GATE Electrical Engineering (EE) 2020 Solved Paper

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Question : 22 of 65

Marks: +1, -0

Consider a signal

x(t)=(\;{1}/{2})^{n} 1[n]

, where

1[n]=0

if

n<0

, and

1[n]=1

if

n ≥ 0 .

The

z^{-}

transform of

x[n-k], k>0

is

\;{z^{-k}}/{1-\;{1}/{2} z^{-1}}

with region of convergence being

$|z|<2$
$|z|>2$
$|z|>1 / 2$
$|z|<1 / 2$

Solution:

x[n]=(\;{1}/{2})^{(n-k)} ⋅ u(n-k)

|z|>\;{1}/{2}

\;{1}/{1-\;{1}/{2} z^{-1}} {<──>}↖{\;\text " ZT. "\;}(\;{1}/{2})^{n} ⋅ {u}[ {n}]

|z|>\;{1}/{2}

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