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 thesystem output and $u(k)$ denotes the input. The independent variable $k$ can be denoted as time, index for a discrete sequences ordifferential operaters (e.g. such as $s$ in Laplace domain or $\\omega$ in frequency domain).

For example, for a simple mass-spring-dashpot system, the systemparameter $p$ can be selected as the mass $m$ , spring constant $k$ anddamping coefficient $d$ . The input $u$ to the said system can be chosenas the force applied to the mass and the output $y$ can be chosen as themass’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 commonlyused in control systems design.
•
Transfer function of Fourier transform variable $\\omega$ , which iscommonly used in communication theory and signal processing.
•
Transfer function of z-transform variable $z^{-1}$ , which iscommonly used in digital signal processing (DSP).
•
State-space equations, which is commonly used in modern controltheory and mechanical systems.

Note that all transfer functions are LTI systems, but not allstate-space equations are LTI systems.