SVCE TIRUPATI

COURSE MATERIAL

SUBJECT SIGNALS & SYSTEMS (EC20APC303)

UNIT III

COURSE B.TECH

DEPARTMENT ECE

SEMESTER II-I

Dr. D. Srinivasulu Reddy

Professor

PREPARED BY
(Faculty Name/s) Mr.V.Mahesh
Assistant Professor

Version V-1

PREPARED / REVISED DATE 22-11-2021

1|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

TABLE OF CONTENTS – UNIT I

S.No. CONTENTS PAGE No.
1 COURSE OBJECTIVES 3
2 PREREQUISITES 3
3 SYLLABUS 3
4 COURSE OUTCOMES 3
5 CO - PO/PSO MAPPING 4
6 LESSON PLAN 4
7 ACTIVITY BASED LEARNING 4
8 LECTURE NOTES 4
8.1 Laplace Transform 4
8.2 Properties Of Roc In Laplace Transform 6
8.3 Properties Of Laplace Transform 7
8.4 Problems 13
8.5 Inverse Laplace transforms 22
8.6 Problems 23
8.7 The S-plane and BIBO stability 25
8.8 Transfer functions 27
8.9 System Response to standard signals 28
8.10 Solution of differential equations with initial 31
conditions.
9 PRACTICE QUIZ 37
10 ASSIGNMENTS 38
11 PART A QUESTIONS & ANSWERS (2 MARKS QUESTIONS) 38
12 PART B QUESTIONS 39
13 SUPPORTIVE ONLINE CERTIFICATION COURSES 41
14 REAL TIME APPLICATIONS 41
15 CONTENTS BEYOND THE SYLLABUS 42

2|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

16 PRESCRIBED TEXT BOOKS & REFERENCE BOOKS 42

17 MINI PROJECT SUGGESTION 42

1. Course Objectives
The objectives of this course is to
 To introduce students to the basic idea of signal and system analysis and its
characterization in time and frequency domains.
 To present Fourier tools through the analogy between vectors and signals.
 To teach concept of sampling and reconstruction of signals.
 To analyze characteristics of linear systems in time and frequency domains.
 To understand Laplace and z-transforms as mathematical tool to analyze continuous
and discrete-time signals and systems.

2. Prerequisites
Students should have knowledge on
1. Differential equations and Integrals
2. Ordinary differential equations
3. Series and expansions

3. Syllabus
UNIT III
Laplace Transform: Definition, ROC, Properties, Inverse Laplace transforms,
the S-plane and BIBO stability, Transfer functions, System Response to
standard signals, Solution of differential equations with initial conditions.
Course outcomes
The outcome of this course is to

1. Understand the limitations of Fourier transform and need for Laplace transform
and develop. (L1)
2. Evaluate response of linear systems to known inputs by using Laplace
transforms. (L2)
3. Classify the systems based on their properties and determine the response
of them.(L4)

3|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

a. CO-PO / PSO Mapping

DSD PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 PO10 PSO1 PSO2

CO1 3 2 3

CO2 3 3 2 2 2

CO3 3 2 2 3

CO4 3 3 3 2 2

b. Lesson Plan
Lecture No. Weeks Topics to be covered References

1 Laplace Transform T1, R1

2 Properties Of Roc In Laplace Transform T1, R1

1
3 Properties Of Laplace Transform T1, R1

4 Problems T1, R1

5 Inverse Laplace transforms T1, R2

6 Problems T1, R1
2
7 The S-plane and BIBO stability T1, R1

8 Transfer functions T1, R1

9 System Response to standard signals T1, R1

3
10 Solution of differential equations with initial T1, R1
conditions.

c. Activity Based Learning

1. Observing CMOS Logic design application to design VLSI concepts.
2. Analyzing TTL families with CMOS logic families to interface CMOS with TTL.

