ECE 341
Lecture 6
1
Signal Analysis
ECE 341
Lecture # 6
Discrete-Time Signal (DTS)& Fourier series
Lecturer:
Email :
Dr. Hesham Abd EL Hakim
hesham.fahem@must.edu.eg
2
Representation of DTS
3
Representation of DTS
4
Representation of DTS
5
Representation of DTS
6
Representation of DTS
7
8
Classification of DTS
The smallest value of N is called fundamental period
9
Classification of DTS
10
11
12
13
Periodic Signal Representation – Fourier Series
Ground Rule: All periodic signals are formed by sum of
sinusoidal waveforms


x t   ao   an cos nt   bn sin nt
1
(1)
1
T/2
2
an 
xt  cos ntdt

T T / 2
T/2
1
ao 
xt dt

T T / 2
(2)
T/2
2
bn 
xt  sin ntdt

T T / 2
(3)
Fourier Series – Parseval’s Identity
Energy is preserved after Fourier Transform
1
T


1
2
2
2
2




x
t
dt

a

a

b

o
n
T / 2
2 1 n
T/2


1
1

(4)
x t   ao   an cos nt   bn sin nt
 xt  dt
T/2
2
T / 2
 ao 
T/2
T / 2

xt dt   an 
1
T/2
T / 2

xt  cos ntdt   bn 
1
T/2
T / 2
xt  sin ntdt
Fourier Series – Parseval’s Identity
2




x
t
dt
T / 2
T/2
 ao 
T/2
T / 2

xt dt   an 
1
T/2
T / 2

xt  cos ntdt   bn 
1

T  T
 ao T   an   bn
2 1
2
1
2

T  T
 ao T   an   bn
2 1
2
1
2
1

T
 xt 
T/2
T / 2
2

1  2
dt  ao   a n  bn2
2 1
2

T/2
T / 2
xt  sin ntdt
Periodic Signal Representation – Fourier Series
-T/2
1
x(t) T/2
-t
t
-T/4
T/2
T/4
2
an 
xt  cos ntdt

T T / 2
-1
t
x(t)
-T/2 to –T/4
-1
-T/4 to +T/4
+1
+T/4 to +T/2
-1
2

T
T /4
T /4
T /2

2
    cos ntdt   cos ntdt   cos ntdt 
T  T / 2
T / 4
T /4

T / 4
T /4
T /2

2   sin nt 
 sin nt 
 sin nt  
 






T  n  T / 2  n  T / 4  n  T / 4 
Periodic Signal Representation – Fourier Series
x(t)
1
-t
t
-T/4
T/2
T/4
2
an 
xt  cos ntdt

T T / 2
-1
t
x(t)
-T/2 to –T/4
-1
-T/4 to +T/4
+1
+T/4 to +T/2
-1
2

T
T / 4
T /4
T /2

2   sin nt 
 sin nt 
 sin nt  
 






T  n  T / 2  n  T / 4  n  T / 4 
8
4
 n T 
 n T 

sin 
sin 


n T
 4  n T  2 
Periodic Signal Representation – Fourier Series
-t
2

T
x(t)
1
t
-T/4
-1
T/4
8
4
 n T 
 nT 
an 
sin 
sin 


n T
 4  n T
 2 
t
x(t)
-T/2 to –T/4
-1
-T/4 to +T/4
+1
+T/4 to +T/2
-1
zero for all n
4
 n  2

sin    sin n 
n  2  n
4
We have, ao  0 , a1  , a2  0 , a3 
,.......

3
4
Periodic Signal Representation – Fourier Series
-t
2

T
x(t)
1
t
-T/4
T/4
-1
t
x(t)
-T/2 to –T/4
-1
-T/4 to +T/4
+1
+T/4 to +T/2
-1
It can be easily shown that bn = 0 for all values of n. Hence,
4
1
1
1

x t    cost  cos3t  cos5t  cos7t  ....

3
5
7

Only odd harmonics are present and the DC value is zero
The transformed space (domain) is discrete, i.e., frequency
components are present only at regular spaced slots.
Periodic Signal Representation – Fourier Series
-T/2
A
x(t) T/2
-t
t
-t/2 t/2
t
x(t)
-t/2 to –t/2
A
-T/2 to - t /2
0
+ t /2 to +T/2
0
t/2
1
1
At
ao 
xt dt 
Adt 


T T / 2
T t / 2
T
T/2
2 T2
2 t2
an   T x t cosntdt   TAcosntdt
T 2
T t 2
t
2
2 A  sin nt 
4A
nt


sin


T  n  t nT
2
2
2

T
Periodic Signal Representation – Fourier Series
-T/2
A
x(t) T/2
-t
t
-t/2 t/2
t
2
2 A  sin nt 
4A
nt
an 

sin


T  n  t nT
2
t
x(t)
-t/2 to –t/2
A
-T/2 to - t /2
0
+ t /2 to +T/2
0
2

T
2
It can be easily shown that bn = 0 for all values of n. Hence, we have
At 2 At
x t  

T
T
sin nt / 2 
1 nt / 2 cosnt

Periodic Signal Representation – Fourier Series
At 2 At
x t  

T
T
Note:
sin  y  y  0
Hence: an  0
At
for
sin nt / 2 
1 nt / 2 cosnt

y  nt 2  k
for
nt
2 k
 k  n  
2
t
k 1,2 ,3 ,...
T

0
2
t
4
t