---
title: "Causality condition for CT LTI system"
subject: "Signals and Systems"
chapter: 5
key: "semester_4/signals_and_systems:causal-ct-lti-condition"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/causal-ct-lti-condition
site: BCT Engineering
---

# Causality condition for CT LTI system

Signals and Systems · Chapter 5: Continuous-Time Systems

- Asked in: 2077 Chaitra, 2+3 marks
- Video: [Causal and Non-Causal Systems](https://www.youtube.com/watch?v=mqwUtn5cip8) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/causal-ct-lti-condition
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-5

## Question

What do you mean by causal and non causal systems? Derive the condition for a continuous time LTI system to be causal.

## Short answer

A causal system output depends only on present and past inputs; non-causal uses future inputs. For CT LTI, causality iff h(t)=0 for t<0. Then output integral runs from tau=0 to infinity.

## Answer

#### Causal and non-causal systems
- A **causal** system is one whose output at any time depends only on the present and past values of the input, not on future values. Example: $y(t) = x(t) + 2x(t-1)$. All physical real-time systems are causal.
- A **non-causal** system's output depends on future input values as well. Example: $y(t) = x(t+1)$, $y(t) = x(-t)$. Such systems can be used only on stored (recorded) data, e.g. image processing.

#### Condition for a CT LTI system to be causal
For an LTI system,
$$
y(t) = \int_{-\infty}^{\infty} h(\tau)\,x(t-\tau)\,d\tau
$$
Split the integral at $\tau = 0$:
$$
y(t) = \underbrace{\int_{0}^{\infty} h(\tau)\,x(t-\tau)\,d\tau}_{\text{uses } x \text{ at times} \le t} + \underbrace{\int_{-\infty}^{0} h(\tau)\,x(t-\tau)\,d\tau}_{\text{uses } x \text{ at times} > t}
$$
The second integral uses future inputs $x(t - \tau)$ with $t - \tau > t$. For $y(t)$ to be independent of every future input, this term must vanish for all $x$, which requires
$$
h(t) = 0 \quad \text{for } t < 0
$$
Then
$$
y(t) = \int_{0}^{\infty} h(\tau)\,x(t-\tau)\,d\tau = \int_{-\infty}^{t} x(\tau)\,h(t-\tau)\,d\tau
$$
**Condition: a CT LTI system is causal if and only if its impulse response $h(t) = 0$ for $t < 0$.** (Physically, the system cannot respond before the impulse is applied at $t = 0$.)

Examples: $h(t) = e^{-2t}u(t)$ is causal; $h(t) = e^{-2|t|}$ or $h(t) = \sin(\omega_c t)/\pi t$ is non-causal.

## Check yourself

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

1. A causal system's output at time t depends on:
   A. Future inputs only
   B. All inputs equally
   C. Present and past inputs only
   D. Only the input at t+1
   Answer: C. Causal systems are non-anticipative. (Key point: Causal definition)

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

3. Splitting y(t)=∫h(τ)x(t-τ)dτ at τ=0, which part uses future input?
   A. The integral over τ < 0
   B. The integral over τ > 0
   C. Both parts
   D. Neither part
   Answer: A. For τ<0, x(t-τ) has argument greater than t. (Key point: Splitting the integral)

4. Condition for a CT LTI system to be causal:
   A. h(t) = 0 for t > 0
   B. h(t) = 0 for t < 0
   C. h(0) = 0
   D. h(t) is absolutely integrable
   Answer: B. The future-input term must vanish for all x. (Key point: Result)

## Related in Signals and Systems

- [Condition for LTI system to be causal](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-causality-condition) — Signals and Systems, chapter 6, asked 3×
- [LTI: causality and time variance](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-causality-time-variance) — Signals and Systems, chapter 5, asked 2×
- [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×
- [Commutative property of CT LTI systems](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Acommutative-property-ct) — Signals and Systems, chapter 5
- [Short note: Causal and non-causal systems](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Ashort-note-causal-noncausal) — Signals and Systems, chapter 5
- [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×

## Related in other subjects

- [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
- [Classification of systems; ideal LPF is non-causal](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aclassify-systems-ideal-lpf) — Communication Systems, chapter 1
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- [LTI system output in time domain](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Alti-output-time-domain) — Communication Systems, chapter 1