Write down the differential equation relating input x(l) and output y(t). Is it

Question

Write down the differential equation relating input x(l) and output y(t). Is it possible to obtain the FRF H(I omega ) of S from its System Function H(s) Why/Why Not? Write down the amplitude Y(I omega ) and the phase B(omega) of the Fourier Transform y (I omega ) - of the output y(t) when the input x(t): x(t) = delta (t) - e -t U(t) is applied to the system 5, that is, x(t) = delta (t) - e -t U(t) rightarrow [S] rightarrow y(t)

Explanation / Answer

clearly y(t) is a convolution of two functions h(t) and x(t) where, h(t) = e^-t*sin(t)*u(t) Now, taking Laplace transform on both sides, and applying convolution property of Laplace transforms, Y(s) = H(s)*X(s) where, H(s) = 1/{ (s+1)^2 + 1 } => Y(s) = X(s)/{ (s+1)^2 + 1 } => { (s+1)^2 + 1 }Y(s) = X(s) => { s^2 + 2s + 2 } Y(s) = X(s) => s^2Y(s) + 2sY(s) + 2Y(s) = X(s) Now, taking inverse Laplace transform y''(t) + 2y'(t) + y(t) = x(t) Yes, it is possible to obtain the FRF from H(s) as the given system is LINEAR Now, x(t) = del(t) - e^-t*u(t) => X(s) = 1 - (1/s+1) => X(s) = s/(s+1) Now, Y(s) = H(s)X(s) => Y(s) = [s/(s+1)]*[1/{ (s+1)^2 + 1 }] => Y(s) = s/( (s+1)*(s^2 + 2s + 2) ) (there are two different conventions used for calculating Fourier transform, I'm using Y(iw) = Y(s=iw) ) Y(iw) = iw/( (iw+1)(-w^2 + i2w +2) ) Y(iw) = iw/(iw+1)(2 -w^2 + i2w) now, |Y(iw)| = w/sqrt( (1+w^2)*( (2-w^2)^2 + (2w)^2 ) ) = w/sqrt( (1+w^2)(5w^4 - 4w^2 + 4) ) and, theta(w) = (pi/2) - arctan(w/1) - arctan(2w/(2-w^2))

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