Chapter 4
The Fourier Series and
Fourier Transform
Fourier Series Representation of
Periodic Signals
• Let x(t) be a CT periodic signal with period
T, i.e., x(t + T ) = x(t ), ∀t ∈ R
• Example: the rectangular pulse train
The Fourier Series
• Then, x(t) can be expressed as
x(t ) =
∞
∑ce
k =−∞
k
jkω 0 t
, t∈
where ω 0 = 2π / T is the fundamental
frequency (rad/sec) of the signal and
T /2
1
− jkω o t
(
)
ck =
x
t
e
dt , k = 0, ±1, ±2,…
∫
T −T / 2
c0 is called the constant or dc component of x(t)
Dirichlet Conditions
•
A periodic signal x(t), has a Fourier series
if it satisfies the following conditions:
1. x(t) is absolutely integrable over any
period, namely
a +T
∫ | x(t ) | dt < ∞,
∀a ∈
a
2. x(t) has only a finite number of maxima
and minima over any period
3. x(t) has only a finite number of
discontinuities over any period
Example: The Rectangular Pulse Train
• From figure
T = 2, so ω 0 = 2π / 2 = π
• Clearly x(t) satisfies the Dirichlet conditions and
thus has a Fourier series representation
Example: The Rectangular Pulse
Train – Cont’d
∞
1
1
x(t ) = + ∑
(−1)|( k −1) / 2| e jkπ t , t ∈
2 k =−∞ kπ
k odd
Trigonometric Fourier Series
• By using Euler’s formula, we can rewrite
x(t ) =
∞
∑
k =−∞
as
ck e jkω0t , t ∈
∞
x(t ) = c0 + ∑ 2 | ck |cos(kω 0t + ∠ck ), t ∈
k =1
dc component
k-th harmonic
as long as x(t) is real
• This expression is called the trigonometric
Fourier series of x(t)
Example: Trigonometric Fourier
Series of the Rectangular Pulse Train
• The expression
∞
1
1
x(t ) = + ∑
(−1)|( k −1) / 2| e jkπ t , t ∈
2 k =−∞ kπ
k odd
can be rewritten as
1
x(t ) = +
2
∞
∑
k =1
k odd
2
π⎞
⎛
cos ⎜ kπ t + ⎣⎡(−1)( k −1) / 2 − 1⎦⎤ ⎟ , t ∈
kπ
2⎠
⎝
Gibbs Phenomenon
• Given an odd positive integer N, define the
N-th partial sum of the previous series
1
xN (t ) = +
2
N
∑
k =1
2
π⎞
⎛
cos ⎜ kπ t + ⎡⎣(−1)( k −1) / 2 − 1⎤⎦ ⎟ , t ∈
2⎠
kπ
⎝
k odd
• According to Fourier’s theorem,
theorem it should be
lim | xN (t ) − x(t ) |= 0
N →∞
Gibbs Phenomenon – Cont’d
x3 (t )
x9 (t )
Gibbs Phenomenon – Cont’d
x21 (t )
x45 (t )
overshoot:
overshoot about 9 % of the signal magnitude
(present even if N → ∞)
Parseval’s Theorem
• Let x(t) be a periodic signal with period T
• The average power P of the signal is defined
as
T /2
P=
1
2
(t )dt
x
∫
T −T / 2
∞
• Expressing the signal as x(t ) = ∑ ck e jkω t , t ∈
k =−∞
it is also
0
P=
∞
∑ |c
k =−∞
k
|2
Fourier Transform
• We have seen that periodic signals can be
represented with the Fourier series
• Can aperiodic signals be analyzed in terms of
frequency components?
• Yes, and the Fourier transform provides the
tool for this analysis
• The major difference w.r.t. the line spectra of
periodic signals is that the spectra of
aperiodic signals are defined for all real
values of the frequency variable ω not just
for a discrete set of values
Frequency Content of the
Rectangular Pulse
x(t )
xT (t )
x(t ) = lim xT (t )
T →∞
Frequency Content of the
Rectangular Pulse – Cont’d
• Since xT (t ) is periodic with period T, we
can write
xT (t ) =
∞
∑
k =−∞
ck e jkω0t , t ∈
where
T /2
1
− jkω o t
(
)
ck =
x
t
e
dt , k = 0, ±1, ±2,…
∫
T −T / 2
Frequency Content of the
Rectangular Pulse – Cont’d
• What happens to the frequency components
of xT (t ) as T → ∞ ?
