Can we say that a Linear Constant Coefficient Difference Equation can always represent a Linear Shift Invarient system ? Are there any conditions which need to be satisfied additionally by these kind of equations to be able to do that?
Can Linear Constant coefficient Difference Equations always represent an LTI system?
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2 Answers
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The above link is a pdf that has the answer to your question.
It is not necessary that a linear constant coefficient difference equation must represent an LTI system. It will represent an LTI system if and only if the solution satisfies the initial rest condition, namely if $x[n] = 0$ for $n
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1Just as a note, this initial rest condition is called *causality*. So, all causal LTI discrete time systems can be represented as a difference equation. – 2012-08-24
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0@svenkatr Thnaks for the link. I guess the condition mentioned here represents a causal LTI system. Is it possible to represent an LTI system in general with the help of these equations (without imposing the additional constraint of causality) ? – 2012-08-25
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0@bubble - causality is a necessary condition as well. – 2012-08-25
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0@svenkatr Cant get you. Necessary for what ? – 2012-08-27
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WHat about final rest condition? (i.e if x(t) = 0, for t > t0 then y(t) is also 0 for t > t0 ) whether now the differential equation satisfies LSI or not?
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0Welcome to MSE! In its current form, this is probably better as a comment as opposed to an answer (I realize you don't yet have enough reputation). perhaps you can improve it to make it an answer. Regards – 2013-05-23
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0Thank you.Initial rest condition is not necessary condition for linearity. But it is essential for Time invariance to satisfy.For linearity alone, zero initial condition like y(1) = 0 (which is not initial rest condition) is enough. – 2013-05-25