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Linearity of y[n]=x²[n]

Short answer

The system y[n]=x^2[n] is non-linear. Superposition fails: for x1=x2=1, T[x1+x2]=4 while T[x1]+T[x2]=2, and homogeneity also fails because T[2x]=4 not 2T[x].

Determine whether the system y[n] = x²[n] is linear or non-linear.

A system is linear if it obeys superposition: T{a₁x₁[n] + a₂x₂[n]} = a₁T{x₁[n]} + a₂T{x₂[n]} for all inputs and constants.

System: y[n] = x²[n].

Step 1. Output for x₁ and x₂ separately: y₁[n] = x₁²[n], y₂[n] = x₂²[n]. Weighted sum of outputs:

a₁y₁[n] + a₂y₂[n] = a₁x₁²[n] + a₂x₂²[n]

Step 2. Output for the combined input x₃[n] = a₁x₁[n] + a₂x₂[n]:

y₃[n] = (a₁x₁[n] + a₂x₂[n])² = a₁²x₁² + 2a₁a₂x₁x₂ + a₂²x₂²

Step 3. Compare: y₃[n] ≠ a₁y₁[n] + a₂y₂[n] (extra cross term 2a₁a₂x₁x₂, and a₁², a₂² instead of a₁, a₂).

Counter-example: x₁ = x₂ = 1 (a₁ = a₂ = 1): y₃ = (1 + 1)² = 4, but y₁ + y₂ = 1 + 1 = 2. Also homogeneity fails: x = 2 gives 4, not 2·T{1} = 2.

Answer: The system y[n] = x²[n] is non-linear.

✓ Answer checked by Claude Opus 5.5 (high effort)

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