---
title: "LTI: causality and time variance"
subject: "Signals and Systems"
chapter: 5
key: "semester_4/signals_and_systems:lti-causality-time-variance"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/lti-causality-time-variance
site: BCT Engineering
---

# LTI: causality and time variance

Signals and Systems · Chapter 5: Continuous-Time Systems

- Asked in: 2082 Chaitra, 4 marks; 2075 Baisakh, 2+4 marks (asked 2 times)
- Video: [Linear, Time-Invariant, and Causal Systems](https://www.youtube.com/watch?v=mxEbLLs0sm8) — Barry Van Veen
- 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-time-variance
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-5

## Question

Define linear time invariant system. Explain causality and time variance properties of continuous time system.

## Short answer

An LTI system is linear (superposition holds) and time-invariant (a delay in input causes same delay in output). Causality means output depends only on past/present inputs; for LTI, h(t)=0 for t<0. Time-invariance test: if x(t-t0) → y1(t), y1(t) equals y(t-t0).

## Answer

#### LTI system
A **linear time-invariant (LTI) system** is one that is both
- **linear:** obeys superposition, $a x_1(t) + b x_2(t) \to a y_1(t) + b y_2(t)$, and
- **time-invariant:** a delay in the input gives the same delay in the output, $x(t - t_0) \to y(t - t_0)$.

An LTI system is completely described by its impulse response $h(t)$, and $y(t) = x(t) * h(t)$.

#### Causality
A system is **causal** if the output at any time $t_0$ depends only on present and past inputs ($t \le t_0$), never on future inputs. All real-time physical systems are causal.
- Causal: $y(t) = x(t-1)$, $y(t) = \int_{-\infty}^{t} x(\tau)d\tau$.
- Non-causal: $y(t) = x(t+1)$, $y(t) = x(-t)$ (at $t = -1$ it needs $x(1)$).
- For an LTI system: causal $\iff h(t) = 0$ for $t < 0$.

#### Time variance
A system is **time-invariant** if its behaviour does not change with time: if $x(t) \to y(t)$ then $x(t - t_0) \to y(t - t_0)$ for every $t_0$. Otherwise it is **time-variant**.

**Test:** find $y_1(t)$, the output for $x(t - t_0)$, and compare with $y(t - t_0)$.

**Example 1:** $y(t) = x(2t)$.
$$
\begin{aligned}
y_1(t) &= x(2t - t_0) \\
y(t - t_0) &= x(2(t - t_0)) = x(2t - 2t_0)
\end{aligned}
$$
They differ, so the system is **time-variant**.

**Example 2:** $y(t) = x(t)^2$: $y_1(t) = x(t - t_0)^2 = y(t - t_0)$, so it is **time-invariant**.

| Property | Condition | Example |
|---|---|---|
| Causal | output uses $t \le t_0$ only | $y = x(t-2)$ |
| Non-causal | output uses future input | $y = x(t+2)$ |
| Time-invariant | $x(t-t_0) \to y(t-t_0)$ | $y = 3x(t)$ |
| Time-variant | shift not preserved | $y = t\,x(t)$ |

## Check yourself

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

1. An LTI system is one that is:
   A. Linear and time-invariant
   B. Linear and causal
   C. Causal and stable
   D. Memoryless and linear
   Answer: A. LTI means superposition holds and delay of input gives the same delay of output. (Key point: LTI definition)

2. A system is causal if:
   A. Output depends on future inputs
   B. Output at t0 depends only on inputs at t <= t0
   C. Output is always zero
   D. The system is linear
   Answer: B. Causal systems cannot use future input; for LTI, h(t) = 0 for t < 0. (Key point: Causality)

3. Is y(t) = x(2t) time-invariant?
   A. Yes; both equal x(2t - t0)
   B. Yes; scaling never matters
   C. No; y1(t) = x(2t - t0) differs from y(t - t0) = x(2t - 2t0)
   D. No; because it is non-causal only
   Answer: C. The shifted-input output and shifted output differ, so it is time-variant. (Key point: Time variance test)

4. Which system is non-causal?
   A. y(t) = x(t+1)
   B. y(t) = x(t-1)
   C. y(t) = ∫_{-∞}^{t} x(τ)dτ
   D. y(t) = 3x(t)
   Answer: A. x(t+1) needs the future input at t+1. (Key point: Non-causal example)

## Related in Signals and Systems

- [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×
- [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
- [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×
- [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 properties; is y[n] = e^(x[n]) linear](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-properties-exponential-system) — Signals and Systems, chapter 6
- [LTI: causality and stability](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-causality-stability) — Signals and Systems, chapter 5

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- [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
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