Chapter 4
The Fourier Series and
Fourier Transform
Representation of Signals in Terms
of Frequency Components
• Consider the CT signal defined by
N
x(t )   Ak cos( k t   k ), t 
k 1
• The frequencies `present in the signal’ are the
frequency  k of the component sinusoids
• The signal x(t) is completely characterized by
the set of frequencies  k , the set of amplitudes
Ak , and the set of phases  k
Example: Sum of Sinusoids
• Consider the CT signal given by
x(t )  A1 cos(t )  A2 cos(4t   / 3)  A3 cos(8t   / 2),
t
• The signal has only three frequency
components at 1,4, and 8 rad/sec, amplitudes
A1 , A2 , A3 and phases 0,  / 3,  / 2
• The shape of the signal x(t) depends on the
relative magnitudes of the frequency
components, specified in terms of the
amplitudes A1 , A2 , A3
Example: Sum of Sinusoids –Cont’d
 A1  0.5

 A2  1
A  0
 3
 A1  1

 A2  0.5
A  0
 3
 A1  1

 A2  1
A  0
 3
Example: Sum of Sinusoids –Cont’d
 A1  0.5

 A2  1
 A  0.5
 3
 A1  1

 A2  0.5
 A  0.5
 3
 A1  1

 A2  1
A 1
 3
Amplitude Spectrum
• Plot of the amplitudes Ak of the sinusoids
making up x(t) vs. 
• Example:

Phase Spectrum
• Plot of the phases  k of the sinusoids
making up x(t) vs. 
• Example:

Complex Exponential Form
• Euler formula: e
• Thus
j
 cos( )  j sin( )
Ak cos( k t   k )    Ak e j (k t  k ) 
real part
whence
N
x(t )     Ak e
k 1
j ( k t  k )
, t 

Complex Exponential Form – Cont’d

• And, recalling that ( z )  ( z  z ) / 2 where
z  a  jb, we can also write
N
1
j ( k t  k )
 j ( k t  k )

, t 
x(t )    Ak e
 Ak e

2
k 1
• This signal contains both positive and
negative frequencies
• The negative frequencies  k stem from
writing the cosine in terms of complex
exponentials and have no physical meaning
Complex Exponential Form – Cont’d
• By defining
Ak j k
ck 
e
2
c k
Ak  j k

e
2
it is also
N
x(t )   ck e jk t  c k e  jk t  
k 1
N

k  N
k 0
ck e jk t , t 
complex exponential form
of the signal x(t)
Line Spectra
• The amplitude spectrum of x(t) is defined as
the plot of the magnitudes | ck | versus 
• The phase spectrum of x(t) is defined as the
plot of the angles ck  arg(ck ) versus 
• This results in line spectra which are defined
for both positive and negative frequencies
• Notice: for k  1, 2,
| ck || c k |
ck  c k
arg(ck )   arg(c k )
Example: Line Spectra
x(t )  cos(t )  0.5cos(4t   / 3)  cos(8t   / 2)
0.
0.
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
jk0t
, t
where  0  2 / T is the fundamental
frequency (rad/sec) of the signal and
T /2
1
 jk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)
The Fourier Series – Cont’d
• The frequencies k 0 present in x(t) are
integer multiples of the fundamental
frequency  0
• Notice that, if the dc term c0 is added to
x(t ) 
N
ce
k  N
k 0
j k t
k
and we set N   , the Fourier series is a
special case of the above equation where all
the frequencies are integer multiples of  0
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 whence  0  2 / 2  
• 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 jk t , t 
2 k  k
k odd
Trigonometric Fourier Series
• By using Euler’s formula, we can rewrite
x(t ) 
as

ce
k 
k
jk0t
, t

x(t )  c0   2 | ck |cos(k 0t  ck ), t 
k 1
dc component
k-th harmonic
• This expression is called the trigonometric
Fourier series of x(t)
Example: Trigonometric Fourier
Series of the Rectangular Pulse Train
• The expression

1
1
|( k 1) / 2| jk t
x(t )   
(1)
e , t
2 k  k
k odd
can be rewritten as
1
x(t )  
2


k 1
k odd
2


( k 1) / 2
cos  k t  (1)
 1  , t 
k
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
k odd
2


