22.02 – Introduction to Applied Nuclear Physics – Spring 2011
Fourier Transform
The Fourier transform is a generalization of the complex Fourier series. The complex Fourier Series is an expansion of a periodic
function (periodic in the interval [−L/2, L/2]) in terms of an infinite sum of complex exponential:
∞
X
An e2iπnx/L
(1)
n=−∞
where the coefficients An are:
An =
1
L
Z
L/2
f (x)e−2iπnx/L dx.
(2)
−L/2
Note that this expansion of a periodic function is equivalent to using the exponential functions un (x) = e2iπnx/L as a basis for the
function vector space of periodic functions. The coefficient of each ”vector” in the basis are given by the coefficient An . Accordingly,
we can interpret equation (2) as the inner product hun (x)|f (x)i.
In the limit as L → ∞ the sum over n becomes an integral. The discrete coefficients An are replaced by the continuous function
F (k)dk where k = n/L. Then in the limit (L → ∞) the equations defining the Fourier series become :
Z ∞
f (x) =
F (k)e2πikx dk
(3)
−∞
F (k) =
Z
∞
f (x)e−2πikx dx.
(4)
−∞
Here,
F (k) = Fx [f (x)](k) =
Z
∞
f (x)e−2πikx dx
−∞
is called the forward Fourier transform, and
f (x) =
Fk−1 [F (k)](x)
=
Z
∞
F (k)e2πikx dk
−∞
is called the inverse Fourier transform.
The notation Fx [f (x)](k) is common but fˆ(k) and f˜(x) are sometimes also used to denote the Fourier transform.
In physics we often write the transform in terms of angular frequency ω = 2πν instead of the oscillation frequency ν (thus for
example we replace 2πk → k). To maintain the symmetry between the forward and inverse transforms, we will then adopt the
convention
Z ∞
1
f (x)e−ikx dx
F (k) = F [f (x)] = √
2π −∞
Z ∞
1
−1
f (x) = F [F (k)] = √
F (k)eikx dk.
2π −∞
Sine-Cosine Fourier Transform
Since any function can be split up into even and odd portions E(x) and O(x),
f (x) =
1
1
[f (x) + f (−x)] + [f (x) − f (−x)] = E(x) + O(x),
2
2
a Fourier transform can always be expressed in terms of the Fourier cosine transform and Fourier sine transform as
Z ∞
Z ∞
Fx [f (x)](k) =
E(x) cos(2πkx)dx − i
O(x) sin(2πkx)dx.
−∞
−∞
1
Properties of the Fourier Transform
• The smoother a function (i.e., the larger the number of continuous derivatives), the more compact its Fourier transform.
• The Fourier transform is linear, since if f(x) and g(x) have Fourier transforms F(k) and G(k) , then
Z
Z ∞
Z ∞
−2πikx
−2πikx
[af (x) + bg(x)]e
dx = a
f (x)e
dx + b
g(x)e−2πikx dx = aF (k) + bG(k).
−∞
−∞
Therefore,
F [af (x) + bg(x)] = aF [f (x)] + bF [g(x)] = aF (k) + bG(k).
• The Fourier transform is also symmetric since F (k) = Fx [f (x)](k) implies F (−k) = Fx [f (−x)](k) .
• The Fourier transform of a derivative f’(x) of a function f(x) is simply related to the transform of the function f(x) itself. Consider
Z ∞
Fx [f ′ (x)](k) =
f ′ (x)e−2πikx dx.
−∞
Now use integration by parts
Z
with
Z
vdu = [uv] −
du = f ′ (x)dx
and
udv
v = e−2πikx
dv = −2πike−2πikx dx,
u = f (x)
then
∞
Fx [f ′ (x)](k) = f (x)e−2πikx −∞ −
Z
∞
−∞
f (x) −2πike−2πikx dx
The first term consists of an oscillating function times f(x) . But if the function is bounded so that
lim
x−→±∞
f (x) = 0
(as any physically significant signal must be), then the term vanishes, leaving
Z ∞
′
Fx [f (x)](k) = 2πik
f (x)e−2πikx dx = 2πikFx [f (x)](k).
−∞
This process can be iterated for the nth derivative to yield
Fx [f (n) (x)](k) = (2πik)n Fx [f (x)](k).
• If f (x) has the Fourier transform Fx [f (x)](k) = F (k) , then the Fourier transform has the shift property
Z ∞
Z ∞
−2πikx
f (x − x0 )e
dx =
f (x − x0 )e−2πi(x−x0 )k e−2πi(kx0 ) d(x − x0 ) = e−2πikx0 F (k),
−∞
−∞
so f (x − x0 ) has the Fourier transform
Fx [f (x − x0 )](k) = e−2πikx0 F (k).
• If f (x) has a Fourier transform Fx [f (x)](k) = F (k), then the Fourier transform obeys a similarity theorem.
Z ∞
Z ∞
f (ax)e−2πikx dx = 1/(|a|)
f (ax)e−2πi(ax)(k/a) d(ax) = 1/(|a|)F (k/a),
−∞
so f (ax) has the Fourier transform
−∞
Fx [f (ax)](k) = |a|−1 F (k/a).
• Any operation on f (x) which leaves its area unchanged leaves F (0) unchanged, since
Z ∞
f (x)dx = Fx [f (x)](0) = F (0).
−∞
2
Table of common Fourier transform pairs.
Function
f (x)
F (k) = Fx [f (x)](k)
Constant
1
δ(k)
δ(x − x0 )
e−2πikx0
Delta function
Cosine
cos(2πk0 x)
1
2 [δ(k
Sine
sin(2πk0 x)
1
2 i[δ(k
+ k0 ) − δ(k − k0 )]
1 k0
2
2
rπ k + k0
π −π2 k2 /a
e
a
e−2πk0 |x|
Exponential function
2
Gaussian
e−ax
Heaviside step function
H(x)
1
[δ(k) − i/(πk)]
2
1
Γ/2
π (x − x0 )2 + (Γ/2)2
Lorentzian function
In two dimensions, the Fourier transform becomes
Z ∞Z
F (x, y ) =
−∞
=
Z
∞
e−2πikx0 e−Γ π|k|
f (kx , ky )e−2πi(kx x+ky y )dkx dky f (kx , ky )
−∞
∞
−∞
Z
∞
F (x, y)e2πi(kx x+ky y )dxdy.
−∞
Similarly, the n -dimensional Fourier transform can be defined for k , x in Rn by
Z ∞ Z ∞
Z ∞ Z
F (x) =
...
f (k)e−2πik·x dn kf (k) =
...
−∞
− k0 ) + δ(k + k0 )]
−∞
−∞
3
∞
−∞
F (x)e2πik·x dn x.
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22.02 Introduction to Applied Nuclear Physics
Spring 2012
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