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Up to this point we×39’ve been dwelling on the responses of systems to standard inputs such as the impulse the step the ramp and the parabola and we#39’ve seen that each one of those responses provides us with a different way onlooking at the system with a different insight into its dynamics, but there's another standard input heaven×39’t studied yet one which improbably as important as all the others together as you see I'm referring to the sinusoidal input do you remember that Wesak that the impulse response that is the response G of T when the input of system is a unit impulse do you remember that we saw that it conveys all the dynamic information of a system that incompletely a characterizes it is does so because if certain input is applied Toto the system let's call it u of T solar we have U of T we can calculate the output Y of T as the convolution between the impulse response and the input itself U of T and in the Laplace transform the expression of these fatwas simply that if this is the Laplace transform of the input the Laplace transform of the output was the transfer function defining the system Gogh s multiplied by the Laplace transform of the input so G of T is a perfect characterization of the dynamic sand here then we would found I'm done#39;know maybe this response well with the sinusoidal function something very similar happens and to illustrate that let me assume that our input U of T is now a periodic signal sole me draw it, so this is now the input to the system at the end the fact that it is periodic means that I can apply Fourier transform no sorry Fourier series and decompose it into an infinite sum of different sinusoidal so for example U of T would be the sum of Done×39’t know maybe this first since oil this would be calledu1 of t plus this second sinusoidu2 of t plus a third one with higher frequency, so we would have to continue to add in infinite terms here and all of them together would be our input U of Now since our system is linear that means that we can get the output of the system when the input is onlyu1 so let's call it y1 of T and the output for the second input Y 2 of T and the outputs to all the rest of sinusoidal as I was saying since the system is linear the sum of all those contributions would be Y of T that is the output of the system when the input is U of T so in order Roget to each one of these components behave to find the response of the system to a sinusoidal input, so we can use the plus for that Y office is the up the transfer function of the system times that the Laplace transform of the input U of But since let#39’s let's say that U of T ISA simple sinusoidal a times the sine omega T so 8 the amplitude and Omega Isis its frequency that means that Harwell that means that U of S is equal to Omega over s square plus Omega squares if we apply fracture a partial fraction decomposition to this multiplication we get all these terms are 1 over s plus P 1 plus R 2 over plus P 2 all those corresponding...
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