An LTI system having transfer function $\dfrac{s^2 +1}{s^2 +2s + 1}$ and input x (t) = sin (t + 1) is in steady state. The output is sampled at a rate $\omega_s$rad/s to obtain the final output {x (k)}. Which of the following is true?
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y (.) is zero for all sampling frequencies $\omega_s$.
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y (.) is nonzero for all sampling frequencies $\omega_s$.
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y (.) is nonzero for $\omega_s$> 2, but zero for $\omega_s$< 2.
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y (.) is zero for$\omega_s$ > 2, but nonzero for $\omega_s$< 2.
A
Correct answer
Explanation
The transfer function H(s) = (s²+1)/(s²+2s+1) has poles at s=-1 (repeated). When input is sin(t+1), the steady-state response is zero because the system cannot generate sin(t) due to pole-zero cancellation (the s²+1 term in numerator cancels the imaginary axis poles that would respond to sin(t)). Thus output y(t) = 0 for all t, and sampling gives y(k) = 0 for any sampling frequency.