Page 460 - Numerical Methods for Chemical Engineering

Convolution and correlation 449

Introducing the change of variable t → s = t − τ, e −iωt = e −iωs e −iωτ , dt = ds,
 
+∞ +∞
1 '  1 ' 
∗ −iωs −iωτ
[G F](ω) = √ g(τ) √ f (s)e ds e dτ
2π  2π 
−∞ −∞
   
+∞ +∞
1 1
 '   ' 
∗ −iωs −iωτ
[G F](ω) = √ f (s)e ds √ g(τ)e dτ
 2π   2π 
−∞ −∞
∗
[G F](ω) = G(ω)F(ω) (9.73)
QED
Correlation of two signals

A similar operation to convolution is the correlation of two functions,
+∞
1 '
C g, f (t) = √ g(τ + t) f (τ)dτ (9.74)
2π
−∞
which measures how similar the signal g is to the signal f after a lag time t. The autocor-
relation of a signal g is its correlation to itself, C g,g (t). Like convolution, the correlation
of two signals can be computed efficiently by FFT methods. The Fourier transform of the
correlation function is

+∞
1 ' −iωt
C g, f (ω) = √ C g, f (t)e dt (9.75)
2π
−∞
Substituting for C g , f (t) and switching the integration order, we have
 
+∞ ' +∞
'
1  1 −iωt 
C g, f (ω) = √ √ g(τ + t) f (τ)e dt dτ (9.76)
2π  2π 
−∞ −∞
Defining s = τ + t, we can write dt = ds in the inner integral, where τ is fixed, and using
e −iωt = e −iω(s−τ) = e −iωs e iωτ , we have
 
+∞ ' +∞
1  1 −iωs  iωτ
'
C g, f (ω) = √ √ g(s) f (τ)e ds e dτ (9.77)
2π  2π 
−∞ −∞
Thus,
   
+∞ +∞
1 −iωs 1 iωτ
 '   ' 
C g, f (ω) = √ g(s)e ds √ f (τ)e dτ (9.78)
 2π   2π 
−∞ −∞
and we have the simple result
C g, f (ω) = G(ω)F(−ω) (9.79)