CHAPTER
3
Fourier Representation
of Signals and LTI
Systems.
1
School
of Computer and Communication Engineering,
UniMAP
EKT 230
3.0 Fourier Representation of
Signals and LTI Systems.
3.1 Introduction.
3.2 Complex Sinusoids and Frequency Response of LTI Systems.
3.3 Fourier Representation of Four Classes of Signals.
3.4 Discrete Time Periodic Signals: Discrete Time Fourier Series.
3.5 Continuous-Time Periodic Signals: Fourier Series.
3.6 Discrete-Time Non Periodic Signals: Discrete-Time Fourier
Transform.
3.7 Continuous-Time Non Periodic Signals: Fourier Transform.
3.8 Properties of Fourier Representation.
3.9 Linearity and Symmetry Properties.
3.10 Convolution Properties.
2
3.1 Introduction.


Signals are represented as superposition's of complex sinusoids
which leads to a useful expression for the system output and
provide a characterization of signals and systems.
Example in music, the orchestra is a superposition of sounds
generated by different equipment having different frequency
range such as string, base, violin and etc. The same example
applied the choir team.

Study of signals and systems using sinusoidal representation
is termed as Fourier Analysis introduced by Joseph Fourier
(1768-1830).

There are four distinct Fourier representations, each applicable to
different class of signals.
3
3.2 COMPLEX SINUSOIDAL AND
FREQUENCY RESPONSE OF LTI
SYSTEM.


The response of an LTI system to a sinusoidal input that lead to
the characterization of the system behavior that is called
frequency response of the system.
An impulse response h[n] and the complex sinusoidal input
x[n]=ejWn.
4
Cont’d…
Discrete-Time (DT)
Derivation:
yn 

 hk xn  k 
k  
yn 

jW  n  k 


h
k
e

k  
factor e jWn
yn  e jWn
out

 j Wk


h
k
e

k  
 
 H e jW e jWn
 Where, the frequency response:
   hk e
H e jW 

k  
 jWk
5
Cont’d…
Continuous-Time (CT)
Derivation:

y t  
j t  


d
e

h


e
j t

 j


d
e

h


 H  j e jt
Frequency Response:
H  j  

 jt


h

e
d


 THE OUTPUT OF A COMPLEX SINUSOIDAL INPUT TO AN LTI SYSTEM IS
6
A COMPLEX SINUSOID OF THE SAME FREQUENCY AS THE INPUT,
MULTIPLIED BY THE FREQUENCY RESPONSE OF THE SYSTEM.
Example 3.3: Frequency Response.
t
1  RC
ht  
e u t 
RC
The impulse response of a system is given as
Find the expression for the frequency response, and plot the
magnitude and phase response.



1
 j
RC


H  j  
e
u

e
d
Solution:

RC
Step 1: Find frequency response, H(j).
Substitute h(t) into equation below,



1 
  j 

1
RC 


e
d
RC 0

H  j  

 jt


h

e
d



1 
  j 

1
1
RC 


e
1
RC 

 j 

RC 

1
1
0  1

1 
RC 
 j 

RC


1
RC

7
1 

 j 

RC 

0
Cont’d…
Step 2: From frequency response, get the magnitude & phase
response.
The magnitude response is,
| H  j  |
1
RC
 1 
2  

 RC 
2
8
Cont’d…
Figure 3.1: Frequency response of the RC circuit (a) Magnitude response.
(b) Phase response.
The phase response is arg{H(j)}= -arctan(RC)
 .
9
3.3 FOURIER REPRESENTATION OF
FOUR CLASS OF SIGNALS.

There are four distinct Fourier representations, class of signals;
(i) Periodic Signals.
(ii) Nonperiodic Signals.
(iii) Discrete-Time Periodic Signals.
(iv) Continuous-Time Periodic Signals.

Fourier Series (FS) applies to continuous-time periodic signals.
Discrete Time Fourier Series (DTFS) applies to discrete-time
periodic signals.
Nonperiodic signals have Fourier Transform (FT) representation.
Discrete Time Fourier Transform (DTFT) applies to a signal
that is discrete in time and non-periodic.



