Real outputs from complex filters

In most applications, we start with a real-valued signal to filter and we need a real-valued output, but in general, a compound filter with a transfer function as above will give a complex-valued output. However, we can construct filters with non-real-valued coefficients which nonetheless give real-valued outputs, so that the analysis that we carry out using complex numbers can be used to predict, explain, and control real-valued output signals. We do this by pairing each elementary filter (with coefficient $P$ or $Q$) with another having as its coefficient the complex conjugate $\overline{P}$ or $\overline{Q}$.

For example, putting two non-recirculating filters, with coefficients $Q$ and $\overline{Q}$, in series gives a transfer function equal to:

$$H(Z) = (1 - {Q}{Z^{-1}}) \cdot (1 - \overline{Q}{Z^{-1}})$$

which has the property that:

$$H(\overline{Z}) = \overline{H(Z)}$$

Now if we put any real-valued sinusoid:

$${X_n} = 2 \, \mathrm{re}(A{Z^n}) = A{Z^n} + \overline{A} {{\overline{Z}}^n}$$

we get out:

$$A \cdot H(Z) \cdot {Z^n} + \overline{A} \cdot \overline{H(Z)} \cdot {{\overline{Z}}^n}$$

which, by inspection, is another real sinusoid. Here we're using two properties of complex conjugates. First, you can add and multiply them at will:

$$\overline{A+B} = \overline{A} + \overline{B}$$

$$\overline{AB} = \overline{A} \cdot \overline{B}$$

and second, anything plus its complex conjugate is real, and is in fact twice its real part:

$$A + \overline{A} = 2 \, \mathrm{re} (A)$$

This result for two conjugate filters extends to any compound filter; in general, we always get a real-valued output from a real-valued input if we arrange that each coefficient $Q_i$ and $P_i$ in the compound filter is either real-valued, or else appears in a pair with its complex conjugate.