---
title: "Check LTI: y(t) = 2x(t) + 3"
subject: "Data Communication"
chapter: 2
key: "semester_5/data_communication:lti-check-2x-plus-3"
source: https://www.bctengineer.com/q/semester_5/data_communication/lti-check-2x-plus-3
site: BCT Engineering
---

# Check LTI: y(t) = 2x(t) + 3

Data Communication · Chapter 2: Signals and Systems

- Asked in: 2079 Baisakh, 5 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/lti-check-2x-plus-3
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-2

## Question

A system's output is given as y(t) = 2x(t) + 3, where x(t) is its input. Determine whether the system is LTI or not.

## Short answer

The system y(t)=2x(t)+3 is not LTI. It is non-linear because the constant offset 3 breaks superposition (zero input gives 3). It is time-invariant: delaying input yields same delayed output.

## Answer

A system is LTI only if it is both linear and time-invariant.

#### Linearity
Let x₁ give y₁ = 2x₁ + 3 and x₂ give y₂ = 2x₂ + 3.

Response to a·x₁ + b·x₂: y = 2(a·x₁ + b·x₂) + 3

Combination of outputs: a·y₁ + b·y₂ = 2(a·x₁ + b·x₂) + 3(a + b)

These are equal only when a + b = 1, so superposition fails in general. (Quick test: x(t) = 0 gives y(t) = 3 ≠ 0.) The system is **non-linear**. It is only incrementally (affine) linear.

#### Time invariance
Delay the input: x(t - t₀) gives y₁ = 2x(t - t₀) + 3.
Delay the output: y(t - t₀) = 2x(t - t₀) + 3.
They are equal, so the system is **time-invariant**.

**Answer: the system is not LTI**, because it is non-linear, although it is time-invariant.

## Check yourself

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

1. y = 2x + 3 is
   A. linear
   B. time-varying
   C. non-linear
   D. unstable
   Answer: C. x=0 gives 3. (Key point: Linearity)

2. y = 2x + 3 is
   A. time-varying
   B. non-causal
   C. unstable
   D. time-invariant
   Answer: D. Shifts match. (Key point: Time invariance)

3. The system is
   A. not LTI
   B. LTI
   C. linear only
   D. causal only
   Answer: A. Non-linear. (Key point: Conclusion)

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