---
title: "Condition for LTI system to be causal"
subject: "Signals and Systems"
chapter: 6
key: "semester_4/signals_and_systems:lti-causality-condition"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/lti-causality-condition
site: BCT Engineering
---

# Condition for LTI system to be causal

Signals and Systems · Chapter 6: Discrete-Time Systems

- Asked in: 2080 Asoj, 2 marks; 2079 Asoj, 5 marks; 2078 Chaitra, 5 marks (asked 3 times)
- Video: [Causal LTI System](https://www.youtube.com/watch?v=xdxHuw7Rs8k) — 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/lti-causality-condition
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6

## Question

What is the condition for LTI system to be causal?

## Short answer

An LTI system is causal if and only if its impulse response is zero for negative time: h[n] = 0 for n < 0 (or h(t) = 0 for t < 0). This ensures the output depends only on present and past inputs.

## Answer

An LTI system is **causal** if and only if its impulse response is zero for negative time:

$$
h[n] = 0 \quad \text{for } n < 0 \qquad \left(\text{CT: } h(t) = 0 \text{ for } t < 0\right)
$$

**Reason:** the output of a DT LTI system is the convolution sum

$$
y[n] = \sum_{k=-\infty}^{\infty} h[k]\,x[n-k] = \underbrace{\sum_{k=-\infty}^{-1} h[k]\,x[n-k]}_{\text{future inputs}} + \underbrace{\sum_{k=0}^{\infty} h[k]\,x[n-k]}_{\text{present and past}}
$$

For $k < 0$, $x[n-k]$ is an input at a time later than $n$ (a future value). A causal system's output cannot depend on future inputs, so every weight $h[k]$ for $k < 0$ must be zero. Then

$$
y[n] = \sum_{k=0}^{\infty} h[k]\,x[n-k] = \sum_{k=-\infty}^{n} x[k]\,h[n-k]
$$

Physically: $h[n]$ is the response to $\delta[n]$ applied at $n = 0$, and a causal system cannot respond before the input arrives.

**Examples:**
- $h[n] = (0.5)^n u[n]$: causal.
- $h[n] = u[n] - u[n-4]$: causal.
- $h[n] = \delta[n+1] + \delta[n]$, i.e. $y[n] = x[n+1] + x[n]$: non-causal (uses the future input $x[n+1]$).
- $h[n] = (0.5)^{|n|}$: non-causal.

**Related terms:** a sequence that is zero for $n < 0$ is called a **causal sequence**. If $h[n] = 0$ for $n > 0$ instead, the system is **anti-causal**. Causality is needed for real-time operation; non-causal systems can still be used for stored data (e.g. image processing).

## Check yourself

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

1. An LTI system is causal if and only if:
   A. h[n] = 0 for n > 0
   B. The sum of |h[n]| is finite
   C. h[n] = 0 for n < 0
   D. h[n] is even
   Answer: C. Causal means no response before the impulse is applied. (Key point: Condition)

2. Why must h[k] = 0 for k < 0?
   A. Because it ensures stability
   B. Because x[n - k] with k < 0 is a future input
   C. Because h[k] would be infinite
   D. Because the system would be time varying
   Answer: B. A causal output cannot depend on future inputs. (Key point: Reasoning)

3. Which system is non-causal?
   A. h[n] = 0.5^n u[n]
   B. h[n] = u[n] - u[n - 4]
   C. h[n] = delta[n+1] + delta[n], i.e. y[n] = x[n+1] + x[n]
   D. h[n] = delta[n]
   Answer: C. It uses the future input x[n+1]. (Key point: Non-causal example)

4. What does h[n] = 0 for n > 0 define?
   A. A causal system
   B. A memoryless system
   C. An unstable system
   D. An anti-causal system
   Answer: D. Anti-causal systems depend only on present and future inputs. (Key point: Anti-causal)

## Related in Signals and Systems

- [Causality condition for CT LTI system](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Acausal-ct-lti-condition) — Signals and Systems, chapter 5
- [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
- [Frequency response of a CT LTI system](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Act-frequency-response) — Signals and Systems, chapter 5, asked 3×
- [Describe LTI system properties](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-properties-describe) — Signals and Systems, chapter 5, asked 3×
- [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×
- [Short note: LTI system](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Ashort-note-lti-system) — Signals and Systems, chapter 5

## Related in other subjects

- [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×
- [Plot sequence; properties of LTI system](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aplot-sequence-lti-properties) — Digital Signal Analysis and Processing, chapter 1
- [Parseval's theorem; output PSD of LTI system](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aparseval-power-lti-psd) — Communication Systems, chapter 1
- [Hilbert transformation; properties of LTI system](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahilbert-define-lti-properties) — Communication Systems, chapter 1