Page 500 - Advanced engineering mathematics

480 CHAPTER 14 The Fourier Integral and Transforms

Convolution has the following properties.

Commutativity If f ∗ g is defined, so is g ∗ f and

f ∗ g = g ∗ f.

Linearity This means that, for numbers α and β and functions f , g and h,
(αf + βg) ∗ h = α( f ∗ h) + β(g ∗ h)

provided that all these convolutions are defined.
For the next three properties of convolution, suppose that f and g are bounded and
continuous on the real line and that f and g are both absolutely integrable. Then
∞ ∞ ∞

( f ∗ g)(t)dt = f (t)dt g(t)dt.
−∞ −∞ −∞
Time Convolution
F[ f ∗ g]= f ˆg.
ˆ
This says that the Fourier transform of the convolution of two functions is the product of the
Fourier transforms of the functions. This is known as the convolution theorem, and a similar
result holds for the Laplace transform. The ramification of convolution for the inverse Fourier
transform is that
−1
ˆ
F [ f (ω) ˆg(ω)](t) = ( f ∗ g)(t).
That is, the inverse Fourier transform of a product of two transformed functions is the convolution
of the functions.

Frequency Convolution

1
ˆ
F[ fg](ω) = ( f ∗ˆg)(ω).
2π
EXAMPLE 14.10
We will compute

1
−1
F .
2
2
(4 + ω )(9 + ω )
We want the inverse transform of a product, knowing the inverse of each factor:

1 1
−1 −2|t|
F = f (t) = e
4 + ω 2 4
and

1 1
−1 −3|t|
F = g(t) = e .
9 + ω 2 6
The inverse version of the convolution theorem tells us that
∞
1 1
−1
e
F (t) = ( f ∗ g)(t) = e −2|t−τ| −3|τ| dτ.
(4 + ω )(9 + ω ) 24
2
2
−∞
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October 14, 2010 16:43 THM/NEIL Page-480 27410_14_ch14_p465-504