Correct Answer - Option 1 :
\(\dfrac{z^2}{(z+1)(z-1)}\)
Concept:
1. Analysis of continuous-time LTI systems can be done using z-transforms.
It is a powerful mathematical tool to convert differential equations into algebraic equations.
2. The bilateral (two-sided) z-transform of a discrete-time signal x(n) is given as
\(Z.T[x(n)] = X(Z) = \Sigma_{n = -\infty}^{\infty} x(n)z^{-n}\)
3. The unilateral (one-sided) z-transform of a discrete-time signal x(n) is given as
\(Z.T[x(n)] = X(Z) = \Sigma_{n = 0}^{\infty} x(n)z^{-n}\)
4. Time shifting property of Z transform
x(n - 1) = z-1x(z)
5. Z transform of u(n)
\(u\left( n \right) \leftrightarrow \frac{1}{{1 - {z^{ - 1}}}}\)
Calculation:
Given,
y(n) + y(n-1) = x(n) ----(1)
Now finding unit step response of equation (1) using above properties we get:
\(Y\left( z \right) + {z^{ - 1}}Y\left( z \right) = \frac{1}{{1 - {z^{ - 1}}}}\)
\(Y\left( z \right)\left[ {1 + {z^{ - 1}}} \right] = \frac{1}{{1 - {z^{ - 1}}}}\)
\(Y\left( z \right) = \frac{{{z^2}}}{{\left( {z + 1} \right)\left( {z - 1} \right)}}\)
Hence option (1) is the correct answer.