• For k = 0 :
c0 = 1/ T
• For k = ±1, ±2,… :
ck =
⎛ kω ⎞ 1
⎛ kω ⎞
sin ⎜ 0 ⎟ =
sin ⎜ 0 ⎟
kω0T
⎝ 2 ⎠ kπ
⎝ 2 ⎠
2
ω 0 = 2π / T
Frequency Content of the
Rectangular Pulse – Cont’d
plots of T | ck |
vs. ω = kω 0
for T = 2,5,10
Frequency Content of the
Rectangular Pulse – Cont’d
• It can be easily shown that
⎛ω
lim Tck = sinc ⎜
T →∞
⎝ 2π
⎞
⎟, ω ∈
⎠
where
sinc(λ )
sin(πλ )
πλ
Fourier Transform of the
Rectangular Pulse
• The Fourier transform of the rectangular
pulse x(t) is defined to be the limit of Tck
as T → ∞ , i.e.,
⎛ω
X (ω ) = lim Tck = sinc ⎜
T →∞
⎝ 2π
⎞
⎟, ω ∈
⎠
| X (ω ) |
arg( X (ω ))
The Fourier Transform in the
General Case
• Given a signal x(t), its Fourier transform
X (ω ) is defined as
X (ω ) =
∞
∫
x(t )e − jω t dt , ω ∈
−∞
• A signal x(t) is said to have a Fourier
transform in the ordinary sense if the above
integral converges
The Fourier Transform in the
General Case – Cont’d
•
The integral does converge if
1. the signal x(t) is “well-behaved”
behaved
2. and x(t) is absolutely integrable,
integrable namely,
∞
∫ | x(t ) | dt < ∞
−∞
•
Note: well behaved means that the signal
has a finite number of discontinuities,
maxima, and minima within any finite time
interval
Example: The DC or Constant Signal
• Consider the signal x(t ) = 1, t ∈
• Clearly x(t) does not satisfy the first
requirement since
∞
∞
−∞
−∞
∫ | x(t ) | dt = ∫ dt =∞
• Therefore, the constant signal does not have
a Fourier transform in the ordinary sense
• Later on, we’ll see that it has however a
Fourier transform in a generalized sense
Example: The Exponential Signal
• Consider the signal x(t ) = e − bt u (t ), b ∈
• Its Fourier transform is given by
X (ω ) =
∞
∫
e − bt u (t )e − jω t dt
−∞
∞
= ∫e
t =∞
− ( b + jω ) t
0
1
⎡⎣e − ( b + jω ) t ⎤⎦
dt = −
b + jω
t =0
Example: The Exponential Signal –
Cont’d
• If b < 0 , X (ω ) does not exist
• If b = 0 , x(t ) = u (t ) and X (ω ) does not
exist either in the ordinary sense
• If b > 0 , it is
1
X (ω ) =
b + jω
amplitude spectrum
1
| X (ω ) |=
b2 + ω 2
phase spectrum
⎛ω ⎞
arg( X (ω )) = − arctan ⎜ ⎟
⎝b⎠
Example: Amplitude and Phase
Spectra of the Exponential Signal
x(t ) = e −10t u (t )
Rectangular Form of the Fourier
Transform
• Consider
X (ω ) =
∞
∫
x(t )e − jω t dt , ω ∈
−∞
• Since X (ω ) in general is a complex
function, by using Euler’s formula
⎛ ∞
⎞
X (ω ) = ∫ x(t ) cos(ω t )dt + j ⎜ − ∫ x(t )sin(ω t )dt ⎟
−∞
⎝ −∞
⎠
∞
R (ω )
X (ω ) = R (ω ) + jI (ω )
I (ω )
Polar Form of the Fourier Transform
• X (ω ) = R (ω ) + jI (ω ) can be expressed in
a polar form as
X (ω ) =| X (ω ) | exp( j arg( X (ω )))
