GATE 2017 EC (EC2) – Question 43
Consider an LTI system with magnitude response
$$\lvert H(f)\rvert=\begin{cases}1-\lvert f\rvert/20,&\lvert f\rvert\le20,\\0,&\lvert f\rvert>20,\end{cases}$$
and phase response $\arg H(f)=-2f$. The input is
$$x(t)=8\cos(20\pi t+\pi/4)$$
$$\qquad+16\sin(40\pi t+\pi/8)$$
$$\qquad+24\cos(80\pi t+\pi/16).$$
The average power of the output y(t) is ________.
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Show answer and explanation
Correct answer: 7.95 to 8.05
Explanation
The three input frequencies are 10, 20 and 40 Hz. The corresponding magnitude gains are $1-10/20=1/2$, zero and zero.
Only the 10 Hz component remains. Its output amplitude is $8(1/2)=4$. The phase response changes the phase but does not change the sinusoid's average power.
A sinusoid of amplitude A has average squared value $A^2/2$. Thus the output power is $4^2/2=$ **8**, within the official interval 7.95–8.05.