Introduction to Wavelets

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Last Edit:--Dlamba 05:11, 6 November 2009 (UTC)

Taking Fourier's torch forward...

I can bet a great deal of money, that as Electrical Engineers, the first person that comes to mind when someone says "SIGNAL PROCESSING" is Fourier.
Jean Baptiste Joseph Fourier (1768 - 1830) laid a rock-solid foundation for signal analysis, when he claimed that all (continuously differentiable) signals can be represented as the sums of sines and cosines.
It is hard to imagine the iPod generation without the work this great man did over 2 centuries ago.
However, the educated world (Electrical Engineers, :D) gradually evolved from their happy continuous perception of life, to the slightly scary, yet extremely promising world of discrete (a.k.a "digital") signals.
In this world, there is no such thing as continuity (obviously), and signals must be represented as discrete sets of zeros and ones. *These "jumps" would make Fourier very unhappy, because Fourier analysis starts breaking down (leakage, spectrogram uncertainty) when we make abrupt cut-offs and chop signals at will.
Fundamentally, sines and cosines are infinite length and continuously differentiable, so to represent a jump (zero time or infinite frequency) one would technically need an infinite number of frequencies. Another concept that, albeit well defined on paper, but one that computers detest, is this whole concept of "infinity".

This whole concept of breaking a signal down into a summation of simpler or "basis" signals is the cornerstone of Fourier theory.
However, as mentioned above, the basis sinusoidal signals do not work very well with discontinuities.
This is where wavelets come in, because they are very good with jumps. (More on this later)
Another problem we see with Fourier analysis, is the uncomfortable trade-off that exists between time and frequency resolution.
In other words, if you take a spectrogram of certain data, making your time window finer(increasing temporal resolution), makes your frequency resolution worse. The inverse is also true, i.e. good spectral resolution, causes worse time resolution.
This is an unhappy decision for signal processors to make, and they often lament, "Why can't I have both?" or "How do I see the tiny variations in frequency while maintaining a reasonably fine resolution in time (and vice versa)"
Unfortunately, it is impossible using a spectrogram; but wavelets mitigate this problem to a great extent.
I like how Amara Graps [1] puts it: In wavelet analysis, the scale that we use to look at the data plays a special role; a way of seeing the forest and the trees, so to speak.

Now that I've delivered my sales pitch on wavelets, comes the hard part of giving some convincing evidence that they work.
Having asked even a few grad students, I am convinced that wavelets are a very difficult concept to grasp.
So before we can use them, let's try to lay a foundation for some of the underlying concepts.

A basis function can be thought of as building block for functions.
Just as a set of unit vectors, (1,0) and (0,1) can represent any vector in 2-D space, a set of basis functions can be chosen to represent any function.
A very important conditions for basis functions is that their dot product must be zero; in other words, they must be orthogonal.
It is intuitive to see different combinations of sines and cosines as the basis functions for a Fourier representation.
In wavelet analysis, the basis functions are more complicated and called "mother wavelets" or simply wavelets.

We noticed in Fourier Analysis, that with a longer spectrogram window, we get good resolution in the frequency domain, but poor resolution in the time domain. Vice versa for short windows.
Here is where wavelets are better. The windows in wavelets vary. To isolate the short time changes (temporal resolution) we choose some short basis functions, and some very long basis functions for detailed spectral information.
The only catch is, we have to choose our basis functions, unlike in Fourier Analysis, where they were all sines and cosines.
And the problem with choosing them is, wavelets offer an infinite set of such functions.