---
title: "Stability of y(t) = x(t²)"
subject: "Data Communication"
chapter: 2
key: "semester_5/data_communication:stability-xt2"
source: https://www.bctengineer.com/q/semester_5/data_communication/stability-xt2
site: BCT Engineering
---

# Stability of y(t) = x(t²)

Data Communication · Chapter 2: Signals and Systems

- Asked in: 2078 Bhadra, 5 marks
- Video: [Stable and Unstable Systems](https://www.youtube.com/watch?v=ngJ2QWvMPbI) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/stability-xt2
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-2

## Question

Determine if the following system is stable LTI: y(t) = x(t²).

## Short answer

y(t)=x(t²) is linear but time-varying, so not LTI. It is BIBO stable: bounded input guarantees bounded output. Non-causal because for |t|>1, t²>t, requiring future input values.

## Answer

y(t) = x(t²)

#### Linearity
The response to a·x₁ + b·x₂ is a·x₁(t²) + b·x₂(t²) = a·y₁ + b·y₂. The system is **linear**.

#### Time invariance
Delay the input: x₁(t) = x(t - t₀) gives y₁(t) = x₁(t²) = x(t² - t₀).
Delay the output: y(t - t₀) = x((t - t₀)²).
Since t² - t₀ ≠ (t - t₀)², the system is **time-varying**.

So the system is **not LTI**, and the LTI stability test through h(t) cannot be applied.

#### BIBO stability
If |x(t)| ≤ M for all t, then |y(t)| = |x(t²)| ≤ M. A bounded input always gives a bounded output, so the system is **BIBO stable**.

#### Causality (for completeness)
For 0 < t < 1, t² < t (a past value is used), but for t > 1, t² > t, so y(t) uses a future value of x; for example, y(2) = x(4). For negative t, t² > t as well, e.g. y(-1) = x(1). So the system is **non-causal**.

**Answer: y(t) = x(t²) is stable (BIBO), but it is not an LTI system (it is linear but time-varying), and it is non-causal.**

## Check yourself

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

1. y = x(t^2) is
   A. time-invariant
   B. non-linear
   C. unbounded
   D. time-varying
   Answer: D. t^2-t0 != (t-t0)^2. (Key point: Time invariance)

2. y = x(t^2) is BIBO
   A. stable
   B. unstable
   C. undefined
   D. marginal
   Answer: A. |y| <= M. (Key point: BIBO)

3. y(2) = x(4) shows
   A. causal
   B. non-causal
   C. anticausal
   D. LTI
   Answer: B. Uses future. (Key point: Causality)

4. It is
   A. LTI
   B. LTI unstable
   C. not LTI
   D. anticausal LTI
   Answer: C. Time-varying. (Key point: LTI)

## Related in Data Communication

- [Stable/unstable systems; test stability](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Astability-define-test-72c) — Data Communication, chapter 2
- [Define LTI system and impulse response](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Alti-impulse-response-define) — Data Communication, chapter 2, asked 2×
- [Linear/TI/stable/memoryless: y(t), y[k]](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Acheck-1-minus-e-4t-and-sin) — Data Communication, chapter 2
- [Range of a, b for stability](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Astability-range-ab) — Data Communication, chapter 2
- [Check LTI: y(t) = 2x(t) + 3](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Alti-check-2x-plus-3) — Data Communication, chapter 2
- [Stability of h(t) = t·e^(-3t)u(t)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Astability-te-3t) — Data Communication, chapter 2

## Related in other subjects

- [LTI system: linearity, stability, TI, causality](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-properties-four) — Signals and Systems, chapter 5, asked 2×
- [Random process; LTI output is WSS if input is WSS](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Arandom-process-lti-wssp) — Communication Systems, chapter 9
- [LTI properties; BIBO stability condition](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-bibo-stability) — Signals and Systems, chapter 6
- [Stability and causality of LTI systems](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Astability-causality-lti) — Digital Signal Analysis and Processing, chapter 1