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Signals and Systems: Introduction
What is a signal?
Signals may describe a wide variety of physical phenomena
...

A signal is represented mathematically as a function of one
or more independent variables
...

Speech Signal

ECG (Electrocardiogram) Signal

Amplitude, mV

1

Amplitude

0
...
5

-1
0

100

200
Time (ms)

300

400

CEN340: Signals and Systems; Ghulam Muhammad

Time (ms)

2

Examples of Signals - 2
x

x

y

y

t

Intensity of the image at location (x, y)
can be expressed as I (x, y)
...


A video has three independent
variables (x, y, and t (time)),
therefore, it is a three dimensional
signal
...


CEN340: Signals and Systems; Ghulam Muhammad

3

Two Basic Types of Signals

Continuous Signal
A continuous-time (CT)
signal is one that is present
at all instants in time or
space, such as oscillating
voltage signal
...
For example
closing stock market average is
a signal that changes only at
discrete points in time (at the
close of each day)
...


Discrete time
CEN340: Signals and Systems; Ghulam Muhammad

5

Notation of Continuous and discretetime Signals
To distinguish between continuous-time and discrete-time signals,
we will use
• The symbol ‘t’ to denote the continuous-time independent
variable and
• ‘n’ to denote the discrete-time independent variable
...
)’
and for discrete-time signals, we will use brackets ‘[
...


n
Independent
variable
Sampling
period

CEN340: Signals and Systems; Ghulam Muhammad

‘n ’ is always
an integer
...

 A system generally establishes a relationship between its
input and its output
...

Systems that operate on continuous-time signal are known as
continuous-time (CT) systems
...

Input Signal

system

Output Signal

CEN340: Signals and Systems; Ghulam Muhammad

8

Examples of Systems
Automatic speech recognition (ASR)
system

An RLC circuit

ASR System

I am Mr
...
Alan S
...
C
...
Most of the signals in this physical world is ……………
...
Choose
the right one
...
Mention four systems other than those mentioned in the slides
...
Mention three signals other than those mentioned in the slides
...
How can we convert a CT signal into a DT signal?
5
...
What do you mean by time-domain signal and spatial-domain signal?

CEN340: Signals and Systems; Ghulam Muhammad

10

MATLAB

• Matlab ® is a software tool for computation in science and engineering
...

• Originally developed as a “Matrix Laboratory” but now used in
applications in almost all areas of science and engineering
...


• http://www
...
com/help/pdf_doc/matlab/getstart
...
mathworks
...
html

CEN340: Signals and Systems; Ghulam Muhammad

11

1
...
2 Signal Power and Energy
Continuous-time (CT) signal
 The total energy over the time internal t1 ≤ t ≤ t2 in a continuous-time
signal x(t) is defined as t 2
2
|
x
(
t
)
|
dt

t1

where |x| denotes the magnitude of the (possibly complex) number x
...
e
...
1
...
e
...
e
...
For example, in continuous
case, if 𝐸∞ < ∞, then
𝐸∞
𝑃∞ = lim
=0
𝑇→∞ 2𝑇
An example of a finite-energy signal is a signal that takes on the value of 1 for
0 ≤ 𝑡 ≤ 1 and 0 otherwise
...


Case 2: Signals with finite average power, i
...
, 𝑷∞ < ∞:
For example, consider the constant signal where 𝑥[𝑛] = 4
...
In this case
both 𝐸∞ and 𝑃∞ are infinite

CEN340: Signals and Systems; Ghulam
Muhammad

15

Some Frequently
Used Signals

CEN340: Signals and Systems; Ghulam Muhammad

16

Example: Power and Energy
Problem 1:

2t
Find P and E for the signal, x 1 (t )  e u (t )

Solution:



E 

 |x




1

(t ) |2 dt 

 |e

2t





u (t ) |2 dt   | e 2t |2 dt
0



1 1
1 
1
1
=  | e 4t | dt     4(0)  4(  )     1  0  
4 e
e
4
4

0

P is zero, because E < 

CEN340: Signals and Systems; Ghulam Muhammad

17

1
...