8. Lecture Notes
8.1INTRODUCTION
The Laplace transform can be used to solve differential equations. Besides
being a different and efficient alternative to variation of parameters and

4|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

undetermined coefficients, the Laplace method is particularly advantageous for

input terms that are piecewise-defined, periodic or impulsive.
The Laplace transform can be alternatively defined as the bilateral Laplace
transform or two-sided Laplace transform by extending the limits of integration
to be the entire real axis. The bilateral Laplace transform is defined as follows:
The direct Laplace transform or the Laplace integral of a function f (t) defined


for 0 ≤ t < ∞ is the ordinary calculus integration problem  e

st
f (t) dt , succinctly
0

denoted L (f (t)) in science and Engineering literature. The L–notation recognizes

that integration always proceeds over t = 0 to t = ∞ and that the integral involves
an integrator 𝑒−𝑠𝑡dt instead of the usual dt. These minor differences distinguish
Laplace integrals from the ordinary integrals found on the inside covers of
calculus texts.
Let f(t) be a function of ‘t’, define for all positive values of ‘t’ then the Laplace
transform of f(t) is denoted by L{f(t)} and it is defined as


L {f (t)} =  e
st
f (t) dt = f (s)  F (s)
0

f (t) = L-1 { f (s)}

Where f (t) is called inverse Laplace transformation of f (s).
The symbol “L” is a transformation of f (t) to f (s) then “L” is called Laplace
transform operation. We know that Fourier transform exists if the signals have
finite energy. But for the signals such as ramp, rising exponents etc.This condition
of finite energy is not satisfied. Thus FT does not exist for such signals. By the use of
Laplace transformation this limitation can be avoided.
WKT in FT the variable s = jw.
But in Laplace transform variable s can be expressed as s = σ+jώ
σ - real part of which represents the attenuation factor.
Jώ- imaginary part, ώ- angular frequency.
Laplace transform exists for almost all signals of practical interest. Some of the
advantages of Laplace transform are as follows: Laplace transform can be used
for the analysis of unstable systems.
There are two types of Laplace transform:
1) Bilateral (or) two sided Laplace transform
2) Unilateral (or) one sided Laplace transform
The unilateral Laplace transform is the special case of Laplace transform and is
defined as,

5|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

The unilateral Laplace transform has the following features:

1. The unilateral Laplace transform simplifies the system analysis

considerably.
2. The signals are restricted to casual signals.
3. There is one to one correspondence between LT and ILT.
4. In view of above advantages Laplace transform means unilateral Laplace
transform.
 Bilateral Laplace Transform
The Laplace transform can be alternatively defined as the bilateral Laplace
transform or two-sided Laplace transform by extending the limits of integration
to be the entire real axis.
The bilateral Laplace transform is defined as follows:

The above equation shows that F(s) is basically a Fourier transform of f(t) e-σt.

8.2 PROPERTIES OF ROC IN LAPLACE TRANSFORM

ROC contains strip lines parallel to jω axis in s-plane.
 If x(t) is absolutely integral and it is of finite duration, then ROC is entire s-
plane.
 If x(t) is a right sided sequence then ROC : Re{s} > σo .
 If x(t) is a left sided sequence then ROC : Re{s} < σo.
 If x(t) is a two sided sequence then ROC is the combination of two
regions.

Figure 1.1 S plane

This is the relationship between Fourier transform and Laplace transform. The
Fourier transform given by the above equation must exist, which is actually

6|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

Laplace transform. For a system to be causal, all poles of its transfer function
must be right half of s-plane.

A system is said to be stable when all poles of its transfer function lay on the left
half of s-plane.

A system is said to be unstable when at least one pole of its transfer function is
shifted to the right half of s-plane.

A system is said to be marginally stable when at least one pole of its transfer
function lies on the jω axis of s-plane.

8.3 PROPERTIES OF LAPLACE TRANSFORM

1. Linearity: Let x1(t) be the two Laplace transform pairs. Then linearity property
states that,

Here a1 and a2 are constants.

