My homework assignment has this question: In the analysis of linear systems inel

Question

My homework assignment has this question:

In the analysis of linear systems inelectrical engineering, we commonly simply solve the system in thesinusoidal steady state at one particular frequency. We thenclaim to understand the system's behavior in the presence of anyinput. Is this claim in principle right? Stateprecisely why or why not.

I am not sure of the answer. I think.. so far.. the onlythings I can think of that are that by using phasors, the outputwill have the same frequency as the input, so the only change inthe output are the amplitude and shift. Another thought isthat the output of a linear system is a scaled version of theinput. I'm not sure how these things can be used to figureout the question though. May I have some help? Thanks!

Explanation / Answer

For Linear Systems the following relation holds :
If x1(t) input produces the output y1(t) and x2(t) inputproduces the output y2(t) then input a1*x1(t) + a2*x2(t) produces output = a1*y1(t) +a2*y2(t)if a1 and a2 are scalars.
Now, while analysing such a system, we look at the response ofthe system when input is a sinusoid of a particularfrequency. We know that any signal can be decomposed into a summation ofsinusoids of different frequencies.This is taught in the FourierTransform class. So any arbitrary signal can be treated as a signal formed bythe summation of a number of sinusoids. x(t) =a1sin(1t)+a2sin(2t)+a3sin(3t)+........ So by linearity, it is sufficient to know the system responseto the sinusoid input. These responses can be added to get theresponse to the input x(t).
Hope it is clear.

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