Linear_time-invariant_system

# Linear time-invariant system  ## Microsims — p5.js ### Linear time-invariant system (p5.js) · `h(t) * x` <div class="microsim-player"> <iframe src="https://editor.p5js.org/sciencenibber/full/_EtxrlbJb" width="100%" height="480" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Linear time-invariant system — p5.js microsim"></iframe> </div> *An LTI system is fully described by its impulse response; its output is just convolution.* **Open in the editor:** [▶ fork this sketch](https://editor.p5js.org/sciencenibber/sketches/_EtxrlbJb) · movement *VI · Control theory & estimation* · library `p5js` ### Related microsims Live sims on neighbouring articles — 6 of them inside this article's own Wikipedia link tree: - [[Control_theory]] *(in tree)* - [[Convolution]] *(in tree)* - [[Filter_design]] *(in tree)* - [[Frequency_domain]] *(in tree)* - [[Frequency_response]] *(in tree)* - [[Impulse_response]] *(in tree)* *Sim hosted off-article; the article owns the reference, not the runtime (WIKI_RULES §10.4). Placed by `g08_place_microsims.py`.*  ## Links (Wikipedia order)  `Aliasing` · `Amplitude` · `Analog-to-digital_converter` · `Anti-aliasing_filter` · `Applied_mathematics` · `BIBO_stability` · [[Block_diagram]] · `Boxcar_function` · `Capacitor` · `Causal_system` · `Circulant_matrix` · `Commutative_property` · `Complex_number` · [[Control_theory]] · [[Convolution]] · `Derivative` · [[Differential_equation]] · `Dirac_delta_function` · `Discrete-time_Fourier_transform` · `Discrete_system` · `Eigenfunction` · `Eve_Riskin` · `Exponential_function` · `Filter_bank` · [[Filter_design]] · `Fourier_series` · `Fourier_transform` · [[Frequency_domain]] · [[Frequency_response]] · `Green's_function` · [[Impulse_response]] · `Inductor` · [[Laplace_transform]] · `Linear_amplifier` · `Linear_system` · `Locus_(mathematics)` · `Low-pass_filter` · `Lp_space` · [[MIMO]] · `Matrix_(mathematics)` · [[Mechanical_engineering]] · `Network_analysis_(electrical_circuits)` · `Nyquist_frequency` · [[Ordinary_differential_equation]] · `Phase_(waves)` · [[Physical_system]] · `RLC_circuit` · `Resistor` · `Sample_and_hold` · [[Sampling_(signal_processing)]] · [[Signal-flow_graph]] · [[Signal_processing]] · `Sinc_function` · `Superposition_principle` · [[System]] · `System_analysis` · `Time-invariant_system` · [[Time_domain]] · [[Transfer_function]] · `Two-sided_Laplace_transform` · `Unit_circle` · [[Wayback_Machine]] · `Wiener–Khinchin_theorem` · `Without_loss_of_generality` · [[Z-transform]] > Signal Processing concept · part of the Signal Processing Portal · movement IX · !09 解析 kaiseki.svg  !480 *Rendered from the live microsim (▶ motion).*  ## See it next [![Convolution|200](Convolution_thumb.png)](Convolution) *→ [[Convolution|Convolution]]*  --- Back to Signal Processing Portal · the room · Semiotic gateway ## What it is A linear time-invariant (LTI) system is a system whose response obeys both linearity (superposition and scaling) and time invariance (a shift of the input produces only an equal shift of the output) — the class of systems that most of signal-processing theory is built around. ## How it works / why it matters Because of these two properties, an LTI system is **completely characterized by its impulse response** $h(t)$: its output for any input is the **convolution** $y(t) = (x * h)(t)$. Feeding it a complex exponential (or sinusoid) returns the same exponential scaled by a complex factor — so exponentials are the system's eigenfunctions, and its behavior is summarized by a **frequency response** (the Fourier transform of $h$) or a **transfer function**. This is why analysis and design collapse to multiplication in the frequency domain, and why filters, circuits, and control loops are all studied through the same LTI framework. ## Signs & universals Instantiates: system · signal · transformation · frequency — a system transforming a signal, fully described in the frequency domain. ## Related An LTI system acts on its input precisely by [[Convolution]] with its impulse response — the central theorem linking the two. Every Digital filter is a designed LTI system, and its frequency response is obtained through the Discrete Fourier transform; the same theory governs the shift-invariant operations of [[Digital_image_processing]] and Audio signal processing.   *Built to the [[WT!P5_js_Microsim_Master_Class|p5.js Master Class]].*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Linear_time-invariant_system) : [Wikitube](https://en.wikitube.io/wiki/Linear_time-invariant_system) ## Previous hub tags Tree parents: [[Control_theory]] · [[Systems_science]]. Legacy hubs: none. --- *Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*