Proof: let us find the Laplace transform of a1f1(t) + a2 f2(t) by applying definition.ie

7|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

2. Time scaling property: It states that

3. Scaling in s domain: It states that

Proof: According to time scaling property,

Let

Replacing b by a we get

4. Time differentiation: It states that

Proof: According to the definition of Laplace transform

8|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

The above integral is evaluated by parts using

The time differentiation twice is proved as follows:

Using the property

We get

In general

9|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

5. Time integration: The time integration property states that if

Proof: We define

Differentiating the above equation we get

6. Time convolution: The time convolution property states that if

Proof:

The inner integral is the LT of x2(t-ς) with a time delay ς.substituting

10|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

In the above equation, we get

7. Complex frequency differentiation: According to this property,

Proof: By definition of LT,

Differentiating both sided with respect to s,

8. Frequency shifting: According to this property,

11|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

9. Conjugation property: According to this property,

Proof: By definition of LT

10. Initial value theorem: According to this theorem,

Proof:

12|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

11. Final value theorem: According to this theorem,

Proof: The LT of d/dt(x (t)) could be written as

Taking s-> 0 on both sides of the above equation, we get

The above theorem is valid if X(s) has no poles in RHP of s-plane.

8.4 Problems:
1. Find the Laplace transform of ramp signal.
The ramp signal is given as,

13|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

By the definition of Laplace transform,

2. Find the Laplace transform of delayed ramp signal.

Solution: If unit ramp function is delayed by time t0 , it is given as,

By the shifting property of Laplace transform

Similarly

3. Find out the Laplace transform of impulse function.

We have evaluated the relationship between unit impulse function and step
function. The differentiation of unit step function gives unit impulse function i.e.,

Taking Laplace transform on both sides,

By differentiation property, the RHS of above equation will be,

14|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

In the above equation

Therefore

If the impulse function is delayed by t0, then its Laplace transform can be
obtained by shifting property as,

4. Determine the LT of an exponential decay which is shown in figure.

Solution: The exponential decay is represented by

Taking LT for the above function we get

15|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

5. Find out the Laplace transform of sine wave.

Solution:
A sine wave is given as,

We know that sin w0t can be represented using Euler’s identity as,

So f(t) becomes

Taking Laplace transform on both sides,

Putting these values in L{f(t)} we get,

16|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

6.Determine the LT of a sine function which is shown in figure

Solution: A sinusoidal function shown in figure is mathematically expressed as

follows:

The given sinusoidal function is written as follows using Euler’s identity.

17|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

7. Determine the LT of a cosine function which is shown in figure

Solution: A cosine function shown in figure is mathematically expressed as

follows:

The given sinusoidal function is written as follows using Euler’s identity

Taking LT for x (t), the following equation is written

8. Find out the Laplace transform of sine wave.

Solution: A sine wave is given as,

We know that sin w0t can be represented using Euler’s identity as,

So f(t) becomes

18|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

Taking Laplace transform on both sides,

9. Find the Laplace transform of damped sine wave.

We know that,

Using the complex shifting property,

Applying the above property we get,

(Or)

19|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

Taking Laplace transform on both sides,

10. Find the Laplace transform of damped cosine wave.

Solution: With the help of Euler’s identity,

Taking Laplace transform on both sides,

11. By applying the complex differentiation property, determine the LT of

Solution: We know that

20|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

According to the complex differentiation property,

12. Determine the LT of

Solution: The given signal x(t) is written in the following form.

13. Find the Laplace transform and ROC of

Solution:

21|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

8.5 Inverse Laplace transform

The time signal x (t) is the Inverse LT of X(s).This is represented by the following
mathematical equation.

