The input-output relationship of a system S is , t R. Write down the Unit Step R

Question

The input-output relationship of a system S is , t R. Write down the Unit Step Response g(t) and the Frequency Response Function H(i omega) of S. Compute the Fourier Transform Y(i omega) of output y(t) - by the method which you claim to be your expertise - when the input x(t) = tetU(-t) is applied to S.

Explanation / Answer

applying Laplace transform on both sides Y(s) = X(s) - (1/(s+1))X(s) Y(s) = (s/(s+1))X(s) => H(s) = s/(s+1) Now, G(s) = (1/s)H(s) G(s) = 1/(s+1) applying inverse Laplace transform g(t) = (e^-t)*u(t) Now, H(iw) = H(s=iw) H(iw) = iw/(1+iw) you can leave it either as this or you can write this as H(iw) = iw*(1-iw)/(1+w^2) H(iw) = w(w+i)/(1+w^2) now, x(t) = t*e^t*u(-t) taking Laplace transform of x(t) X(s) = 1/(1-s)^2 now, Y(s) = H(s)X(s) Y(s) = (s/(s+1))(1/(1-s)^2) Y(s) = s/(s^3-s^2-s+1) now, Y(iw) = Y(s=iw) Y(iw) = iw/(-iw^3 +w^2 - iw +1) Y(iw) = iw/((1+w^2) -i(w + w^3)) Y(iw) = iw/((1+w^2)(1-i)) again you can leave it like this or, Y(iw) = iw*(1+i)/(2*(1+w^2)) Y(iw) = w*(-1+i)/(2*(1+w^2))

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