CHAPTER 3
Fourier
Representation of
Signals and LTI
Systems.
EKT 232
1
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 ect. 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.
2
Fourier Series
Discrete Time Fourier series (DTFS)
x F  n 

k  NF
X  k  e j 2 kn / N F
1
X k  
NF
n0  N F 1

n n0
x  n  e  j 2 kn / N F
where N F is the representation time and the notation

k  NF
means
a summation over any range of consecutive k’s exactly N F in length.
3
Fourier Series
Notice that in
x F  n 

k  NF
X  k  e j 2 kn / N F
the summation is over exactly one period, a finite summation.
This is because of the periodicity of the complex sinusoid,
e j 2 kn / N F
in harmonic number k . That is, if k is increased by any integer
multiple of N F the complex sinusoid does not change.
e  j 2 kn / N F  e
 j 2  k  mN F n / N F
, (m an integer)
This occurs because discrete time n is always an integer.
Fourier Series
In the very common case in which the representation time is
taken as the fundamental period N 0 the DTFS is
x  n 

k  N0
X k e
j 2 kn / N 0
1

 X k  
N0
FS

n N0
x  n  e  j 2 kn / N0
5
CT Fourier Series Definition
The Fourier series representation x F  t  of a signal x(t )
over a time t0  t  t0  TF is
x F t  

j 2 kf F t
X
k
e



k 
where X[k] is the harmonic function, k is the harmonic
number and f F  1/ TF . The harmonic function
can be found from the signal as
1
X k  
TF
t0 TF

x  t  e  j 2 kf F t dt
t0
The signal and its harmonic function form a Fourier series
FS
pair indicated by the notation x  t  
 X  k .
6
CTFS Properties
Let a signal x(t ) have a fundamental period T0 x and let a
signal y(t ) have a fundamental period T0 y . Let the CTFS
harmonic functions, each using a common period TF as the
representation time, be X[k ] and Y[k ]. Then the following
properties apply.
Linearity
FS
 x  t    y  t  
 X  k    Y  k 
7/28/2017
Dr. Abid Yahya
7
CTFS Properties
FS
x  t  t0  
 e j 2 kf0t0 X  k 
Time Shifting
FS
x  t  t0  
 e jk0t0 X  k 
7/28/2017
8
CTFS Properties
FS
Frequency Shifting e j 2 k0 f0t x  t  
 X  k  k0 
(Harmonic Number jk  t
FS
0 0
e
x
t

 X  k  k0 

Shifting)
A shift in frequency (harmonic number) corresponds to
multiplication of the time function
by a complex exponential.
FS
x

t

 X  k 


Time Reversal
7/28/2017
9
CTFS Properties
Change of Representation Time
FS
With TF  T0 x , x  t  
 X k 
FS
With TF  mT0 x , x  t  
 Xm  k 
X  k / m  , k / m an integer
Xm k   
, otherwise
0
(m is any positive integer)
7/28/2017
Dr. Abid Yahya
10
CTFS Properties
Change of Representation Time
7/28/2017
11
CTFS Properties
Time Differentiation
FS


d
FS
x
t

 j 2 kf 0 X  k 




dt
d
FS
x
t

 jk0 X  k 




dt
FS


7/28/2017
12
CTFS Properties
Time Integration
Case 1
Case 1. X  0  0
t
Case 2
X k 

 x    d  
j 2  kf 
FS

0
t
X k 

 x    d  
j  k 
FS

0
Case 2. X  0  0
t
 x    d  is not periodic

7/28/2017
. J. Roberts - All Rights
Reserved
13
CTFS Properties
Multiplication-Convolution Duality
FS
x  t  y  t  
 X k   Y k 
(The harmonic functions, X[ k ] and Y[ k ], must be based
on the same representation period TF .)
FS
x  t  # y  t  
T0 X  k  Y  k 
The symbol # indicates periodic convolution.
Periodic convolution is defined mathematically by
x  t  # y  t    x   y  t    d
T0
x t  y t   x ap t  y t  where x ap t  is any single period of x t 

7/28/2017
14
Fourier Series(DTFS)
x F  n 

k  NF
X  k  e j 2 kn / N F
1
X k  
NF
n0  N F 1

n n0
x  n  e  j 2 kn / N F
where N F is the representation time and the notation

k  NF
means
a summation over any range of consecutive k’s exactly N F in length.
7/28/2017
15
Fourier Series(DTFS)
Notice that in
x F  n 

k  NF
X  k  e j 2 kn / N F
the summation is over exactly one period, a finite summ
This is because of the periodicity of the complex sinusoid
e j 2 kn / N F
in harmonic number k . That is, if k is increased by any integer
multiple of N F the complex sinusoid does not change.
e  j 2 kn / N F  e
 j 2  k  mN F n / N F
, (m an integer)
This occurs because discrete time n is always an integ
7/28/2017
16
Fourier Series(DTFS)
In the very common case in which the representation time is
taken as the fundamental period N 0 the DTFS is
x  n 
7/28/2017

k  N0
X k e
j 2 kn / N 0
1

 X k  
N0
FS

n N0
x  n  e  j 2 kn / N0
17
DTFS Properties
Let a signal x[n] have a fundamental period N 0 x and let a
signal y[n] have a fundamental period N 0 y . Let the DTFS
harmonic functions, each using a common period N F as
the representation time, be X[ k ] and Y[k ]. Then the following
properties apply.
Linearity
FS
 x  t    y  t  
 X  k    Y  k 
7/28/2017
18
DTFS Properties
Time Shifting
7/28/2017
FS
x  n  n0  
 e j 2 kn0 / NF X  k 
19
DTFS Properties
Frequency Shifting
j 2 k0n / N F
FS
e
x
n