…… (1)
Use of the above equation to obtain x (t) from X(s) is really a tedious process.
The alternative is to express X(s) in polynomial form both in the numerator and
the denominator. Both these polynomials are factorized as

……(2)
The points in the s plane at which X(s) = 0 are called zeros. Thus (s+z1), (s+z2),
(s+z3)….. (s+zm) are the zeros of X(s). Similarly, the points in the s-plane at which
X(s) = ∞ are called poles of X(s).
The zeros are identified by a small circle 0 and the poles by a small cross x in the
s- plane. For m<n the degree of the numerator polynomial is less than the
degree of the denominator polynomial. Under this condition X(s) in equation (2)
is written in the following partial fraction form.

…… (3)
In equation (3) A1,A2,…. An are called the residues and are determined by any
convenient method. Once the residues are determined, one can easily obtain
x(t) which is the required inverse LT of X(s).

22|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

8.6 Problems:
1. Find the inverse LT of

Solution: Consider the function

Putting this into partial fraction we get

Taking inverse LT we get

According to time shifting property of LT

23|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

2. Find the inverse LT of

Solution: The given function is written in the following form:

Now consider X2(s) without delay as X3(s)

24|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

8.7.The S-Plane :

Once the poles and zeros have been found for a given Laplace Transform, they can
be plotted onto the S-Plane. The S-plane is a complex plane with an imaginary and real axis
referring to the complex-valued variable z. The position on the complex plane is given
by rejƟ and the angle from the positive, real axis around the plane is denoted by θ. When
mapping poles and zeros onto the plane, poles are denoted by an "x" and zeros by an "o".
The below figure shows the S-Plane, and examples of plotting zeros and poles onto the
plane can be found in the following section.

25|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

S-Plane

It is quite difficult to qualitatively analyze the Laplace transform and Z-transform,

since mappings of their magnitude and phase or real part and imaginary part result in
multiple mappings of 2-dimensional surfaces in 3-dimensional space. For this reason, it is
very common to examine a plot of a transfer function's poles and zeros to try to gain a
qualitative idea of what a system does.

Definition: zeros
1. The value(s) for ss where P(s)=0P(s)=0.
2. The complex frequencies that make the overall gain of the filter transfer function
zero.
Definition: poles
1. The value(s) for ss where Q(s)=0Q(s)=0.
2. The complex frequencies that make the overall gain of the filter transfer function
infinite.
3.
BIBO Stability :

BIBO stability stands for bounded input, bounded output stability. BIBO stablity is the
system property that any bounded input yields a bounded output.

A bounded signal is any signal such that there exists a value such that the absolute value of
the signal is never greater than some value. Since this value is arbitrary, what we mean is
that at no point can the signal tend to infinity, including the end behavior.

26|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

A bounded signal is a signal for which there exists a constant A such that :(|f(t)|<A)

It turns out that a continuous time LTI system with impulse response h(t)h t is BIBO
stable if and only if Continuous-Time Condition for BIBO Stability

∫|h(t)|dt<∞

This is to say that the impulse response is absolutely integrable.Stability is very easy to
infer from the pole-zero plot of a transfer function. The only condition necessary to
demonstrate stability is to show that the iωω-axis is in the region of convergence.
Consequently, for stable causal systems, all poles must be to the left of the imaginary axis.

8.8 Transfer function :

Once the Laplace-transform of a system has been determined, one can use the information
contained in function's polynomials to graphically represent the function and easily observe
many defining characteristics. The Laplace-transform will have the below structure, based
on Rational Functions

H(s)=P(s)/Q(s)

The two polynomials P(s) and Q(s), allow us to find the poles and zeros of the Laplace-
Transform.

27|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

8.9 System Response to standard signal:

The main goal in analysis of any dynamic system is to find its response to a
given input. The system response in general has two components: zero-state response
due to external forcing signals and zero-input response due to system initial
conditions. The Laplace transform will produce both the zero-input and zero-state
components of the system response. We will also present procedures for obtaining the
system impulse, step, and ramp responses. Note that using the Fourier transform, we
have been able to find only the zero-state system response.