where
| X (ω ) |= R 2 (ω ) + I 2 (ω )
⎛ I (ω ) ⎞
arg( X (ω )) = arctan ⎜
⎟
ω
(
)
R
⎝
⎠
Fourier Transform of
Real-Valued Signals
• If x(t) is real-valued, it is
X (−ω ) = X ∗ (ω )
• Moreover
Hermitian
symmetry
X ∗ (ω ) =| X (ω ) | exp(− j arg( X (ω )))
whence
| X (−ω ) |=| X (ω ) | and
arg( X (−ω )) = − arg( X (ω ))
Example: Fourier Transform of the
Rectangular Pulse
• Consider the even signal
• It is τ / 2
2
2
t =τ / 2
⎛ ωτ ⎞
X (ω ) = 2 ∫ (1) cos(ω t )dt = [sin(ω t ) ]t =0 = sin ⎜
⎟
ω
ω
2
⎝
⎠
0
⎛ ωτ ⎞
= τ sinc ⎜
⎟
⎝ 2π ⎠
Example: Fourier Transform of the
Rectangular Pulse – Cont’d
⎛ ωτ ⎞
X (ω ) = τ sinc ⎜
⎝ 2π ⎠
Example: Fourier Transform of the
Rectangular Pulse – Cont’d
amplitude
spectrum
phase
spectrum
Bandlimited Signals
• A signal x(t) is said to be bandlimited if its
Fourier transform X (ω ) is zero for all ω > B
where B is some positive number, called
the bandwidth of the signal
• It turns out that any bandlimited signal must
have an infinite duration in time, i.e.,
bandlimited signals cannot be time limited
Bandlimited Signals – Cont’d
• If a signal x(t) is not bandlimited, it is said
to have infinite bandwidth or an infinite
spectrum
• Time-limited signals cannot be
bandlimited and thus all time-limited
signals have infinite bandwidth
• However, for any well-behaved signal x(t)
it can be proven that lim X (ω ) = 0
ω →∞
whence it can be assumed that
| X (ω ) |≈ 0 ∀ω > B
B being a convenient large number
Inverse Fourier Transform
• Given a signal x(t) with Fourier transform
X (ω ) , x(t) can be recomputed from X (ω )
by applying the inverse Fourier transform
given by
1
x(t ) =
2π
• Transform pair
∞
∫
X (ω )e jω t dω , t ∈
−∞
x(t ) ↔ X (ω )
Properties of the Fourier Transform
x(t ) ↔ X (ω )
y (t ) ↔ Y (ω )
• Linearity:
α x(t ) + β y (t ) ↔ α X (ω ) + β Y (ω )
• Left or Right Shift in Time:
x(t − t0 ) ↔ X (ω )e − jω t0
• Time Scaling:
x(at ) ↔
1 ⎛ω ⎞
X⎜ ⎟
a ⎝a⎠
Properties of the Fourier Transform
• Time Reversal:
x(−t ) ↔ X (−ω )
• Multiplication by a Power of t:
n
d
t n x(t ) ↔ ( j ) n
X (ω )
n
dω
• Multiplication by a Complex Exponential:
x(t )e jω0t ↔ X (ω − ω 0 )
Properties of the Fourier Transform
• Multiplication by a Sinusoid (Modulation):
j
[ X (ω + ω 0 ) − X (ω − ω0 )]
2
1
x(t ) cos(ω 0t ) ↔ [ X (ω + ω 0 ) + X (ω − ω 0 ) ]
2
x(t )sin(ω 0t ) ↔
• Differentiation in the Time Domain:
dn
n
(
)
↔
(
ω
)
x
t
j
X (ω )
n
dt
Properties of the Fourier Transform
• Integration in the Time Domain:
t
∫
−∞
x(τ )dτ ↔
1
X (ω ) + π X (0)δ (ω )
jω
• Convolution in the Time Domain:
x(t ) ∗ y (t ) ↔ X (ω )Y (ω )
• Multiplication in the Time Domain:
x(t ) y (t ) ↔ X (ω ) ∗ Y (ω )
Properties of the Fourier Transform
• Parseval’s Theorem:
∫ x(t ) y(t )dt ↔
if
y (t ) = x(t )
1
2π
∗
X
∫ (ω )Y (ω )dω
2
|
x
(
t
)
|
dt ↔
∫
1
2
|
X
(
)
|
dω
ω
∫
2π
• Duality:
X (t ) ↔ 2π x(−ω )
Properties of the Fourier Transform Summary
Example: Linearity
x(t ) = p4 (t ) + p2 (t )
⎛ 2ω ⎞
⎛ω ⎞
X (ω ) = 4sinc ⎜
⎟ + 2sinc ⎜ ⎟
⎝ π ⎠
⎝π ⎠
Example: Time Shift
x(t ) = p2 (t − 1)
⎛ω ⎞
X (ω ) = 2sinc ⎜ ⎟ e − jω
⎝π ⎠
Example: Time Scaling
p2 (t )
⎛ω ⎞
2sinc ⎜ ⎟
⎝π ⎠
⎛ω ⎞
sinc ⎜
⎟
⎝ 2π ⎠
p2 (2t )
a > 1 time compression ↔ frequency expansion
0 < a < 1 time expansion ↔ frequency compression
Example: Multiplication in Time
x(t ) = tp2 (t )
X (ω ) = j
d ⎛
d ⎛ sin ω ⎞
ω cos ω − sin ω
⎛ ω ⎞⎞
=
=
2sinc
j
2
j
2
⎜
⎟
⎜
⎟
⎟
dω ⎜⎝
dω ⎝ ω ⎠
ω2
⎝ π ⎠⎠
Example: Multiplication in Time –
Cont’d
X (ω ) = j 2
ω cos ω − sin ω
ω2
Example: Multiplication by a Sinusoid
x(t ) = pτ (t ) cos(ω 0t )
X (ω ) =
sinusoidal
burst
1⎡
⎛ τ (ω + ω 0 ) ⎞
⎛ τ (ω − ω 0 ) ⎞ ⎤
τ
sinc
+
τ
sinc
⎜
⎟
⎜
⎟⎥
2 ⎢⎣
2π
2π
⎝
⎠
⎝
⎠⎦
Example: Multiplication by a
Sinusoid – Cont’d
X (ω ) =
1⎡
⎛ τ (ω + ω 0 ) ⎞
⎛ τ (ω − ω 0 ) ⎞ ⎤
τ
sinc
+
τ
sinc
⎜
⎟
⎜
⎟⎥
2 ⎢⎣
2π
2π
⎝
⎠
⎝
⎠⎦
⎧ω 0 = 60 rad / sec
⎨
⎩τ = 0.5
Example: Integration in the Time
Domain
⎛ 2|t |⎞
v(t ) = ⎜1 −
⎟ pτ (t )
τ ⎠
⎝
x(t ) =
dv(t )
dt
Example: Integration in the Time
Domain – Cont’d
• The Fourier transform of x(t) can be easily
found to be
⎛
⎛ τω ⎞ ⎞⎛
⎛ τω ⎞ ⎞
X (ω ) = ⎜ sinc ⎜
j
2sin
⎟ ⎟⎜
⎜
⎟⎟
4
4
π
⎝
⎠
⎝
⎠⎠
⎝
⎠⎝
• Now, by using the integration property, it is
V (ω ) =
1
τ
⎛ τω ⎞
X (ω ) + π X (0)δ (ω ) = sinc 2 ⎜
⎟
2
jω
⎝ 4π ⎠
Example: Integration in the Time
Domain – Cont’d
τ
⎛ τω ⎞
V (ω ) = sinc 2 ⎜
⎟
2
⎝ 4π ⎠
Generalized Fourier Transform
• Fourier transform of δ (t )
− jω t
δ
(
t
)
e
dt = 1
∫
⇒
δ (t ) ↔ 1
• Applying the duality property
x(t ) = 1, t ∈
↔ 2πδ (ω )
generalized Fourier transform
of the constant signal x(t ) = 1, t ∈
Generalized Fourier Transform of
Sinusoidal Signals
cos(ω 0t ) ↔ π [δ (ω + ω 0 ) + δ (ω − ω 0 ) ]
sin(ω 0t ) ↔ jπ [δ (ω + ω 0 ) − δ (ω − ω 0 ) ]
Fourier Transform of Periodic Signals
• Let x(t) be a periodic signal with period T;
as such, it can be represented with its
Fourier transform
x(t ) =
∞
∑
k =−∞
ck e jkω0t
ω 0 = 2π / T
• Since e jω0t ↔ 2πδ (ω − ω 0 ), it is
X (ω ) =
∞
∑ 2π c δ (ω − kω )
k =−∞
k
0
Fourier Transform of
the Unit-Step Function
• Since
t
u (t ) =
∫ δ (τ )dτ
−∞
using the integration property, it is
t
u (t ) =
∫ δ (τ )dτ ↔
−∞
1
+ πδ (ω )
jω
Common Fourier Transform Pairs