10
Cont’d…
Time
Property
Periodic
Nonperiodic
Continuous (t)
Fourier Series (FS)
[Chapter 3.5]
Fourier Transform (FT)
[Chapter 3.7]
Discrete (n)
Discrete Time Fourier
Series (DTFS)
[Chapter 3.4]
Discrete Time Fourier
Transform (DTFT)
[Chapter 3.6]
Table 3.1: Relation between Time Properties of a Signal an the
Appropriate Fourier Representation.
11
Cont’d…
(1) Discrete
Time Periodic
Signal
Fourier
Representatio
n
(2) Continuous
Time Periodic
Signal
(3) Continuous
Time Non
Periodic Signal
(4) Discrete
Time Non 12
Periodic Signal
3.3.1 Periodic Signals:
Fourier Series Representation
 If x[n] is a discrete-time signal with fundamental period N then
x[n] of DTFS is represents as,
ˆxn   Ak e jkWo n
k
 where Wo= 2p/N is the fundamental frequency of x[n].
 The frequency of the kth sinusoid in the superposition is kWo
13
Cont’d…
 If x(t) is a continuous-time signal with fundamental period T,
x(t) of FS is represents as
xˆ t    Ak e
jkot
k
 where o= 2p/T is the fundamental frequency of x(t). The
frequency of the kth sinusoid is ko and each sinusoid has a
common period T.
 A sinusoid whose frequency is an integer multiple of a fundamental
frequency is said to be a harmonic of a sinusoid at the
fundamental frequency. For example ejk0t is the kth harmonic of
ej0t.
 The variable k indexes the frequency of the sinusoids, A[k] is the
14
function of frequency.
3.3.2 Nonperiodic Signals:
Fourier-Transform Representation.
 The Fourier Transform representations employ complex sinusoids
having a continuum of frequencies.
 The signal is represented as a weighted integral of complex
sinusoids where the variable of integration is the sinusoid’s
frequency.
 Continuous-time sinusoids are used to represent continuous
signal in FT.
1
xˆ t  
2p

jt


X
j

e
d


where X(j)/(2p) is the “weight” or coefficient applied to a
sinusoid of frequency  in the FT representation
15
Cont’d…
 Discrete-time sinusoids are used to represent discrete time signal
in DTFT.
 It is unique only a 2p interval of frequency. The sinusoidal
frequencies are within 2p interval.
1
xˆn 
2p
p
p X e e
jW
jWn
d

16
3.4 Discrete Time Periodic Signals:
Discrete Time Fourier Series.
 The Discrete Time Fourier Series representation;
N 1
xn   X k e jkWo n
k 0
 where x[n] is a periodic signal with period N and fundamental
frequency Wo=2p/N. X[k] is the DTFS coefficient of signal x[n].
1
X k  
N
N 1
 jkW o n


x
n
e

n 0
 The relationship of the above equation,
xn X k 
DTFS ;Wo
17
Cont’d…




Given N value of x[n] we can find X[k].
Given N value of X[k] we can find x[n] vise-versa.
The X[k] is the frequency-domain representation of x[n].
DTFS is the only Fourier representation that can be numerically
evaluated using computer, e.g. used in numerical signal analysis.
18
Example 3.1: Determining DTFS Coefficients.
Find the frequency-domain representation of the signal in Figure 3.2
below.
Figure 3.2: Time Domain Signal.
Solution:
Step 1: Determine N and W0.
The signal has period N=5, so W0=2p/5.
Also the signal has odd symmetry, so we sum over n = -2 to n = 2 from
equation
19
Cont’d…
Step 2: Solve for the frequency-domain, X[k].
From step 1, we found the fundamental frequency, N =5, and we sum
over n = -2 to n = 2 .
1 N 1
X k    xne  jkWo n
N n 0
1 2
X k    xne  jk 2pn / 5
5 n  2
1
 x 2e jk 4p / 5  x 1e jk 2p / 5  x0e j 0  x1e  jk 2p / 5  x2e  jk 4p / 5
5

20

Cont’d…
From the value of x{n} we get,
1  1 jk 2p / 5 1  jk 2p / 5 
X k   1  e
 e

5 2
2

1
 1  j sin k 2p / 5
5
Step 3: Plot the magnitude and phase of DTFS.
From the equation, one period of the DTFS coefficient X[k], k=-2 to
k=2, in the rectangular and polar coordinate as
1
sin 4p / 5
X  2   j
 0.232e  j 0.531
5
5
1
sin 2p / 5
X  1   j
 0.276e  j 0.760
5
5
21
Cont’d…
X 0 
1
 0.2e j 0
5
1
sin 2p / 5
X 1   j
 0.276e j 0.760
5
5
1
sin 4p / 5
X 2   j
 0.232e j 0.531
5
5
The above figure shows the magnitude and phase of X[k] as a
function of frequency index k.
22
Figure 3.3: Magnitude and phase of the DTFS coefficients for the signal in Fig. 3.2.
Cont’d…
Just to Compare, for different range of n.
Calculate X[k] using n=0 to n=4 for the limit of the sum.