It is important to point out that the Laplace transform is very convenient for
dealing with the system input signals that have jump discontinuities (and delta
impulses). Note that in general the linear system differentiates input signals. The delta
impulse inputs can come from the system differentiation of input signals that have
jump discontinuities.

Definition of the generalized derivative, which indicates that at the point of a

jump discontinuity, the generalized derivative generates the impulse delta signal.
Furthermore, for the same reason, a signal that is continuous and differentiable for all
t>0, but has a jump discontinuity at t=0, for example e^(-t)u(t), will generate an
impulse delta signal (after being differentiated) at t=0. The same is true for the signal
u(t)sin(t)(after being differentiated twice), which is continuous but not differentiable at
t=o. Note that

Using the Laplace transform as a method for solving differential equations that
represent dynamics of linear time invariant systems can be done in a straight forward
manner despite delta impulses generated by the system differentiation of input signals.

8.9.1 System Transfer Function and Impulse Response

Let us take the Laplace transform of both sides of a linear differential equation that
describes the dynamical behavior of an nth order linear system

28|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

Using the time derivative property of the Laplace transform we have

where I(s) contains terms coming from the system initial conditions
Recall That

It can be easily shown that in general

29|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

8.9.3 System Response

The input signal is applied to the system at t=0, and we are interested in finding the
complete system response—the response due to both system initial conditions and
input signals. From the original derivations we have

which produces the solution Y(s) in the frequency domain of the original differential
equation. To get the time domain solution, we must use the inverse Laplace transform,

If the initial conditions are set to zero, then I(s)=0. The quantity

defines the system transfer function. The transfer function can also be written as

Example:

Let the system transfer function be given by

30|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

8.10 Solution of differential equations with initial conditions:

It’s now time to get back to differential equations. We’ve spent the last three
sections learning how to take Laplace transforms and how to take inverse Laplace
transforms. These are going to be invaluable skills for the next couple of sections so
don’t forget what we learned there.

Before proceeding into differential equations we will need one more formula. We
will need to know how to take the Laplace transform of a derivative. First recall
that f(n) denotes the nth derivative of the function ff. We now have the following fact.

Since we are going to be dealing with second order differential equations it will be
convenient to have the Laplace transform of the first two derivatives.

Notice that the two function evaluations that appear in these formulas, y(0) and y′(0),
are often what we’ve been using for initial condition in our IVP’s. So, this means that if
we are to use these formulas to solve an IVP we will need initial conditions at t=0.

While Laplace transforms are particularly useful for nonhomogeneous differential

equations which have Heaviside functions in the forcing function we’ll start off with a
couple of fairly simple problems to illustrate how the process works.

Example1: Solve

Solution:

The first step in using Laplace transforms to solve an IVP is to take the transform of
every term in the differential equation.

31|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

Using the appropriate formulas from our table of Laplace transforms gives us the
following.

Plug in the initial conditions and collect all the terms that have a Y(s) in them.

Solve for Y(s).

At this point it’s convenient to recall just what we’re trying to do. We are trying to
find the solution, y(t), to an IVP. What we’ve managed to find at this point is not the
solution, but its Laplace transform. So, in order to find the solution all that we need to
do is to take the inverse transform.

Before doing that let’s notice that in its present form we will have to do partial
fractions twice. However, if we combine the two terms up we will only be doing partial
fractions once. Not only that, but the denominator for the combined term will be
identical to the denominator of the first term. This means that we are going to partial
fraction up a term with that denominator no matter what so we might as well make the
numerator slightly messier and then just partial fraction once.

This is one of those things where we are apparently making the problem messier,
but in the process we are going to save ourselves a fair amount of work!

Combining the two terms gives,

The partial fraction decomposition for this transform is,

32|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

Setting numerators equal gives,

Picking appropriate values of ss and solving for the constants gives,

Plugging in the constants gives,

Finally taking the inverse transform gives us the solution to the IVP.

Example2: Solve

Solution:

As with the first example, let’s first take the Laplace transform of all the terms in the
differential equation. We’ll the plug in the initial conditions to get,

Now solve for Y(s).

33|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

Now, as we did in the last example we’ll go ahead and combine the two terms together
as we will have to partial fraction up the first denominator anyway, so we may as well
make the numerator a little more complex and just do a single partial fraction. This will
give,

The partial fraction decomposition is then,

Setting numerator equal gives,

In this case it’s probably easier to just set coefficients equal and solve the resulting
system of equation rather than pick values of ss. So, here is the system and its
solution.

We will get a common denominator of 125 on all these coefficients and factor that out
when we go to plug them back into the transform. Doing this gives,

Notice that we also had to factor a 2 out of the denominator of the first term and fix up
the numerator of the last term in order to get them to match up to the correct entries
in our table of transforms.
34|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

Taking the inverse transform then gives,

Example3: Solve

Solution:

Take the Laplace transform of everything and plug in the initial conditions.

Now solve for Y(s) and combine into a single term as we did in the previous two
examples.

Now, do the partial fractions on this. First let’s get the partial fraction decomposition.

Now, setting numerators equal gives,

Setting coefficients equal and solving for the constants gives,

Now, plug these into the decomposition, complete the square on the denominator of
the second term and then fix up the numerators for the inverse transform process.

35|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

Finally, take the inverse transform,

36|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

9. Practice Quiz
1. Laplace transform of sin(at) u(t) is?
a)a ⁄ a2+s2
b)1 ⁄ a2 +s2
c) s ⁄ a2+s2
d) a ⁄ s2
2. Find the laplace transform of et Sin(t).
a) 1/a2+(s−1)2
b) a/a2+(s−1)2
c) 1/a2+s2
d) 2/a2+(s−1)2
3. The laplace transform of u(t) is
a)0
b)1/s
c) 1/s2
d) 1/s3
4.The laplace transform of u(-t) is
a)1/s
b)1/s2
c) -1/s
d) -1/s2
5.The laplace transform of δ(t) is
a)0
b)1
c) 2
d) 3

37|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

10. Assignments
S.No Question BL CO

1 State and prove the any five Properties Laplace Transform 2 4

2 Find the Laplace transform for any 5 standard signals. 2 4

3 Prove initial and final value theorems of Laplace transform. 2 4

4 Find the inverse laplace transform of the following. 2 4

i)s/[(s+2)2+1] ii)s/[(s-b)2+a2]

11. Part A- Question & Answers

S.No Question & Answers BL CO

1 Define laplace transform

Laplace transform is the another mathematical tool 1 4

used for analysis of continuous time signals and systems.

It is defined as F(s) = f(t) e-st dt

2 What are the methods for evaluating inverse Laplace
transform. 4
1
The two methods for evaluating inverse laplace
transform are (i). By Partial fraction expansion method.

(ii). By convolution integral.

3 Define dynamic behavior?

If there is a change between input and output is known 1 4

as dynamic behavior. the mismatch occurs at load and
output terminal.
4
State the sufficient condition for the existence of laplace 4
1
transform.

i)x(t) should be continuous in the given closed interval

[a,b] where a>0

ii)x(t) should be of exponential order.

38|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

5 What is the relationship between fourier transform and

laplace transform.
1 4
X(s)=X(jw) when s=jw
This states that laplace transform is same as fourier
transform when s=jw.
6 What is the use of Laplace Transform?

Laplace transform is another mathematical tool used for

analysis of signals and systems. Laplace transform is
used for analysis of unstable systems.
7 What are the types of Laplace transform?

i)Bilateral or two sided laplace transform. 4

ii)Unilateral or single sided laplace transform. 1

8
State the properties of Laplace transform.
1 4
i)Linearity property

ii)Time shifting property

iii)Differentiation in time domain

iv)Convolution in time domain

12. Part B- Questions

S.No Question BL CO

State and prove the any five Properties Laplace

1 2 4
Transform

Discuss about the Properties of the ROC of Laplace

2 2 4
transform

3 Find the Laplace transform for any 5 standard signals 2 4

State and prove initial and final value theorems of

4 2 4
Laplace transform?

