Signals & Systems (CNET - 221)
Chapter-4
Mr. ASIF ALI KHAN
Department of Computer Networks
Faculty of CS&IS
Jazan University
Chapter Objective
Following are the objectives of Chapter-III




Continuous and Discrete LTI Systems
Representation of signal in terms of impulses
Unit Impulse Signal response & Convolution
LTI System Properties
PAGE : 192
Examples : 3.2, 3.3, 3.4, 3.5
Course Description-Chapter-4
Fourier Series
4.1 Introduction Fourier Series Representation Of Continuous- Time Signals
4.2 Fourier Series Representation of Continuous -Time Periodic Signals
4.2.1 Linear Combination of Harmonically related complex Exponentials
4.2.2 Determination of the Fourier Series Representation of a Continuous-Time
Periodic Signal
4.3 Convergence of the Fourier Series
4.4 Properties of Continuous-Time Fourier Series
Linearity, Time Shifting , Time Reversal , Time Scaling , Multiplication ,
Conjugation and Conjugate Symmetry, Parseval's Relation
4.5 Fourier Series Representation of Discrete-Time Periodic Signals
4.5.1 Linear Combination of harmonically related complex exponentials
4.5.2 Determination of the Fourier Series Representation of a Periodic Signal
Fourier Series
Fourier series is just a means to represent a periodic signal as an
infinite sum of sine wave components.
Fourier Series-Decomposition
Example (Square Wave)
f(t)
1
-6 -5 -4 -3 -2 -

2 3
4 5
2 
2 
1

a0 
1
dt

1
a

cos
ntdt

sin
nt
 0 n  1,2,
n


0
0
0
2
2
n
2 / n n  1,3,5,
2 
1
1

bn 
sin ntdt  
cos nt 0  
(cos n  1)  

0
n  2,4,6,
2
n
n
0
1 2
1
1

f (t )    sin t  sin 3t  sin 5t  
2 
3
5

1.5
1
0.5
0
-0.5
Harmonics
Harmonics…….Continued
Harmonics…….Continued
Complex Exponentials
e
e
jn0t
 jn0t
 cos n0t  j sin n0t
 cos n0t  j sin n0t

1 jn0t
cos n0t  e
 e  jn0t
2




1 jn0t
j jn0t
 jn0t
 jn0t
sin n0t 
e
e
 e
e
2j
2

Complex Form of the Fourier Series
Complex Form of the Fourier Series
Complex Form of the Fourier Series
Complex Form of the Fourier Series
Complex Frequency Spectra
Example
f(t)
A
t
T

T
2

d
2
d
2
T
2
T
Example
A/5
-120
-150
-80
-100
 nd 
sin 

Ad
T


cn 
T  nd 


 T 
-40
-50
0
40
50
80
100
120
150
1
1 d 1
, T ,

20
4 T 5
2
0 
 8
T
d
Example
A/10
-120
-80
-40
-300
-200
-100
 nd 
sin 

Ad
 T 
cn 
T  nd 


T


0
40
80
120
100
200
300
1
1 d 1
, T ,

20
2 T 5
2
0 
 4
T
d
Convergence of the CTFS
Convergence of the CTFS
Convergence of the CTFS
CTFS Properties
CTFS Properties………Continued
CTFS Properties………Continued
CTFS Properties………Continued
CTFS Properties………Continued
CTFS Properties………Continued
CTFS Properties………Continued
CTFS Properties………Continued
CTFS Properties………Continued
CTFS Properties………Continued
CTFS Properties………Continued
CTFS Properties………Continued
Some Common CTFS Pairs
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
x(t ) 
 Expressing the signal as
it is also
P

k 

 |c
k 

k
2
|
ck e jk0t , t 
Videos
1. https://www.youtube.com/watch?v=7Z3LE5uM6Y&list=PLbMVogVj5nJQQZbah2uRZIRZ_9kfoqZyx
2. Signals & Systems Tutorial
https://www.youtube.com/watch?v=yLezP5ziz0U&list=PL56ED47
DCECCD69B2