PFE to Real, Second-Order Sections

When all coefficients of $A(z)$ and $B(z)$ are real (implying that $H(z)=B(z)/A(z)$ is the transfer function of a real filter), it will always happen that the complex one-pole filters will occur in complex conjugate pairs. Let $(r,p)$ denote any one-pole section in the PFE of Eq.(6.7). Then if $p$ is complex and $H(z)$ describes a real filter, we will also find $(\overline{r},\overline{p})$ somewhere among the terms in the one-pole expansion. These two terms can be paired to form a real second-order section as follows:

$$\begin{eqnarray*} H(z) &=& \frac{r}{1-pz^{-1}} + \frac{\overline{r}}{1-\overline{p}z^{-1}}\\ [5pt] &=& \frac{r-r\overline{p}z^{-1}+\overline{r}-\overline{r}pz^{-1}}{(1-pz^{-1})(1-\overline{p}z^{-1})}\\ [5pt] &=& \frac{2\mbox{re}\left\{r\right\}-2\mbox{re}\left\{r\overline{p}\right\}z^{-1}}{1-2\mbox{re}\left\{p\right\}z^{-1}+ \left\vert p\right\vert^2 z^{-2}} \end{eqnarray*}$$

Expressing the pole $p$ in polar form as $p=Re^{j\theta}$ , and the residue as $r=Ge^{j\phi}$ , the last expression above can be rewritten as

$$H(z) \eqsp 2G\frac{\cos(\phi)-R\,\cos(\phi-\theta)z^{-1}}{1-2R\,\cos(\theta)z^{-1}+ R^2 z^{-2}}.$$

The use of polar-form coefficients is discussed further in the section on two-pole filters (§B.1.3).

Expanding a transfer function into a sum of second-order terms with real coefficients gives us the filter coefficients for a parallel bank of real second-order filter sections. (Of course, each real pole can be implemented in its own real one-pole section in parallel with the other sections.) In view of the foregoing, we may conclude that every real filter with $M<N$ can be implemented as a parallel bank of biquads.^7.6 However, the full generality of a biquad section (two poles and two zeros) is not needed because the PFE requires only one zero per second-order term.

To see why we must stipulate $M<N$ in Eq.(6.7), consider the sum of two first-order terms by direct calculation:

$$H_2(z) \eqsp \frac{r_1}{1-p_1z^{-1}} + \frac{r_2}{1-p_2z^{-1}} \eqsp \frac{(r_1 + r_2) - (r_1 p_2 + r_2 p_1) z^{-1}}{(1-p_1z^{-1})(1-p_2z^{-1})}$$ (7.9)

Notice that the numerator order, viewed as a polynomial in $z^{-1}$ , is one less than the denominator order. In the same way, it is easily shown by mathematical induction that the sum of $N$ one-pole terms $r_i/(1-p_iz^{-1})$ can produce a numerator order of at most $M=N-1$ (while the denominator order is $N$ if there are no pole-zero cancellations). Following terminology used for analog filters, we call the case $M<N$ a strictly proper transfer function.^7.7 Thus, every strictly proper transfer function (with distinct poles) can be implemented using a parallel bank of two-pole, one-zero filter sections.