Question

Q3. Consider a causal LTI system whose input and output are related by the following differential equation: dy(t) +4y(t) x(t) dt and the input is x(t) cos(2Tut) sin(4t), find: (a) The transfer function of the system H(ja) (5 m) (5 m) (10 m) (5 m) (b) The impulse response of the system h(t) (c) The output y(t) (d) The power of y(t)

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Answer 1

Transfer function = Laplace transform of output y (t)/Laplace transform of input x (t) given differential equation is d y (t)/dt + 4 y (t) = x (t) taking Laplace transform S Y (s) + 4 Y (s) = X (s) Y (s) [s + 4] = X (s) H (s) = Y (s)/X (s) = 1/s + 4 in frequency domain (replacing S = j w) = H (j w) = 1/j w + 4 (ii) H (j w) = 1/4 + j w taking inverse Fourier transform impulse response is h (t) = e^- 4 t u (t) (since e^- a t u (t) rightarrow F T 1/a + j w) impulse response h (t) = e^- 4 t u (t) (iii) given input x (t) = cos 2 pi t + sin 4 pi t for sinusoidal input output is
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