequency domains, since it lower-bounds the time bandwidth product of any basis function by ΔTΔΩ>= 1 where (ΔT)2 and 4,r (A0) 2 are the variances of the time function and its Fourier transform respectively [43][44]. An important parameter of a window function is its size (or scale). The selection of an appropriate window size poses a fundamental problem in signal analysis. Thus, by varying the window function used, one can trade the resolution in time for the resolution in frequency. An intuitive way to achieve this is to have short time duration high frequency basis functions, and long time duration low frequency ones. Fortunately, the wavelet transform provides for this desired feature and is defined as, where a E R+ ,b E R . Here a, and b are the scale and shift variables respectively, and they are continuous variables. Depending on the scaling parameter a, the wavelet function ψ(t) dilates or contracts in time causing the corresponding contraction or dilation in the frequency domain. Therefore a flexible time-frequency resolution is achievable with the wavelet transform. Another significant difference of these transforms is that the STFT is never a real function on the time-frequency plane regardless of the choice of co(t), but the wavelet transform is real if the basic wavelet ψ(t) is chosen to be real.
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65<br />
3.7 Wavelet Transforms<br />
Analysis <strong>of</strong> signals using appropriate basis functions is one <strong>of</strong> the fundamental problems<br />
in the signal processing field. Fourier proposed the complex sinusoids as the basis<br />
functions for signal decomposition [41]. The Fourier transform <strong>of</strong> a finite energy<br />
continuous time signal f (t) , (i.e. f (t) E L2 ) is defined as:<br />
The strength <strong>of</strong> the standard Fourier analysis is that it allows the decomposition<br />
<strong>of</strong> a signal into its individual frequency components and establishes the relative intensity<br />
<strong>of</strong> each frequency component. Because <strong>of</strong> the infinite durations <strong>of</strong> these basis functions,<br />
any time-local information (e.g. an abrupt change in the signal) is spread over the whole<br />
frequency spectrum. Therefore, this transform cannot reflect any time-localized<br />
characteristic <strong>of</strong> f (t) into the frequency domain. It only provides the frequency<br />
behavior <strong>of</strong> f (t) in the interval -00< t
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