We will focus on a very limited but important class of signal
transformations that involves the modifications of the
independent variable, i
...
, the time axis
...

Signals could be termed as delayed or advanced in this case
...
Several receivers placed at different locations receive the time
shifted signals due to the transmission time they take while passing
through a medium (air, water or rock etc
...
For example, if the original
signal is some audio recording, then the time reversed signal would be the audio recording
played backward
...
g
...
If we think of the signal 𝑥(𝑡ሻ
as audio recording, then 𝑥(2𝑡ሻ is the audio recording played at twice the speed and
𝑥(𝑡Τ2ሻ is the recording played at half of the speed
...
It has the
following effects on the original signal:







The general shape of the signal is preserved
...

The signal is linearly compressed if 𝛼 > 1
...

The signal is advanced (shifted in time) if 𝛽 > 0
...

CEN340: Signals and Systems; Ghulam Muhammad

23

Example: Time Shift: (1)
𝑖𝑓 𝑡 < 0
1 𝑖𝑓 0 ≤ 𝑡 < 1
𝑥 𝑡 =൞
2 − 𝑡 𝑖𝑓 1 ≤ 𝑡 < 2
0
𝑖𝑓 𝑡 ≥ 2
0

The signal 𝑥 𝑡 + 1 can be obtained by shifting the given signal to the left by one unit

CEN340: Signals and Systems; Ghulam Muhammad

24

Example: Time Shift: (2)
The signal 𝑥 −𝑡 + 1 can be obtained using the mathematical definition
or figure of the original signal 𝑥 𝑡
...

𝒕
−𝟐
−𝟏
...
𝟓
𝟎
𝟎
...
𝟓
𝟐
𝟐
...
0
2
...
0
1
...
0
0
...
0
−0
...
0
−1
...
0

𝒙 −𝒕 + 𝟏
0
0
0
0
...


CEN340: Signals and Systems; Ghulam Muhammad

25

MATLAB Drill - 1
In MATLAB®, the original signal can be written as an inline function
...

>>g = inline(' ((t>=0)&(t<1)) + (2-t)
...
001:3;
>>subplot(3,1,1),

plot(t,

g(t)),

axis([-3

3

-0
...
1]),

-0
...
1]),

title('Original Signal')
>>subplot(3,1,2),

plot(t,

g(t+1)),

axis([-3

3

title('Time-Shifted Signal')
>>subplot(3,1,3),plot(t,

g(-t+1)),axis([-3

3

-0
...
1]),

title('Time-Reversed Signal')

CEN340: Signals and Systems; Ghulam Muhammad

26

MATLAB Drill – 1: continued
Original Signal
1

0
...
5

0
-3

-2

-1

0
Time-Reversed Signal

1

0
...
2
...

A typical example is that of a sinusoidal signal 𝑥 𝑡 = sin(𝑡ሻ for −∞ < 𝑡 < +∞
...
It can be noticed that for any time 𝑡:
𝒔𝒊𝒏 𝒕 + 2𝝅 = 𝒔𝒊𝒏(𝒕ሻ

𝒔𝒊𝒏 𝒕 + 𝒎2𝝅 = 𝒔𝒊𝒏(𝒕ሻ
where 𝑚 is a positive number
...

A discrete-time signal 𝑥[𝑛] is periodic with period 𝑁, where 𝑁is a positive
integer, if it is unchanged by a time-shift of 𝑁, i
...
, if
𝑥 𝑛 = 𝑥[𝑛 + 𝑁] for all values of 𝑛
...

