WAVELET TRANSFORM
What is WAVELET TRANSFORM?
Wavelet transform provides ‘TIME-FREQUENCY’ representation of a signal, it uses MULTI-RESOLUTION technique by which different frequencies are analyzed with different resolutions and it is mostly used to analyze NON-STATIONARY SIGNALS.
Why TIME-FREQUENCY representation?
We know that most signals in practice are TIME-DOMAIN signals and when you plot it you get TIME-AMPLITUDE representation. But for some applications the actual information is hidden in the frequency content of the signal. To find the frequency content we use FOURIER TRANSFORM. So, FT gives us FREQUENCY-AMPLITUDE representation of the time-domain signal.
So, we know that no frequency information is present in time-domain signal and no time related information is present in frequency-domain signal. But what if we require both time and frequency information at the same time???
Problem with NON-STATIONARY signals
FT tells us how much of each frequency exists in a signal, BUT it does not tell us when in time these contents exist. Now if the signal is STATIONARY i.e. its frequency content do not change in time then we do not need to know when these contents occur because they are present at ALL TIMES!
But this is not the case for NON STATIONARY signals i.e. signal whose frequency response changes with time. e.g. Biomedical signals like ECG, EEG,EMG .
Here if you are just concerned with what frequency contents are present then you can use FT but what if you want to simultaneously know ‘When’ these frequency contents are present. Hence, there is a need for TIME-FREQUENCY representation of the signal i.e. to get time and frequency information simultaneously.
Need for MULTI-RESOLUTION technique
So, many transforms were developed for time frequency representation. One of them was Short time Fourier Transform. In STFT signal is divided into short segments and stationary condition for non-stationary signal is assumed. Then a window function equal to the length of the segment is selected. It is multiplied with the signal. Then its Fourier transform is taken. The window is then shifted and the process is repeated. One can say STFT is windowed FT. This gives time-freq representation of the signal. I wont go into detail it will take lot of time. So, what was the problem with STFT...RESOLUTION!!!
STFT analyses signals using windows of finite length, which covers only a portion of the signal. So you just know “a band of frequency” that exist in a signal not the exact frequency components that exists in the signal. So frequency resolution is poor. Narrower this window, poorer frequency resolution but if the window is shorter you will be able to resolve your signal better in time i.e. good time resolution. Conversely, wider the window, good freq resolution but poor time resolution.
To solve this resolution problem wavelet transform was born. Finally!!!
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. But here the width of the window is changed for each single frequency component . This is called MULTI-RESOLUTION analysis i.e. it analyzes the signals at different freq with different resolution.
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.
PHEW! tired of typing. Kuch jyada ho gaya :P . So, this was all about WHY we need wavelet transform. I will write about HOW we use it, some other day (Provided I don’t get busy with something else or start feeling lazy ).Aaj ke liya itna kafi hai.
Reason for posting this:
For those who want to learn about wavelet transform: Just explained the basics of WT without any mathematical equations , hope you understood :). Refer the tutorial on wavelets by Robi Polikar(do a google search). His explanation is best (thousands time better than mine). I particularly liked this line from his tutorial “most of these books and articles are written by math people, for the other math people; still most of the math people don't know what the other math people are talking about”.So you know what to expect from his tutorial:).
For me: I still need to learn a lot about wavelet. Unfortunately there are very few people who really understand it. I am still searching for them. Whatever I have posted above is what I have understood about wavelets. But somewhere my interpretation about WT may be wrong or I may have missed something imp. So, I want people to correct me or add something to it.
For those who already know wavelets perfectly: Well, then lets have a discussion on it
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