Example: the Fourier Transform of a
rectangle function: rect(t)
F (ω ) =
1/ 2
∫
exp(−iωt )dt =
−1/ 2
1
[exp(−iωt )]1/−1/2 2
−iω
1
[exp(−iω / 2) − exp(iω/2)]
−iω
1 exp(iω / 2) − exp(−iω/2)
=
(ω/2)
2i
sin(ω/2)
=
(ω/2)
=
F (ω ) = sinc(ω/2)
F(w)
Imaginary
Component = 0
w
Example: the Fourier Transform of a
Gaussian, exp(-at2), is itself!
F {exp(−at 2 )} =
∞
2
exp(
−
at
) exp(−iωt ) dt
∫
−∞
∝ exp(−ω / 4a)
2
The details are a HW problem!
exp( − ω 2 / 4a)
exp( − at 2 )
∩
0
t
0
w
Fourier Series & The Fourier Transform
What is the Fourier Transform?
Fourier Cosine Series for even
functions and Sine Series for odd
functions
The continuous limit: the Fourier
transform (and its inverse)
The spectrum
Some examples and theorems
f (t ) =
1
2π
∞
∞
∫
−∞
F (ω ) exp(iω t ) dω
F (ω ) =
∫
−∞
f (t ) exp(−iω t ) dt
Source: Prof. Rick Trebino, Georgia Tech
What do we hope to achieve with the
Fourier Transform?
Plane waves have only
one frequency, w.
Light electric field
We desire a measure of the frequencies present in a wave. This will
lead to a definition of the term, the spectrum.
Time
This light wave has many
frequencies. And the
frequency increases in
time (from red to blue).
It will be nice if our measure also tells us when each frequency occurs.
Lord Kelvin on Fourier’s theorem
Fourier’s theorem is not
only one of the most
beautiful results of
modern analysis, but it
may be said to furnish an
indispensable instrument
in the treatment of nearly
every recondite question
in modern physics.
Lord Kelvin
Joseph Fourier
Fourier was
obsessed with the
physics of heat and
developed the
Fourier series and
transform to model
heat-flow problems.
Joseph Fourier 1768 - 1830
Anharmonic waves are sums of sinusoids.
Consider the sum of two sine waves (i.e., harmonic
waves) of different frequencies:
The resulting wave is periodic, but not harmonic.
Essentially all waves are anharmonic.
Fourier
decomposing
functions
Here, we write a
square wave as
a sum of sine
waves.
sin(wt)
sin(3wt)
sin(5wt)
Any function can be written as the
sum of an even and an odd function.
E(-x) = E(x)
E ( x) ≡ [ f ( x) + f (− x)] / 2
O(-x) = -O(x)
O( x) ≡ [ f ( x) − f (− x)] / 2
⇓
f ( x) = E ( x) + O( x)
Fourier Cosine Series
Because cos(mt) is an even function (for all m), we can write an even
function, f(t), as:
f(t) =
∞
F
∑
π
1
m
cos(mt)
m =0
where the set {Fm; m = 0, 1, … } is a set of coefficients that define the
series.
And where we’ll only worry about the function f(t) over the interval
(–π,π).
The Kronecker delta function
δ m,n
⎧1 if m = n
≡⎨
⎩0 if m ≠ n
Finding the coefficients, Fm, in a Fourier Cosine Series
Fourier Cosine Series:
f (t ) =
∞
F cos(mt )
∑
π
1
m
m =0
To find Fm, multiply each side by cos(m’t), where m’ is another integer, and integrate:
π
∫
f (t ) cos(m ' t ) dt =
π
∞
F
∑
∫
π
1
m
m=0
−π
cos(mt ) cos(m ' t ) dt
−π
π
⎧π if m = m '
cos(mt ) cos(m ' t ) dt = ⎨
≡ π δ m,m '
⎩ 0 if m ≠ m '
−π
∫
But:
π
So:
∫
−π
1
f (t ) cos(m ' t ) dt =
π
∞
∑F π δ
m
m,m '
ß only the m’ = m term contributes
m=0
Dropping the ’ from the m:
π
Fm =
∫
−π
f (t ) cos(mt ) dt
ß yields the
coefficients for
any f(t)!
Fourier Sine Series
Because sin(mt) is an odd function (for all m), we can write
any odd function, f(t), as:
∑
∞
f (t) =
1
π
Fm′ sin(mt)
m= 0
where the set {F’m; m = 0, 1, … } is a set of coefficients that define
the series.
where we’ll only worry about the function f(t) over the interval (–
π,π).
