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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 T-tw 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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