Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Fourier series
Fourier series of a function f (t) with period T and corresponding
angular frequency ω = 2π/T :
∞
a0 X
f (t) =
+
(an cos(nωt) + bn sin(nωt)) ,
2
n=1
Fourier series is a linear sum of cosine and sine functions with
discrete frequencies that are integer multiples of the
frequency of f (t).
This gives rise to a discrete frequency spectrum given by
the Fourier coefficients (=frequency amplitudes).
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Complex Fourier series
Complex representation of Fourier series of a function f (t) with
period T and corresponding angular frequency ω = 2π/T :
f (t) =
∞
X
cn e inωt , where
n=−∞

 (an − ibn )/2, n > 0,
a0 /2
n = 0,
cn =

(a|n| + ib|n| )/2 n < 0
Note that the summation goes from −∞ to ∞.
Now have a negative frequencies as well.
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Positive and negative frequencies
Positive frequencies (n > 0): e inωt = cos(nωt) + i sin(nωt)
Negative frequencies (n < 0): e −i|n|ωt = cos(|n|ωt) − i sin(|n|ωt)
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Fourier series demonstration
Applet for Fourier Series:
http://www.westga.edu/~jhasbun/osp/Fourier.htm
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
The Fourier transform
For non-periodic (or periodic) functions f (t), we can define the
Fourier transform
Z ∞
1
Fourier transform F[f (t)] = g (ω) = √
f (t)e −iωt dt
2π −∞
Z ∞
1
−1
inverse transform F [g (ω)] = f (t) = √
g (ω)e iωt dω,
2π −∞
where g (ω) corresponds to a continuous frequency spectrum.
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Properties of the Fourier transform
Linearity
Shifting
Scaling
Power spectrum
Parseval’s theorem
δ-function
F[f (t) + g (t)] = F[f (t)] + F[g (t)]
F[f (t − t0 )] = e −iωt0 g (ω)
F−1 [g (ω − ω0 )] = e iω0 t f (t)
1
F[f (αt)] = |α|
g (ω/α)
1
−1
F [g (βt)] = |β| f (t/β)
F[f (t)] (F[f (t)])∗ = |F[f (t − t0 )]|2 = |g (ω)|2
cannot reconstruct f(t) from power spectrum
since phase information is lost.
(t −Rt0 )]|2 = |F[f (t)]|2
R ∞Note: |F[f
∞
2
(t)| dt = −∞ |g (ω)|2 dω
−∞ |f
| {z }
| {z }
Power
Power
R ∞ ixt spectrum
1
δ(x) = 2π
−∞ e dt
δ(−x) = δ(x)
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Fourier Transform - Symmetry properties
The Fourier transform obeys certain symmetries. Consider the
Fourier transform and its complex conjugate:
Z ∞
1
g (ω) = √
f (t)e −iωt dt
2π −∞
Z ∞
1
∗
g (ω) = √
f ∗ (t)e iωt dt
2π −∞
If f (t) is real (f ∗ (t) = f (t)), then
Z ∞
1
∗
g (ω) = √
f (t)e −i(−ω)t dt = g (−ω)
2π −∞
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Fourier Transform - Symmetry properties
g ∗ (ω) = g (−ω) means the real part of the transform Re(g (ω)) is
even, while Im(g (ω)) is odd:
g (ω) = Re(g (ω)) + iIm(g (ω))
g (−ω) = Re(g (−ω)) + iIm(g (−ω))
g ∗ (ω) = Re(g (ω)) − iIm(g (ω))
Conversely, if f (t) is purely imaginary, then
Z ∞
1
∗
√
g (ω) = −
f (t)e −i(−ω)t dt = −g (−ω)
2π −∞
Hence, g (−ω) = −g ∗ (ω), which means that Re(g (ω)) is odd,
while Im(g (ω)) is even.
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Fourier Transform - Symmetry properties
Let f (t) be even: f (−t) = f (t), then can use scaling property to
show:
F[f (−t)] = F[f ((−1)t)] =
1
g (ω/(−1))
| − 1|
= g (−ω)
But,
F[f (−t)] = F[f (t)] = g (ω)
Therefore, g (ω) = g (−ω). The transform is even too. Similarly,
can show that if f (t) is odd, then g (ω) is odd as well.
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Fourier Transform - Symmetry properties
The symmetry properties of the Fourier transform can be
summarized as follows:
f (t)
f (t)
f (t)
f (t)
f (t)
f (t)
f (t)
f (t)
real
imaginary
even
odd
real and even
real and odd
imaginary and even
imaginary and odd
Re(g (ω)) even and Im(g (ω)) odd
Re(g (ω)) odd and Im(g (ω)) even
g (ω) even
g (ω) odd
g (ω) real and even
g (ω) imaginary and odd
g (ω) imaginary and even
g (ω) real and odd
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Fourier Series vs. Fourier transform
Let’s compare Fourier series, complex representation of Fourier
series and Fourier transform:
Consider f (t) = sin(ωt), where ω = 2π/T .
What are its real Fourier coefficients?
∞
f (t) =
a0 X
+
(an cos(nωt) + bn sin(nωt)) ,
2
n=1
By inspection, an = 0, b1 = 1. bn = 0 for n 6= 1.
(1)
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
More formally, substitute f (x) into the equations below (lecture 3,
eqns. (15,16))
an =
bn =
2
T
2
T
Z
T /2
f (t) cos(nωt)dt
(2)
f (t) sin(nωt)dt
(3)
−T /2
Z T /2
−T /2
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Therefore,
an =
2
T
Z
T /2
sin(ωt) cos(nωt) dt
{z } | {z }
−T /2 |
odd
even
= 0
(5)
Z
bn =
=
(4)
T /2
2
sin(ωt) sin(nωt)dt
T −T /2
Z
2 1 π
sin(x) sin(nx)dx ,
T ω −π
|
{z
}
(6)
(7)
πδ1n
=
2 T
πδ1n
T 2π
where x = ωt. Therefore, bn = δ1n .
(8)
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Now let’s find the Fourier coeffcients of f (t) = sin(ωt) for the
complex representation of the Fourier series:
∞
X
f (t) =
cn e inωt ,
(9)
n=−∞
where,
1
cn =
T
Z
T /2
f (t)e −inωt dt
Substituting f (t), we obtain
Z
1 T /2
cn =
sin(ωt)(cos(nωt) − i sin(nωt))dt
T −T /2

=
1
T
(10)
−T /2

Z T /2
Z T /2



sin(ωt) cos(nωt)dt −i
sin(ωt) sin(nωt))dt 

 −T /2

−T /2
|
{z
}
=0
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
cn
i
= −
2π
Z
−π
|
=
π
− 2i
+ 2i
sin(x) sin(nx)) dx
{z
}
πδ1n
n=1
n = −1
Hence, c1 = −i/2 and c−1 = i/2, all other cn are zero.
Note: A single harmonic (sin or cos) is represented by two Fourier
coefficients in the complex Fourier series.
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Now consider the Fourier transform of f (t) = sin(Ωt). Rewrite as
iΩt
−iΩt
f (t) = e −e
:
2i
Z ∞
1
g (ω) = √
f (t)e −iωt dt
2π −∞
Z ∞ iΩt
1
e − e −iΩt −iωt
= √
e
dt
2i
2π −∞


=
2π
√
2i 2π
Z ∞
 1 Z ∞

1

i(Ω−ω)t
−i(Ω+ω)t 
e
dt −
e
dt 

2π −∞

 2π −∞
|
{z
} |
{z
}
√
=
δ(Ω−ω)
i 2π
(δ(Ω + ω) − δ(Ω − ω))
2
Note that δ(x) = δ(−x) .
δ(−(Ω+ω))=δ(Ω+ω)
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Therefore,
√
i 2π
(δ(Ω + ω) − δ(Ω − ω))
F[sin(Ωt)] =
2
yields two δ-function peaks at ω = ±Ω with imaginary amplitudes.
This is analogous to the two complex Fourier coefficients we
obtained earlier.
Similarly, one can obtain
√
F[cos(Ωt)] =
2π
(δ(Ω + ω) + δ(Ω − ω))
2
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
What is the Fourier transform of a constant function f (t) = C ?
Z ∞
1
g (ω) = √
f (t)e −iωt dt
2π −∞
Z ∞
1
Ce −iωt dt
= √
2π −∞
Z ∞
2πC 1
= √
e −iωt dt
2π
2π
−∞
|
{z
}
√
=δ(−ω)=δ(ω)
= C 2πδ(ω)
δ-function peak at ω = 0. Zero frequency mode.
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Fourier transform of periodic functions
Fourier transform can operate on non-periodic, but also periodic
functions, which we can express in terms of Fourier series. Let f (t)
be a periodic function with angular frequency Ω:
∞
f (t) =
a0 X
+
(an cos(nΩt) + bn sin(nΩt)) ,
2
n=1
Therefore,
#
∞
a0 X
+
(an cos(nΩt) + bn sin(nΩt)) ,
F[f (t)] = F
2
"
n=1
Now use linearity of transform
F[f (t)] = F[a0 /2] +
∞
X
n=1
(an F[cos(nΩt)] + bn F[sin(nΩt)])
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Fourier transform of periodic functions
F[f (t)]
√
a0 2π
=
δ(ω)
2
√
∞
2π X
an (δ(nΩ + ω) + δ(nΩ − ω))
+
2
n=1
√
∞
i 2π X
+
bn (δ(nΩ + ω) − δ(nΩ − ω))
2
n=1
Therefore, Fourier transforms of periodic functions yield a sum of
δ-functions located at integer multiples of the frequency Ω.
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Gaussian peak:
f (t) = e −αt
2
F[f (t)] =
=
=
=
Z ∞
1
2
√
e −αt e −iωt dt
2π −∞
Z ∞
iω 2
iω 2
1
t+( 2α
−α t 2 + iω
) −( 2α
)
α
√
dt
e
2π −∞
Z ∞
iω 2
iω 2
1
√
e −α(t+ 2α ) e α( 2α ) dt
2π −∞
Z ∞
ω2
1
2
√ e − 4α
e −αx dx
2π
| −∞ {z
}
√
=
=
ω2
1
√ e − 4α
2α
π/α
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Table of common Fourier transforms
f (t)
δ(t)
constant C
sin(Ωt)
cos(Ωt)
2
Gaussian (α > 0): e −αt
Exponential (α > 0): e −α|x|
g (ω)
√1
√ 2π
C 2πδω
√
i 2π δ(ω+Ω)−δ(ω−Ω)
2
√ δ(ω+Ω)+δ(ω−Ω)
2π
2
2
√1 e −ω /(4α)
2α
q
α
2
π α2 +ω 2