( k 1) / 2
cos  k t  (1)
 1  , t 
k
2

• According to Fourier’s 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: 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
1
2
P
x (t )dt

T T / 2
• Expressing the signal as x(t ) 
it is also
P

 |c
k 
k
2
|


k 
ck e jk0t , t 
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  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 ) 

ce
k 
k
jk0t
, t
where
T /2
1
 jk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
1
c0 
T
• For k  0
 k 0  1
 k 0 
ck 
sin 
sin 

 , k  1, 2,
k 0T
 2  k
 2 
2
 0  2 / T
Frequency Content of the
Rectangular Pulse – Cont’d
plots of T | ck |
vs.   k 0
for T  2,5,10
Frequency Content of the
Rectangular Pulse – Cont’d
• It can be easily shown that

lim Tck  sinc 
T 
 2

,  

where
sinc( )
sin( )

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.,

X ( )  lim Tck  sinc 
T 
 2
| X ( ) |

,  

arg( X ( ))
Fourier Transform of the Rectangular
Pulse – Cont’d
• The Fourier transform X ( ) of the
rectangular pulse x(t) can be expressed in
terms of x(t) as follows:

1
 jko t
ck   x(t )e
dt , k  0, 1, 2,
T 
x(t )  0 for t  T / 2 and t  T / 2
whence

Tck 
 x(t )e

 jko t
dt , k  0, 1, 2,
Fourier Transform of the Rectangular
Pulse – Cont’d
• Now, by definition X ( )  lim Tck and,
T 
since k 0   as T  
X ( ) 

 x(t )e
 j t
dt ,  

• The inverse Fourier transform of X ( ) is
1
x(t ) 
2

 X ( )e

j t
d , t 
The Fourier Transform in the General
Case
• Given a signal x(t), its Fourier transform
X ( ) is defined as
X ( ) 

 x(t )e

 j t
dt ,  
• 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”
2. and x(t) is absolutely 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
 bt
• Consider the signal x(T )  e u (t ), b 
• Its Fourier transform is given by
X ( ) 

e
 bt
u (t )e
 j t
dt


 e
0
t 
 ( b  j ) t
1
 ( b  j ) t


dt  
e


b  j
t 0
Example: The Exponential Signal –
Cont’d
• If b  0 , X ( ) does not exist
• If b  0 , x(t )  u(t ) and X ( ) does not
exist either in the ordinary sense
• If b  0 , it is
1
X ( ) 
b  j
amplitude spectrum
1
| X ( ) |
b2   2
phase spectrum
 
arg( X ( ))   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 ( ) 

 x(t )e
 j t
dt ,  

• Since X ( ) in general is a complex
function, by using Euler’s formula
 

X ( )   x(t ) cos( t )dt  j    x(t )sin( t )dt 

 


R ( )
X ( )  R( )  jI ( )
I ( )
Polar Form of the Fourier Transform
• X ( )  R( )  jI ( ) can be expressed in
a polar form as
X ( ) | X ( ) | exp( j arg( X ( )))
where
| X ( ) | R ( )  I ( )
2
2
 I ( ) 
arg( X ( ))  arctan 

 R( ) 
Fourier Transform of
Real-Valued Signals
• If x(t) is real-valued, it is
X ( )  X ( )

• Moreover
Hermitian
symmetry
X ( ) | X ( ) | exp( j arg( X ( )))

whence
| X ( ) || X ( ) | and
arg( X ( ))   arg( X ( ))
Fourier Transforms of
Signals with Even or Odd Symmetry
• Even signal: x(t )  x(t )

X ( )  2 x(t ) cos(t )dt
0
• Odd signal: x(t )   x(t )

X ( )   j 2 x(t )sin(t )dt
0
Example: Fourier Transform of the
Rectangular Pulse
• Consider the even signal
• It is  / 2
2
2
t  / 2
  
X ( )  2  (1) cos( t )dt  sin( t ) t 0  sin 



2


0
  
  sinc 

2



Example: Fourier Transform of the
Rectangular Pulse – Cont’d
  
X ( )   sinc 

 2 
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 ( ) is zero for all   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 ( )  0
 
whence it can be assumed that
| X ( ) | 0   B
B being a convenient large number
Inverse Fourier Transform
• Given a signal x(t) with Fourier transform
X ( ) , x(t) can be recomputed from X ( )
by applying the inverse Fourier transform
given by
1
x(t ) 
2