1
X k   x 0e j 0  1e  j 2p / 5  2e  jk 4p / 5  3e  jk 6p / 5  4e  jk 8p / 5
5
1 1
1

X k   1  e  jk 2p / 5  e  jk 8p / 5 
5 2
2


This expression is different from which we obtain from using n=-2 to
n=2. Note that,
e  jk 8p / 5  e  jk 2p e jk 2p / 5
 e jk 2p / 5
n=-2 to n=2 and n= 0 to n=4, yield equivalent expression for the DTFS
coefficient.
23
.
3.5 Continuous-Time Periodic
Signals: Fourier Series.
 Continues time periodic signals are represented by Fourier series
(FS). The fundamental period is T and fundamental frequency o
=2p/T.

xt  
jko t
X
(
k
)
e

k  
 X[k] is the FS coefficient of signal x(t).
T
1
X (k )   xt e  jkot dt
T0
 Below is the relationship of the above equation,
xt   X (k )
FS ;o
24
Cont’d…
 The FS coefficient X(k) is known as frequency-domain
representation of x(t).
25
Example 3.2: Direct Calculation of FS Coefficients.
Determine the FS coefficients for the signal x(t) depicted in Figure 3.4.
Solution:
Figure 3.4: Time Domain Signal.
Step 1: Determine T and 0.
1 is T=2, so  =2p/2 = p. On the interval 0<=t<=2,
The period ofX (kx(t)
0
)  e e
dt
2
one period of 1x(t) is expressed as x(t)=e-2t, so it yields
2
 2t
 jkpt
0
2
X (k ) 
e
2
  2  jkp t
dt
0
Step 2: Solve for X[k].
T
1
X (k )   xt e  jko dt
T0
26
Cont’d…
Step 3: Plot the magnitude and phase spectrum.
1
X (k ) 
e 2 jkp t
2(2  jkp )
2
0
1
X (k ) 
e  4 e  jk 2p  1
4  2 jkp


1  e 4
X (k ) 
4  jk 2p
- From the above equation for example k=0,
X(k)= (1-e-4)/4 = 0.245.
- Since e-jk2p=1 Figure 3.5 shows the magnitude spectrum|X(k)| and
27
the phase spectrum arg{X(k)}.
Cont’d…
Figure 3.5: Magnitude and Phase Spectra..
28
3.6 Discrete-Time Non Periodic
Signals: Discrete-Time Fourier
Transform.
 The DTFT representation of time domain signal,
1
xn 
2p
p
 
jW
jWn
X
e
e
dW

p
 X[k] is the DTFT of the signal x[n].
 

X e jW   xne  jWn

 Below is the relationship of the above equation,
 
xn X e
DTFT
jW
29
Cont’d…
 The FT, X[j], is known as frequency-domain representation
of the signal x(t).
1
xn 
2p
p
p X e e
jW
jWn
dW

 The above equation is the inverse FT, where it map the frequency
domain representation X[j] back to time domain.

30
3.7 Continuous-Time Nonperiodic
Signals: Fourier Transform.
 The Fourier transform (FT) is used to represent a continuous
time nonperiodic signal as a superposition of complex sinusoids.
 FT representation,
1
xt  
2p
where,

jt


X
j

e
d


X  j  

 jt


x
t
e


 Below is the relationship of the above equation,
xt   X  j 
FT
31
Example 3.4: FT of a Real Decaying Exponential.
Find the Fourier Transform (FT) of x(t) =e-at u(t).
Solution:
The FT does not converge for a<=0, since x(t) is not absolutely
integrable, that is

 at
e
 dt  ,
a0
0
for a  0, we have

X  j    e  at u (t )e  jt dt
0

  e ( a  j ) t dt
0
1

e  ( a  j ) t
a  j

1
a  j

0
32
Cont’d…
Converting to polar form, we find that the magnitude and phase of X(jw)
are respectively given by
X  j  
1
a
2


1
2 2
and
 
argX  j    arctan  
a
This is shown in Figure 3.6 (b) and (c).
33
Cont’d…
Figure 3.6: (a) Real time-domain exponential signal. (b) Magnitude spectrum.34
(c) Phase spectrum.
 .
3.8 Properties of Fourier
Representations.
 The four Fourier representations discussed in this chapter are
summarized in Table 3.2.
 Attached Appendix C is the comprehensive table of all properties.
35
Cont’d…
Time
Domain
C
O
N
T
I
N
U
O
U
S
(t)
Periodic
(t,n)
Fourier Transform
Fourier Series
x t  

 X k e
jk 0 t
k  
Fourier
Series
X


k xt 
T

0
T
2p
has period
t2p
0 xt 
t
Discrete Time Fourier Series
N 1
xn   X k e
jkW 0 n
k 0
1 N 1
X k    xne  jkW 0 n
N n 0
xn and X k  have period N
2p
W0 
n

1


x
t

Fourier Transform2p
dt
1X k e jk t
x t e  jk0t


k T

0
1
0t
x t  X k has
  xt e  jkperiod
dt
T 0
0 
D
I
S
C
R
E
T
E
(t)
Non periodic
(t,n)
xt  
jt