CEN340: Signals and Systems; Ghulam Muhammad

31

Periodic Signals - Example
cos(t ) if t  0
x (t )  
 sin(t ) if t  0
Since, cos(2+t) = cos(t) and sin(2+t) = sin(t), considering t < 0 and t  0
separately, the signal repeats itself in every interval of 2
...
Therefore, x(t) is not periodic
...
2
...
e
...

Even continuous-time Signal
Even Discrete-time Signal

𝒙 −𝒕 = 𝒙(𝒕ሻ
𝒙[−𝒏] = 𝒙[𝒏]

CEN340: Signals and Systems; Ghulam Muhammad

33

Odd Signals
A signal 𝑥(𝑡ሻ or 𝑥[𝑛] is defined as an odd signal if,
Odd continuous-time Signal

𝒙 −𝒕 = −𝒙(𝒕ሻ

Odd Discrete-time Signal

𝒙 −𝒏 = −𝒙[𝒏]

As a special case, the odd signal must be zero at 𝑡 = 0 or 𝑛 = 0
...


Signal
Continuous-time
Signal 𝒙(𝒕ሻ

Discrete-time
Signal 𝒙[𝒏]

Component
Even Part
Odd Part
Even Part

Odd Part

Mathematical Form
1
2
1
𝒪𝑑 𝑥 𝑡 =
2
1
ℰ𝑣 𝑥[𝑛] =
2
1
𝒪𝑑 𝑥[𝑛] =
2
ℰ𝑣 𝑥 𝑡

CEN340: Signals and Systems; Ghulam Muhammad

=

𝑥 𝑡 + 𝑥(−𝑡ሻ
𝑥 𝑡 − 𝑥(−𝑡ሻ
𝑥[𝑛] + 𝑥[−𝑛]
𝑥[𝑛] − 𝑥[−𝑛]

35

Decomposing a Signal into Even and Odd Parts
Example

=

+

CEN340: Signals and Systems; Ghulam Muhammad

36

1
...

Real Exponential Signals
In this case both 𝐶 and 𝑎 are real numbers, and 𝑥(𝑡ሻ is called a real exponential
...

More, specifically, we consider:
𝑥 𝑡 = 𝑒 𝑗𝜔0𝑡
An important property of this signal is that it is periodic
...
If, 𝜔0 = 0, then 𝑥 𝑡 = 1, which is periodic for any value of 𝑇
...
If, 𝜔0 ≠ 0, then the fundamental period 𝑇0 of 𝑥 𝑡 , i
...
the smallest value of
𝑇 for which the above equation holds, is
2𝜋
𝑇0 =
𝜔0
CEN340: Signals and Systems; Ghulam Muhammad

38

Periodic Signals
Replacing the value of 𝑇with this 𝑇0 , and using Euler’s formula, that is,
𝑒 𝑗𝜔0𝑇 = cos(𝜔0 𝑇ሻ + 𝑗sin(𝜔0 𝑇ሻ
We get
𝑒 𝑗𝜔0 𝑇 = cos 2𝜋 + 𝑗 sin 2𝜋 = 1 + 𝑗0 = 1
Therefore, the signal 𝑥 𝑡 is a periodic signal
...


Sinusoidal Signal:
𝑥 𝑡 = 𝐴 co s( 𝜔0 𝑡 + 𝜙ሻ

Continuous-Time Sinusoidal Signal

x(t)=A cos(0t+)
A

T0 

2

0

A cos()

t

CEN340: Signals and Systems; Ghulam Muhammad

39

Sinusoid Signals
𝐴 cos 𝜔0 𝑡 + 𝜙 = 𝐴

𝑒𝑗

𝜔0 𝑡+𝜙

+ 𝑒 −𝑗(𝜔0𝑡+𝜙ሻ
𝐴 𝑗𝜙 𝑗𝜔 𝑡 𝐴 −𝑗𝜙 −𝑗𝜔 𝑡
0
= 𝑒 𝑒 0 + 𝑒
𝑒
2
2
2

𝐴 cos 𝜔0 𝑡 + 𝜙 = 𝐴 ℜℯ 𝑒 𝑗(𝜔0𝑡+𝜙ሻ
𝐴 sin 𝜔0 𝑡 + 𝜙 = 𝐴 ℑ𝓂 𝑒 𝑗(𝜔0𝑡+𝜙ሻ
The fundamental period 𝑇0 of a continuous-time sinusoidal or a periodic
complex exponential signal, is inversely proportional to the 𝜔0 , which is
called the fundamental frequency
...
Alternatively, if

