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 Invarient when it satifies the two basic criteria implied in its name, one it must be linear and two it must be time invarient. A Linear system is charterized by two propeties superposition (additvity) and scaling (homegeneity). The superpostion 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,

$x_{1}(t)$

$x_{2}(t)$

$y_{1}(t)=H(x_{1}(t))$

$y_{2}(t)=H(x_{2}(t))$

$Ay_{1}(t)+By_{2}(t)=H(Ax_{2}(t)+Bx_{1}(t))$

for any scalar values of A and B.

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

$x(t-T)$

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