Finding the coefficients, F’m, in a Fourier Sine Series
f (t ) =
Fourier Sine Series:
∞
F ′ sin(mt )
∑
π
1
m
m =0
To find Fm, multiply each side by sin(m’t), where m’ is another integer, and integrate:
π
∫
f (t ) sin(m ' t ) dt =
π
F ′ sin(mt ) sin(m ' t ) dt
∑
∫
π
1
m
m =0
−π
But:
∞
−π
π
⎧π if m = m '
sin(mt ) sin(m ' t ) dt = ⎨
≡ π δ m,m '
⎩ 0 if m ≠ m '
−π
∫
So:
π
∫ f (t ) sin(m ' t ) dt
=
−π
∞
F′ π δ
∑
π
1
m
m,m '
ß only the m’ = m term contributes
m =0
π
Dropping the ’ from the m:
Fm′
=
∫ f (t ) sin(mt ) dt
−π
ß yields the coefficients
for any f(t)!
Fourier Series
So if f(t) is a general function, neither even nor odd, it can be
written:
f (t ) =
1
∞
F
∑
π
m =0
m
1
∞
F ′ sin(mt )
∑
π
cos(mt ) +
m =0
even component
m
odd component
where
Fm =
∫
f (t) cos(mt) dt and
Fm′ =
∫
f (t) sin(mt) dt
We can plot the coefficients of a Fourier Series
1
Fm vs. m
.5
0
5
10
25
15
20
30
m
We really need two such plots, one for the cosine series and another
for the sine series.
Discrete Fourier Series vs.
Continuous Fourier Transform
Let the integer
m become a
real number
and let the
coefficients,
Fm, become a
function F(m).
Fm vs. m
F(m)
m
Again, we really need two such plots, one for the cosine series and
another for the sine series.
The Fourier Transform
Consider the Fourier coefficients. Let’s define a function F(m) that
incorporates both cosine and sine series coefficients, with the sine
series distinguished by making it the imaginary component:
F(m) º Fm – i F’m =
∫
f (t ) cos(mt ) dt − i
∫
f (t ) sin(mt ) dt
Let’s now allow f(t) to range from –¥ to ¥, so we’ll have to integrate
from –¥ to ¥, and let’s redefine m to be the “frequency,” which we’ll
now call w:
∞
The Fourier
F (ω ) =
f (t ) exp ( −iω t ) dt
Transform
−∞
F(w) is called the Fourier Transform of f(t). It contains equivalent
information to that in f(t). We say that f(t) lives in the time domain,
and F(w) lives in the frequency domain. F(w) is just another way of
looking at a function or wave.
∫
The Inverse Fourier Transform
The Fourier Transform takes us from f(t) to F(w).
How about going back?
Recall our formula for the Fourier Series of f(t) :
f (t ) =
1
π
∞
∑
Fm cos(mt ) +
m =0
1
π
∞
∑
Fm' sin(mt )
m =0
Now transform the sums to integrals from –¥ to ¥, and again replace
Fm with F(w). Remembering the fact that we introduced a factor of i
(and including a factor of 2 that just crops up), we have:
1
f (t ) =
2π
∞
∫
−∞
F (ω ) exp(iω t ) dω
Inverse
Fourier
Transform
The Fourier Transform and its Inverse
The Fourier Transform and its Inverse:
∞
F (ω ) =
∫ f (t ) exp(−iωt ) dt
FourierTransform
−∞
f (t )
=
1
2π
∞
∫ F (ω ) exp(iωt ) dω
Inverse Fourier Transform
−∞
So we can transform to the frequency domain and back.
Interestingly, these transformations are very similar.
There are different definitions of these transforms. The 2π can
occur in several places, but the idea is generally the same.
Fourier Transform Notation
There are several ways to denote the Fourier transform of a
function.
If the function is labeled by a lower-case letter, such as f,
we can write:
f(t) ® F(w)
If the function is already labeled by an upper-case letter, such as E,
we can write:
E (t ) → F {E (t )} or: E (t ) → E%( )
ω
∩
Sometimes, this symbol is
used instead of the arrow:
The Spectrum
We define the spectrum, S(w), of a wave E(t) to be:
S (ω ) ≡ F {E (t )}
2
This is the measure of the frequencies present in a light wave.
