conept map gate.pdf

1
SIGNALS AND SYSTEMS
For
Graduate Aptitude Test in Engineering
SIGNALS AND SYSTEMS
For
Graduate Aptitude Test in Engineering
By
Dr. N. Balaji
Professor of ECE
JNTUK

2
Session: 2
Topic : Classification of Signals and Systems
Date : 12.05.2020
Session: 2
Topic : Classification of Signals and Systems
Date : 12.05.2020
By
Dr. N. Balaji
Professor of ECE

3
Syllabus
Syllabus
• Continuous-time signals: Fourier series and Fourier transform representations,
sampling theorem and applications; Discrete-time signals: discrete-time Fourier
transform (DTFT), DFT, FFT, Z-transform, interpolation of discrete-time signals;
• LTI systems: definition and properties, causality, stability, impulse response,
convolution, poles and zeros, parallel and cascade structure, frequency response,
group delay, phase delay, digital filter design techniques.

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Contents
Contents
 Unit Impulse, Unit Step, Unit Ramp functions and their Properties
 Example Problem on properties of the functions.
 Classification of Systems
 Causal and Non-causal Systems
 Linear and Non Linear Systems
 Time Variant and Time-invariant Systems
 Stable and Unstable Systems
 Static and Dynamic Systems
 Invertible and non-invertible Systems
 Solved Problems of previous GATE Exam

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Unit Impulse Function
 One of the more useful functions in the study of linear systems is an Unit Impulse Function.
 An ideal impulse function is a function that is zero everywhere but at the origin, where it is
infinitely high. However, the area of the impulse is finite.
• The unit impulse function,
= undefined for t=0 and has the following special property
Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff

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• A consequence of the delta function is that it can be approximated by a narrow pulse as
the width of the pulse approaches zero while the area under the curve =1.
→

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;

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Representation of Impulse Function
Representation of Impulse Function
• The area under an impulse is called its strength or weight. It is represented graphically
by a vertical arrow. An impulse with a strength of one is called a unit impulse.

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Unit Impulse Train
Unit Impulse Train
• The unit impulse train is a sum of infinitely uniformly- spaced impulses and is given by

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• is an even signal
• It is a neither energy nor power signal.
• Weight/strength of impulse
• Area of weighted impulse
= weight of impulse
Properties of an Impulse Function
Properties of an Impulse Function

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• Eg:-
• Scaling property of impulse:-
= )
Scaling Property of an Impulse Function
Scaling Property of an Impulse Function

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Multiplication property of an Impulse Function

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Sampling Property of an Impulse Function
Sampling Property of an Impulse Function
• The Sampling Property

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Example Problem based on Sampling property
Example Problem based on Sampling property

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Proof of Sampling Property
Proof of Sampling Property

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Example problem on Sampling Property

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Example Problem
Example Problem

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Derivatives of impulse function
Derivatives of impulse function

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Example problem for Derivatives of an Impulse function
Example problem for Derivatives of an Impulse function

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Unit Step Function
Unit Step Function

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Unit Ramp Function
Unit Ramp Function
     
, 0
ram p u u
0 , 0
t
t t
t d t t
t
 
 

 
  
 

 

•The unit ramp function is the integral of the unit step function.
•It is called the unit ramp function because for positive t, its slope is one amplitude
unit per time.

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Relation among Ramp, Step and Impulse Signals
Relation among Ramp, Step and Impulse Signals
Acknowledgement : Lecture slides from http://DrSatvir.in

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Sinusoidal and Exponential Signals
Sinusoidal and Exponential Signals
 Sinusoids and exponentials are important in signal and system analysis because
they arise naturally in the solutions of the differential equations.
 Sinusoidal Signals can expressed in either of two ways :
cyclic frequency form- A sin (2Пfot) = A sin(2П/To)t
radian frequency form- A sin (ωot)
ωo = 2Пfo = 2П/To
To = Time Period of the Sinusoidal Wave

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Sinusoidal and Exponential Signals Contd.
Sinusoidal and Exponential Signals Contd.
x(t) = A sin (2Пfot+ θ)
= A sin (ωot+ θ)
x(t) = Aeat Real Exponential
= Aejω̥t = A[cos (ωot) +j sin (ωot)] Complex Exponential
θ = Phase of sinusoidal wave
A = amplitude of a sinusoidal or exponential signal
fo = fundamental cyclic frequency of sinusoidal signal
ωo = radian frequency
Sinusoidal signal

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x(t) = e-at x(t) = eαt
Real Exponential Signals and damped Sinusoidal

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Signum Function
Signum Function
   
1 , 0
sg n 0 , 0 2 u 1
1 , 0
t
t t t
t

 
 
   
 
 
 
 
Precise Graph Commonly-Used Graph
The Signum function, is closely related to the unit-step
function.

