Question

Question - 18: Signals And Systems:

Find the \(\color{red}{\text{Laplace Transform}}\) of following functions:

\(\color{green}{a) \qquad f(t) = 1}\)

\(\color{blue}{b) \qquad f(t) = e^{-a|t|}}\)

Answer 1

Laplace Transform of a function is given as:
\[\large \color{green}{F(s) = \int\limits_{-\infty}^{\infty} f(t) \cdot e^{-st} \cdot dt}\]

Answer 2

For first part:
\[\large F(s) = \int\limits_{-\infty}^{\infty} e^{-st} \cdot dt\]

Answer 3

This integral does not seem to be Converging..
May be, I have to find this by using Inverse Laplace Transform..

Answer 4

@dan815 @eliassaab

Answer 5

the integral goes from 0 to infinity

Answer 6

For Bilateral Laplace, limit is from \(-\infty\) to \(\infty\) no?

Answer 7

yes

Answer 8

So?

Answer 9

the integral will not converge then for the 1st one

Answer 10

you should also use " bilateral Laplace transform" in the title of your question since the regular Laplace transform is an integral on the positive reals

Answer 11

I want to know both..

Answer 12

Actually, the thing is, my teacher has taught Bilateral first and then he gave some questions for homework, and then after it, he taught unilateral laplace.
May be he was checking that what result we make out from that homework..

Answer 13

So, for bilateral laplace, it is coming out to be Not-defined..

Answer 14

for the first one...that is correct

Answer 15

Let me try for the second one for bilateral laplace.

Answer 16

\[\implies \int\limits_{- \infty}^{0} e^{at} e^{-st} \cdot dt + \int\limits_{0}^{\infty} e^{-at}e^{-st} \cdot dt\]

Answer 17

\[\implies \frac{1}{a-s} + \frac{1}{a+s}\]

Answer 18

you need to put a bound on the possible values of a,s

Answer 19

\[\frac{-2a}{s^2 - a^2}\]

Answer 20

For first one, Re(s) < a.. Right?

Answer 21

yes

Answer 22

For second one, Re(s) > -a

Answer 23

yes

Answer 24

\[\frac{-2a}{s^2 - a^2} , \quad -a < \Re\{s\} < a\]

Answer 25

looks fine

Answer 26

One more thing, when we use Unilateral, these conditions come or not?

Answer 27

For second one for unilateral:
\[F(s) = \frac{1}{s+a}\]

Answer 28

the condition for the second integral will still apply

Answer 29

Here also we need to define ROC?

Answer 30

Okay, got it.. Thank you @Zarkon it really helped a lot to understand. :)

Answer 31

One more thing..

Answer 32

I have a doubt in Fourier transform, can you clarify that?

Answer 33

I haven't done Fourier analysis in a long time, but I can look

Answer 34

Suppose you have two functions, that are totally same but the dc value of two are different..

Answer 35

and you evaluate their Fourier Transform by using Time-Differentiation Property//

Answer 36

On differentiating, the dc value of both will become zero, and you will get same transform for both, what is the condition to be used here to have different transforms?

Answer 37

Like one function is : f(t), and other is f(t)+ (1/2)..
When you use differentiation property, for second 1/2 will become 0, and you have both functions as same..

Answer 38

getting my doubt?

Answer 39

I don't remember. That is something I would have to refresh myself on.

Last time I did stuff like this was around 2001

Answer 40

I know the same condition for Fourier Series but don't know for Fourier Transform..

Answer 41

Okay, you have all the time.. :)

Answer 42

2001?? I was like in 5th grade at that time.. :P

Answer 43

I guess I'm a little older than you ;)

Answer 44

you sure you are "little" older than me?? :P

Answer 45

I'm in my 40's

Answer 46

That is not little for a person who is in his 20's..

Answer 47

ic ;)

Answer 48

you were born around the time I graduated from high school :)

Answer 49

If you are planning to read for my doubt, then do read Time-integration property too..

Answer 50

I guess it is something to do with that..

Answer 51

ok...I'll see if I can find something. though I only have my home library and the internet to work from. I'm done in my office till next year.

Answer 52

yeah for sure.. 1991...

Answer 53

yep...that is the year I graduated HS :)

Answer 54

:)

Answer 55

I was born in 1972

Answer 56

I am not going to tell you the date otherwise you will say I was graduated on that date. :P

Answer 57

I doubt that

Answer 58

It was 3rd June 1991, Monday..

Answer 59

I don't remember the day anyway.. it was a long time ago :)

Answer 60

:)

Answer 61

I'm sure the graduation was not on a Monday

Answer 62

It was really nice of you for sharing such information.. :)

Answer 63

yeah we don't graduate on Monday which is just after Sunday. :P

Answer 64

you too

Answer 65

Thanks for all your help..
Good day..!! :)

Answer 66

later