linear time invariant system

A linear time invariant system (LTI) is a linear dynamical system $T(p)$ ,

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 mass-spring-dashpot 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 $y(k) = T x(k)$ , then $$ y(k+\delta_k) = T x(k + \delta_k) $$

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 z-transform variable $z^{-1}$ , which is commonly used in digital signal processing (DSP).
• State-space equations, which is commonly used in modern control theory and mechanical systems.

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