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Rectangular Pulse or Gate Function
Rectangular Pulse or Gate Function
Rectangular pulse,
 
1 / , / 2
0 , / 2
a
a t a
t
t a

 

 




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The Unit Triangle Function
The Unit Triangle Function
A triangular pulse whose height and area are both one but its base width is not one, is called unit triangle function. The
unit triangle is related to the unit rectangle through an operation called convolution.

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Sinc Function
Sinc Function
 The unit Sinc function is
related to the unit Rectangle
function through the Fourier
Transform.
 It is used for noise removal in
signals
 
 
sin
sinc
t
t
t




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Sinc Function
Sinc Function

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Sampling Function
Sampling Function

34 Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff

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Introduction to System
Introduction to System
• Systems process input signals to produce output signals
• A system is a combination of elements that processes one or
more signals to accomplish a function and produces output.
system
output signal
input signal

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Types of Systems
Types of Systems
• Causal and non-causal
• Linear and Non Linear
• Time Variant and Time-invariant
• Stable and Unstable
• Static and Dynamic
• Invertible and non-invertible Systems

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Causal, Anti Causal and Non-Causal Signals
Causal, Anti Causal and Non-Causal Signals
• Causal signals are signals that are zero
for all negative time(or spatial
positions).
• Anticausal are signals that are zero for
all positive time.
• Non-causal signals are signals that
have nonzero values in both positive
and negative time.
Causal signal

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Causal and Non-causal Systems
• Causal system : A system is said to be causal if the present
value of the output signal depends only on the present
and/or past values of the input signal.
• Examples: 1. y[n]=x[n]+1/2x[n-1]
2. y(t) = x(t)
3. y(t) = x(t-1)
4. y(t) = x(t) + x(t-1)

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• Non-causal system : A system is said to be Non-causal if the present
value of the output signal depends also on the future values of the
input signal.
• Example: 1. y[n]=x[n+1]+1/2x[n-1]
2. y(t) = x(t+1)
3. y(t) = x(t) + x(t+1)
4. y(t) = x(t-1) + x(t+1)
5. y(t) = x(t-1) + x(t) + x(t+1)

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Exercise on Causal and Non-causal Systems
Exercise on Causal and Non-causal Systems
Q) Check whether the following are casual or non-casual
system.
1. y(t) = x(2t) 7. y(t) =
2. y(t) = x(-t) 8. y(t) =
3. y(t)= x(sin t) 9.y(t) =
4. y
5. y(t) = odd [x(t)]
6. y(t) = sin (t+2) x(t-1)

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Solution to the Problems
1. y(t) = x(2t)
Substitute t=1 in the above then y(1) = x(2)
Hence the given System is Non-Casual
2. y(t) = x(-t)
Substitute t=1 in the above then y(-1) = x(1)
Hence the given System is System is non-casual
3. y(t) = x(sin t)
Substitute t= - in the above y(-Π) = x(0) (- Π = -3.14)
System is non-casual

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4. y
substitute t= -1 y(-1) = x(-2), which is past value of input.
0, substitute t = 1 y(1) = x(0), which is past.
System is Casual.
5. y(t) = odd x(t)
y(t) =
x(t) – x(−t)
substitute t=-1 then y(-1) =
x(−1) – x(1)
, which
is dependent on future value. Hence the given System is non-casual

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Solution to the Problems on Causal and non-causal System
Solution to the Problems on Causal and non-causal System
6. y(t) = sin(t+2) x(t-1)
put t = 1
y(1) = sin(3) x(0)
Constant coefficient Past value
System is casual
7. y(t) =
y(t) =
Present output depends on present
and past values
Hence, the system is casual
8. y(t) =
y(t) =
Present output depends on future values
also
Hence, the system is non-casual
9. y(t) =
y(t) =
Present output depends on future values
also. Hence, the system is non-casual

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Linear and Non Linear Systems
Linear and Non Linear Systems
• A system is said to be linear if it satisfies the principle of superposition or if it satisfies the
properties of Homogeneity and Additivity.
• Consider a system where an input of x1[t] produces an output of y1[t]. Further suppose
that a different input, x2[t], produces another output, y2[t]. The system is said to
be additive, if an input of x1[t] + x2[t] results in an output of y1[t] + y2[t], for all possible
input signals.
• Homogeneity means that a change in the input signal's amplitude results in a
corresponding change in the output signal's amplitude. In mathematical terms, if an input
signal of x[t] results in an output signal of y[t], an input of cx[t] results in an output
of cy[t], for any input signal and c is a constant.

