General Digital Filter
Refers to a filter defined in terms of the [see page 4, general-difference-equation]. Such practical
filters are defined in some order based on the degree to which the filter approximates
it's idealised form.
The general difference equation is defined as: \[ Y(Z) = Y(z) \sum_^{p}{a_i {z}^{-i}} + X(z) \sum_^{q}{b_i {z}^{-i}} \] With:
| Variable | Meaning |
|---|---|
| \( p, q \) | Number of poles and zeros respectively |
| \( a_i \) | The output of the filter at time \( i \) |
| \( b_i \) | The input to the filter at time \( i \) |
Which can be re-arranged into a form matching the z-plane transfer function.
\begin{align*} H(z) &= \frac{\text{zeros}}{\text{poles}} \\
&= \frac{\sum\_{i=0}^{q}{b\_i {z}^{-1}}}{1 - \sum\_{i=1}^{p}{a\_i {z}^{-1}}} \\\\\end{align*}
Take note that this definition is defined both in terms of the previous outputs of the filter \( a_i \) (forward paths) and the current inputs to the filter \( b_i \) (feedback paths). Also note that zeros are defined entirely by the current inputs of the filter and poles by the previous outputs.