# 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, $x_1(t)\!$ $x_2(t)\!$ $y_1(t) = H(x_1(t))\!$ $y_2(t) = H(x_2(t))\!$ $Ay_1(t) + By_2(t) = H(Ax_2(t) + Bx_1(t))\!$

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, $x(t - T)\!$ $y(t - T) = H(x(t - T))\!$