X
j

e
dw



1
jt 


X
j

e
dw

2p X
xt e  jt dt
  j  


X  j    xt e

 jt
Discrete Time Fourier Transform
dt

p
jW
j Wn
Discrete xnTime
  1 Fourier
dW
 X e e Transform
xn 
p
2p
p
 
1 jW jW jWnjWn
ne dW
X e X e xe
2p p jW 
 
X e 
 
X e

has
period
X e jW   xne  jWn
jW
2p

has
period
P
E
R
I
O
D
I
C
(k,W)
2p
Table 3.2: The Four Fourier Representations.
Discrete (k)
N
O
N
P
E
R
I
O
D
I
C
(k,w)
Continuous (,W)
36
Freq.
Domain
3.9 Linearity and Symmetry
Properties.
 All four Fourier representations involve linear operations.
z t   axt   by t  Z  j   aX  j   bY  j 
FT
FS;
o  Z k   aX k   bY k 
z t   axt   by t 

DTFT






z n   axn   byn 
 Z  e jW   aX  e jW   bY  e jW 






DTFS;W
o  Z k   aX k   bY k 
z n   axn   byn 
 The linearity property is used to find Fourier representations of
signals that are constructed as sums of signals whose
representation are already known.
37
Example 3.5: Linearity in the Fourier Series.
Given z(t) which is the periodic signal. Use the linearity property to
determine the FS coefficients Z[k].
3
1
z t   xt   y t 
2
2
Solution:
From Example 3.31(Text) we have
1
 kp 
xt   X k  
Sin

kp
 4 
1
 kp 
FS ; 2p
y t   Y k  
Sin 

kp
 2 
FS ; 2p
The linearity property implies that
FS ; 2p
z t  
 Z k  
.
3
 kp
Sin
2kp
 4
1

 kp 
Sin


 2kp
 2 
38
3.9.1 Symmetry Properties:
Real and Imaginary Signal.
Representation
Real-Valued
Time Signal
Imaginary-Valued
Time Signal
FT
X*(j) = X(-j)
X*(j) = -X(-j)
FS
X*[k] = X[-k]
X*[k] = -X[-k]
DTFT
X*(ej) = X(e-j)
X*(ej) = -X(e-j)
DTFS
X*[k] = X[-k]
X*[k] = -X[-k]
Table 3.3: Symmetry Properties for Fourier Representation of
Real-and Imaginary–Valued Time Signals.
39
3.9.2 Symmetry Properties:
Even and Odd Signal.

X
*
 j    x t e  j ( t ) dt

 Change of variable  = -t.
X *  j  

 j


x

e
d


 X  j 
 The only way that the condition X*(j) =X(j) holds is for the
imaginary part of X(j) to be zero.
40
Cont’d…
 If time signal is real and even, then the frequency-domain
representation of it also real.
 If time signal is real and odd, then the frequency-domain
representation is imaginary.
41
3.10 Convolution Property.


The convolution of the signals in the time domain transforms to
multiplication of their respective Fourier representations in the
frequency domain.
The convolution property is a consequences of complex sinusoids
being eigen functions of LTI system.
3.10.1 Convolution of Non periodic Signals.
3.10.2 Filtering.
3.10.3 Convolution of Periodic Signals.
42
3.10.1 Convolution of Non Periodic
Signals.
 Convolution of two non periodic continuous-time signals x(t) and
h(t) is defines as
y t   ht * xt 



xt    
1
2p

 X  j e
j t  
d

h xt   d .

 express x(t-) in term of FT:
43
Cont’d…
Substitute into convolution integral yields,
 1
y t    h 

 2p

1
y t  
2p
1

2p


 X  j e

jt  j
e

d d



 j
j t
h e d  X  j e d

 H  j X  j e

jt

d

44
Cont’d…
 Convolution of h(t) and x(t) in the time domain corresponds to
multiplication of Fourier transforms, H(j) and X(j) in the
frequency domain that is;
y t   ht * xt  Y  j   X  j H  j .
FT
 Discrete-time nonperiodic signals, if
 
hn H e 
DTFT
xn
 X e jW
DTFT
jW
and
then
 
   
DTFT
yn  hn* xn
 Y e jW  X e jW H e jW .
45
Example 3.6: Convolution in Frequency Domain.
Let x(t)= (1/(pt))sin(pt) be the input of the system with impulse
response h(t)= (1/(pt))sin(2pt). Find the output y(t)=x(t)*h(t)
Solution:
From Example 3.26 (text, Simon) we have,
1,   p
xt   X  j   
0,   p
FT
1,   2p
ht   H  j   
0,   2p
FT
Since
FT
y t   xt * ht  
Y  j   X  j H  j 
it
follows that
1,
Y  j   
0,
We conclude that
.
 p 

  p 
1
y t     sin pt .
 pt 
46