 X  k  k0 


(Harmonic Number
Shifting)
Conjugation
FS
x*  n 
 X*   k 
Time Reversal
FS
x  n 
 X  k 
7/28/2017
20
DTFS Properties
Time Scaling
Let z  n   x  an  , a  0
If a is not an integer, some values of z[n] are undefined
and no DTFS can be found. If a is an integer (other than
1) then z[n] is a decimated version of x[n] with some
values missing and there cannot be a unique relationship
between their harmonic functions. However, if
 x  n / m  , n / m an integer
z  n  
, otherwise
0
then
Z  k   1/ m  X  k  , N F  mN 0
7/28/2017
21
DTFS Properties
Change of Representation Time
FS
With N F  N 0 x , x  n  
 X k 
FS
With N F  qN 0 x , x  n  
 Xq k 
X  k / q  , k / q an integer
Xq k   
, otherwise
0
(q is any positive integer)
7/28/2017
22
DTFS Properties
FS
x
n
y
n

 Y k  # X k    Y q X k  q




Multiplicationq  N0
Convolution
FS
x
n
#
y
n

 N0 Y k  X k 




Duality
FS
First Backward x  n   x  n  1 
 1  e  j 2 k / N F  X  k 
Difference
7/28/2017
Dr. Abid Yahya
23
The Fourier Transform
Extending the CTFS
• The CTFS is a good analysis tool for systems
with periodic excitation but the CTFS cannot
represent an aperiodic signal for all time
• The continuous-time Fourier transform
(CTFT) can represent an aperiodic signal for
all time
7/28/2017
Dr. Abid Yahya
25
Definition of the CTFT
Forward
X  f   F  x t  


Inverse
f form
x  t  e  j 2 ft dt x  t   F

X  j   F  x  t   

 X  f    X  f  e
-1
 j 2 ft
x t  e
Inverse
 form
 jt
dt x  t   F

1
X
j





 2
-1

 jt
X
j

e
d




Commonly-used notation:
F
x  t  
X f 
7/28/2017
df

Forward


F
or x  t   X  j 
26
Some Remarkable Implications
of the Fourier Transform
The CTFT expresses a finite-amplitude, real-valued,
aperiodic signal which can also, in general, be time-limite
as a summation (an integral) of an infinite continuum of
weighted, infinitesimal-amplitude, complex sinusoids, eac
of which is unlimited in time. (Time limited means “havin
non-zero values only for a finite time.”)
7/28/2017
27
The Discrete-Time Fourier
Transform
Extending the DTFS
• Analogous to the CTFS, the DTFS is a good
analysis tool for systems with periodic
excitation but cannot represent an aperiodic
signal for all time
• The discrete-time Fourier transform
(DTFT) can represent an aperiodic signal for
all time
7/28/2017
Dr. Abid Yahya
29
Definition of the DTFT
F Form
Inverse
x  n   X  F  e
1
Inverse
1
x  n 
2
7/28/2017

2
j 2 Fn
dF  X  F  
F
Forward

 j 2 Fn
x
n
e



n
 Form
Forward
F
X  e j  e jn d 
 X  e j  

 jn
x
n
e



n
30
The Four Fourier Methods
7/28/2017
31
Relations Among Fourier
Methods
Multiplication-Convolution Duality
Continuous Time
Discrete Time
Continuous Time
Discrete Time
7/28/2017
Discrete Frequency
Continuous Frequency
FS
x  t  y  t  
 X k   Y k 
F
x  t  y  t  
X f Y f 
FS
F
x  n  y  n  
 Y  k  # X  k  x  n  y  n  
XF # YF 
Discrete Frequency
Continuous Frequency
FS
x  t  # y  t  
 T0 X  k  Y  k 
F
x  t   y  t  
X  f Y  f 
FS
F
x  n  # y  n  
 N 0 Y  k  X  k  x  n   y  n  
XF YF 
Dr. Abid Yahya
32
Relations Among Fourier
Methods
Time and Frequency Shifting
Discrete Frequency
Continuous Time
FS
x  t  t0  
 X k e
Discrete Time
FS
x  n  n0  
 X k e
 j  k0 t0
 j  k 0 n0
Discrete Frequency
 j  k00 t
FS

 X  k  k0 
 j  k0  0  n
FS

 X  k  k0 
Continuous Time
x t  e
Discrete Time
x  n e
7/28/2017
Dr. Abid Yahya
Continuous Frequency
F
x  t  t0  
 X  j  e  jt0
F
x  n  n0  
 X  e j  e  jn0
Continuous Frequency
F
x  t  e  j0t 
 X  j    0  

F
x  n  e  j0n 
X e
j  0 
33

,
Tutorials
1. Compute the CTFS:
x
(
t)

4
c
o
s
(
5
0
0

t)TF 1/50
7/28/2017
34
2. Find the frequency-domain representation of the signal in Figure
3.1 below.
Figure 3.1: Time Domain Signal.
Solution:
Step 1: Determine N and 0.
The signal has period N=5, so 0=2/5.
Also the signal has odd symmetry, so we sum over n = -2 to n = 2
from equation
35
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
X k  
N
N 1
 xne
 jk o n
n 0
1 2
X k    xne  jk 2n / 5
5 n  2
1
 x 2e jk 4 / 5  x 1e jk 2 / 5  x0e j 0  x1e  jk 2 / 5  x2e  jk 4 / 5
5


36
Cont’d…
From the value of x{n} we get,
1  1 jk 2 / 5 1  jk 2 / 5 
X k   1  e
 e

5 2
2

1
 1  j sin k 2 / 5
5
37