---
title: "Properties of continuous time systems"
subject: "Signals and Systems"
chapter: 5
key: "semester_4/signals_and_systems:ct-system-properties"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ct-system-properties
site: BCT Engineering
---

# Properties of continuous time systems

Signals and Systems · Chapter 5: Continuous-Time Systems

- Asked in: 2069 Bhadra, 5 marks
- 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_4/signals_and_systems/ct-system-properties
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-5

## Question

Write about the following properties of continuous time system: (a) Linearity (b) Causality (c) Memory (d) Stability (e) Time invariance.

## Short answer

Linearity means superposition: ax1+bx2 gives ay1+by2. Causality means output at t depends only on present and past inputs. Memoryless systems depend only on the current input; systems with memory use past or future inputs. BIBO stability requires bounded input to give bounded output. Time invariance means input shift x(t-t0) gives output shift y(t-t0).

## Answer

Continuous-time systems are classified by the following basic properties.

#### (a) Linearity

A system is linear if it obeys superposition (additivity + homogeneity): $a x_1(t) + b x_2(t) \rightarrow a y_1(t) + b y_2(t)$.
- Linear: $y(t) = 2x(t)$, $y(t) = \frac{dx}{dt}$.
- Non-linear: $y(t) = x^2(t)$, $y(t) = x(t) + 1$.

#### (b) Causality

Output at time $t$ depends only on present and past inputs, not future ones. It is non-anticipative and can work in real time.
- Causal: $y(t) = x(t) + x(t-2)$.
- Non-causal: $y(t) = x(t+1)$. For LTI: causal if $h(t) = 0$ for $t < 0$.

#### (c) Memory

A **memoryless (static)** system's output depends only on the input at the same instant. A system **with memory (dynamic)** uses past (or future) inputs.
- Memoryless: $y(t) = 5x(t)$, a resistor $v = Ri$.
- With memory: $y(t) = \int_{-\infty}^{t} x(\tau)d\tau$, a capacitor. For LTI: memoryless only if $h(t) = K\delta(t)$.

#### (d) Stability

**BIBO stable:** every bounded input gives a bounded output.
- Stable: $y(t) = e^{x(t)}$, $h(t) = e^{-t}u(t)$.
- Unstable: $y(t) = t\,x(t)$, an ideal integrator. For LTI: $\int|h(t)|dt < \infty$.

#### (e) Time invariance

A time shift in the input causes the same shift in the output: $x(t-t_0) \rightarrow y(t-t_0)$.
- Time invariant: $y(t) = x(t-3)$.
- Time varying: $y(t) = t\,x(t)$, $y(t) = x(2t)$.

## Check yourself

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

1. Which system is memoryless?
   A. y(t) = integral of x from -inf to t
   B. y(t) = x(t) + x(t - 2)
   C. y(t) = 5x(t), like a resistor v = Ri
   D. y(t) = x(t + 1)
   Answer: C. The output depends only on the input at the same instant. A capacitor (integral) has memory. (Key point: Memoryless system)

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

3. Which system is non-linear?
   A. y(t) = 2x(t)
   B. y(t) = dx/dt
   C. y(t) = x(t) + 1
   D. y(t) = x(t) + x(t - 1)
   Answer: C. A constant offset gives a non-zero output for zero input and breaks superposition. (Key point: Linearity)

4. Which conditions on h(t) are given for LTI memoryless and stable systems?
   A. Memoryless if h(t) = K delta(t); stable if the integral of |h| is finite
   B. Memoryless if h(t) = K u(t); stable if h(t) is bounded
   C. Memoryless if h(t) = 0 for t < 0; stable if h(t) is even
   D. Memoryless if h(t) = K; stable if h(t) = 0 for t > 0
   Answer: A. Only an impulse at the origin gives output depending on the present input alone; BIBO stability needs absolute integrability. (Key point: LTI conditions on h)

## Close variants set in other exams

- [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×

## 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×
- [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
- [LTI: causality and stability](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-causality-stability) — Signals and Systems, chapter 5
- [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×
- [CTFT linearity and time shifting](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actft-linearity-time-shifting) — Signals and Systems, chapter 3
- [DTFS linearity and time shift](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-linearity-time-shift) — Signals and Systems, chapter 2

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