Fourier Transforms and
Convolutions
By: 1/C Nick Culver
Who Was Fourier??
 Noticeably
gifted by age 14
 Priest or Mathematician?
 Math it is
 Taught at the Collège de
France
 Joined Napoleon’s Army
 OIC of discoveries in Egypt
 Poisson and Biot
What is Fourier Analysis?
 Fourier
analysis allows a system to
be separated or decomposed into
components made up of simpler
inputs.
 For example, a function f(t) is a
function in time but via a Fourier
transform becomes f(w) , where
omega is a frequency
Convolution:
 Steward
describes convolution as
“the distribution of one function in
accordance with the law specified by
another function (85).”
Convolution cont.
 Overlaps
that are a result of
spreading and smearing of a function
Definition of Convolution
Definition of convolution and theorem
If f and g are two functions, with f 1 and g
1
finite, the convolution of f and g is
denoted b y f ∗g and is defined b y:
f ∗ g()
x =
−
f(
s)g(
x − s)
ds pg 173
PROOF (
The Convolution Theorem and its Applications)
∗ (()
f x = g()
x ) ( f−()(
x g u x)dx
−
=
−
−
f ()(
x g u − x)dx e2π isu du
Let w = u − x
∗ (()
f x = g()
x)
−
−
f ()()
x g w e2π is(x +w)dx dw
Convolution Theorem
Convolution Theorem : The Fourier Transform of the convolution of two functions f and g is the
multiplication of the Fourier transform of f with the Fourier transform of g.
ﾈ
f ∗ g is ()
fﾈ ()
g
Convolution Theorem (
I nverse): The inverse Fourier transform of fﾈgﾈ is the convolution of f ∗ g
PROOF
Using the inverse Fourier Transform:
ﾈ ()
= −1 (
fﾈg)
x
=
−
[
−
−2iπ ux
−
fﾈ()()
u gﾈ u e
du
f(
s)e−2iπ us ds] gﾈ()
u e−2iπ us du
Now interchange the integrals and add the exponents: pg.177
ﾈ ()
= −1 (
fﾈg)
x
−
f(
s)
[ gﾈ()
u e−2iπ u (x−s)du ] ds
−
This is the integral of gﾈ()
u , with the variable (
x-s)so we get g(
x-s)for that integral.
ﾈ ()
Thus, = −1 (
∗fﾈg)
x f g
Change the sum exponentials to a product of exponentials
=
=
−
−
−
f ()
x e2π isx g()
w e2π iswdx dw
f ()
x e2π isx dx
-
g()
w e2π iswdw
w is a dummy variab le, so replace w with x
∗ (()
f x = g()
x)
−
f ()
x e2π isx dx
Thus, ∗ (()
f x = g()
x)
-
g()
x e2π isx dw
(()
f x )=(()
g x ) ()
fﾈ ()
gﾈ
Parseval’s Equality
 Energy
Conservation Statement
Theorem : (
Walker 102)
nπ x
We define ￥a n sin(
as the Fourier sine series for the function g,
L )
n= 1
(
finite)
, if and only if ￥a n
where g <
2
is convergent.Thus introducing
n= 1
Parseval'sequality :
L
2
￥a
n= 1
2
n
=
L
0
g(
x)dx = g
2
2
In physics, the relation is normally written as:
Given: f (
x)=
1
2π
−
g(
x)=
1
2π
−
fﾈ(
x)eikx dx and fﾈ(
x)=
1
2π
−
ﾈ x)=
gﾈ(
x)eimx dx and g(
1
2π
−
Then Parseval'sequality becomes:
−
g(
x)
=f(
x)dx
−
f(
x)e−ikx dx (
Prof.Tankersley)
g(
x)e−imx dx
gﾈ(
k) fﾈ(
k)dk (
where g(
x)and f(
x)is the comple conjugate)