---
title: "Basic properties of systems"
subject: "Data Communication"
chapter: 2
key: "semester_5/data_communication:system-properties"
source: https://www.bctengineer.com/q/semester_5/data_communication/system-properties
site: BCT Engineering
---

# Basic properties of systems

Data Communication · Chapter 2: Signals and Systems

- Asked in: 2074 Asoj, 5 marks; 2071 Chaitra, 4 marks; 2071 Shrawan, 8 marks (asked 3 times)
- Video: [Basic System Properties](https://www.youtube.com/watch?v=npsZ2SBeVpA) — 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/system-properties
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-2

## Question

Explain the basic properties of systems (linearity, time invariance, causality, stability, memory) with examples.

## Short answer

A system is linear if superposition holds (y=3x); time-invariant if a time shift in input only shifts output (y=x(t)+x(t-1)); causal if output depends only on present/past inputs (y=x(t)+x(t-2)); BIBO stable if bounded input gives bounded output; memoryless if output depends only on current input (y=5x).

## Answer

A system maps an input x(t) to an output y(t). Its basic properties are as follows.

#### 1. Linearity
A system is linear if it obeys superposition (additivity and homogeneity): the response to a·x₁ + b·x₂ is a·y₁ + b·y₂.
- Linear: y(t) = 3x(t). Non-linear: y(t) = x²(t), and y(t) = 2x(t) + 3 (because of the constant).

#### 2. Time invariance
A time shift in the input only shifts the output by the same amount: x(t - t₀) gives y(t - t₀).
- Time-invariant: y(t) = x(t) + x(t - 1). Time-varying: y(t) = t·x(t).

#### 3. Causality
The output at time t depends only on present and past inputs, not on future ones.
- Causal: y(t) = x(t) + x(t - 2). Non-causal: y(t) = x(t + 1).
- An LTI system is causal if h(t) = 0 for t < 0.

#### 4. Stability (BIBO)
A bounded input always gives a bounded output. For an LTI system: ∫|h(t)| dt < ∞.
- Stable: h(t) = e^(-t)u(t). Unstable: h(t) = u(t) (integrator).

#### 5. Memory
A memoryless (static) system has an output that depends only on the present input; a system with memory depends on past or future values.
- Memoryless: y(t) = 5x(t). With memory: y(t) = x(t - 1), or an integrator.

Another property is invertibility: distinct inputs give distinct outputs, so the input can be recovered from the output.

## Check yourself

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

1. Which is non-linear?
   A. y = 3x
   B. y = 2x + 3
   C. y = x(t) + x(t-1)
   D. y = 5x
   Answer: B. Constant breaks superposition. (Key point: Linear)

2. Which is time-varying?
   A. y = x(t) + x(t-1)
   B. y = 3x
   C. y = t x(t)
   D. y = x(t-2)
   Answer: C. Multiplier depends on t. (Key point: Time-varying)

3. Which is non-causal?
   A. y = x(t-2)
   B. y = x(t)
   C. y = 5x
   D. y = x(t+1)
   Answer: D. Uses a future input. (Key point: Causal)

4. Which impulse response is unstable?
   A. u(t)
   B. e^(-t)u(t)
   C. t e^(-3t)u(t)
   D. e^(-2t)u(t)
   Answer: A. Integrator is not absolutely integrable. (Key point: Stability)

## Related in Data Communication

- [Invertibility; properties of y[n], y(t)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Ainvertibility-system-properties) — Data Communication, chapter 2
- [Causality, stability; energy and power signals](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Acausality-stability-energy-power) — Data Communication, chapter 2
- [Check properties of y(t) = 2x(t+1)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Acheck-2x-t-plus-1) — Data Communication, chapter 2
- [Recursive/nonrecursive systems; stability](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Arecursive-nonrecursive-stability) — Data Communication, chapter 2
- [Stable/unstable systems; test stability](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Astability-define-test-72c) — Data Communication, chapter 2
- [Properties of continuous-time Fourier series](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afourier-series-properties) — Data Communication, chapter 3

## 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×
- [Properties of continuous time systems](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Act-system-properties) — Signals and Systems, chapter 5, asked 2×
- [Memory and memoryless systems; causality](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Amemory-causality-dt) — 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