Question

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. \(\frac{{Z{(Z-1)}}}{{Z + 1}}\)

Answer 1

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)

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

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

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

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