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The general form of diff equation for an LTi system is :

$$ y(n) = - \sum_{k=1}^N a_k \cdot y(n-k) + \sum_{k=0}^M b_k \cdot x(n-k) $$

My questions are:

1) why is the first term negative ?
2) why limits of $k$ are from $1$ to $N$ and not $0$ to $N$ ?
3) what does $M$ represent if $N$ represents the order of the system ?

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1 Answer 1

  1. The first sum is the feedback from previous outputs. If the feedback wouldn't be negative you would always add more positive values to your output, and the transfer function would diverge to infinity; your system would be unstable. Feedback must be negative.

  2. If you would also consider k = 0 then your first term would be $a_0 \cdot y(n)$. Which is the output you're actually calculating right now, you can't use that in your calculation because it doesn't exist yet.

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