Question

Question - 19 : Signals And Systems

Using the \(\color{blue}{\text{Convolution Property of Bilateral Laplace Transform}}\), find the response of the system having Impulse Response and Input Signal as:

\(a) \large \quad \color{green}{f(t) = e^{\frac{|t|}{4}}}\)

\(b) \large \quad \color{red}{h(t) = e^{-3t} \cdot u(t)}\)

Answer 1

Convolution Property is given as:
\[\text{f(t)*h(t) = F(s) H(s)} \quad \quad * \rightarrow Convolution\]

Answer 2

I just need to check the answer..

Answer 3

I have calculated it using all my brain..

Answer 4

haha

Answer 5

ok umm soo u want a * a and b * b right

Answer 6

Nope..

Answer 7

or f(t) * h(t)?

Answer 8

yep//

Answer 9

ok

Answer 10

I got it as:
\[\large y(t) = f(t)*h(t) = \frac{4}{13}e^{\frac{t}{4}} \cdot u(-t) + \frac{4}{11} e^{\frac{-t}{4}} \cdot u(t) - \frac{8}{143}e^{-3t} \cdot u(t)\]

Answer 11

I got this weird answer.. ROC is : \(\frac{-1}{4} <\Re\{s\} < \frac{1}{4}\)

Answer 12

ummm i see

Answer 13

what happens if u try to take L {e^-3t u(t)}

Answer 14

I can give some more details I got while walking through the solution..

Answer 15

isnt there a way to check out answer

Answer 16

\[H(s) = \frac{1}{s+3},\quad \Re\{s\} > -3\]

Answer 17

Ha ha ha, For that I have come here.. :P

Answer 18

\[e^{-at} u(t) \longleftrightarrow \frac{1}{s+a} , \quad \sigma > -a\]

Answer 19

@hartnn

Answer 20

you didn't use convolution property ?

Answer 21

L(f*g) = FG

Answer 22

http://www.wolframalpha.com/input/?i=inverse+laplace+transform+of+%284%2F%5B%284s-1%29%28s%2B1%29%5D%29

Answer 23

@hartnn I have got that result by using convolution property only.. I just need to check my answer..

Answer 24

How you got that as your Multiplication Result?

Answer 25

this is the property
L(f*g) = FG
Laplace of convolution is the product of laplaces.

Answer 26

you have that right
and how did you calculate laplaces of
f and h

Answer 27

I think my question is wrong..

Answer 28

The bilateral transform of \(e^{\frac{|t|}{4}}\) does not exist I think, @zarkon right?