39|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

5 Find laplace transform of 𝑥(𝑡) = 𝑒−𝑎𝑡𝑢(𝑡) − 𝑒−𝑏𝑡𝑢(−𝑡) 2 4

6 Find laplace transform of 𝑥(𝑡) = 𝑒−5𝑡𝑢(𝑡 − 1) 2 4

1
7 Find inverse Laplace transform of X(𝑠) = 2 4
(𝑠+1)(𝑠+2)(𝑠+3)

8 Find inverse Laplace transform of X(𝑠) =(s+1)/[(s+2)2+1] 2 4

9 Verify initial value theorem for 2 4

i)x(t)=4-2e3t ii)x(t)=3e-5t

40|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

13. Supportive Online Certification Courses

1. nptel.ac.in/courses/112104121/1
2. http://nptel.ac.in/courses/117101055/14

14. Real Time Applications

S.No Application CO

1 Traffic Signal System 1

Traffic signal systems have been employed since the early

twentieth century as a method of continuously managing traffic
flow and saturation at intersections and to promote smooth and
safe automobile transportation. Due to the continuous increase of
traffic congestion in urban areas, there is a need for further
evaluation and implementation of traffic signal systems. This study
reviews and consolidates information on a wide variety of signal
systems, detection devices and communications components in
order to provide a cogent understanding of current technology in
the United States. Additionally, the assessment focuses on
operational functions of various systems, and thereby, establishes
what the systems’ capabilities are when utilized to their fullest
extent. Current signal system practices are reviewed to compare
existing technologies, and postmodern technologies are
investigated. Recommendations for further investigation of traffic
signal systems are also provided.

2 Digital Signal Processing

1
Digital signal processing (DSP) is concerned with the
representation of the signals by a sequence of numbers or
symbols and the processing of these signals. Digital signal
processing and analog signal processing are subfields of signal
processing. The analog waveform is sliced into equal segments
and the waveform amplitude is measured in the middle of each
segment. The collection of measurements makes up the digital
representation of the waveform. Converting a continuously
changing waveform (analog) into a series of discrete levels.

41|SS-UNIT-III
BTECH_ECE-SEM_ 2 1
SVCE TIRUPATI

15. Contents Beyond Syllabus

 Partial Differential equation: We use it to solve higher order partial
differential equations by the method of separation of variables.
 Approximation Theory: We use Fourier series to write a function as a
trigonometric polynomial.

16. Prescribed Text Books & Reference Books

Text Books:
1. A.V. Oppenheim, A.S. Willsky and S.H. Nawab, “Signals and Systems”, 2
nd Edition, PHI, 2009
2. 2. Simon Haykin and Van Veen, “Signals & Systems”, 2 nd Edition,
Wiley, 2005.

References:
1. BP Lathi, “Principles of Linear Systems and Signals”, 2 nd Edition, Oxford
University Press, 015.
2. Matthew Sadiku and Warsame H. Ali, “Signals and Systems A primer with
MATLAB”, CRC Press, 2016.
3. 3. Hwei Hsu, “Schaum's Outline of Signals and Systems”, 4 th Edition, TMH,
2019.

17. Mini Project Suggestion

1. Generate various signals and sequences (Periodic and aperiodic),
such as Triangular, Sinusoidal, Ramp, Sinc using MATLAB
2. Generate various signals and sequences (Periodic and aperiodic),
such as Unit Impulse, Unit Step, Square, Saw tooth using MATLAB

42|SS-UNIT-III
BTECH_ECE-SEM_ 2 1