Regarding the Continous Wavelet Transform 'a' parameter

[tomizzo]

Hi there,

I've recently been doing some studying into time-frequency analysis. I've covered some of the basic materials regarding the Short-Time Fourier Transform (STFT) along with the concepts of temporal and frequency resolution (along with the uncertainty principle of course).

I've now transitioned into studying the Continuous Wavelet Transform (CWT) and having some difficulties fully understanding the definition. Referring to the formal definition (found here - https://upload.wikimedia.org/math/9/1/3/913e2714d24c67d2d31d89baff7c4979.png), the CWT is simply the inner product between some signal and a wavelet function which itself is a function of a variable 'a' and 'b'. The variables 'a' and 'b' correspond to dilation and translation of mother wavelet function respectively.

In the context of time-frequency analysis, 'a' allows us to change the window width as a function of the frequency of interest and 'b' allows us to shift the window in time.

So here's my question: Where exactly is the complex exponential (sinusoid) in the transformation? Unlike the Fourier transform which explicitly contains a complex exponential that is used for computing an inner product, the CWT is defined in terms of an abstract wavelet function with the parameters 'a' and 'b'.

So here's my assumption, but I'm curious if someone could correct me if I'm mistaken:

Time-frequency analysis is simply an application of the CWT. Unlike the Fourier transform which is directly tied to frequency analysis, the CWT could be used for other applications. However, if we were to wish to use the CWT in the context of time-frequency analysis, we must use a wavelet mother function that incorporates a complex exponential.

Assuming we were able to identify an appropriate wavelet function to use for time-frequency analysis, the 'b' parameter would be used to time shift the wavelet (similar to time shifting a window). This seems straightforward.

Now correct me if I'm wrong, but the 'a' parameter must somehow control the 'width' of the wavelet along with a corresponding complex exponential. I'm thinking this 'a' parameter must somehow be related to a decay rate of a Gaussian window. Specifically, I'm thinking the 'a' parameter is inversely related to the window such that small frequencies, we get a large window (i.e. slow decay).

My overall question: in the context of time frequency analysis, is the 'a' parameter used to control both the complex exponential AND the window width?

So if any could offer any help or insight on my assumptions, that would be awesome! I'm also curious if there is a table/reference on the web for some common wavelet functions.

Thank you!

 

[chiro]

Hey tomizzo.

The wavelet transform doesn't exist on a subset of the real line - it exists across it.

The Fourier transform exists for some finite length (according to the length of the first harmonic) where-as the wavelet transform has the same "intuition" (something harmonic analysis studies) but it exists across L^2(R).

This difference in covering the whole real line (as opposed to the length of the fundamental harmonic) complicates things quite a bit mathematically and technically but the result is that you have the wavelet and other transforms that exist across the entire real line.

The inner product intuition is a good one to have.

Also - wavelet's often have to be derived from their relations (this is the case for the Daubechies wavelet) because of these complications.

There is a lot of interesting theory (I took a class in wavelet's a very long time ago) but the difference between a finite interval and the entire real line being consistent with the inner product axioms (which are a function of Hilbert spaces) changes things a lot.

It's (analogously) a lot like the difference between infinities and finite quantities - making it consistent requires one to add new constraints that didn't exist when the subset of this new space was considered.