Introduction Week#1-2 Signal and Systems F23.ppt

MIMO technology is used for Wi-Fi networks and cellular fourth-generation (4G) Long-Term Evolution (LTE) and fifth-generation (5G) technology in a wide range of markets, including law enforcement, broadcast TV production and government

1. Department of Electrical Engineering,
GCUF.
Signal and Systems 1
Signal and Systems
Week # 1
Instructor: Engr.M.Ubaidullah

2. Course outline
Signal and Systems 2

3. Signal and Systems 3

4. Signal and Systems 4

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IntroductionIntroduction
CHAPTER
1.1 What is a signal?1.1 What is a signal?
A signal is formally defined as a function of one or more variables that A signal is formally defined as a function of one or more variables that
conveys information on the nature of a physical phenomenon.conveys information on the nature of a physical phenomenon.
1.2 What is a system?1.2 What is a system?
Figure 1.1 (p. 2)
Block diagram representation of a system.
A system is formally defined as an entity that manipulates one or more A system is formally defined as an entity that manipulates one or more
signals to accomplish a function, thereby yielding new signals.signals to accomplish a function, thereby yielding new signals.
1.3 Overview of Specific Systems1.3 Overview of Specific Systems

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1.3.1 Communication systems1.3.1 Communication systems
1. Analog communication system: modulator + channel + demodulator1. Analog communication system: modulator + channel + demodulator
Elements of a communication system Fig. 1.2Fig. 1.2

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IntroductionIntroduction
CHAPTER
Figure 1.2 (p. 3)Figure 1.2 (p. 3)
Elements of a communication system. The transmitter changes the message
signal into a form suitable for transmission over the channel. The receiver
processes the channel output (i.e., the received signal) to produce an estimate
of the message signal.

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Modulation:Modulation:
2. Digital communication system: 2. Digital communication system:
sampling + quantization + codingsampling + quantization + coding  transmitter transmitter  channel channel  receiver receiver

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Two basic modes of communication:Two basic modes of communication:
1.1.BroadcastingBroadcasting
2.2.Point-to-point communicationPoint-to-point communication
Radio, televisionRadio, television
Telephone, deep-space
communication
Fig. 1.3Fig. 1.3

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IntroductionIntroduction
CHAPTER
Figure 1.3 (p. 5)
(a) Snapshot of Pathfinder
exploring the surface of Mars.
(b) The 70-meter (230-foot)
diameter antenna located at
Canberra, Australia. The
surface of the 70-meter
reflector must remain accurate
within a fraction of the signal’s
wavelength. (Courtesy of Jet
Propulsion Laboratory.)
Example of Tx and Rx working

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IntroductionIntroduction
CHAPTER

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1.3.2 Control systems1.3.2 Control systems
Figure 1.4 (p. 7)
Block diagram of a feedback control system. The controller drives the plant,
whose disturbed output drives the sensor(s). The resulting feedback signal
is subtracted from the reference input to produce an error signal e(t), which,
in turn, drives the controller. The feedback loop is thereby closed.

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Reasons for using control system: 1. Response, 2. RobustnessReasons for using control system: 1. Response, 2. Robustness

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Closed-loop control system: Closed-loop control system: Fig. 1.4.Fig. 1.4.
1.Single-input, single-output (SISO) system
2.Multiple-input, multiple-output (MIMO) system
Controller: digital
computer
(Fig. 1.5Fig. 1.5.)

9. TEE-364 Digital Communication 9
What is SISO and MIMO Systems?
A fan speed control 
is a very common example of
a Single Input Single Output system. A single input
in terms of voltage is supplied to the system which
in turn results in the fan working.
MIMO technology is used for Wi-Fi networks and
cellular fourth-generation (4G) Long-Term Evolution
(LTE) and fifth-generation (5G) technology in a wide
range of markets, including law enforcement,
broadcast TV production and government.

10. TEE-364 Digital Communication 10
In practical control problems; there are typically
a number of process variables which must be
controlled and a number which can be
manipulated.
Example: product quality and throughput must
usually be controlled.

