---
title: "Linearity of y(t) = x²(t)"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:linearity-x2"
source: https://www.bctengineer.com/q/semester_6/communication_systems/linearity-x2
site: BCT Engineering
---

# Linearity of y(t) = x²(t)

Communication Systems · Chapter 1: Introduction

- Asked in: 2080 Chaitra, 3 marks
- Video: [Linear and Non-Linear Systems (Solved Problems) | Part 1](https://www.youtube.com/watch?v=QcAY1oPozhw) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/linearity-x2
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

Determine whether the signal y(t) = x²(t) is linear or not.

## Short answer

Squaring gives y(ax1+bx2)=a²x1²+b²x2²+2ab x1x2, which is not a y1+b y2. Superposition fails, so the system y(t)=x²(t) is non-linear.

## Answer

A system is **linear** if it satisfies superposition: $a x_1(t) + b x_2(t) \to a y_1(t) + b y_2(t)$.

Given: $y(t) = x^2(t)$

For individual inputs: $y_1(t) = x_1^2(t)$, $y_2(t) = x_2^2(t)$.

For the combined input $x_3(t) = a x_1(t) + b x_2(t)$:

$$
\begin{aligned}
y_3(t) &= \left[a x_1(t) + b x_2(t)\right]^2 \\
&= a^2x_1^2(t) + b^2x_2^2(t) + 2ab\,x_1(t)x_2(t)
\end{aligned}
$$

Required for linearity: $a y_1(t) + b y_2(t) = a x_1^2(t) + b x_2^2(t)$.

Clearly $y_3(t) \ne a y_1(t) + b y_2(t)$ (the coefficients become $a^2$, $b^2$ and a cross term $2ab\,x_1x_2$ appears).

Simple check with homogeneity: doubling the input, $x \to 2x$, gives $y = 4x^2$, i.e. the output becomes four times, not twice.

**Answer:** $y(t) = x^2(t)$ is a **non-linear** system (a square-law device, such as that used in square-law modulators and detectors).

## Check yourself

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

1. For y(t) = x^2(t), the response to a x1 + b x2 is:
   A. a x1^2 + b x2^2
   B. a^2 x1^2 + b^2 x2^2 + 2ab x1 x2
   C. (a + b) x1 x2
   D. a^2 x1^2 + b^2 x2^2 only
   Answer: B. Squaring the sum creates a cross term 2ab x1 x2 and squared coefficients, so it differs from a y1 + b y2. (Key point: Expansion of combined input)

2. If the input is doubled in y = x^2, the output becomes:
   A. Twice as large
   B. Half as large
   C. Four times as large, so homogeneity fails
   D. Unchanged
   Answer: C. (2x)^2 = 4x^2, not 2x^2, so the system is not homogeneous and therefore non-linear. (Key point: Homogeneity check)

3. What is the classification of the square-law system y(t) = x^2(t)?
   A. Linear
   B. Non-linear
   C. Linear but time-variant
   D. Linear only for positive inputs
   Answer: B. It violates superposition. Square-law devices are used in modulators and detectors because of this non-linearity. (Key point: Conclusion)

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