Which one of the following statements is correct for the given system?

\(y(n) = {x^2}(n) + \frac{1}{{{x^2}(n - 1)}}\)

For Linear System :

System must satisfy additivity and homogeneity.

for input x₁(n) : 

\(y_1(n) = {x_1^2}(n) + \frac{1}{{{x_1^2}(n - 1)}}\)

for input x₂(n)

\(y_2(n) = {x_2^2}(n) + \frac{1}{{{x_2^2}(n - 1)}}\)

for input x(n) = x₁(n) + x₂(n)

\(y(n) = {[x_1(n)+x_2(n)]^2} + \frac{1}{{[x_1(n-1)+x_2(n-1)]^2}}\)

y(n) \(\neq\) y₁(n)+y₂(n) 

Hence Not Linear.

The present output depends on present and past values of input hence causal system.

Time Invariance :

For delayed input x(n-t)

\(y_1(n) = {x_1^2}(n-t) + \frac{1}{{{x_1^2}(n - t-1)}}\)

For delayed output y (n-t)

\(y(n-t) = {x_1^2}(n-t) + \frac{1}{{{x_1^2}(n - t-1)}}\)

y(n-t) = y₁(n)

Hence Time/Shift-Invariant System.