---
title: "Check properties of y(t) = t·x(t-1)"
subject: "Data Communication"
chapter: 2
key: "semester_5/data_communication:check-t-xt-1"
source: https://www.bctengineer.com/q/semester_5/data_communication/check-t-xt-1
site: BCT Engineering
---

# Check properties of y(t) = t·x(t-1)

Data Communication · Chapter 2: Signals and Systems

- Asked in: 2069 Asar, 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_5/data_communication/check-t-xt-1
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-2

## Question

Determine whether or not the given system is linear, time invariant, causal and with or without memory: y(t) = t·x(t-1).

## Short answer

It is linear, time-variant, causal, and with memory (output depends on past input x(t-1)).

## Answer

y(t) = t·x(t - 1)

- **Linear?** Response to a·x₁ + b·x₂ is t[a·x₁(t-1) + b·x₂(t-1)] = a·y₁ + b·y₂. **Linear.**
- **Time-invariant?** Delay the input by t₀: y₁(t) = t·x(t - 1 - t₀). Delay the output: y(t - t₀) = (t - t₀)·x(t - t₀ - 1). These differ because of the multiplier t. **Time-variant.**
- **Causal?** The output at t depends on x(t - 1), the input one second earlier. No future values are used. **Causal.**
- **Memory?** The output at time t depends on a past input, x(t - 1), so the system **has memory** (it is dynamic).

**Answer: linear, time-variant, causal, with memory.**

## Check yourself

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

1. y = t x(t-1) is
   A. Nonlinear
   B. Linear only if t>1
   C. Nonlinear due to t
   D. Linear
   Answer: D. t is a fixed multiplier. (Key point: Linearity)

2. Why is it time-variant?
   A. The multiplier t makes the delayed output (t-t0)x(t-t0-1)
   B. The shift is 1
   C. It has memory
   D. It is nonlinear
   Answer: A. Explicit time dependence. (Key point: Time invariance)

3. It is causal because
   A. It is linear
   B. It uses x(t-1), a past value
   C. It is time-variant
   D. It uses x(t+1)
   Answer: B. No future input. (Key point: Causality)

4. It has memory because
   A. It is causal
   B. It is linear
   C. Output depends on the input 1 s earlier
   D. It has t
   Answer: C. Dependence on past input. (Key point: Memory)

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