GATE 2017 EE – Question 61
Consider a causal and stable LTI system with rational transfer function $H(z)$, whose corresponding impulse response begins at $n = 0$. Furthermore, $H(1) = \frac{5}{4}$. The poles of $H(z)$ are $p_k = \frac{1}{\sqrt{2}}\exp\left(j\frac{(2k - 1)\pi}{4}\right)$ for $k = 1, 2, 3, 4$. The zeros of $H(z)$ are all at $z = 0$. Let $g[n] = j^nh[n]$. The value of $g[8]$ equals ________. (Give the answer up to three decimal places.)
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Correct answer: 0.09 to 0.1
Explanation
The four poles are the roots of $z^4 = \left(\frac{1}{\sqrt{2}}\right)^4 e^{j\pi} = -\frac{1}{4}$, so $H(z) = \frac{Kz^4}{z^4 + 1/4} = \frac{K}{1 + \frac{1}{4}z^{-4}}$. From $H(1) = \frac{K}{1.25} = \frac{5}{4}$ we get $K = \frac{25}{16}$. Expanding gives $h[4m] = K\left(-\frac{1}{4}\right)^m$ and zero at other times. So $h[8] = K \times \frac{1}{16}$. Then $g[8] = j^8h[8] = \frac{25}{256} = 0.098$.