THE WAVELET TUTORIAL. PART IV. by. ROBI POLIKAR. MULTIRESOLUTION ANALYSIS:THE DISCRETE WAVELET TRANSFORM. The wavelet transform or wavelet analysis is probably the most recent solution to an interesting tutorial aimed at engineers by Robi Polikar from Iowa State. ROBI POLIKAR. FUNDAMENTAL CONCEPTS. &. AN OVERVIEW OF THE WAVELET THEORY. Welcome to this introductory tutorial on wavelet transforms.
|Published:||21 March 2017|
|PDF File Size:||40.67 Mb|
|ePub File Size:||42.5 Mb|
Now, let's look at its STFT: The "x" and "y" axes are time and frequency, respectively.
Please, ignore the numbers on the axes, since they are normalized in some respect, which is not of any interest to wavelet transform robi polikar at this time.
Just examine the shape of the time-frequency representation. First of all, note that the graph is symmetric with respect to midline of the wavelet transform robi polikar axis.
The symmetric part is said to be associated with negative frequencies, an odd concept which is difficult to comprehend, fortunately, it is not important; it suffices to know that STFT and FT are symmetric.
THE WAVELET TUTORIAL
What is important, are the four peaks; note that there are four peaks corresponding to four different frequency components. Also note wavelet transform robi polikar, unlike FT, these four peaks are located at different time intervals along the time axis.
Remember that the original signal had four spectral components located at different times. Now we have a true time-frequency representation of the signal.
We not only know what frequency components are present in the signal, but we also know where they are located in time. The implicit problem of the STFT is not obvious wavelet transform robi polikar the above example.
Of course, an example that would work nicely was chosen on purpose to demonstrate the concept.
THE WAVELET TUTORIAL PART I by ROBI POLIKAR
This principle originally applied to the momentum and location of moving particles, can be applied to time-frequency information of a signal.
Simply, this principle states that one cannot know the exact time-frequency representation of a signal, i.
- Wavelet Tutorial - Part 4
- Wavelet Tutorial - Part 1
- Wavelet Tutorial - Part 4
- THE FOURIER TRANSFORM
What one can know are the time intervals in which certain wavelet transform robi polikar of frequencies exist, which is a resolution problem.
The problem with the STFT has something to do with the width of the window function that is used. To be technically correct, this width of the window function is known as the support of the window.
Wavelet Tutorial - Part 1
If the window function is narrow, than it is known as compactly supported. This terminology is more often used in the wavelet wavelet transform robi polikar, as we will see later. Here is what happens: Recall that in the FT there is no resolution problem in the frequency domain, i.
Conversely, the time resolution in the FT, and the frequency resolution in the time domain are zero, since we have no information about them. Now, in STFT, our window is of finite length, thus it covers only a portion of wavelet transform robi polikar signal, which causes the frequency resolution to get poorer.
What I mean by getting poorer is that, we wavelet transform robi polikar longer know the exact frequency components that exist in the signal, but we only know a band of frequencies that exist: In FT, the kernel function, allows us to obtain perfect frequency resolution, because the kernel itself is a window of infinite length.
THE WAVELET TUTORIAL PART II by ROBI POLIKAR
In STFT is window is of finite length, and we no longer have perfect frequency resolution. You may ask, why don't we make the length of the window in the STFT infinite, just like as it is in the FT, to get perfect frequency resolution? To make a long story real short, we are faced with the following dilemma: If we use a wavelet transform robi polikar of infinite length, we get the FT, which gives perfect frequency resolution, but no time information.