Electronics and Communication Engineering (ECE) Exam > Electronics and Communication Engineering (ECE) Questions > Obtain the Laplace Transform of the signal x ...
Start Learning for Free
Obtain the Laplace Transform of the signal x (t) =e^(-at) Cos (wot) u(-t), and indicate its ROC.?
Most Upvoted Answer
Obtain the Laplace Transform of the signal x (t) =e^(-at) Cos (wot) u(...
Laplace Transform of the Signal
To find the Laplace Transform of the signal x(t) = e^(-at) cos(w0t) u(-t), we will follow the steps below:
Step 1: Definition of Laplace Transform
- The Laplace Transform is defined as:
L{x(t)} = ∫[0, ∞] e^(-st) x(t) dt
However, since x(t) is multiplied by the unit step function u(-t), we will consider the integral from -∞ to 0.
Step 2: Applying the Transform
- The Laplace Transform becomes:
L{x(t)} = ∫[-∞, 0] e^(-st) e^(-at) cos(w0t) dt
- This can be simplified to:
L{x(t)} = ∫[-∞, 0] e^(-(s + a)t) cos(w0t) dt
Step 3: Evaluating the Integral
- The integral can be evaluated using the standard formula for the Laplace Transform of cos:
L{cos(ωt)} = s / (s^2 + ω^2)
Thus, the result for our function becomes:
L{x(t)} = (s + a) / ((s + a)^2 + w0^2)
ROC (Region of Convergence)
- The ROC for this Laplace Transform is determined by the exponential decay factor e^(-at).
- Since x(t) is defined for t < 0="" (due="" to="" u(-t)),="" the="" roc="" will="" be="" re{s}="" />< />
Conclusion
- The Laplace Transform of x(t) = e^(-at) cos(w0t) u(-t) is:
L{x(t)} = (s + a) / ((s + a)^2 + w0^2)
- The ROC is Re{s} < -a.="" />
To find the Laplace Transform of the signal x(t) = e^(-at) cos(w0t) u(-t), we will follow the steps below:
Step 1: Definition of Laplace Transform
- The Laplace Transform is defined as:
L{x(t)} = ∫[0, ∞] e^(-st) x(t) dt
However, since x(t) is multiplied by the unit step function u(-t), we will consider the integral from -∞ to 0.
Step 2: Applying the Transform
- The Laplace Transform becomes:
L{x(t)} = ∫[-∞, 0] e^(-st) e^(-at) cos(w0t) dt
- This can be simplified to:
L{x(t)} = ∫[-∞, 0] e^(-(s + a)t) cos(w0t) dt
Step 3: Evaluating the Integral
- The integral can be evaluated using the standard formula for the Laplace Transform of cos:
L{cos(ωt)} = s / (s^2 + ω^2)
Thus, the result for our function becomes:
L{x(t)} = (s + a) / ((s + a)^2 + w0^2)
ROC (Region of Convergence)
- The ROC for this Laplace Transform is determined by the exponential decay factor e^(-at).
- Since x(t) is defined for t < 0="" (due="" to="" u(-t)),="" the="" roc="" will="" be="" re{s}="" />< />
Conclusion
- The Laplace Transform of x(t) = e^(-at) cos(w0t) u(-t) is:
L{x(t)} = (s + a) / ((s + a)^2 + w0^2)
- The ROC is Re{s} < -a.="" />
Attention Electronics and Communication Engineering (ECE) Students!
To make sure you are not studying endlessly, EduRev has designed Electronics and Communication Engineering (ECE) study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Electronics and Communication Engineering (ECE).
|
Explore Courses for Electronics and Communication Engineering (ECE) exam
|
|
Similar Electronics and Communication Engineering (ECE) Doubts
Top Courses for Electronics and Communication Engineering (ECE)View all
Obtain the Laplace Transform of the signal x (t) =e^(-at) Cos (wot) u(-t), and indicate its ROC.?
Question Description
Obtain the Laplace Transform of the signal x (t) =e^(-at) Cos (wot) u(-t), and indicate its ROC.? for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about Obtain the Laplace Transform of the signal x (t) =e^(-at) Cos (wot) u(-t), and indicate its ROC.? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Obtain the Laplace Transform of the signal x (t) =e^(-at) Cos (wot) u(-t), and indicate its ROC.?.
Obtain the Laplace Transform of the signal x (t) =e^(-at) Cos (wot) u(-t), and indicate its ROC.? for Electronics and Communication Engineering (ECE) 2024 is part of Electronics and Communication Engineering (ECE) preparation. The Question and answers have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about Obtain the Laplace Transform of the signal x (t) =e^(-at) Cos (wot) u(-t), and indicate its ROC.? covers all topics & solutions for Electronics and Communication Engineering (ECE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Obtain the Laplace Transform of the signal x (t) =e^(-at) Cos (wot) u(-t), and indicate its ROC.?.
Solutions for Obtain the Laplace Transform of the signal x (t) =e^(-at) Cos (wot) u(-t), and indicate its ROC.? in English & in Hindi are available as part of our courses for Electronics and Communication Engineering (ECE).
Download more important topics, notes, lectures and mock test series for Electronics and Communication Engineering (ECE) Exam by signing up for free.
Here you can find the meaning of Obtain the Laplace Transform of the signal x (t) =e^(-at) Cos (wot) u(-t), and indicate its ROC.? defined & explained in the simplest way possible. Besides giving the explanation of
Obtain the Laplace Transform of the signal x (t) =e^(-at) Cos (wot) u(-t), and indicate its ROC.?, a detailed solution for Obtain the Laplace Transform of the signal x (t) =e^(-at) Cos (wot) u(-t), and indicate its ROC.? has been provided alongside types of Obtain the Laplace Transform of the signal x (t) =e^(-at) Cos (wot) u(-t), and indicate its ROC.? theory, EduRev gives you an
ample number of questions to practice Obtain the Laplace Transform of the signal x (t) =e^(-at) Cos (wot) u(-t), and indicate its ROC.? tests, examples and also practice Electronics and Communication Engineering (ECE) tests.
|
|
Explore Courses for Electronics and Communication Engineering (ECE) exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
x
![]()
For Your Perfect Score in Electronics and Communication Engineering (ECE)
The Best you need at One Place
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup