ةبيط ةعماج
EE 372 – Communication Theory and Systems I Lecture 3: Review of Signals
Omar Siddiqui
Department of Electrical Engineering College of Engineering
Taibah University
Signals
Periodic and aperiodic
Deterministic or Random Analog and Digital (amplitude)
Continuous or Discrete (time)
Signal Classification
Energy and Power
Signal Classification: Analog Vs Digital (in amplitude)
Analog: The signal amplitude can take any value at each instant of time
Amplitude t
Digital: The signal amplitude can take only specific values at each instant of time
Time: continuous, Amplitude: continuous
t
Amplitude
Time: continuous, Amplitude: Discrete
Signal Classification: Continuous or Discrete (in time)
Continuous-time Signal: x(t) where t can take any real value
V(t)
Voltage
Time (t)
Discrete time Signal:x[n] where n {...-3,-2,-1,0,1,2,3...}
n can only be integer
Example: cos(t)
V(t)
Voltage
Time (t) Example: cos(n)
Signal Classification: Deterministic or Random
Deterministic Signal: The value of the signal can be predicted.
- Usually they have a mathematical form
V(t)
Voltage
Random Signal: The signal values cannot be predicted - No Mathematical form
Example: cos(t)
V(t)
Voltage
Signal Classification: Periodic and Aperiodic
Aperiodic Signal: It does not repeat itse. The time period can be defined as:
Examples: cos(t) sin(t)
V(t) g(t) = g(t+To)
Periodic Signal: It repeats itself with a time period of To such that:
o
T
Signal Classification: Energy and Power Signals
Energy Signal: A Signal with finite energy an energy signal
dt t
g
)
2(
Power Signal: A Signal with finite power is a power signal
dt t
T g
T
T T
2 /
2 /
)
21 (
lim
0
Some Basic Signal Operations
Time Shifting
) ( t T g )
(t
g g ( t T )
Time Scaling
) (t
g g ( t 2 ) g (t / 2 )
Some Basic Signal Operations
Time Inversion (folding)
) ( t g )
(t g
) (t
g g ( t )
The Unit Step Function
0 0
0 ) 1
( t
t t u
Definition
1
) (t u
0 t
The Unit Step Function (To resolve the discontinuity)
- There is a discontinuity at time t = 0
- To resolve the dicontinuity, the u(t) can be assumed to be an increasing function
- u(t) can be defined as:
) ( lim
)
( t
0u t
u
1
) (t u
1
) (t u
0 t
t
The Unit impulse function
Lets take the derivative of u(t). Because of the sudden jump at t=0, the rate of change
approaches infinite
0 0 1
) 0
( dt
du
The derivative of u(t) is called an impulse function and is represented by an arrow
1
) (t u
0 t
Sudden jump
1
)
(t
0 t
dt t t du ( )
) (
The Unit impulse function as a limiting form of a ‘pulse’
Start from the Limiting Unit step function and find its derivative assuming approaches zero
dt t t du( )
) (
dt
t
t du ( )
lim )
(
0
) ( lim )
(
0
u t t
u
But
1
) (t u
1
) (t u
1
) (t u
1
dt t du( )
1
1 dt
t du( )
1 1
dt t
du( ) 1
1
) (t u
dt 1 t
du )( Area
Sifting Properties of Impulse Function Multiplication of a function by an Impulse
) ( ) ( t t
0
)
(t )
(t 0
x =
)
(t )
( t T
=
)
(T
) (
)
( t t T
( T ) ( t T )
0
) 0
( (t ) )
0
( (t )
) (
)
( T t T )
(T
Sifting Properties of Impulse Function Integration of a function by an Impulse
)
(t )
( t T
0
=
0 )
(T
) (
)
( t t T
( T ) ( t T )
) (
)
( T t T
dt T t
t ) ( )
(
dt T t
t ) ( )
(
T T T
The Trigonometric Fourier Series
Trigonometric Form of Fourier Series: Any periodic function can be decomposed into sum of sine and cosine functions with frequencies that are multiples of the fundamental frequency
1
) 2
sin(
) 2
cos(
) (
n
o n
o n
o
a nf t b nf t
a t
g
Where the coefficients ao, an, bn are the integrals given by:
2 /
2 /
) 1 o (
o
T
o T
o g t dt
a T
2 /
2 /
) 2
cos(
) 2 o (
o
T
T
o o
n g t nf t dt
a T
2 /
2 /
) 2
sin(
) 2 o (
o
T
T
o o
n g t nf t dt
b T
Graphical Representation
= + +
The Compact Trigonometric Fourier Series
The compact form is obtained by combining the sine and cosine terms
1
) 2
sin(
) 2
cos(
) (
n
o n
o n
o a nf t b nf t
a t
g
1
) 2
cos(
n
n o
n
o C nf t
C
) 2
cos( o n
n nf t
C
) 2
sin(
) 2
cos( nf t b nf t
an o n o
2 2
n n
n a b
C
n n
n a
1 b
tan
o
o a
C
Example of the Compact Trigonometric Fourier Series
Example 2.8: Find the trigonometric Fourier series
Solution
Example of the Compact Trigonometric Fourier Series
Example 2.8: Find the compact trigonometric Fourier series
bn=0
2 2
n n
n a b
C
ao=1/2
C 2
o
o a
C
2
1
Co
To find the compact form:
...
