10 34 Numerical Methods Applied to Chemical Engineering Professor William H Green Lecture 34 Fourier Transforms and Fast Fourier Transforms FFT Fourier Analysis Transforms f x c m m x m sin mx 2 m m m x 2 m cos mx m Basis Set Methods m e imx Convolution Integral g h g h t d F g h F g F h Correlation g h g t h d F Correlation g h F g F h complex conjugate Quantum Mechanics x eikx state of definite momentum px k eim definite angular momentum J m Spectroscopy E h pulsed NMR time domain measurement of I FTIR I l Figure 1 Diagram showing the path of light in an FTIR Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY These methods are powerful but require a computer to interpret the results Scattering Experiments X ray stattering neutron scattering light scattering Fourier Series if f t f t 2P f t an P half period M 1 m t m t ao a m cos bm sin 2 P P m 1 1 2P n t f t cos dt P 0 P bn 1 2P n t f t sin dt P 0 P O M 2 effort Euler s Formula ei cos i sin f t cm e im t P m 1 F 2 1 f t f t e 2 cn i t 1 P f t e 2 P P dt F e i t d in t P dt If you do not know P compute Fourier Transform instead of Fourier Series Plot F 2 vs where F is power density Inverse Fourier Transform Discrete Fourier Transform f tk tk 0 T f k 1 t k 1 N evenly spaced time points N Fn f k 1 t e i 2 n 1 k 1 N k 1 n 1 2 Fn F T 1 N f t Fn e i 2 n 1 t T N n 1 O N 2 effort 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 34 Page 2 of 3 Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY Fast Fourier Transform FFT N 2k Fn N N f k 1 t e i 2 n 1 k 1 N f k 1 t e i 2 n 1 k 1 N k even Fn e i 2 n 1 N FnN e k odd N 2 i 2 n 1 N l 1 f 2l 1 t e i 2 n 1 l 1 N 2 N 2 N 2 N 2 f 2l 2 t e i 2 n 1 l 1 N 2 l 1 Fn offset Fn Can do this iteratively One can split each FnM into even and odd series O N log 2 N effort This is much less than O N2 That is why this transform is called Fast MatLAB fft and ifft 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 34 Page 3 of 3 Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY
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