View source for Linear Time Invarient System

*[[Signals and systems|Signals and Systems]]
==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,

::<math>x_1(t)</math>
::<math>x_2(t)</math>
::<math>y_1(t) = H(x_1(t))</math>
::<math>y_2(t) = H(x_2(t))</math>
::<math>Ay_1(t) + By_2(t) = H(Ax_2(t) + Bx_1(t))</math>
for any scalar values of A and B.

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

::<math>x(t - T)</math>
::<math>y(t - T) = H(x(t - T))</math>