# FTIntro - Exploring Fourier Transform Techniques with...

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Functions satisfying the Dirichlet conditions (one of which is periodicity) can be written as a sum of a series of sines and cosines, called a Fourier series. For example the sum f(t) below approximates a square wave. range for variable n n .. , 1 3 9 fundamental frequency ν 1 definition of function as sum of sines f( ) t n . n 1 sin( ) . . . . n 2 π ν t range of t for graph t .. , 0 .01 5 0 2 4 6 1 0 1 f( ) t t Press F9 to show the graph. In the reverse of this process, given a waveform, one can determine its frequency makeup by Fourier analysis. The Fourier Transform The Fourier transform relates a function to another function of a conjugate variable. The product of the variable and its conjugate is unitless. If t and ϖ are conjugate variables, then the Fourier transform of f(t) is F( ) ϖ . ( ) . 2 π 1 2 dt . f( ) t e . . i ϖ t ϖ is the angular frequency, 2 π ν . The inverse Fourier transform of F( ϖ ) gives f(t) back: f( ) t . ( ) . 2 π 1 2 d ϖ . F( ) ϖ e . . i ϖ t For example, if f(t) is a waveform, then F(w) is the contribution of angular frequency w to the waveform. In chemistry, the most common conjugate variable pairs are time/frequency as in this exapmle, and length/wavenumber, which occurs in FTIR. Created: 1997
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## This note was uploaded on 11/07/2009 for the course FOFL 4531 taught by Professor Haiza during the Spring '09 term at Carlos Albizu.

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FTIntro - Exploring Fourier Transform Techniques with...

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