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 mφ m ( x )
φ m = sin(mx)
∂2
φ m = λ mφ m
∂x 2
φ m = cos(mx)
m
Basis Set Methods
φ m = e imx
Convolution Integral
∞
g *h ≡
∫ g (τ )h(t − τ )dτ
−∞
Fˆ ( g * h) = Fˆ ( g ) Fˆ ( h)
∞
Correlation (g,h) ≡
∫ g (t + τ )h(τ )dτ
−∞
[
]
*
Fˆ (Correlation ( g , h) ) = Fˆ ( g ) Fˆ ( h)
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:
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
⎛ mπt ⎞
⎛ mπt ⎞⎤
ao + ∑ ⎢a m cos⎜
⎟ + bm sin ⎜
⎟⎥
2
⎝ P ⎠
⎝ P ⎠⎦
m =1 ⎣
1 2P
⎛ nπt ⎞
f (t ) cos⎜
⎟dt
∫
P 0
⎝ P ⎠
bn =
1 2P
⎛ nπt ⎞
f (t ) sin ⎜
⎟dt
∫
P 0
⎝ P ⎠
O( M 2 ) effort
Euler’s Formula
eiθ = cos(θ) + i sin(θ)
f (t ) =
∞
∑ cm e
imπt
P
m = −∞
1
F (ω ) =
2π
1
f (t ) =
∞
∫∞ f (t )e
2π
∫
∞
∞
cn =
−iωt
1 P
f (t )e
2 P ∫− P
dt
F (ω )e iωt dω
inπt
P
dt
If you do not know P, compute Fourier
Transform instead of Fourier Series
Plot |F(ω)|2 vs. ω, where F(ω) is power density
(Inverse Fourier Transform)
Discrete Fourier Transform
f(tk)
tk = 0 … T
f((k-1)Δt)
k = 1 … N evenly spaced time points
N
Fn = ∑ f ((k − 1)Δt )e −i 2π ( n −1)( k −1) / N
k =1
⎛ (n − 1)2π ⎞
Fn ≈ F ⎜
⎟
T
⎝
⎠
1 N
f (t ) ≈ ∑ Fn e i 2π ( 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)Δt )e i 2π ( n−1)(k −1) / N + ∑ f ((k − 1)Δt )e i 2π ( n−1)(k −1) / N
k even
Fn = e i 2π ( n−1) / N
FnN
=e
k odd
N /2
∑
i 2π ( n −1) / N
l =1
f ((2l − 1)Δt )e i 2π ( n −1)(l −1) /( N / 2) +
(N / 2)
(N / 2)
N /2
∑ f ((2l − 2)Δt )e i 2π (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].