Linear Time Invarient System: Difference between revisions
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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, |
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::<math>x_1(t)</math> |
::<math>x_1(t)\!</math> |
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::<math>x_2(t)</math> |
::<math>x_2(t)\!</math> |
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::<math>y_1(t) = H(x_1(t))</math> |
::<math>y_1(t) = H(x_1(t))\!</math> |
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::<math>y_2(t) = H(x_2(t))</math> |
::<math>y_2(t) = H(x_2(t))\!</math> |
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::<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> |
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for any scalar values of A and B. |
for any scalar values of A and B. |
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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, |
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::<math>x(t - T)</math> |
::<math>x(t - T)\!</math> |
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::<math>y(t - T) = H(x(t - T))</math> |
::<math>y(t - T) = H(x(t - T))\!</math> |
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<br> |
<br> |
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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,
for any scalar values of A and B.
Time invariance of a system means that for any input 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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