The discrete time system described by y(n) = [x(n)]² is

Concept:

Linearity: Necessary and sufficient condition to prove the linearity of the system is that the linear system follows the laws of superposition i.e. the response of the system is the sum of the responses obtained from each input considered separately.

y{ax1[t] + bx2[t]} = a y{x1[t]} + b y{x2[t]}

Conditions to check whether the system is linear or not.

• The output should be zero for zero input
• There should not be any non-linear operator present in the system.

Causal system:

If O/P of the system is independent of the future value of input then the system is said to be causal. Causal systems are practical or physically reliable systems.

Analysis:

y(n) = [x(n)]2 

Linearity check:

x₁(n) → x₁(n)²

x₂(n) → x₂(n)²

x₃(n) = [x1(n) + x₂(n)] →  [x1(n) + x2(n)]² = x₁(n)² + x₂(n)² + 2x₁(n) x₂(n)

≠ x1(n)² + x₂(n)²

∴ Non - linear

Causality check:

y(0) = x(0)²

y(1) = x(1)²

y(-1) = x(-1)²

∴ the system is causal.