Networks, Signals and Systems · LTI Systems · 2 marks · Numerical answer
Consider the parallel combination of LTI systems shown. Their impulse responses are $h_1(t)=2\delta(t+2)-3\delta(t+1)$ and $h_2(t)=\delta(t-2)$. If x(t) is a unit step, the energy of y(t) is ________.
Parallel systems add their outputs. Convolution with each shifted impulse gives $$y(t)=2u(t+2)-3u(t+1)+u(t-2).$$ Thus y is zero before −2, equals 2 on [−2,−1), equals −1 on [−1,2), and returns to zero from t=2 onward.
Integrate its squared magnitude over these intervals: $E=2^2(1)+(-1)^2(3)=$ **7**. Endpoint values do not affect the integral.
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