---
title: "Causality of y(t) = t·x(t)"
subject: "Data Communication"
chapter: 2
key: "semester_5/data_communication:causality-t-xt"
source: https://www.bctengineer.com/q/semester_5/data_communication/causality-t-xt
site: BCT Engineering
---

# Causality of y(t) = t·x(t)

Data Communication · Chapter 2: Signals and Systems

- Asked in: 2078 Kartik, 4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/causality-t-xt
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-2

## Question

Test the given function y(t) = t·x(t) for causality, non-causality and anti-causality.

## Short answer

y(t)=t·x(t) is causal and memoryless; output depends solely on the present input. It is not non-causal. Anticausal only in the trivial sense of not using future inputs.

## Answer

y(t) = t·x(t)

- **Causal**: the output at time t depends only on the present input x(t) (multiplied by t); no future value is needed. So the system is **causal**.
- **Non-causal**: it would need x(t + τ) with τ > 0, which is not so; hence it is not non-causal.
- **Anticausal**: an anticausal system needs only present and future values. The output depends on the present value only, so it technically also satisfies the anticausal definition (every memoryless system is both causal and anticausal). It is not a strict anticausal system, since it uses no future input.

Check: at t = 2, y(2) = 2x(2); at t = -3, y(-3) = -3x(-3). Each uses only the same instant.

**Answer: causal (and memoryless); not non-causal; anticausal only in the trivial sense of a memoryless system.** (The system is time-varying but this does not affect causality.)

## Check yourself

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

1. y = t x(t) depends on
   A. present input only
   B. future input
   C. past input
   D. all inputs
   Answer: A. Same instant. (Key point: Dependence)

2. y = t x(t) is
   A. non-causal
   B. causal
   C. strictly anticausal
   D. unstable
   Answer: B. No future value. (Key point: Causal?)

3. Does t multiplier affect causality?
   A. Yes, non-causal
   B. Yes, anticausal
   C. No
   D. Yes, unstable
   Answer: C. Time variation is separate. (Key point: Time varying)

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