Sidebar- What are Basis Functions?

What are Basis Functions?

It is simpler to explain a basis function if we move out of the realm of analog (functions) and into the realm of digital (vectors) (*). Every two-dimensional vector (x,y) is a combination of the vector (1,0) and (0,1). These two vectors are the basis vectors for (x,y). Why? Notice that x multiplied by (1,0) is the vector (x,0), and y multiplied by (0,1) is the vector (0,y). The sum is (x,y).

The best basis vectors have the valuable extra property that the vectors are perpendicular, or orthogonal to each other. For the basis (1,0) and (0,1), this criteria is satisfied.

Now let's go back to the analog world, and see how to relate these concepts to basis functions. Instead of the vector (x,y), we have a function f(x). Imagine that f(x) is a musical tone, say the note A in a particular octave. We can construct A by adding sines and cosines using combinations of amplitudes and frequencies. The sines and cosines are the basis functions in this example, and the elements of Fourier synthesis. For the sines and cosines chosen, we can set the additional requirement that they be orthogonal. How? By choosing the appropriate combination of sine and cosine function terms whose inner product add up to zero. The particular set of functions that are orthogonal and that construct f(x) are our orthogonal basis functions for this problem.

What are Scale-Varying Basis Functions?

A basis function varies in scale by chopping up the same function or data space using different scale sizes. For example, imagine we have a signal over the domain from 0 to 1. We can divide the signal with two step functions that range from 0 to 1/2 and 1/2 to 1. Then we can divide the original signal again using four step functions from 0 to 1/4, 1/4 to 1/2, 1/2 to 3/4, and 3/4 to 1. And so on. Each set of representations code the original signal with a particular resolution or scale.

Reference

(*) G. Strang, "Wavelets," American Scientist, Vol. 82, 1992, pp. 250-255.

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Last Modified by Amara Graps on 12 May 2004.
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