📄 an introduction to wavelets what are basis functions.htm
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<H3><FONT size=8>S</FONT>idebar- <FONT size=6>W</FONT>hat are <FONT
size=6>B</FONT>asis <FONT size=6>F</FONT>unctions?</H3>
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<H3>What are Basis Functions?</H3>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 <EM>(x,y)</EM> is a combination of
the vector <EM>(1,0)</EM> and <EM>(0,1).</EM> These two vectors are the basis
vectors for <EM>(x,y)</EM>. Why? Notice that <EM>x</EM> multiplied by
<EM>(1,0)</EM> is the vector <EM>(x,0)</EM>, and <EM>y</EM> multiplied by
<EM>(0,1)</EM> is the vector <EM>(0,y)</EM>. The sum is <EM>(x,y)</EM>.
<P>The best basis vectors have the valuable extra property that the vectors
are perpendicular, or orthogonal to each other. For the basis <EM>(1,0)</EM>
and <EM>(0,1),</EM> this criteria is satisfied.
<P>Now let's go back to the analog world, and see how to relate these concepts
to basis functions. Instead of the vector <EM>(x,y)</EM>, we have a function
<EM>f(x)</EM>. Imagine that <EM>f(x)</EM> is a musical tone, say the note
<EM>A</EM> in a particular octave. We can construct <EM>A</EM> 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 <EM>and</EM> that
construct <EM>f(x)</EM> are our orthogonal basis functions for this problem.
<H3>What are Scale-Varying Basis Functions?</H3>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.
<P>
<H4>Reference</H4>(*) G. Strang, "Wavelets," </EM>American Scientist,</EM>
Vol. 82, 1992, pp. 250-255.
<P>
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<H5>You may <A
href="http://www.amara.com/ftpstuff/IEEEwavelet.ps.gz">download</A> this
paper: "Introduction to Wavelets" </H5>
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<ADDRESS>Last Modified by <A href="mailto:amara@amara.com">Amara
Graps</A> on 8 October 1997.<BR>© Copyright Amara Graps, 1995-1997.
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