--- title: "Describe LTI system properties" subject: "Signals and Systems" chapter: 5 key: "semester_4/signals_and_systems:lti-properties-describe" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/lti-properties-describe site: BCT Engineering --- # Describe LTI system properties Signals and Systems · Chapter 5: Continuous-Time Systems - Asked in: 2071 Magh, 6 marks - Video: [#113 Properties of LTI System in terms of impulse response and SLOVED PROBLEM // EC Academy](https://www.youtube.com/watch?v=uORgvV8IcMY) — EC 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-properties-describe - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-5 ## Question For the LTI system, describe following properties: (a) linearity (b) causality (c) stability (d) time invariance. ## Short answer Linearity: responds to a weighted sum with the same weighted sum of outputs. Causality: h(t)=0 for t<0, output depends only on past/present inputs. Stability: BIBO stable if ∫|h(t)|dt < ∞. Time invariance: a delay in input causes identical delay in output. ## Answer An **LTI system** is linear and time invariant; it is fully described by its impulse response $h(t)$, and $y(t) = x(t) * h(t)$. #### (a) Linearity A system is linear if it obeys **superposition**: $$ a\,x_1(t) + b\,x_2(t) \rightarrow a\,y_1(t) + b\,y_2(t) $$ It combines additivity ($x_1 + x_2 \rightarrow y_1 + y_2$) and homogeneity ($a x \rightarrow a y$). Zero input must give zero output. - Linear: $y(t) = 5x(t)$, $y(t) = \int_{-\infty}^{t} x(\tau)d\tau$. - Non-linear: $y(t) = x^2(t)$, $y(t) = x(t) + 3$. #### (b) Causality The output at any time depends only on the present and past inputs. For an LTI system: $$ h(t) = 0 \quad \text{for } t < 0 $$ so that $y(t) = \int_0^{\infty} h(\tau)x(t-\tau)d\tau$ uses only $x$ up to time $t$. - Causal: $h(t) = e^{-3t}u(t)$. - Non-causal: $h(t) = \delta(t+1)$, i.e. $y(t) = x(t+1)$. #### (c) Stability A system is **BIBO stable** if every bounded input gives a bounded output. Since $|y(t)| \le M_x\int|h(\tau)|d\tau$ for $|x| \le M_x$, the condition for an LTI system is: $$ \int_{-\infty}^{\infty} |h(t)|\,dt < \infty $$ - Stable: $h(t) = e^{-t}u(t)$ (area 1). - Unstable: $h(t) = u(t)$, $h(t) = e^{t}u(t)$. #### (d) Time invariance A time shift in the input gives the same time shift in the output: $$ x(t-t_0) \rightarrow y(t-t_0) $$ The system's characteristics do not change with time (constant parameters). - Time invariant: $y(t) = x(t-1)$, $y(t) = \frac{dx}{dt}$. - Time varying: $y(t) = t\,x(t)$, $y(t) = x(2t)$. | Property | LTI test on $h(t)$ | |---|---| | Linearity | Holds for every convolution system | | Causality | $h(t) = 0$ for $t < 0$ | | Stability | $\int \lvert h(t)\rvert dt$ finite | | Time invariance | Holds for every convolution system | ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. A system is linear if it obeys which rule? A. Superposition: a x1 + b x2 -> a y1 + b y2, with zero output for zero input B. x(t - t0) -> y(t - t0) C. h(t) = 0 for t < 0 D. Bounded input gives bounded output Answer: A. Linearity combines additivity and homogeneity, so a zero input must give zero output. (Key point: Linearity) 2. Which impulse response corresponds to a causal LTI system? A. delta(t + 1) B. e^(-3t) u(t) C. e^(-|t|) D. u(t + 2) Answer: B. Causality requires h(t) = 0 for t < 0; e^(-3t)u(t) satisfies this. (Key point: Causality of LTI) 3. What is the BIBO stability condition for an LTI system? A. h(t) = 0 for t < 0 B. h(t) is bounded C. The integral of h(t) from -inf to inf is zero D. The integral of |h(t)| over all t is finite Answer: D. Since |y| <= Mx times the integral of |h|, absolute integrability guarantees a bounded output. (Key point: Stability condition) 4. Which system is time varying? A. y(t) = x(t - 1) B. y(t) = dx/dt C. y(t) = t x(t) D. y(t) = 5x(t) Answer: C. The gain t changes with time, so a shifted input does not give a simply shifted output. (Key point: Time invariance) ## Close variants set in other exams - [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× - [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 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× - [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 ## 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