we increase the value of the magnitude of 𝜔0 ,
we increase the rate of oscillations and hence
decrease the period 𝑇0
...
The total average power is however
remains 1, as by definition,
𝑇

1
1
2
න 𝑒 𝑗𝜔0𝑡 𝑑𝑡 = lim
2𝑇 = 1
𝑇→∞ 2𝑇
𝑇→∞ 2𝑇

𝑃∞ = lim

−𝑇

CEN340: Signals and Systems; Ghulam Muhammad

42

Harmonics of a Periodic Complex Exponential
We have noted that,

𝑒 𝑗𝜔𝑇0 = 1

which implies that 𝜔𝑇0 is a multiple of 2𝜋, i
...
,
𝜔𝑇0 = 2𝜋𝑘 where 𝑘 = 0, ±1, ±2, ⋯
This shows that 𝜔 must be an integer multiple of 𝜔0 , i
...
, the fundamental frequency
...


CEN340: Signals and Systems; Ghulam Muhammad

43

Expressing Two Complex Exponentials into a
Product of One Complex Exp
...
5t e  j 0
...
5t  2e j 2
...
5t )
The magnitude of x(t) is:

| x(t ) | 2 | cos(0
...


CEN340: Signals and Systems; Ghulam Muhammad

44

General Complex Exponential Signals
The general complex exponential signals are of the form
𝑥 𝑡 = 𝐶𝑒 𝑎𝑡
Where both 𝐶 and 𝑎 are complex numbers
...
For 𝑟 = 0, the real and imaginary parts of a complex exponential are sinusoidal
...
For 𝑟 > 0 , they correspond to sinusoidal signals multiplied with growing
exponential
...
For 𝑟 < 0, they correspond to sinusoidal signals multiplied with decreasing
exponentials
...
3
...
This could also be written as
𝑥[𝑛] = 𝐶𝑒 𝛽𝑛
where 𝛼 = 𝑒 𝛽
Real Exponential Signals
In this case both 𝐶 and 𝛼 are real numbers, and 𝑥[𝑛] is called a real exponential
...

CEN340: Signals and Systems; Ghulam Muhammad

47

Example: Real Exponential Signals

x[n] = Cn
(a)  > 1; (b) 0 <  < 1; (c) -1 <  < 0; (d)  < -1
What will happen if (i)  = 1, and (ii)  = -1?
CEN340: Signals and Systems; Ghulam Muhammad

48

Discrete-Time Sinusoid Signals
𝑥[𝑛] = 𝑒 𝑗𝜔0𝑛 = cos 𝜔0 𝑛 + 𝑗 si n( 𝜔0 𝑛൯
Therefore, a discrete-time sinusoid signal can be written as:
𝐴 cos 𝜔0 𝑛 + 𝜙 = 𝐴

𝑒𝑗

𝜔0 𝑛+𝜙

+ 𝑒 −𝑗(𝜔0𝑛+𝜙ሻ
𝐴 𝑗𝜙 𝑗𝜔 𝑛 𝐴 −𝑗𝜙 −𝑗𝜔 𝑛
0
= 𝑒 𝑒 0 + 𝑒
𝑒
2
2
2

Using real and imaginary parts, we find:
𝐴 cos 𝜔0 𝑛 + 𝜙 = 𝐴 ℜℯ 𝑒 𝑗(𝜔0𝑛+𝜙ሻ
𝐴 sin(𝜔0 𝑛 + 𝜙ሻ = 𝐴 ℑ𝓂 𝑒 𝑗(𝜔0𝑛+𝜙ሻ
Both the shaded signals have infinite total energy, but finite average power
...


