linear system analysis method

[whit]

asin(wt)+bcos(wt)

Waht make sense that a linear system with Acos(wt) input and
Bcos(wt+th) output can be solved in complex plane with e^(jwt)
input and Be^(jwt+th) output?

 

[afonso]

Can you explain better your doubt, please?


PS.: Do you know that Ae^(jwt) = Acos(wt) + jAsin(wt)?

 

[me2please]

Don't quite understand what your question is but I'll try .

Acos(wt) ------> L --------> L[Acos(wt)] = Bcos(wt+th)

For any linear systems
                     a  ----> L ------> L(a)
then           a+b -----> L ------>  L(a+b)  =  L(a) + L(b)  "property of linear system".
 
We know that   Acos(wt) ------> L --------> L[Acos(wt)] = Bcos(wt+th)
Then,            from    Asin(wt) = Acos(wt-90)

              Acos(wt) + iAsin(wt) ---> L ----> L[Acos(wt)] + L[iAsin(wt)]
                                                           = L[Acos(wt)] + iL[Acos(wt-90)]
                                                           = Bcos(wt+th) + iBcos(wt-90+th)
                                                           = Bcos(wt+th) + iBsin(wt+th)
So
             e^(jwt) ---> L -----> L[e^(jwt)] = Be^(jwt+th)
is
             Acos(wt) + iAsin(wt) ---> L ----> Bcos(wt+th) + iBsin(wt+th)

         REAL :     Acos(wt)  ---> L ----> Bcos(wt+th)
and      IMAG :    Asin(wt)   ----> L ----> Bsin(wt+th)

It's like you pass 2 cosince waves instead of 1 into linear system.  These 2 waves(real and imaginary) don't interfere with each other.  At the output if you pick real part, you get the same thing as 1 cosine input(shown in red above).

 

[devrimaksin]

This is the property of the linear system,
for a given input signal frequency, if the
system is really linear, all the signal within the
system will at the same frequency (magnitude
and phase changes of course)

 

[whit]

me2please

What I expected is just what you explained。

 

[djalli]

A system is linear if function characterizing it is linear.

Which means:

a) function is homogenious.
b) function is additive.

If a) and b) is true for the function characterizing the system then system is linear. Always do a test. If a) and b) are true then you do not need to worry. Sytem is linear.