9 , 7 , 5 , 3 ,
1 n
for Cn 0 for n 2,4,6,8,10...
Example of the Compact Trigonometric Fourier Series
n n
n a
1 b
tan
n n
/ 2 tan 1 0
n n
/ 2 tan 1 0
,...
9 , 5 ,
1
n for
,...
11 , 7 ,
3 forn
x y
0
0
n
n
or
2
1 1 1 1
bn=0
Example of the Compact Trigonometric Fourier Series
2 2 ) 1
(t
w cos2fot 3cos6 ( )
1
fo t cos10fot 5
1 cos14 ( )
7
1
fo t cos18 ...
9
1
fot
Spectrum of the Compact Fourier Series
Cn n2 2
1 Co
...
9 , 7 , 5 , 3 ,
1 n for
0
Cn for n 2,4,6,8,10...
In terms of
Demonstration of how the harmonics add up
2 2 ) 1
(t
w cos2fot cos2 (3 )( ) 3
1
fo t cos2 (5fo)t 5
1
cos2 (7 )( )
7
1
fo t cos2 (9 ) ...
9
1
fo t
0 1
2
1
0 f
Fourier Components N
1
3
3 30
2 3f0 3
1
0 2
2
0 1
0 2
2
0
1
2 A
3
2 A
=
N
n
o n
o a nf t
a
1
) 2
cos(
ao
=
ao
0 2
2
1
1
First
0 2
2
01
2
1
0 f
1
1
2 A
First
01
2
1
0 f
1
2 A
+
0 2
2
1
1
Third
3 30
2 3f0 3
1 3 1
2
A 505
2 5f0 5
5
2 A
Demonstration of how the harmonics add up
2 2 ) 1
(t
w cos2fot cos2 (3 )( ) 3
1
fo t cos2 (5fo)t 5
1
cos2 (7 )( )
7
1
fo t cos2 (9 ) ...
9
1
fo t
Fourier Components
N
n
o n
o a nf t
a
1
) 2
cos(
N
0 2
2
01
2
1
0 f
5
+
0 2
2
3 30
2 3f0 3
0 2
2
5
50
2 5f0 5
+
7
0 2
2
0 2
2
0
2
2
0
1 0
1
1 1
1
1
2
A 5
2 A
ao
=
=
ao
Third Fifth
+ 3rd + 5th
7 70
2 7f0 7
+
1
1 7
2 A
Seventh First
0 2
2
01
2
1
0 f
1
1
2 A
First
1 A 2 19
Gibb’s Phenomenon
0 2
2
0 1
0 2
2
Gibb’s Phenomenon
The Exponential Fourier Series
0
2 0
0
) 1 (
T
t f jn
n
g t e dt
D T
A periodic function x(t) can be decomposed into a linear combination of the harmonically related exponentials:
t jn n
n
e D t
g ( ) 0
The constants (weight of each harmonic) can be calculated as:
t f jn n
n
e D t
g ( ) 2 0
The Exponential Fourier Series
Examples 2.7 and 2.10
Find the trignometric and exponential Fourier series of the following function
Solution of Trigonometric Series Solution of Exponential Series
Reading from the book
Chapter 2: Signals and Signal Space, Pages 20 - 67
B.P. Lathi, “Modern Digital and analog Communication Systems”, 4th Edition
