---
title: "Causality of y(t) = x²(t) + x(t + 2)"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:causality-x2-xt2"
source: https://www.bctengineer.com/q/semester_6/communication_systems/causality-x2-xt2
site: BCT Engineering
---

# Causality of y(t) = x²(t) + x(t + 2)

Communication Systems · Chapter 1: Introduction

- Asked in: 2081 Chaitra, 4 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_6/communication_systems/causality-x2-xt2
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

Determine whether the signal y(t) = x²(t) + x(t + 2) is causal or not.

## Short answer

The term x(t+2) requires knowledge of the input two seconds into the future at every t. Thus the output depends on future values, making the system non-causal (anticipative).

## Answer

A system is **causal** if the output at any time $t$ depends only on present and past values of the input, never on future values, i.e. $y(t_0)$ depends only on $x(t)$ for $t \le t_0$.

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

Check the output at a few time instants:

| $t$ | $y(t)$ | Input values needed |
|---|---|---|
| 0 | $x^2(0) + x(2)$ | $x(0)$ present, $x(2)$ future |
| 1 | $x^2(1) + x(3)$ | $x(3)$ future |
| $-2$ | $x^2(-2) + x(0)$ | $x(0)$ future |

- The term $x^2(t)$ uses the **present** input, which is allowed.
- The term $x(t+2)$ uses the input **2 seconds ahead**, i.e. a future value, at every instant $t$.

Since the output at time $t$ needs $x(t+2)$, which has not yet arrived, the system cannot be built to work in real time.

**Answer:** $y(t) = x^2(t) + x(t+2)$ is **non-causal** (anticipative). (Note also that, because of the $x^2(t)$ term, it is a non-linear system.)

## Check yourself

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

1. A system is causal if the output at time t depends on:
   A. Only past and future inputs
   B. Only future inputs
   C. Only present and past inputs
   D. Only the present input, never past ones
   Answer: C. Causal systems cannot use future input values. (Key point: Causality definition)

2. In y(t) = x^2(t) + x(t+2), which term makes the system non-causal?
   A. x^2(t), since squaring is non-linear
   B. Both terms equally
   C. Neither, the system is causal
   D. x(t+2), since it uses the input 2 seconds ahead
   Answer: D. x(t+2) needs a future input value at every instant; x^2(t) uses only the present input, which is allowed. (Key point: Culprit term)

3. What does the non-causality of this system imply?
   A. It cannot be built to work in real time
   B. It must be unstable
   C. It must be time-variant
   D. It is also linear
   Answer: A. The output at time t needs x(t+2), which has not arrived yet. The x^2 term actually makes it non-linear. (Key point: Real-time implementation)

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