Example: the Fourier Transform of a
rectangle function: rect(t)
F (ω ) =
1/ 2
∫
exp(−iωt )dt =
−1/ 2
1
[exp(−iωt )]1/−1/2 2
−iω
1
[exp(−iω / 2) − exp(iω/2)]
−iω
1 exp(iω / 2) − exp(−iω/2)
=
(ω/2)
2i
sin(ω/2)
=
(ω/2)
=
F (ω ) = sinc(ω/2)
F(w)
Imaginary
Component = 0
w
Example: the Fourier Transform of a
Gaussian, exp(-at2), is itself!
F {exp(−at 2 )} =
∞
2
exp(
−
at
) exp(−iωt ) dt
∫
−∞
∝ exp(−ω / 4a)
2
The details are a HW problem!
exp( − ω 2 / 4a)
exp( − at 2 )
∩
0
t
0
w
The Dirac delta function
Unlike the Kronecker delta-function, which is a function of two
integers, the Dirac delta function is a function of a real variable, t.
⎧∞ if t = 0
δ (t ) ≡ ⎨
⎩ 0 if t ≠ 0
d(t)
t
The Dirac delta function
⎧∞ if t = 0
δ (t ) ≡ ⎨
⎩ 0 if t ≠ 0
It’s best to think of the delta function as the limit of a series of
peaked continuous functions.
fm(t) = m exp[-(mt)2]/√p
d(t)
f3(t)
f2(t)
f1(t)
t
Dirac d-function Properties
∞
∫ δ (t ) dt = 1
t
−∞
∞
∞
−∞
−∞
∫ δ (t − a) f (t ) dt = ∫ δ (t − a) f (a) dt = f (a)
∞
∫ exp(±iωt ) dt = 2π δ (ω )
−∞
∞
∫ exp[±i(ω − ω ′)t ]
−∞
d(t)
dt = 2π δ (ω − ω ′)
The Fourier Transform of d(t) is 1.
∞
∫ δ (t ) exp(−iωt ) dt = exp(−iω [0]) = 1
−∞
d(t)
1
w
t
0
∞
And the Fourier Transform of 1 is 2pd(w):
∫ 1 exp(−iωt ) dt = 2π δ (ω )
−∞
2pd(w)
1
t
0
w
The Fourier transform of exp(iw0 t)
F
{exp(iω0 t )}
∞
=
∫
∞
=
∫
exp(iω0 t ) exp(−i ω t ) dt
−∞
exp(−i [ω − ω0 ] t ) dt = 2π δ (ω − ω0 )
−∞
exp(iw0t)
Im
Re
0
0
F {exp(iw0t)}
t
t
0
w0 w The function exp(iw0t) is the essential component of Fourier analysis.
It is a pure frequency.
The Fourier transform of cos(w0 t)
F
{cos(ω0t )}
1
=
2
1
=
2
∞
∞
∫
∞
=
∫ cos(ω t ) exp(−i ω t ) dt
0
−∞
[exp(i ω0 t ) + exp(−i ω0 t )] exp(−i ω t ) dt
−∞
∫ exp(−i [ω − ω ]t ) dt
0
−∞
+
1
2
∞
∫ exp(−i [ω + ω ] t ) dt
0
−∞
= π δ (ω − ω0 ) + π δ (ω + ω0 )
F {cos(ω0t )}
cos(w0t)
0
t
-w0 0 +w0 w Fourier Transform Symmetry Properties
Expanding the Fourier transform of a function, f(t):
∞
∫ [Re{ f (t)} + i Im{ f (t)}] [cos(ωt) − i sin(ωt)] dt
F (ω ) =
−∞
Expanding more, noting that:
∞
∫ O(t ) dt = 0
if O(t) is an odd function
−∞
∞
∫
F (ω ) =
−∞
∞
+ i
∫
−∞
= 0 if Re{f(t)} is odd
¯
Re{ f (t )} cos(ω t ) dt +
= 0 if Im{f(t)} is odd
¯
Im{ f (t )} cos(ω t ) dt − i
­
Even functions of w
∞
∫
−∞
∞
∫
= 0 if Im{f(t)} is even
¯
Im{ f (t )} sin(ω t) dt
= 0 if Re{f(t)} is even
¯
Re{ f (t )} sin(ω t) dt
­
Odd functions of w
−∞
¬Re{F(w)}
¬Im{F(w)}
The Modulation Theorem:
The Fourier Transform of E(t) cos(w0 t)
F
{E (t ) cos(ω0t )}
1
=
2
∞
∞
=
∫
E (t ) cos(ω0t ) exp(−i ω t ) dt
−∞
∫ E (t ) ⎡⎣exp(i ω t ) + exp(−i ω t )⎤⎦ exp(−i ω t ) dt
1
1
=
E (t ) exp(−i [ω − ω ] t ) dt +
E (t ) exp(−i [ω + ω ] t ) dt
2∫
2∫
∞
0
−∞
0
∞
0
0
−∞
−∞
1 %
1 %
F {E (t ) cos(ω0t )} = E (ω − ω0 ) +
E (ω + ω0 )
2
2
Example:
E (t ) cos(ω0t )
F
{E(t ) cos(ω0t )}
E(t) = exp(-t2)
t
-w0
0
w0
w
Scale Theorem
F { f (at )} = F (ω /a) / a
The Fourier transform
of a scaled function, f(at):
∞
Proof:
∫
F { f (at )} =
f (at ) exp( −iω t ) dt
−∞
Assuming a > 0, change variables: u = at
∞
F { f (at )} =
∫
f (u ) exp(−iω [ u /a]) du / a
−∞
∞
=
∫ f (u) exp(−i [ω /a] u) du / a
−∞
= F (ω /a) / a
If a < 0, the limits flip when we change variables, introducing a
minus sign, hence the absolute value.
