Introduction
There are two main families of digital filters, distinguished by their impulse responses. For a filter to be considered, its impulse response must be both causal and stable, leading to:
\(h\left(n\right)=0\ \ \ \ for\ \ n<0\) , (causality).
\(\sum_{0}^{N}\left|h(n)\right|<\infty\) , (stability).
These conditions ensure that the filter's response is causal, meaning it only depends on past and present inputs, and stable, indicating that the filter's output remains bounded for finite inputs.
Recurrence Equation of a Digital Filter :
Generally, one can express the recurrence equation of a digital filter as follows:
\(\left(n\right)=-\sum_{k=1}^{N}{a_k\ y\left(n-k\right)+\sum_{k=1}^{M}{b_k\ x\left(n-k\right)\ \ }}\) (1)
If all coefficients \(a_k\) are zero, the filter is recursive, or FIR (Finite Impulse Response). It suffices for one \(a_k\) coefficient to be non-zero for the filter to be non-recursive, or IIR (Infinite Impulse Response).