CEN340: Signals and Systems; Ghulam Muhammad

49

Example: Discrete-Time Sinusoid Signals

CEN340: Signals and Systems; Ghulam Muhammad

50

Discrete-Time Complex Exponential Signals
The general discrete-time complex exponential signals are of the form
𝑥[𝑛] = 𝐶𝛼 𝑛

where both 𝐶 and 𝛼 are complex numbers
...
For 𝛼 = 1, the real and imaginary parts of a complex exponential are sinusoidal
...
For 𝛼 > 1, they correspond to sinusoidal signals / sequences multiplied with
growing exponential
...
For 𝛼 < 1, they correspond to sinusoidal signals / sequences multiplied with
decreasing exponentials
...

But also there are many important differences
...
The larger the magnitude of 𝜔0 , the higher is the rate of oscillations in the
signal;

2
...


To see the difference for the first property, consider the discrete-time
complex exponential:
𝑒 𝑗(𝜔0+2𝜋ሻ𝑛 = 𝑒 𝑗2𝜋𝑛 𝑒 𝑗𝜔0𝑛 = 𝑒 𝑗𝜔0𝑛
This shows that the exponential at 𝜔0 + 2𝜋 is the same as that at frequency 𝜔0
CEN340: Signals and Systems; Ghulam Muhammad

53

Discrete-Time Complex Exponential Signals
In case of continuous-time exponential, the signals 𝑒 𝑗𝜔0𝑡 are all distinct for
distinct values of 𝜔0
...
In fact, the signal with frequency
𝜔0 is identical to signals with frequencies 𝜔0 ± 2𝜋, 𝜔0 ± 4𝜋 and so on
...
The most commonly used 2𝜋 intervals
are 0 ≤ 𝜔0 ≤ 2𝜋 or the interval −𝜋 ≤ 𝜔0 ≤ 𝜋
...
If 𝜔0 is increased from 0 to 2𝜋, the rate of
oscillations first increase and then decreases
...

CEN340: Signals and Systems; Ghulam Muhammad

54

Discrete-Time Complex Exponential Signals
x[n] = cos( n/8)

x[n] = cos(0*n)

x[n] = cos( n/4)

1

1

1

0
...
5

20

0

30

-1

0

10

20

10

20

30

x[n] = cos(2 n)
1

10

30

Start
decreasing
from here

x[n] = cos(15 n/8)
1

0

20

0

1

-1

10

1

0

10

0

x[n] = cos(3 n/2)

1

0

-1

x[n] = cos( n)

1

-1

10

30

0

0

CEN340: Signals and Systems; Ghulam Muhammad

10

20

30

55

Periodicity of Discrete-Time Complex
Exponential Signals
𝑒 𝑗𝜔0(𝑛+𝑁ሻ = 𝑒 𝑗𝜔0𝑛

𝑒 𝑗𝜔0𝑁 = 1
It is true if 𝜔0 𝑁 is a multiple of 2𝜋
𝜔0 𝑚
𝜔0 𝑁 = 2𝜋𝑚
=
2𝜋 𝑁

It means that the discrete-time signal 𝑒 𝑗𝜔0 𝑛 is periodic only when
𝒆𝒋𝝎𝟎 𝒕
Distinct signals for distinct values of 𝝎𝟎
...

Fundamental frequency 𝝎𝟎
...


𝒆𝒋𝝎𝟎 𝒏
Identical signals for values of 𝜔0 separated
by multiples of 2𝜋
...

