---
title: "Linearity of y[n]=x²[n]"
subject: "Digital Signal Analysis and Processing"
chapter: 1
key: "semester_7/digital_signal_analysis_and_processing:linearity-check-square"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/linearity-check-square
site: BCT Engineering
---

# Linearity of y[n]=x²[n]

Digital Signal Analysis and Processing · Chapter 1: Discrete Time Signals and Systems

- Asked in: 2074 Chaitra, 3 marks
- Video: [Check for Linearity, Causality, Time Invariance, and Dynamicity of a System (DSP Lecture-54)](https://www.youtube.com/watch?v=Ddf5Z6XBP00) — Engineers choice tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/linearity-check-square
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-1

## Question

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

## 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].

## Answer

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**.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Linearity requires T{a1x1 + a2x2} =
   A. T{x1} T{x2}
   B. a1 a2 T{x1}
   C. T{x1} + T{x2} + a1
   D. a1 T{x1} + a2 T{x2}
   Answer: D. Additivity and homogeneity together. (Key point: Superposition)

2. For y = x^2, the combined input a1x1 + a2x2 yields
   A. a1x1^2 + a2x2^2
   B. a1a2x1x2
   C. a1^2x1^2 + 2a1a2x1x2 + a2^2x2^2
   D. x1^2 + x2^2
   Answer: C. Squaring a sum creates a cross term. (Key point: Combined input)

3. With x1 = x2 = 1, the combined output is 4 but sum of outputs is
   A. 4
   B. 2
   C. 1
   D. 0
   Answer: B. 1 + 1 = 2 not 4, so the system is non-linear. (Key point: Counter-example)

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