Linear Time Invarient System

LTI systems

LTI System theory is a powerful and widely used concept in electrical engineering. It has applictions in circuit anlysis, control theory , and our main topic of interest signal processing.

LTI system properties

A system is considered to be a Linear Time Invariant when it satisfies the two basic criteria implied in its name, one it must be linear and two it must be time invariant. A Linear system is characterized by two properties superposition (additivity) and scaling (homogeneity). The superposition principal says that for any linear system a linear combination of solutions to the system is also a solution to the same linear system. The principal of scaling implies that if you adjust your scale an input by N amount, your output will also be adjusted by N amount. An example of a linear system then would be,

$x_1(t)$

$x_2(t)$

$y_1(t)=H(x_1(t))$

$y_2(t)=H(x_2(t))$

$A{y}_1(t)+B{y}_2(t)=H(A{x}_2(t)+B{x}_1(t))$

for any scalar values of A and B.

Time invariance of a system means that for any input $x(t)$ by some amount of time T the out put will also be adjusted by that amount of time. This implies that for,

$x(t-T)$

$y(t-T)=H(x(t-T))$