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 things
suppose 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.