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Linearity Condition
Linearity Condition
 For the system to be linear, it should satisfy two properties
then
Then Additivity
B Scaling (or) Homogeneity
or
b Super position

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For linearity:
1.Output should be zero for zero input.
2.There should not be any nonlinear operation
Example : The functions like Sin, Cos, tan, Cot, Sec,
cosec, Log, Exponential, Modulus, Square, Cube, Root,
Sampling function(), sinc(), Sgn() etc.…. have nonlinear
operations.
Linearity Condition
Linearity Condition

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Problem based on Linearity and Non-Linearity
Problem based on Linearity and Non-Linearity
1. y(t) = x(t) + 2
If input is (t), then (t) is output
If input is (t) then (t) is output
If input is (t) + (t) then the output must be (t) + (t)
(t) (t) = (t) + 2
(t) (t) = (t) + 2
(t) + (t) (t) + (t)
(t) + 2 + (t) + 2
= (t) + (t) + 4
(t) + (t) ≠ (t) + (t) + 4
Hence the system is non-linear

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2.
Check whether the given system is linear or nonlinear

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Time Invariant and Time Variant Systems
Time Invariant and Time Variant Systems
• A system is said to be time invariant if a time delay or time advance of
the input signal leads to a identical time shift in the output signal.

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Time Variant and Time in Variant System
Time Variant and Time in Variant System

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Stable and Unstable Systems
• A system is said to be bounded-input bounded- output stable
(BIBO stable) if every bounded input results in a bounded
output.

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Stable and Unstable Systems Contd.
Example
y[n]=1/3(x[n]+x[n-1]+x[n-2])
3
y[n] 
1
x[n] x[n 1] x[n  2]

1
(| x[n]|  | x[n 1]|  | x[n  2]|)
3
x x x x

1
(M  M  M )  M
3

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Example: The system represented by y(t) = A x(t) is
unstable ; A˃1
Reason: let us assume x(t) = u(t), then at every instant
u(t) will keep on multiplying with A and
hence it will not result in a bounded value and it may
tend to infinite value.

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1. y(t) = x(t) +2
put x(t) = 10
y(t) = 10 + 2
= 12
As input is bounded value, output is also a bounded value.
Hence System is Stable
2. y(t) = t x(t)
put x(t) = 10
y(t) = 10t
As ‘t’ can be any value between -∞ to ∞,
y(t) is unbounded. Hence System is Unstable

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Problems on Stable and Unstable Systems
Problems on Stable and Unstable Systems
3. y(t) =
put x(t) = 2
y(2) =
When ‘t’ is 0 and Π, then sin(t) has values of sin 0 = 0 and
Sin(Π) = 0 respectively.
Therefore, y(t) = i.e., y(t) is unstable as output is not bounded

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Static Systems
Static Systems
• A static system is memoryless system
• It has no storage devices
• Its output signal depends on present values of the input
signal
• For example

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Dynamic Systems
Dynamic Systems
• A dynamic system possesses memory
• It has the storage devices
• A system is said to possess memory if its output signal
depends on past values and future values of the input signal

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Example: Static or Dynamic?

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Answer:
• The system shown above is RC circuit
• R is memoryless
• C is memory device as it stores charge because of which
voltage across it can’t change immediately
• Hence given system is dynamic or memory system

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Examples
Examples

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Exercise Problems
Exercise Problems
Check whether given system is Static or Dynamic
1. y(t) = x(t) + x(t-1)
2. y(t) = x(-t)
3. y(t) = x(sin t)
4. y(t) = x(t-1)
5. y(t) = Even [x(t)]
6. y(t) = Real [x(t)]

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Invertible & Non-invertible Systems
Invertible & Non-invertible Systems
• If a system is invertible if it has an Inverse System. Otherwise it is
non-invertible system
• Example: y(t)=2x(t)
– System is invertible must have inverse, that is:
– For any x(t) we get a distinct output y(t)
– Thus, the system must have an Inverse
• x(t)=1/2 y(t)=z(t)
y(t)
System
Inverse
System
x(t) x(t)
y(t)=2x(t)
System
(multiplier)
Inverse
System
(divider)
x(t) x(t)

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Check whether the following Systems are invertible

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69

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Previous Gate Questions
Previous Gate Questions

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Gate 2013 question
Gate 2013 question

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Gate 2013 solution
Gate 2013 solution

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Gate 2011 question
Gate 2011 question

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Gate 2011 solution
Gate 2011 solution

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Gate 2010 question
Gate 2010 question

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Gate 2010 solution
Gate 2010 solution

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Gate 2008 question
Gate 2008 question

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Gate 2008 solution
Gate 2008 solution

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Gate 2005 question
Gate 2005 question

80
Gate 2005 solution
Gate 2005 solution

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Gate 2004 question
Gate 2004 question

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Gate 2004 solution
Gate 2004 solution

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Gate 2004 question
Gate 2004 question

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Gate 2004 solution
Gate 2004 solution

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Sources and Reference Material
Sources and Reference Material
Sources:
i) Lecture slides of Michael D. Adams and
ii) Lecture slides of Prof. Paul Cuff
iii) Solved Problems from Standard Textbooks.
 Disclaimer: The material presented in this presentation is taken
from various standard Textbooks and Internet Resources and the
presenter is acknowledging all the authors.

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