

Wavelets
Name: existing
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995
Question:
What is wavelet analysis? It sounds like the use of fourier series to do
analysis. Is that it ? I am a senior in high school and a concept oriented
response as well as possibly a more technical response.
Replies:
I am just learning about wavelets. The idea seems to be that if you have a
signal that is a function of time, call it f(t), take a "window" of the
signal (I will explain shortly), and then take a fourier transform of the
windowed signal. The art is to pick an optimum window.
By a "window" I mean take the signal between times Ttw and t+tw where
2xtw is the width of the signal, and T, the center of the window, is a
variable.
The fun part of doing science is that we are always studying new
things.
jlu
Yes, wavelets are one of the great ideas that was really only recently
invented  by a geophysicist in fact. Fourier transforms are the standard
tools for examining what sort of frequencies are present in a particular
time signal (or spatial frequencies in a function that varies in space).
For example, a pure musical tone has a pressure wave that varies periodical
ly with time, producing a Fourier transform that shows a spike at a funda
mental frequency (the inverse of the period) and at harmonics (multiples of
the fundamental). All well and good. But real music (or most signals of
real interest) is not like that. Real music consists of sequences of notes,
all with different fundamental frequencies, and there is no overall period
at all  music that repeats itself over and over is boring, at least to most
of us! Basically, as jlu said, the wavelet approach does consider the
frequency dependence, as with Fourier analysis, but only over a short time
interval. A "wavelet" has both a distribution in frequency and in time. In
fact, musical notation is exactly one form of wavelet representation, and
the process of turning musical notation on a page into music in the ear is
actually the process of producing a wavelet transform. The frequency of
each note is determined by its vertical position within its clef, its
position in time is determined by its horizontal position on the page, and
its duration (the "level" of the wavelet  you can have wavelets of many
different resolutions) by the type of note. Hope that helps!
asmith
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Update: June 2012

