Laplace transform: why do we have to convert a function in frequency domain?

can any one tell the importance of laplace tranformation interms that why we need to convert a particular function in frequency domain when we can solve it in time domain vice versa ??

Re: Laplace transform???

sometimes you get more information of a signal if you process in Freq domain. Based on what you want to do or what you r looking in a signal you may have to process either in freq or time domain or some times in both.(a function is representing a singal, i assume).
e.g. we are trying to detect a small sine wave in the presence of large signals. In a time domain you will just see a single waveform(combination of several singals). In freq domain the same signal can be viewed as separate components the small components are easy to see because
they are not masked by larger ones

Re: Laplace transform???

Additionally to chitturi_sb;
In time domain generally we have differential equations which may be difficult to solve. Instead just apply Laplace transform, then do much more easier calculation on the signals in "s" domain. Then do the inverse laplace to get the result in time domain.
ie convolution in time domain, just multiplicaiton in laplace domain.
As a summary helps to recude computational comlexity.

Hope helps

laplace transform help in analysis the signal when designing filter ,doing that in time domain would be to hectic

I think the most important idea behind this comes from the system we are working on.
They are at first linear systems.
In my opinion , the linear systems do two thingssuppose the input signal is x(t) the output signal is y(t))
1.
It multiply a coefficient by the value of the signal at every instant.
the coefficient can be a function of t, but not a function of x(t).

eg: y(t)=t*x(t) is a linear system , as the t is just a function of t.


2.
the second thing that a linear system do is to add the scaled value of every instant together .
as for the case above , y(t)=t*x(t), is a memoryless system, it means the coefficients for
x(t+t') (t' is any real number except zero.) are all zero.
Generally , this is not true.
eg. When the coefficients for x(t+t') (t'<0) are not zero, we often get differential or integral systems.


this explains why y(t)=x(t)+1 is not a linear system:
1 is not changing with x(t), this means it's not a form of f(t)*x(t).



this is the thing about linear systems.
and if the coefficient is not even a function of t , it's a time invariant system.

Combine the two together , we have the LTI system.



then
We may want to find a way to easily analyze the LTI (linear time invariant) systems regardless of what the input signal is.
suppose every input signal x(t) can be represented in this form : sum of (some function which has nothing to do with t ) * (some function which has something to do with t .)
let me write it in a more beautiful form :Σ Ak*Fk(t) (k= ...-2,-1,0,1,2......)

IF Hk(t) is the response of the system to Fk(t), because it is a linear system , we have the response of the system to Σ Ak*Fk(t) is in this form :
Σ Ak*Hk(t) .


Well , Maybe the most amazing part is that we can find a special Fk(t) ,and its response to the LTI system is
Hk'(?)*Fk(t) (the ? in Hk'(?) means it's a function which has nothing to do with t).

And the the response of the system to Σ Ak*Fk(t) can be represented like this :
Σ Ak*Hk'(?) *Fk(t)

Hk'(?) has nothing to do with t .
Unluckily , Fk(t) may has something to do with "?".
but luckily "?" has nothing to do with t.
if every signal can be not only represented in the form of Σ Ak*Fk(t) ,but also with the same set of
Fk(t) , Hk'(?) has nothing to do with the input signal (Actually , this will lead to an integral rather than a sum , here ,I still use the sum form . it's simple for analysis.)


It's clear Hk'(?) is a characteristic the system has.
It can be used to describe the system regardless of what the input signal is .
If our assumption(every signal can be represented in the form of Σ Ak*Fk(t), actually a lot of . ) is true ,
this can be a very good way to see the systems.
It's very clear that it's just a coefficient Modifier.


now , every LTI system can be analyzed in this way , a function Hk'(?) of ?.

"?" is an independent variable , it can be anything ....
But why is it frequency

Actually, the Fk(t) we choose is exponential function with the base e.
I think we get the idea of frequency largely from the "sine waves "
Luckily ,a sine function and a cosine function with frequency w make up the e^（jwt）
and Hk'(?) here has something to do with the "w",or in this case , ? is just w..
When we choose this Fk(t) we can give the "?" a special meaning :the frequency of the component sine wave frequency.

suppose you have a square wave with frequency of X ,but in frequency domain , you see it consists a lot of frequency components , the reason is above.

At last we can have do a lot of research into the Hk'(w),well,you got a lot of interesting and useful properties ..
But I think , at the beginning ,we just want to find a more general way to describe a system.