Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Notes
Fourier series
Fourier series of a periodic function f (t) with period T and
corresponding angular frequency ω = 2π/T :
∞
f (t) =
a0 X
+
(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
Notes
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=−∞
cn =

 (an − ibn )/2, n > 0,
a0 /2
n = 0,

(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
Notes
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
Notes
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Notes
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
inverse transform F−1 [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
Notes
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
F−1 [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
|f
(t)|
dt
= −∞ |g (ω)|2 dω
−∞ | {z }
| {z }
Power
spectrum
R ∞Power
1
δ(x) = 2π −∞ e ixt 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
Notes
Fourier transform of a derivative
Fourier transform can be used to solve differential equations.
Consider
Z +∞
1
F[f 0 (t)] = √
f 0 (t)e −iωt dt
2π −∞
(1)
where f 0 (t) = df /dt. Integrating by parts, we obtain
e −iωt
iω
F[f 0 (t)] = √ f (t)|∞
−∞ + √
2π
2π
Z
+∞
f (t)e −iωt dt
(2)
−∞
Since f (t) vanishes as t → ±∞, the first term vanishes and we
have
F[f 0 (t)] = iωF[f (t)]
(3)
Can show that for nth derivative F[f n (t)] = (iω)n F[f (t)]
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 −∞
Notes
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Notes
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
Notes
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
Notes
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
Notes
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.
(4)
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Notes
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 ,
(5)
n=−∞
where,
T /2
Z
1
T
cn =
f (t)e −inωt dt
(6)
−T /2
Substituting f (t), we obtain
Z
1 T /2
cn =
sin(ωt)(cos(nωt) − i sin(nωt))dt
T −T /2

=

Z T /2
Z T /2

1 


sin(ωt) cos(nωt)dt −i
sin(ωt) sin(nωt))dt 

T  −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
|
Notes
− 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
Notes
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 −∞


Z ∞
 1 Z ∞

1


e i(Ω−ω)t dt −
e −i(Ω+ω)t dt 

2π −∞
 2π −∞

|
{z
} |
{z
}
=
2π
√
2i 2π
=
√
i 2π
(δ(Ω + ω) − δ(Ω − ω))
2
δ(Ω−ω)
δ(−(Ω+ω))=δ(Ω+ω)
Note that δ(x) = δ(−x) .
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Therefore,
F[sin(Ωt)] =
√
i 2π
(δ(Ω + ω) − δ(Ω − ω))
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
Notes
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Notes
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
Notes
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,
"
F[f (t)] = F
#
∞
a0 X
+
(an cos(nΩt) + bn sin(nΩt)) ,
2
n=1
Now use linearity of transform
F[f (t)] = F[a0 /2] +
∞
X
(an F[cos(nΩt)] + bn F[sin(nΩt)])
n=1
Fourier Series and Transform - summary
Fourier Transform - Symmetry properties
Fourier Series and Transform - Comparison
Fourier Transform example - non-periodic function
Notes
Fourier transform of periodic functions
√
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
F[f (t)]
=
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
Notes
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
−α t 2 + iω
t+( 2α
) −( 2α
)
α
√
e
dt
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
Notes
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|t|
g (ω)
√1
2π
√
C 2πδ(ω)
√
i 2π δ(ω+Ω)−δ(ω−Ω)
2
√ δ(ω+Ω)+δ(ω−Ω)
2π
2
2
√1 e −ω /(4α)
2α
q
2
α
π α2 +ω 2
Notes
Notes
Notes