GATE 2024 IN – Question 22
A filter is defined by $y[n]-0.5y[n-2]=a\,x[n-4]$, where $x[n],y[n]$ are input and output. If $|H(\pi/2)|=0.5$, the value of $a$ is ______ (two decimal places).
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Correct answer: 0.70 to 0.80
Explanation
Taking the frequency response gives $H(\omega)=ae^{-j4\omega}/(1-0.5e^{-j2\omega})$. At $\omega=\pi/2$, the denominator is $1.5$ and the delay has unit magnitude.
Therefore $|a|/1.5=0.5$ and $|a|=0.75$. The official accepted answer is the positive value **0.75**; magnitude alone does not determine its sign.