Digital Realisation Structures
A representation of a digital filter that could be implemented in a programming language. See slides.
$$ \begin{align} H(z) &= \frac{Y(z)}{X(z)} = \frac{ \sum_{i=0}^{N} a_{i}z^{-i} }{ 1 + \sum_{i=1}^{N} b_{i} z^{-i} } \ \ y(n) &= \sum_{i=0}^{N} a_{i} \cdot x(n-i) - \sum_{i=1}^{N}b_{i} \cdot y(n-1) \end{align} $$ $n$: time step $x(n)$: input in the discrete time domain $y(n)$: output in discrete time domain
- Seperate $X(z)$ and $Y(z)$ terms the their own sides of the equation.
- Take the inverse z-transformation of each part of the equation.
- Isolate $y(n)$
- Find constants and draw the realisationsstructure
Direct Type 1
See slides.
This is the intuitive way to implement a discrete transfer function.
Direct Type 2
See slides.
We split up the transfer funktion with an intermediate function $W(z)$. Note that this does not change the transfer function. $$ >H(z) = >\frac{W(z)}{X(z)} \cdot \frac{Y(z)}{W(z)} = >\underbrace{\frac{1}{1 + \sum_{i=1}^{N} b_{i}z^{-1}}}_{H_{1}} >\cdot >\underbrace{\sum_{i=0}^{N} a_{i} z^{-i}}_{H_{2}} >$$ The new transfer functions can be described as follows: $$ >\begin{align} >H_{1}(z) &= \frac{W(z)}{X(z)} = \frac{1}{1 + \sum_{i=1}^{N} b_{i}z^{-1}} \ >H_{2}(z) &= \frac{Y(z)}{W(z)} = \sum_{i=0}^{N} a_{i} z^{-i} >\end{align} >$$
These two transfer functions can be realised like this:
The memory xells ($z^{-1}$) now share the same values and can be combined:
This is the better way of implementing a filter, as it stores half as many values resulting in a less expensive filter implementation.
