FTIntro - Exploring Fourier Transform Techniques with...

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Exploring Fourier Transform Techniques with Mathcad Document 1: Introduction to the Fourier Transform by Mark Iannone Department of Chemistry Millersville University Millersville, PA 17551-0302 miannone@marauder.millersv.edu © Copyright 1999 by the Division of Chemical Education, Inc., American Chemical Society. All rights reserved. For classroom use by teachers, one copy per student in the class may be made free of charge. Write to JCE Online, jceonline@chem.wisc.edu, for permission to place a document, free of charge, on a class Intranet. Notes 1. The Automatic Calculation option under the Math menu should NOT be checked. 2. F9 causes Mathcad to calculate graphs and formulas up to your present position in the document. 3. CTRL R refreshes the window in case some part becomes illegible. . Objectives After completing the exercises suggested in this document, the student should be able to 1. sketch the FT of a simple waveform; 2. demonstrate or describe why sampling leads to aliasing; 3. determine the Nyquist frequency and predict aliasing; 4. estimate the resolution from the data acquisition time. Created: 1997 Updated: January 1999 FTIntro.mcd Author: M. Iannone 1
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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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