--- title: "LTI system: linearity, stability, TI, causality" subject: "Signals and Systems" chapter: 5 key: "semester_4/signals_and_systems:lti-properties-four" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/lti-properties-four site: BCT Engineering --- # LTI system: linearity, stability, TI, causality Signals and Systems · Chapter 5: Continuous-Time Systems - Asked in: 2073 Magh, 2+8 marks - Video: [Properties Of Systems | Example 1](https://www.youtube.com/watch?v=NYtQ542JRw0) — EnggClasses - 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-properties-four - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-5 ## Question What is LTI system? For LTI system, describe the properties: (a) linearity (b) stability (c) time invariance and (d) causality. ## Short answer LTI system is linear and time-invariant, described by h(t). Linearity means superposition. Stability is BIBO iff ∫|h(t)|dt<∞. Time invariance means shifted input gives shifted output. Causality requires h(t)=0 for t<0. ## Answer A **Linear Time-Invariant (LTI) system** is a system that is both linear (obeys superposition) and time invariant (its behaviour does not change with time). An LTI system is completely described by its impulse response $h(t)$, and its output for any input is the convolution $y(t) = x(t) * h(t)$. #### (a) Linearity A system is linear if it satisfies **superposition** = additivity + homogeneity: $$ a\,x_1(t) + b\,x_2(t) \;\rightarrow\; a\,y_1(t) + b\,y_2(t) $$ - **Additivity:** $x_1 + x_2 \rightarrow y_1 + y_2$. - **Homogeneity (scaling):** $a x \rightarrow a y$. - A consequence: zero input gives zero output. - Examples: $y(t) = 3x(t)$, $y(t) = \frac{dx}{dt}$ are linear; $y(t) = x^2(t)$ and $y(t) = 2x(t) + 1$ are not. - Convolution itself is a linear operation, so an LTI system is linear in its input. #### (b) Stability A system is **BIBO stable** (bounded-input bounded-output) if every bounded input $|x(t)| \le M_x < \infty$ gives a bounded output $|y(t)| \le M_y < \infty$. For an LTI system: $$ |y(t)| = \left|\int h(\tau)x(t-\tau)d\tau\right| \le M_x \int_{-\infty}^{\infty}|h(\tau)|\,d\tau $$ So the condition is that the impulse response is **absolutely integrable**: $$ \int_{-\infty}^{\infty} |h(t)|\,dt < \infty $$ - $h(t) = e^{-2t}u(t)$: area $= 1/2$, stable. - $h(t) = u(t)$ (integrator): area infinite, unstable. #### (c) Time invariance A system is time invariant if a time shift of the input causes the same time shift of the output: $$ x(t) \rightarrow y(t) \;\Rightarrow\; x(t - t_0) \rightarrow y(t - t_0) $$ - Test: find the output for the delayed input, then compare with $y(t-t_0)$. - $y(t) = x(t-2)$ is time invariant. - $y(t) = t\,x(t)$ is time varying, because the delayed input gives $t\,x(t-t_0)$ but $y(t-t_0) = (t-t_0)x(t-t_0)$. - Systems with constant coefficients are time invariant. #### (d) Causality A system is causal if its output at time $t$ depends only on present and past inputs ($\tau \le t$), not on future inputs. For an LTI system, $y(t) = \int h(\tau)x(t-\tau)d\tau$. To avoid using future values $x(t-\tau)$ with $\tau < 0$: $$ h(t) = 0 \quad \text{for } t < 0 $$ Then $y(t) = \int_{0}^{\infty} h(\tau)\,x(t-\tau)\,d\tau$. - $h(t) = e^{-t}u(t)$ is causal. - $h(t) = e^{-|t|}$ is non-causal (it is stable, though). | Property | General condition | LTI condition | |---|---|---| | Linearity | Superposition holds | Always (convolution) | | Time invariance | Shifted input gives shifted output | Always | | Causality | No future inputs used | $h(t)=0$, $t<0$ | | Stability | Bounded in, bounded out | $\int \lvert h(t)\rvert dt < \infty$ | ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. LTI BIBO stable iff: A. h(t)=0 for t<0 B. ∫h(t)dt=0 C. ∫|h(t)|dt<∞ D. h is even Answer: C. Absolutely integrable. (Key point: BIBO stability) 2. Which is not linear? A. y=3x B. y=dx/dt C. y=0 D. y=2x+1 Answer: D. Zero input gives nonzero output. (Key point: Linearity) 3. y(t)=t x(t) is time varying because: A. Delayed input gives t x(t-t0) but y(t-t0)=(t-t0)x(t-t0) B. It's nonlinear C. It's noncausal D. It's unstable only Answer: A. The two differ. (Key point: Time-varying) ## 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: 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× - [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× - [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 - [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 ## 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 - [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 - [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 - [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