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Homelinear time invariant system
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linear time invariant system
A linear time invariant system (LTI) is a linear dynamical system $T(p)$,
$\displaystyle y(k)$  $\displaystyle=T(p)\;u(k),$ 
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 $y_{1}=Tx_{1}$ and $y_{2}=Tx_{2}$, 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).
Note that all transfer functions are LTI systems, but not all statespace equations are LTI systems.
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