Difference between revisions of "Linear Time Invarient System"

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(LTI system properties)
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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,
 
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 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,
 
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>x(t - T)\!</math>
::<math>y(t - T) = H(x(t - T))</math>
+
::<math>y(t - T) = H(x(t - T))\!</math>
   
 
<br>
 
<br>

Revision as of 22:31, 12 October 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,

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))\!



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