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Timestamp: 2019-04-21 03:14:46+00:00

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when the poles are distinct. The matlab function residuez7.5 will find poles and residues computationally, given the difference-equation (transfer-function) coefficients.
Note that in Eq. (6.8), there is always a pole-zero cancellation at . That is, the term is always cancelled by an identical term in the denominator of , which must exist because has a pole at . The residue is simply the coefficient of the one-pole term in the partial fraction expansion of at . The transfer function is , in the limit, as .
A more elaborate example of a partial fraction expansion into complex one-pole sections is given in §3.12.1.
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.
Notice that the numerator order, viewed as a polynomial in , 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 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.
Thus, the impulse response of every strictly proper LTI filter (with distinct poles) can be interpreted as a linear combination of sampled complex exponentials. Recall that a uniformly sampled exponential is the same thing as a geometric sequence. Thus, is a linear combination of geometric sequences. The term ratio of the th geometric sequence is the th pole, , and the coefficient of the th sequence is the th residue, .
The FIR part (a finite-order polynomial in ) is also easily inverted by inspection.
The case of repeated poles is addressed in §6.8.5 below.
When , we define . 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).
where 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.
In the first form, the 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 ( ) can be used to match additional terms in the impulse response after the FIR part has ``died out''.
In summary, an arbitrary digital filter transfer function with distinct poles can always be expressed as a parallel combination of complex one-pole filters, together with a parallel FIR part when . 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 . 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.
Numerical examples of partial fraction expansions are given in §6.8.8 below. Another worked example, in which the filter 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.
for a pole having multiplicity 2.
and the three residues associated with the pole are 1, 2, and 4.
Note that is a first-order polynomial in . Similarly, a pole repeated three times corresponds to an impulse-response component that is an exponential decay multiplied by a quadratic polynomial in , and so on. As long as , the impulse response will eventually decay to zero, because exponential decay always overtakes polynomial growth in the limit as goes to infinity.
So What's Up with Repeated Poles?
In the previous section, we found that repeated poles give rise to polynomial amplitude-envelopes multiplying the exponential decay due to the pole. On the other hand, two different poles can only yield a convolution (or sum) of two different exponential decays, with no polynomial envelope allowed. This is true no matter how closely the poles come together; the polynomial envelope can occur only when the poles merge exactly. This might violate one's intuitive expectation of a continuous change when passing from two closely spaced poles to a repeated pole.
Note that the result is symmetric in and . If , then becomes proportional to for large , while if , it becomes instead proportional to .
which is the first-order polynomial amplitude-envelope case for a repeated pole. We can see that the transition from ``two convolved exponentials'' to ``single exponential with a polynomial amplitude envelope'' is perfectly continuous, as we would expect.
In §5.6 (page ), a filter was defined to be stable if its impulse response decays to 0 in magnitude as time goes to infinity. In §6.8.5, we saw that the impulse response of every finite-order LTI filter can be expressed as a possible FIR part (which is always stable) plus a linear combination of terms of the form , where is some finite-order polynomial in , and is the th pole of the filter. In this form, it is clear that the impulse response always decays to zero when each pole is strictly inside the unit circle of the plane, i.e., when . Thus, having all poles strictly inside the unit circle is a sufficient criterion for filter stability. If the filter is observable (meaning that there are no pole-zero cancellations in the transfer function from input to output), then this is also a necessary criterion.
This characterization of stability is pursued further in §8.4, and yet another stability test (most often used in practice) is given in §8.4.1.
in which the FIR and IIR parts have overlapping impulse responses.
See Sections J.5 and J.6 starting on page for listings of residuez, residued and related discussion.
Figure 6.4: Use of residued to perform a partial fraction expansion of an IIR filter transfer function .
That this must work can be seen by looking at Eq. (6.21) and noting that the impulse-response of the remainder (the strictly proper part) does not begin until time , so that the first two samples of the impulse-response come only from the FIR part.
In summary, we may conveniently use convolution and deconvolution to perform polynomial multiplication and division, respectively, such as when converting transfer functions to various alternate forms.
When carrying out a partial fraction expansion on a transfer function having a numerator order which equals or exceeds the denominator order, a necessary preliminary step is to perform long division to obtain an FIR filter in parallel with a strictly proper transfer function. This section describes how an FIR part of any length can be extracted from an IIR filter, and this can be used for PFEs as well as for more advanced applications .

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