 X ( )e
j t
d , t 

• Transform pair
x(t )  X ( )
Properties of the Fourier Transform
x(t )  X ( )
y(t )  Y ( )
• Linearity:
 x(t )   y(t )   X ( )  Y ( )
• Left or Right Shift in Time:
x(t  t0 )  X ( )e
• Time Scaling:
 j t0
1  
x(at )  X  
a a
Properties of the Fourier Transform
• Time Reversal:
x(t )  X ( )
• Multiplication by a Power of t:
n
d
t x(t )  ( j )
X ( )
n
d
n
n
• Multiplication by a Complex Exponential:
x(t )e
j0t
 X (  0 )
Properties of the Fourier Transform
• Multiplication by a Sinusoid (Modulation):
j
x(t )sin(0t )   X (  0 )  X (  0 ) 
2
1
x(t ) cos(0t )   X (  0 )  X (  0 ) 
2
• Differentiation in the Time Domain:
n
d
n
x(t )  ( j ) X ( )
n
dt
Properties of the Fourier Transform
• Integration in the Time Domain:
t
1
 x( )d  j X ( )   X (0) ( )
• Convolution in the Time Domain:
x(t )  y(t )  X ( )Y ( )
• Multiplication in the Time Domain:
x(t ) y(t )  X ( )  Y ( )
Properties of the Fourier Transform
• Parseval’s Theorem:
1
 x(t ) y(t )dt  2
X

( )Y ( )d
1
2
if y(t )  x(t )  | x (t ) | dt 
| X ( ) | d

2
2
• Duality:
X (t )  2 x( )
Properties of the Fourier Transform Summary
Example: Linearity
x(t )  p4 (t )  p2 (t )
 2 
 
X ( )  4sinc 
  2sinc  
  
 
Example: Time Shift
x(t )  p2 (t  1)
    j
X ( )  2sinc   e
 
Example: Time Scaling
p2 (t )
p2 (2t )
 
2sinc  
 
 
sinc 

2



a  1 time compression  frequency expansion
0  a  1 time expansion  frequency compression
Example: Multiplication in Time
x(t )  tp2 (t )
d 
d  sin  
 cos   sin 
  
X ( )  j
2sinc     j 2

  j2

d 
d   
2
  
Example: Multiplication in Time –
Cont’d
 cos   sin 
X ( )  j 2
2

Example: Multiplication by a Sinusoid
x(t )  p (t ) cos( 0t )
sinusoidal
burst
1
  (   0 ) 
  (   0 )  
X ( )   sinc 
   sinc 

2
2
2




Example: Multiplication by a
Sinusoid – Cont’d
1
  (   0 ) 
  (   0 )  
X ( )   sinc 
   sinc 

2
2
2




0  60 rad / sec

  0.5
Example: Integration in the Time
Domain
 2|t |
v(t )  1 
 p (t )
 

dv(t )
x(t ) 
dt
Example: Integration in the Time
Domain – Cont’d
• The Fourier transform of x(t) can be easily
found to be

   
   
X ( )   sinc 
  j 2sin 

 4  
 4 

• Now, by using the integration property, it is
1

2   
V ( ) 
X ( )   X (0) ( )  sinc 

j
2
 4 
Example: Integration in the Time
Domain – Cont’d

  
V ( )  sinc 

2
 4 
2
Generalized Fourier Transform
• Fourier transform of  (t )
 j t

(
t
)
e
dt

1


 (t )  1
• Applying the duality property
x(t )  1, t 
 2 ( )
generalized Fourier transform
of the constant signal x(t )  1, t 
Generalized Fourier Transform of
Sinusoidal Signals
cos(0t )    (  0 )   (  0 )
sin(0t )  j  (  0 )   (  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 ) 

ce
jk 0 t
k
k 
 0  2 / T
• Since e j0t  2 (  0 ), it is
X ( ) 

 2 c  (  k )
k 
k
0
Fourier Transform of
the Unit-Step Function
• Since
t
u (t ) 
  ( )d

using the integration property, it is
t
1
u (t )    ( )d 
  ( )
j

Common Fourier Transform Pairs