linear time invariant system
A linear time invariant system (LTI) is a linear dynamical system $T(p)$,
$y(k)$  $=T(p)u(k),$ 
with parameter $p$ that is time independent. $y(k)$ denotes the system output and $u(k)$ denotes the input. The independent variable $k$ can be denoted as time, index for a discrete sequences or differential operaters (e.g. such as $s$ in Laplace domain or $\omega $ in frequency domain).
For example, for a simple massspringdashpot system, the system parameter $p$ can be selected as the mass $m$, spring constant $k$ and damping coefficient $d$. The input $u$ to the said system can be chosen as the force applied to the mass and the output $y$ can be chosen as the mass’s displacement.
LTI system has the following properties.
 Linearity:

If ${y}_{1}=T{x}_{1}$ and ${y}_{2}=T{x}_{2}$, then
$$T\{\alpha {x}_{1}+\beta {x}_{2}\}=\alpha {y}_{1}+\beta {y}_{2}$$  Time Invariance:

If $y(k)=Tx(k)$, then
$$y(k+{\delta}_{k})=Tx(k+{\delta}_{k})$$  Associative:

$${T}_{1}\cdot ({T}_{2}\cdot {T}_{3})=({T}_{1}\cdot {T}_{2})\cdot {T}_{3}$$  Commutative:

$${T}_{1}\cdot {T}_{2}={T}_{2}\cdot {T}_{1}$$
A LTI system can be represented with the following:

•
Transfer function of Laplace transform^{} variable $s$, which is commonly used in control systems design.

•
Transfer function of Fourier transform^{} variable $\omega $, which is commonly used in communication theory and signal processing.

•
Transfer function of ztransform^{} variable ${z}^{1}$, which is commonly used in digital signal processing (DSP).

•
Statespace equations, which is commonly used in modern control theory and mechanical systems.
Note that all transfer functions are LTI systems, but not all statespace equations are LTI systems.
Title  linear time invariant system 

Canonical name  LinearTimeInvariantSystem 
Date of creation  20130322 14:22:25 
Last modified on  20130322 14:22:25 
Owner  Mathprof (13753) 
Last modified by  Mathprof (13753) 
Numerical id  11 
Author  Mathprof (13753) 
Entry type  Definition 
Classification  msc 93A10 
Synonym  LTI 
Related topic  Controllability 
Related topic  Observability 
Related topic  SystemDefinitions 