Fundamental frequency 𝜔0 /𝑚
...
1

1  j
e
2
 (1/ 2)  cos( )  j sin( ) 

 (1/ 2)(1  j 0)

 (1/ 2)(1  j 0)

 (1/ 2)  j (0)

 (1/ 2)  j (0)
1
...
4 (a)

Let, x[n] be a signal with x[n] = 0 for n < -2 and n > 4
...


x[n-3] means shifting the signal towards right by 3 samples
...
8

n > 4  n+ 3 > 4+3 (=7)

0
...
4
0
...


t

CEN340: Signals and Systems; Ghulam Muhammad

59

Workout – (4)

1
...
For the signal x(1 – t),
determine the value of t for which it is guaranteed to be zero
...
8

Amplitude

Amplitude

0
...
6

0
...
2

0
-5

Left shift by 1
and then reflect

0
...
4

0
...


CEN340: Signals and Systems; Ghulam Muhammad

60

Workout – (5)

1
...

EVEN{ x[n]} = 0
...
5 (u[n] – u[n-4] + u[-n] – u[-n-4])
u[n]

1

u[n-4]
1

0
...
5

0
...
4

0
-5

0

5

10

1

0
-5

t

0

5

u[-n] – u[-n-4]

u[-n-4]

t

1

1

0
...
5 (x[n] + x[-n])

0
...
5

0
...
5

0
...
6

Amplitude

0
...
8

Amplitude

u[n] – u[n-4]

1

10

0
...
4
0
...
6
0
...
2
0
-10

-5

0
t

5

10

0
-10

-5

0
t

5

10 0
-4

-2

0

2

4

0
t

5

10

6

t

Zero for n > 3 and n <-3
CEN340: Signals and Systems; Ghulam Muhammad

61

Workout – (6)

1
...
5t), determine the values of the independent variable
at which the even part of the signal is guaranteed to be zero
...
5

0

-0
...


CEN340: Signals and Systems; Ghulam Muhammad

62

1
...

x(t) = A e-at cos(t + ) = -2 = 2 × 1 × (-1) = 2 e-0t cos(0t + )
A = 2, a = 0,  = 0, and  = 
The above problem when the signal is

x (t )  2e j  /4 cos  3t  2 




x (t )  2e j  /4 cos  3t  2   2  cos  j sin  cos  3t  2 
4
4


1
Real part = 2 cos cos  3t  2   2 
 cos 3t  cos 3t
4
2
 1 e 0t  cos(3t  0)

A = 1, a = 0,  = 3, and  = 0
CEN340: Signals and Systems; Ghulam Muhammad

63

Workout – (8)

1
...


x (t )  je j 10t
 j (cos10t  j sin10t )  j cos10t  sin10t
 j sin 10t   / 2   cos 10t   / 2 
e

10t  /2 

Fundamental period:

T0 

2
| 0 |



2 

10 5

CEN340: Signals and Systems; Ghulam Muhammad

64

Workout – (9)

1
...


x (t )  2 cos(10t  1)  sin(4t  1)
T0 

2 

10 5

T0 

2 

4
2

Fundamental period:
LCM ( /5, /2) = LCM (, ) / HCF (5, 2) = /1 = 
a c
LCM  ,
b d

CEN340: Signals and Systems; Ghulam Muhammad


  LCM (a , c ) / HCF (b , d )


65

1
...


x [n ]  1  e j 4 n /7  e j 2 n /5
 2 
N0 m


 0
N 0 (first part)  1
 2 
N 0 (second part)  m 
  m (7 / 2)  7
 4 / 7 
 2 
N 0 (third part)  m 
  m (5 / 2)  5
 2 / 5 
N 0  LCM (1, 7,5)  35
CEN340: Signals and Systems; Ghulam Muhammad

66

Acknowledgement
The slides are prepared based on the following textbook:
• Alan V
...
Willsky, with S
...
, 1997
...
Anwar M
...
Abdul Wadood Abdul Waheed, faculty member, College of Computer
and Information Sciences, King Saud University

CEN340: Signals and Systems; Ghulam Muhammad

67



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