Linear Time Invarient System: Difference between revisions

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*[[Signals and systems|Signals and Systems]]
==LTI 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 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===
===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 you scale you input by N amount your output will also be adjusted by N amount. An example of a Linear system then would be,
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_1(t)\!</math>
::<math>x_2(t)</math>
::<math>x_2(t)\!</math>
::<math>y_1(t) = H(x_1(t))</math>
::<math>y_1(t) = H(x_1(t))\!</math>
::<math>y_2(t) = H(x_2(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>
::<math>Ay_1(t) + By_2(t) = H(Ax_2(t) + Bx_1(t))\!</math>
for any scalar values of A and B.
for any scalar values of A and B.


Time invarience of a system means that for adjust any input <math>x(t)</math> by some amout of time T the out put will also be adjusted by that amount of time. This impies that for,
Time invariance of a system means that if any input <math>x(t)</math> is shifted 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>H(x(t))=y(t)\!</math>
::<math>y(t - T) = H(x(t - T))</math>
::<math>H(x(t - T))=y(t-T)\!</math>

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Latest revision as of 16:08, 5 November 2006

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,

for any scalar values of A and B.

Time invariance of a system means that if any input is shifted by some amount of time T the out-put will also be adjusted by that amount of time. This implies that for,



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