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WAVELET TRANSFORM USING MATLAB
Wavelet transform is a very beautiful tool for signal processing which gives us high degree of freedom and flexibility.
The time domain representation of the signal gives us information that how the amplitude of any signal is varying with respect to time
Or
At different time instances what are the amplitudes of any signal.
But it gives no idea about the different frequency components presents in any given signal
The time resolution of time domain representation is very high but the frequency resolution of time domain representation is zero.
Now with the help of Fourier transform we can convert time domain signal into frequency domain signal.
The frequency domain representation of the signal gives us an information that how the amplitude of any signal is varying with respect to frequency
Or
What are the amplitudes of different frequency components presents in any signal
So frequency resolution of time domain signal is very high but its time resolution is zero
So Fourier transform is again not able to provide complete information about any signal particularly for non stationary signals (in which frequency changes with respect to time) Fourier transform is not a suitable tool.
S.T.F.T. (Sort time Fourier transform) is used to solve the above problem of Fourier transform. In sort time Fourier transform we simply multiply the signal with a window function of small length.
The concept behind S.T.F.T. is that any non stationary signal may be considered stationary for short time interval. So S.T.F.T. solve the problem of F.T. but up to certain extent and it gives a three dimensional information about the signal (about time, frequency and amplitude of any signal)
But the main drawback with S.T.F.T. is that in S.T.F.T. the size of window remains constant and any single window size cannot be suitable for all frequency components present in any signal.
Finally Wavelet transform solve the above problem of S.T.F.T. the wavelet transform used different time scale for the analysis of different frequency components presents in any signal and gives complete three dimensional information about any signal i.e. at which time interval what different frequency component present in any signal and what are the amplitude of these different frequency components.
Hence the wavelet transform has multi resolution properties i.e. it uses different time scales (scale is inversely proportional to the frequency) for the analysis of different frequency components.
High time scale is used for the analysis of low frequency component and low scale is used for the analysis of high frequency components.
Again Wavelet Transform is most suitable tool for studying the local behavior of any signal such as discontinuities or spikes
In Fourier transform (F.T.) or Short time Fourier transform(S.T.F.T.) any time domain signal is converted into sinusoid of different amplitudes and frequencies whereas in Wavelet transform signal is converted into shifted and scaled version mother wavelets.
The sine waves and cosine waves are very smooth and are predictable whereas the wavelets are not smooth and are unpredictable so wavelets are more suitable tool for studying the local behavior such as spikes or discontinuity present in any signal.
There are well defined families of standard wavelet and we have also freedom of defining our own wavelet according to our requirement.
So we can say that wavelet transform is an ultimate tool for signal processing
The time domain representation of the signal gives us information that how the amplitude of any signal is varying with respect to time
Or
At different time instances what are the amplitudes of any signal.
But it gives no idea about the different frequency components presents in any given signal
The time resolution of time domain representation is very high but the frequency resolution of time domain representation is zero.
Now with the help of Fourier transform we can convert time domain signal into frequency domain signal.
The frequency domain representation of the signal gives us an information that how the amplitude of any signal is varying with respect to frequency
Or
What are the amplitudes of different frequency components presents in any signal
So frequency resolution of time domain signal is very high but its time resolution is zero
So Fourier transform is again not able to provide complete information about any signal particularly for non stationary signals (in which frequency changes with respect to time) Fourier transform is not a suitable tool.
S.T.F.T. (Sort time Fourier transform) is used to solve the above problem of Fourier transform. In sort time Fourier transform we simply multiply the signal with a window function of small length.
The concept behind S.T.F.T. is that any non stationary signal may be considered stationary for short time interval. So S.T.F.T. solve the problem of F.T. but up to certain extent and it gives a three dimensional information about the signal (about time, frequency and amplitude of any signal)
But the main drawback with S.T.F.T. is that in S.T.F.T. the size of window remains constant and any single window size cannot be suitable for all frequency components present in any signal.
Finally Wavelet transform solve the above problem of S.T.F.T. the wavelet transform used different time scale for the analysis of different frequency components presents in any signal and gives complete three dimensional information about any signal i.e. at which time interval what different frequency component present in any signal and what are the amplitude of these different frequency components.
Hence the wavelet transform has multi resolution properties i.e. it uses different time scales (scale is inversely proportional to the frequency) for the analysis of different frequency components.
High time scale is used for the analysis of low frequency component and low scale is used for the analysis of high frequency components.
Again Wavelet Transform is most suitable tool for studying the local behavior of any signal such as discontinuities or spikes
In Fourier transform (F.T.) or Short time Fourier transform(S.T.F.T.) any time domain signal is converted into sinusoid of different amplitudes and frequencies whereas in Wavelet transform signal is converted into shifted and scaled version mother wavelets.
The sine waves and cosine waves are very smooth and are predictable whereas the wavelets are not smooth and are unpredictable so wavelets are more suitable tool for studying the local behavior such as spikes or discontinuity present in any signal.
There are well defined families of standard wavelet and we have also freedom of defining our own wavelet according to our requirement.
So we can say that wavelet transform is an ultimate tool for signal processing
43 pages, Kindle Edition
Published February 28, 2016
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