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FIR Part of a PFE

When $ M\geq N$ in Eq.$ \,$(6.7), we may perform a step of long division of $ B(z)/A(z)$ to produce an FIR part in parallel with a strictly proper IIR part:

$\displaystyle H(z) \isdefs \frac{B(z)}{A(z)} \eqsp F(z) + \sum_{i=1}^{N} \frac{r_i}{1-p_iz^{-1}} \protect$ (7.10)

where

\begin{eqnarray*} B(z) &=& b_0 + b_1 z^{-1}+ b_2z^{-2}+ \cdots + b_M z^{-M}\\ A... ...=& f_0 + f_1z^{-1}+ f_2z^{-2}+ \cdots + f_K z^{-K}, \quad K=M-N. \end{eqnarray*}

When $ M<N$, we define $ F(z)=0$. This type of decomposition is computed by the residuez function (a matlab function for computing a complete partial fraction expansion, as illustrated in §6.8.8 below).

An alternate FIR part is obtained by performing long division on the reversed polynomial coefficients to obtain

$\displaystyle H(z) \eqsp F(z) + z^{-(K+1)}\sum_{i=1}^{N} \frac{r_i}{1-p_iz^{-1}}, \protect$ (7.11)

where $ K=M-N\geq 0$ is again the order of the FIR part. This type of decomposition is computed (as part of the PFE) by residued, described in §J.6 and illustrated numerically in §6.8.8 below.

We may compare these two PFE alternatives as follows: Let $ A_N$ denote $ A(z)$, $ F_K\isdeftext F(z)$, and $ B_M\isdeftext B(z)$. (I.e., we use a subscript to indicate polynomial order, and `$ (z)$' is omitted for notational simplicity.) Then for $ K=M-N\geq 0$ we have two cases:

\begin{eqnarray*} (1) && H(z) \eqsp F_K + \frac{B^\prime_{N-1}}{A_N} \eqsp \frac... ..._N} \eqsp \frac{F_K A_N + z^{-(K+1)}B^{\prime\prime}_{N-1}}{A_N} \end{eqnarray*}

In the first form, the $ B^\prime_{N-1}$ coefficients are ``left justified'' in the reconstructed numerator, while in the second form they are ``right justified''. The second form is generally more efficient for modeling purposes, since the numerator of the IIR part ( $ B^{\prime\prime}_{N-1}(z)$) can be used to match additional terms in the impulse response after the FIR part $ F_K(z)$ has ``died out''.

In summary, an arbitrary digital filter transfer function $ H(z)$ with $ N$ distinct poles can always be expressed as a parallel combination of complex one-pole filters, together with a parallel FIR part when $ M\geq N$. When there is an FIR part, the strictly proper IIR part may be delayed such that its impulse response begins where that of the FIR part leaves off.

In artificial reverberation applications, the FIR part may correspond to the early reflections, while the IIR part provides the late reverb, which is typically dense, smooth, and exponentially decaying [86]. The predelay (``pre-delay'') control in some commercial reverberators is the amount of pure delay at the beginning of the reverberator's impulse response. Thus, neglecting the early reflections, the order of the FIR part can be viewed as the amount of predelay for the IIR part.

Example: The General Biquad PFE

The general second-order case with $ M=N=2$ (the so-called biquad section) can be written when $ b_0\ne 0$ as

$\displaystyle H(z) \eqsp g\frac{1 + b_1 z^{-1}+ b_2 z^{-2}}{1 + a_1 z^{-1}+ a_2 z^{-2}}. $

To perform a partial fraction expansion, we need to extract an order 0 (length 1) FIR part via long division. Let $ d=z^{-1}$ and rewrite $ H(z)$ as a ratio of polynomials in $ d$:

$\displaystyle H(d^{-1}) \eqsp g\frac{b_2 d^2 + b_1 d + 1 }{a_2 d^2 + a_1 d + 1} $

Then long division gives % For typesetting long division --- NEEDED WITHIN THE MAKEIMAGE ENV? % (raw TeX,... ...} & & b_1-\frac{b_2}{a_2}a_1 & 1-\frac{b_2}{a_2} & \end{array}\end{displaymath}
yielding

$\displaystyle H(d^{-1}) \eqsp g\frac{b_2}{a_2} + g\frac{\left(b_1-\frac{b_2}{a_2}a_1\right)d+ \left(1-\frac{b_2}{a_2}\right)}{a_2d^2 + a_1d + 1} $

or

$\displaystyle H(z) \eqsp g\frac{b_2}{a_2} + g\frac{\left(1-\frac{b_2}{a_2}\right) +\left(b_1-\frac{b_2}{a_2}a_1\right)z^{-1}}{1 + a_1z^{-1}+ a_2z^{-2}}. $

The delayed form of the partial fraction expansion is obtained by leaving the coefficients in their original order. This corresponds to writing $ H(z)$ as a ratio of polynomials in $ z$:

$\displaystyle H(z) \eqsp g\frac{z^2 + b_1 z + b_2 }{z^2 + a_1 z + a_2} $

Long division now looks like % For typesetting long division --- NEEDED WITHIN THE MAKEIMAGE ENV?\begin{dis... ...rule width 22\digitwidth}} & & b_1-a_1 & b_2-a_2 & \end{array}\end{displaymath}
giving

$\displaystyle H(z) \eqsp g + z^{-1}g\frac{(b_1-a_1) + (b_2-a_2)z^{-1}}{1 + a_1 z^{-1}+ a_2 z^{-2}}. $

Numerical examples of partial fraction expansions are given in §6.8.8 below. Another worked example, in which the filter $ y(n) = x(n) + 0.5^3 x(n-3) - 0.9^5 y(n-5)$ is converted to a set of parallel, second-order sections is given in §3.12. See also §9.2 regarding conversion to second-order sections in general, and §G.9.1 (especially Eq.$ \,$(G.22)) regarding a state-space approach to partial fraction expansion.

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Alternate PFE Methods
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Inverting the Z Transform