Can we say that a linear constant coefficient difference equation can always represent a linear shift invariant system? Are there any conditions which need to be satisfied additionally by these kind of equations to be able to do that?
2026-04-04 15:05:42.1775315142
Can One Represent Any LTI system via a Linear Constant Coefficient Difference Equation?
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http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/lecture-notes/MITRES_6_007S11_lec06.pdf
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 $n<n_0$, then $y[n] = 0$ for $n<n_0$