11. TEE-364 Digital Communication 11

12. TEE-364 Digital Communication 12
Mathematical formulation of 2X2 MIMO System
What will be the
mathematical
expression for SISO
system?

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IntroductionIntroduction
CHAPTER
Analog Versus Digital Signal ProcessingAnalog Versus Digital Signal Processing
Digital approach has two advantages over analog approach:
1.Flexibility
2.Repeatability
3.Reproducibility
1.4 Classification of Signals1.4 Classification of Signals
1. Continuous-time and discrete-time signals
Continuous-time signals: x(t)
Discrete-time signals:( ), 0, 1, 2, .......
s
x n x nT n    (1.1)
Fig. 1-11.Fig. 1-11.
Fig. 1-12.Fig. 1-12.
Parentheses ( )
‧
Brackets [ ]
‧
where t = nT
s

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IntroductionIntroduction
CHAPTER
Figure 1.11 (p. 17)
Continuous-time signal.
Figure 1.12 (p. 17)
(a) Continuous-time signal x(t). (b) Representation of x(t) as a
discrete-time signal x[n].

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Symmetric about vertical axis
IntroductionIntroduction
CHAPTER
2. Even and odd signals
Even signals:( ) ( ) for allx t x t t  (1.2)
Odd signals:( ) ( ) for allx t x t t  (1.3)
Antisymmetric about origin
Example 1.1Example 1.1
Consider the signal
sin ,
( )
0 , otherwise
t
T t T
x t T
 
   
 


Is the signal x(t) an even or an odd function of time?
<Sol.><Sol.>
sin ,
( )
0 , otherwise
sin ,
=
0 , otherwise
= ( ) for all t
t
T t T
x t T
t
T t T
T
x t


 
    
  


 
    




odd function

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Show whether the following functions
are even or Odd?

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IntroductionIntroduction
CHAPTER

◆
Conjugate symmetric:
A complex-valued signal x(t) is said to be conjugate symmetric if
( ) ( )x t x t

  (1.6)
( ) ( ) ( )x t a t jb t 
*
( ) ( ) ( )x t a t jb t 
( ) ( ) ( ) ( )a t jb t a t jb t    
Let
( ) ( )
( ) ( )
a t a t
b t b t
 
 
3. Periodic and nonperiodic signals (Continuous-Time Case)
Periodic signals:( ) ( ) for allx t x t T t  (1.7)
0 0 0
, 2 , 3 , ......T T T T
0
Fundamental periodT T and
1
f
T

Fundamental frequency:
(1.8)
Angular frequency:
2
2f
T

   (1.9)
Refer to
Fig. 1-13
Problem 1-2
Figure 1.13
(p. 20)
(a) One example
of continuous-
time signal.
(b) Another
example of a
continuous-time
signal.

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Signal and Systems

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IntroductionIntroduction
CHAPTER

◆
Example of periodic and nonperiodic signals: Fig. 1-14Fig. 1-14.
Figure 1.14 (p. 21)
(a) Square wave with amplitude A = 1 and period T = 0.2s.
(b) Rectangular pulse of amplitude A and duration T
1
.

◆
Periodic and nonperiodic signals (Discrete-Time Case)
 for integerx n x n N n  (1.10)
N = positive integer
Fundamental frequency of
x[n]:2
N

  (1.11)

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IntroductionIntroduction
CHAPTER
Figure 1.15 (p. 21)
Triangular wave alternative between –1 and +1 for Problem 1.3.
Figure 1.16 (p. 22)
Discrete-time square
wave alternative
between –1 and +1.

◆
Example of periodic and nonperiodic signals:
Fig. 1-16 and Fig. 1-17Fig. 1-16 and Fig. 1-17.

