Question

The discrete time system described by y(n) = x(n²) is
1. causal, linear and time varying 
2. causal, non-linear and time varying 
3. non-causal, linear and time-variant 
4. non-causal, non-linear and time-variant 

Answer 1

Correct Answer - Option 3 : non-causal, linear and time-variant 

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 the output 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.

Time invariant:

A system is time invariant if a time shift in the input signal results in identical time shift in the output signal.

y(n) = x(n) and y(n - k) = x (n – k)

Analysis:

y(n) = x(n²)

Linearity check:

x1(n) → x1(n²)

x2(n) → x2(n²)

x3(n) = [x1(n) + x2(n)] →  [x1(n²) + x2(n²)] 

= x1(n)2 + x2(n)2

∴ the system is linear

Causality check:

y(0) = x(0²)

y(-1) = x(1)

y(2) = x(2²) = x(4)

∴ the system is non-causal.

Time-variance check:

y (n) =  x (n²)

y (n-k) = x((n- k)²)

y (n,k) = x(n²-k)

y(n,k) is the output when input is shifted by k unit

y(n,k) ≠  y(n-k)

Given system is time-variant.