Truncated Infinite Impulse Response (TIIR) Digital Filters

An FIR filter can be constructed as the difference of two IIR filters:

General FIR filter

• Coefficients: $\{h_0,\ldots,h_{N}\}$
• Implementation (convolution):
  $$y(n) = (h \ast x)(n) = \sum_{m=0}^{N} h_m x(n-m)$$

• Transfer function:
  $$\begin{eqnarray*} H_{\rm FIR}(z)&\mathrel{\stackrel{\mathrm{\Delta}}{=}}&h_0 +h_1 z^{-1}+\ldots +h_{N} z^{-N}\\ &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& z^{-N} C(z), \end{eqnarray*}$$
  

where $C(z)$ is the $N$ -th degree polynomial in $z$ formed by the $h_k$

General $P$ -th order IIR filter

• Difference equation
  $$y(n)= -\sum_{k=1}^P a_k y(n-k)+ \sum_{\ell=0}^P b_\ell x(n-\ell)$$

• Transfer function
  $$\begin{eqnarray*} H_{\rm IIR}(z)&\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \frac { b_0 +b_1 z^{-1} +\ldots +b_P z^{-P} } { 1+ a_1 z^{-1}+\ldots +a_P z^{-P} } \\ &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \frac { b_0 z^P +b_1 z^{P-1} +\ldots +b_P } { z^P+ a_1 z^{P-1} +\ldots +a_P }\\ &\mathrel{\stackrel{\mathrm{\Delta}}{=}}&\frac{B(z)}{A(z)}\\ &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& h_0+h_1 z^{-1}+h_2 z^{-2} +\ldots, \end{eqnarray*}$$
  

  where

  $$\begin{eqnarray*} A(z)&\mathrel{\stackrel{\mathrm{\Delta}}{=}}&z^P+a_1 z^{P-1}+\ldots+a_{P} \qquad \hbox{(monic)} \\ B(z)&\mathrel{\stackrel{\mathrm{\Delta}}{=}}&b_0 z^P + b_1 z^{P-1} +\ldots+ b_{P} \end{eqnarray*}$$