Linear Time Invarient System

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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 if any input x(t) is shifted by some amount of time T the out-put will also be adjusted by that amount of time. This implies that for,

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



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