Time Invariance | Lecture Note - Edubirdie

University:
Massachusetts Institute of Technology
Course:
18.03SC | Differential Equations
Academic year:

2024
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153

Pages:

1
Author:

grimering0qp

Time Invariance In the case of constant coefﬁcient operators p( D ), there is an important and useful relationship between solutions to p( D ) x = q(t) for input signals q(t) which start at different times t. The following result shows why these operators are called “Linear Time Invariant” (or LTI). Translation invariance. If p( D ) is a constant-coefﬁcient differential oper ator and p( D ) x = q(t), then p( D )y = q(t − c), where y ( t ) = x ( t − c ). This is the “time invariance” of p( D ). Here is an example of its use. √ Example. Suppose that we know that x p (t) = 2 sin(t/2 − π/4) is a solu tion to the DE .. . 2x + x + x = sin(t/2) (1) Find a solution y p to .. . 2x + x + x = sin(t/2 − π/3) Solution. By translation-invariance, we have immediately that √ √ y p = 2 sin(t/2 − π/4 − π/3) = 2 sin(t/2 − 7π/12). (2)

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