2.0 WAVELET THEORY
The transform of a signal is just another form of representing the signal. It does not change the information content present in the signal. The Wavelet Transform provides a time-frequency representation of the signal. It was developed to overcome the short coming of the Short Time Fourier Transform (STFT), which can also be used to analyze non-stationary signals. While STFT gives a constant resolution at all frequencies, the Wavelet Transform uses multi-resolution technique by which different frequencies are analyzed with different resolutions. A wave is an oscillating function of time or space and is periodic. In contrast, wavelets are localized waves. They have their energy concentrated in time or space and are suited to analysis of transient signals. While Fourier Transform and STFT use waves to analyze signals, the Wavelet Transform uses wavelets of finite energy.
Fig2.1 Demonstration of a wave
Fig 2.2 Demonstration of a wavelet
The wavelet analysis is done similar to the STFT analysis. The signal to be analyzed is multiplied with a wavelet function just as it is multiplied with a window function in STFT, and then the transform is computed for each segment generated. However, unlike STFT, in Wavelet Transform, the width of the wavelet function changes with each spectral component. The Wavelet Transform, at high frequencies, gives good time resolution and poor frequency resolution, while at low frequencies; the Wavelet Transform gives good frequency resolution and poor time resolution. 2.2 DISCRETE WAVELET TRANSFORM:
The Wavelet Series is just a sampled version of CWT and its computation may consume significant amount of time and resources, depending on the resolution required. The Discrete Wavelet Transform (DWT), which is based on sub-band coding, is found to yield a fast computation of Wavelet Transform. It is easy to implement and reduces the computation time and resources required. In CWT, the signals are analyzed using a set of basis functions which relate to each other by simple scaling and translation. In the case of DWT, a time-scale representation of the digital signal is obtained using digital filtering techniques. The signal to be analyzed is passed through filters with different cutoff frequencies at different scales. It is known that mathematical transformations like the Fourier transform, wavelet transform, etc. are applied to signals in the time domain to obtain further information not readily available from the original signal .We use the wavelet transform for analyzing signals that contain discontinuities and sharp spikes (coincidentally, these kind of signals are commonly found in nature). A discrete wavelet transform can be described as a fast, linear operation that operates on a data vector whose length is an integer power of two and thus transformed into a numerically different vector of the same length. The wavelet transform is invertible, and in fact, orthogonal. That is, the inverse transform, when viewed as a big matrix, is the transpose of the transform. In other words, a discrete wavelet transform is a rotation in function space, from the input space (or time) domain where the basis functions are the unit vectors or delta functions in the limit, to a different domain. We refer to the basis functions as "mother functions" and wavelets. Generally speaking, wavelets are mathematical functions that split a given signal in the time domain into different frequency components. The interesting point to note about the wavelet basis functions is that unlike sines and cosines they are localized in space. However, these functions are also localized in terms of frequency and characteristic scale. These characteristics allow different sets of wavelets to make different trade-offs between how compactly they are localized in space and their smoothness. 2.2.1 DWT AND FILTER BANKS:
Filters are one of the most widely used signal processing functions. Wavelets...
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