---
title: "System properties; is y(t) = sin[x(t)] time-variant"
subject: "Signals and Systems"
chapter: 5
key: "semester_4/signals_and_systems:system-properties-sin-x"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/system-properties-sin-x
site: BCT Engineering
---

# System properties; is y(t) = sin[x(t)] time-variant

Signals and Systems · Chapter 5: Continuous-Time Systems

- Asked in: 2079 Chaitra, 2+5 marks; 2070 Bhadra, 2+3 marks (asked 2 times)
- Video: [Properties Of Systems | Example 1](https://www.youtube.com/watch?v=NYtQ542JRw0) — EnggClasses
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/system-properties-sin-x
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-5

## Question

What are the properties of system? Determine whether the given system is time-variant or not. y(t) = sin[x(t)]

## Short answer

The system is time-invariant because a delay in input yields the same delay in output: sin[x(t - t0)] = y(t - t0). It is memoryless, causal, BIBO stable, nonlinear, and non-invertible.

## Answer

#### Properties of systems
1. **Memory:** memoryless if $y(t)$ depends only on $x(t)$ at the same instant (e.g. $y = 2x(t)$); otherwise has memory (e.g. $y = x(t-1)$).
2. **Causality:** causal if $y(t)$ depends only on present and past inputs.
3. **Linearity:** satisfies superposition: $a x_1 + b x_2 \to a y_1 + b y_2$.
4. **Time invariance:** $x(t - t_0) \to y(t - t_0)$.
5. **Stability (BIBO):** every bounded input gives a bounded output.
6. **Invertibility:** distinct inputs give distinct outputs, so an inverse system exists.

#### Is $y(t) = \sin[x(t)]$ time-variant?
Let the input be delayed: $x_1(t) = x(t - t_0)$. The output for this input is
$$
y_1(t) = \sin[x_1(t)] = \sin[x(t - t_0)]
$$
The original output delayed by $t_0$ is
$$
y(t - t_0) = \sin[x(t - t_0)]
$$
Since $y_1(t) = y(t - t_0)$, the system is **time-invariant** (not time-variant). The operation $\sin[\cdot]$ does not depend on $t$ explicitly.

Other properties, for completeness:

| Property | Result | Reason |
|---|---|---|
| Time invariance | Time-invariant | shown above |
| Linearity | Non-linear | $\sin(x_1 + x_2) \ne \sin x_1 + \sin x_2$ |
| Memory | Memoryless | uses only $x(t)$ |
| Causality | Causal | no future input |
| Stability | Stable | $\lvert y(t)\rvert \le 1$ for any input |
| Invertibility | Not invertible | $x$ and $x + 2\pi$ give same $y$ |

## Check yourself

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

1. Is y(t) = sin[x(t)] time-invariant?
   A. No; sin depends on t
   B. Yes; y1(t) = sin[x(t-t0)] = y(t-t0)
   C. No; it is nonlinear so it cannot be invariant
   D. Yes; because it is linear
   Answer: B. The operation has no explicit t dependence, so delay commutes. (Key point: Time invariance)

2. Is the system linear?
   A. No; sin(x1 + x2) != sin x1 + sin x2
   B. Yes; superposition holds
   C. Yes; it is memoryless
   D. No; it is non-causal
   Answer: A. Superposition fails for sin. (Key point: Linearity)

3. Is it BIBO stable?
   A. No; sin oscillates
   B. No; unbounded for large x
   C. Yes; |y(t)| <= 1 for any input
   D. Only for x < π
   Answer: C. Output is always bounded by 1. (Key point: Stability)

4. Is it invertible?
   A. Yes; arcsin undoes it for all x
   B. Yes; it is memoryless
   C. No; it is non-causal
   D. No; x and x + 2π give the same output
   Answer: D. Different inputs can produce the same output. (Key point: Invertibility)

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