---
title: "Linearity of y(t) = 2t²x(t) + 3t x(t − 4)"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:linearity-2t2x-3tx"
source: https://www.bctengineer.com/q/semester_6/communication_systems/linearity-2t2x-3tx
site: BCT Engineering
---

# Linearity of y(t) = 2t²x(t) + 3t x(t − 4)

Communication Systems · Chapter 1: Introduction

- Asked in: 2081 Chaitra, 4 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-2t2x-3tx
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

Determine whether the signal y(t) = 2t²x(t) + 3t x(t − 4) is linear or not.

## Short answer

The output is a linear combination of x(t) and x(t−4) with time-dependent multipliers. Superposition holds: weighting and summing inputs yields the same weighted sum of outputs, so the system is linear (but time-varying).

## Answer

A system is **linear** if it obeys superposition: for inputs $x_1(t)\to y_1(t)$ and $x_2(t)\to y_2(t)$, the input $a x_1(t) + b x_2(t)$ must give $a y_1(t) + b y_2(t)$ for all constants $a$, $b$.

Given: $y(t) = 2t^2 x(t) + 3t\,x(t-4)$

**Step 1: outputs for the individual inputs**

$$
\begin{aligned}
y_1(t) &= 2t^2 x_1(t) + 3t\,x_1(t-4) \\
y_2(t) &= 2t^2 x_2(t) + 3t\,x_2(t-4)
\end{aligned}
$$

**Step 2: output for the combined input** $x_3(t) = a x_1(t) + b x_2(t)$

$$
\begin{aligned}
y_3(t) &= 2t^2\left[a x_1(t) + b x_2(t)\right] + 3t\left[a x_1(t-4) + b x_2(t-4)\right] \\
&= a\left[2t^2 x_1(t) + 3t\,x_1(t-4)\right] + b\left[2t^2 x_2(t) + 3t\,x_2(t-4)\right] \\
&= a\,y_1(t) + b\,y_2(t)
\end{aligned}
$$

**Step 3: compare.** The response to the weighted sum equals the weighted sum of responses, so superposition (additivity and homogeneity) holds. Also, zero input gives zero output.

The multipliers $2t^2$ and $3t$ depend on time, which makes the system **time-varying**, but they do not affect linearity, because the input appears only to the first power and there is no constant term.

**Answer:** $y(t) = 2t^2x(t) + 3t\,x(t-4)$ is a **linear** system (it is, however, time-variant).

## Check yourself

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

1. To test linearity you check that the input a x1(t) + b x2(t) produces:
   A. y1(t) y2(t)
   B. a y1(t) + b y2(t)
   C. (a + b) y1(t)
   D. y1(t - a) + y2(t - b)
   Answer: B. Linearity means additivity and homogeneity: the weighted sum of inputs gives the same weighted sum of outputs. (Key point: Superposition test)

2. What is the conclusion for y(t) = 2t^2 x(t) + 3t x(t-4)?
   A. Non-linear, because of the t^2 term
   B. Non-linear, because of the delay x(t-4)
   C. Linear, and also time-invariant
   D. Linear, but time-variant
   Answer: D. Superposition holds, but the multipliers 2t^2 and 3t depend on time so the system is time-varying. (Key point: Result for the given system)

3. Why do the coefficients 2t^2 and 3t not make the system non-linear?
   A. The input appears only to the first power with no constant term
   B. They are constants in disguise
   C. Time functions are ignored in linearity tests
   D. Delayed inputs always make a system linear
   Answer: A. Linearity concerns how the output depends on the input; a time-varying gain still scales and adds inputs correctly. (Key point: Time-dependent coefficients)

## Close variants set in other exams

- [Linearity of y(t) = x²(t)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Alinearity-x2) — Communication Systems, chapter 1, asked 2×

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