PFE to Real, Second-Order Sections

When all coefficients of and are real (implying that 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 denote any one-pole section in the PFE of Eq. $\,$ (6.7). Then if is complex and describes a real filter, we will also find $(\overline{r},\pc)$ somewhere among the terms in the one-pole expansion. These two terms can be paired to form a real second-order section as follows:

$\begin{eqnarray*} H(z) &=& \frac{r}{1-pz^{-1}} + \frac{\overline{r}}{1-\pc z^{-1... ...box{re}\left\{p\right\}z^{-1}+ \left\vert p\right\vert^2 z^{-2}} \end{eqnarray*}$

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

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

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 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 in Eq. $\,$ (6.7), consider the sum of two first-order terms by direct calculation:

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

(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

one-pole terms $r_i/(1-p_i z^{-1})$ can produce a numerator order of at most

(while the denominator order is

if there are no pole-zero cancellations). Following terminology used for analog filters, we call the case

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.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Introduction to Digital Filters
    Transfer Function Analysis
       Partial Fraction Expansion
          PFE to Real, Second-Order Sections