# Unit step response of the system described by the equation y(n) + y(n - 1) - x(n) is

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Unit step response of the system described by the equation y(n) + y(n - 1) - x(n) is
1. $\frac{{{Z^2}}}{{(Z + 1)(Z - 1)}}$
2. $\frac{{{Z}}}{{(Z + 1)(Z - 1)}}$
3. $\frac{{{Z+1}}}{{Z - 1}}$
4. $\frac{{Z{(Z-1)}}}{{Z + 1}}$

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Correct Answer - Option 1 : $\frac{{{Z^2}}}{{(Z + 1)(Z - 1)}}$

y(n) + y(n – 1) = x(n)

Taking Z transform we get

Y(Z) + Z-1 Y(Z) = X(Z)

$\frac{{Y\left( Z \right)}}{{X\left( Z \right)}} = \frac{1}{{1 + {Z^{ - 1}}}}$

Now,

x(n) = u(n)

$Y\left( Z \right) = \frac{1}{{\left( {1 + {Z^{ - 1}}} \right)}} \times \left( Z \right)$

$= \frac{1}{{\left( {1 + {Z^{ - 1}}} \right)}} \times \frac{1}{{\left( {1 - {Z^{ - 1}}} \right)}}$

$= \left( {\frac{Z}{{Z + 1}}} \right)\left( {\frac{Z}{{Z - 1}}} \right)$

$= \frac{{{Z^2}}}{{\left( {Z + 1} \right)\left( {Z - 1} \right)}}$