A causal stable LTI system has y[n]= y[n-1]+ x[n]. Its impulse response satisfies n=0 h[n]=2. Find the relation between ,.

GATE 2014 EC (EC2) – Question 54

Networks, Signals and Systems · LTI Systems · 2 marks · Multiple choice

A causal stable LTI system has $y[n]=\alpha y[n-1]+\beta x[n]$. Its impulse response satisfies $\sum_{n=0}^\infty h[n]=2$. Find the relation between $\alpha,\beta$.

$\alpha=1-\beta/2$
$\alpha=1+\beta/2$
$\alpha=2\beta$
$\alpha=-2\beta$

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Correct answer: (A) $\alpha=1-\beta/2$

Explanation

With initial rest, h[n]=βαⁿu[n]. Stability and the nonzero specified sum require |α|<1, so Σh[n]=β/(1−α)=2. Rearranging gives **α=1−β/2,A**. The convergence requirement matters when using the geometric-series formula.