NATIONAL UNIVERSITY OF
TECHNOLOGY
COMPUTER ENGINEERING DEPARTMENT
LAB # 11
Laplace Transform
Submitted By : Muhammad Junaid Waseem(F23604037)
Tayyeba Fatima(F23604053)
Huzaifa khalid (F23604008)
Submitted To : Maam Maryam Mehmood
Submitted On: 4/06/25
Objectives:
1. Understanding the Definition and Mathematical Foundation of the Laplace Transform: Grasping the
mathematical basis and formulation of the Laplace transform.
2. Examining Key Properties: Exploring properties such as linearity, time-shifting, scaling, and convolution,
crucial for signal analysis.
3. Analyzing the Region of Convergence (ROC): Studying the ROC and its impact on system stability and
causality.
4. Learning Methods to Compute the Inverse Laplace Transform: Investigating techniques for reverting back to
the time-domain representation.
5. Investigating Applications in Control Systems and Signal Processing: Applying the Laplace transform to solve
real-world engineering problems.
Table of Contents
Objectives: .................................................................................................................................................................. 2
Table of Contents ........................................................................................................................................................ 2
LAB TASK .................................................................................................................................................................... 3
Task 1: .................................................................................................................................................................... 3
Solution : ............................................................................................................................................................. 3
Task 2: .................................................................................................................................................................... 5
Solution:.............................................................................................................................................................. 5
Output: ............................................................................................................................................................... 5
Conclusion .................................................................................................................................................................. 6
LAB TASK
Task 1:
Find the Inverse Laplace transform of the following:
Solution :
%part 1
syms t s
F =(10*(s+2))/(s*(s^2)+4*s+5);
ilaplace(F)
simplify(ans)
pretty(ans)
% part 2
syms t s
F= (s^2 + 1)/(4*s^3 + 4*s^2 + 2*s +1)
ilaplace(F)
simplify(ans)
pretty(ans)
Output:
%part 1
(10*exp(-t))/7 - (10*exp(t/2)*(cos((19^(1/2)*t)/2) - (17*19^(1/2)*sin((19^(1/2)*t)/2))/19))/7
symsum(exp(t*root(s3^3 + s3^2 + s3/2 + 1/4, s3, k))/(12*root(s3^3 + s3^2 + s3/2 + 1/4, s3, k)^2 + 8*root(s3^3 + s3^2
+ s3/2 + 1/4, s3, k) + 2), k, 1, 3) + symsum((exp(t*root(s3^3 + s3^2 + s3/2 + 1/4, s3, k))*root(s3^3 + s3^2 + s3/2 + 1/4,
s3, k)^2)/(12*root(s3^3 + s3^2 + s3/2 + 1/4, s3, k)^2 + 8*root(s3^3 + s3^2 + s3/2 + 1/4, s3, k) + 2), k, 1, 3)
Task 2:
Consider the outputs of Lab Task 1 and compute their Laplace transform and verify
your answer by comparing the output of the Laplace function with the functions given in Task 1
i.e., F(s).
Solution:
%part 1
f=(10*exp(-t))/7 - (10*exp(t/2)*(cos((19^(1/2)*t)/2) - (17*19^(1/2)*sin((19^(1/2)*t)/2))/19))/7;
f=(exp(t*root(s3^3 + s3^2 + s3/2 + 1/4, s3, k))/(2*(6*root(s3^3 + s3^2 + s3/2 + 1/4, s3, k)^2 + 4*root(s3^3 + s3^2
+ s3/2 + 1/4, s3, k) + 1)), k, 1, 3) + ((exp(t*root(s3^3 + s3^2 + s3/2 + 1/4, s3, k))root(s3^3 + s3^2 + s3/2 + 1/4, s3,
k)^2)/(2(6*root(s3^3 + s3^2 + s3/2 + 1/4, s3, k)^2 + 4*root(s3^3 + s3^2 + s3/2 + 1/4, s3, k) + 1)), k, 1, 3);
F=laplace(f,t,s)
simplify(ans)
pretty(ans)
%part 2
f=(exp(t*root(s3^3 + s3^2 + s3/2 + 1/4, s3, k))/(2*(6*root(s3^3 + s3^2 + s3/2 + 1/4, s3, k)^2 + 4*root(s3^3 + s3^2
+ s3/2 + 1/4, s3, k) + 1)), k, 1, 3) + ((exp(t*root(s3^3 + s3^2 + s3/2 + 1/4, s3, k))root(s3^3 + s3^2 + s3/2 + 1/4, s3,
k)^2)/(2(6*root(s3^3 + s3^2 + s3/2 + 1/4, s3, k)^2 + 4*root(s3^3 + s3^2 + s3/2 + 1/4, s3, k) + 1)), k, 1, 3);
F=laplace(f,t,s)
simplify(ans)
pretty(ans)
Output:
Conclusion
In this lab, we successfully explored the concept and applications of the Laplace transform, a vital tool in control
systems and signal processing. Through a series of tasks, we achieved the following objectives:
1. Understanding the Definition and Mathematical Foundation of the Laplace Transform: We reviewed
the mathematical basis and conditions for applying the Laplace transform.
2. Examining Key Properties: We investigated properties such as linearity, time-shifting, scaling, and
convolution, and how they facilitate system analysis and manipulation.
3. Analyzing the Region of Convergence (ROC): We explored the ROC's impact on system stability and
causality, highlighting its significance in practical applications.
4. Learning Methods to Compute the Inverse Laplace Transform: We discussed techniques to revert signals
from the Laplace domain back to the time domain.
5. Investigating Applications: We applied the Laplace transform to real-world problems in control systems and
signal processing.
Through hands-on tasks, we computed convolutions and analyzed poles and zeros of given Laplace transform
functions, reinforcing our understanding of theoretical concepts with practical applications. This lab provided a
comprehensive understanding of the Laplace transform, equipping us with the skills necessary to analyze and design
systems effectively.
