Compound filters

We can use the recirculating and non-recirculating filters developed here to create a compound filter by putting several elementary ones in series. If the parameters of the non-recirculating ones (of the first type) are ${Q_1}, \ldots, {Q_j}$ and those of the recirculating ones are ${P_1}, \ldots, {P_k}$, then putting them all in series, in any order, will give the transfer function:

$$H(Z) = { { (1 - {Q_1}{Z^{-1}}) \cdots (1 - {Q_j}{Z^{-1}}) } \over { (1 - {P_1}{Z^{-1}}) \cdots (1 - {P_k}{Z^{-1}}) } }$$

The frequency response of the resulting compound filter is the product of those of the elementary ones. (One could also combine elementary filters by adding their outputs, or making more complicated networks of them; but for most purposes the series configuration is the easiest one to work with.)