Fourier Analysis / Transforms Quantum Mechanics Spectroscopy Scattering Experiments Fourier Series Euler’s Formula Discrete Fourier Transform Fast Fourier Transform (FFT)10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #34: Fourier Transforms and Fast Fourier Transforms (FFT). Fourier Analysis / Transforms Basis Set Methods ()imxmmmmmmmmmemxxmxxcxf===∂∂==∑φφφλφφφ )cos( )sin()(22 Convolution Integral ∫∞∞−−≡τττdthghg )()(* )(ˆ)(ˆ)*(ˆhFgFhgF = Correlation (g,h) ∫∞∞−+≡τττdhtg )()(()[*)(ˆ)(ˆ),(nCorrelatioˆhFgFhgF =] Å complex conjugate Quantum Mechanics Ψ(x) = eikx Å state of definite momentum px = ħk Ψ(φ) = eimΦ Å definite angular momentum JΦ = ħm Spectroscopy ν = ΔE/h pulsed NMR: time domain measurement of I(ν) FTIR: Il 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) % P = half-period effort )(sin)(1cos)(1sincos21)(220201MOdtPtntfPbdtPtntfPaPtmbPtmaatfPnPnMmmmo∫∫∑⎟⎠⎞⎜⎝⎛=⎟⎠⎞⎜⎝⎛=⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛+==ππππ Euler’s Formula eiθ = cos(θ) + i sin(θ) If you do not know P, compute Fourier Transform instead of Fourier Series ∫∞∞−= dtetfFtiωπω)(21)( Plot |F(ω)|2 vs. ω, where F(ω) is power density ∫∑−∞−∞===PPPtinnmPtimmdtetfPcectfππ)(21)( ∫∞∞=ωωπωdeFtfti)(21)( (Inverse Fourier Transform) Discrete Fourier Transform f(tk) tk = 0 … T f((k-1)Δt) k = 1 … N evenly spaced time points ()effort )(1)(2)1()1(21/)1(21/)1)(1(2NOeFNtfTnFFetkfFNnTtninnNkNknin∑∑=−=−−−≈⎟⎠⎞⎜⎝⎛−≈Δ−=πππ 10.34, Numerical Methods Applied to Chemical Engineering Lecture 34 Prof. William Green 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() ()() ( )()()2/n2/offset /)1(22/1)2//()1)(1(22/1)2//()1)(1(2/)1(2even odd /)1)(1(2/)1)(1(2)22()12()1()1(NNnNniNnNlNlniNlNlniNninNkNkNkniNkninFFeFetlfetlfeFetkfetkfF+=Δ−+Δ−=Δ−+Δ−=−=−−=−−−−−−−∑∑∑∑ππππππ Can do this iteratively. One can split each into even and odd series. MnF()NNO2log 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 Lecture 34 Prof. William Green 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
View Full Document
Unlocking...