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IntroductionIntroduction
CHAPTER
Figure 1.17 (p. 22)
Aperiodic discrete-time signal
consisting of three nonzero samples.
4. Deterministic signals and random signals
A deterministic signal is a signal about which there is no uncertainty with
respect to its value at any time.
Figure 1.13 ~ Figure 1.17Figure 1.13 ~ Figure 1.17
A random signal is a signal about which there is uncertainty before it
occurs.
Figure 1.9Figure 1.9
5. Energy signals and power signals
Instantaneous power:
2
( )
( )
v t
p t
R

2
( ) ( )p t Ri t
(1.12)
(1.13)
If R = 1  and x(t) represents a current or a
voltage, then the instantaneous power is
2
( ) ( )p t x t (1.14)

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IntroductionIntroduction
CHAPTER
The total energy of the continuous-time signal x(t) is
2 22
2
lim ( ) ( )
T
T
T
E x t dt x t dt


  
  
(1.15)
Time-averaged, or average, power is
22
2
1
lim ( )
T
T
T
P x t dt
T

 

22
2
1
( )
T
TP x t dt
T

2
[ ]
n
E x n

 

21
lim [ ]
2
N
n
n N
P x n
N
 

 
1
2
0
1
[ ]
N
n
P x n
N



(1.16)
For periodic signal, the time-averaged power is
(1.18)

◆
Discrete-time case:
Total energy of x[n]:
(1.17)
Average power of x[n]:
(1.19)
(1.20)

★
Energy signal:
If and only if the total energy of the signal satisfies the condition
0E 

★
Power signal:
If and only if the average power of the signal satisfies the condition
0P 

28. Signal and Systems 28

29. Department of Electrical Engineering,
GCUF.
Signal and Systems 29
Signal and Systems
Week # 2
Instructor: Engr.M.Ubaidullah

30. Signal and Systems 30

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•Unit Impulse Function/ Dirac delta
functions
–The unit impulse function is one of the most important
functions that we will be using extensively in this course.
Yet, it is one of the most difficult functions to understand
and use.
–A couple of definitions exist for the unit impulse function
(t), which is sometimes also called the Dirac delta
function. The following are two definitions:
•Graphical Definition:
–The rectangular pulse shape shown below approaches the
unit impulse function as  approaches 0 (notice that the
area under the curve is always equal to 1).
(t)
t

  


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•Mathematical Definition:
–The unit impulse function (t) satisfies the
following conditions:
–1.(t) = 0 if t  0,
(therefore it is non-zero only at t = 0).
– 2.
(so, all of the area under it is concentrated at t
= 0) .
1)(


dtt
.

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•Properties of the Unit Impulse Function:
–a)Multiplication of a function by the unit
impulse response:
–b)Sampling of a function using the unit
impulse response:
–c)Obtaining the unit step function from the
unit impulse function

0 0 0
( ) ( ) ( ) ( )f t t t f t t t   
)()()()()()()( TfdtTtTfdtTtTfdtTttf  







)(
0,1
0,0
)(
tu
t
t
d
t










)(
)(
t
dt
tdu


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Compact Notation for some useful
Functions

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Compact Notation for some useful
Functions
2) Unit triangle function:

53. Unit Ramp is defined as:
•A signal whose magnitude increases same
as time . It can be obtained by
 
integrating
unit step.
53

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Spectrum:
F.T

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Example 3.6
According to Euler formula:
and
As

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Example 3.6
Spectrum:

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IntroductionIntroduction
CHAPTER
1.6 Elementary Signals1.6 Elementary Signals

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1.6.1 Exponential Signals1.6.1 Exponential Signals( )
at
x t Be (1.31)
B and a are real parameters
1.Decaying exponential, for which a < 0
2.Growing exponential, for which a > 0
Figure 1.28 (p. 34)
(a) Decaying exponential form of continuous-time signal. (b) Growing exponential
form of continuous-time signal.

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IntroductionIntroduction
CHAPTER
Figure 1.30 (p. 35)
(a) Decaying exponential form of discrete-time signal. (b) Growing
exponential form of discrete-time signal.

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IntroductionIntroduction
CHAPTER

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1.6.4 Exponential Damped Sinusoidal Signals1.6.4 Exponential Damped Sinusoidal Signals
( ) sin( ), 0
t
x t Ae t

  

   (1.48)
Example for A = 60,
 = 6, and  = 0: Fig.1.35Fig.1.35.
Figure 1.35 (p. 41)
Exponentially damped
sinusoidal signal Ae
at

sin(t), with A = 60 and 
= 6.

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