F(w)
f(t)
The Scale
Theorem
in action
The shorter
the pulse,
the broader
the spectrum!
This is the essence
of the Uncertainty
Principle!
Short
pulse
t
w
t
w
t
w
Mediumlength
pulse
Long
pulse
The Fourier
Transform of a
sum of two
functions
f(t)
F(w)
g(t)
F {a f (t ) + b g (t )} =
aF { f (t )} + bF {g (t )}
G(w)
t
f(t)+g(t)
Also, constants factor out.
w
t
t
w
F(w) +
G(w)
w
Shift Theorem
The Fourier transform of a shifted function, f (t − a) :
F
{ f (t − a)} = exp(−iωa)F (ω)
Proof :
∞
F
{ f (t − a )} = ∫
f (t − a ) exp(−iωt )dt
−∞
Change variables : u = t − a
∞
∫
f (u ) exp(−iω[u + a ])du
−∞
∞
= exp(−iω a ) ∫ f (u ) exp(−iωu )du
−∞
= exp(−iω a ) F (ω )
Fourier Transform with respect to space
If f(x) is a function of position,
∞
F (k ) = ∫ f ( x) exp(−ikx) dx
−∞
x
F {f(x)} = F(k)
We refer to k as the spatial frequency.
k
Everything we’ve said about Fourier transforms between the t and w
domains also applies to the x and k domains.
The 2D Fourier Transform
F
(2){f(x,y)}
f(x,y)
= F(kx,ky)
=
∫∫
If
f(x,y) = fx(x) fy(y),
f(x,y) exp[-i(kxx+kyy)] dx dy
then the 2D FT splits into two 1D FT's.
But this doesn’t always happen.
x
F
y
(2){f(x,y)}
The Pulse Width
Dt
There are many definitions of the
"width" or “length” of a wave or pulse.
t
The effective width is the width of a rectangle whose height and
area are the same as those of the pulse.
f(0)
Effective width ≡ Area / height:
Δteff
∞
1
≡
f (t ) dt
∫
f (0) −∞
Dteff
(Abs value is
unnecessary
for intensity.)
Advantage: It’s easy to understand.
Disadvantages: The Abs value is inconvenient.
We must integrate to ± ∞.
0
t
The Uncertainty Principle
The Uncertainty Principle says that the product of a function's widths
in the time domain (Δt ) and the frequency domain (Δω) has a minimum.
Define the widths
assuming f(t) and
F(w) peak at 0:
∞
∞
1
Δt ≡
f (t ) dt
∫
f (0) −∞
∞
1
Δω ≡
F (ω ) dω
∫
F (0) −∞
∞
1
1
F (0)
Δt ≥
f (t ) dt =
f (t ) exp(−i[0] t ) dt =
∫
∫
f (0) −∞
f (0) −∞
f (0)
∞
∞
1
1
2π f (0)
Δω ≥
F
(
ω
)
d
ω
=
F
(
ω
)
exp(
i
ω
[0])
d
ω
=
∫
∫
F (0) −∞
F (0) −∞
F (0)
Combining results:
f (0) F (0)
Δω Δt ≥ 2π
F (0) f (0)
(Different definitions of the widths and the
Fourier Transform yield different constants.)
or:
Δω Δt ≥ 2π
Δν Δt ≥ 1