Partial Fraction Expansion

An important tool for inverting the z transform and converting among digital filter implementation structures is the partial fraction expansion (PFE). The term partial fraction expansion'' refers to the expansion of a rational transfer function into a sum of first and/or second-order terms. The case of first-order terms is the simplest and most fundamental:

 (7.7)

where

and . (The case is addressed in the next section.) The denominator coefficients are called the poles of the transfer function, and each numerator is called the residue of pole . Equation (6.7) is general only if the poles are distinct. (Repeated poles are addressed in §6.8.5 below.) Both the poles and their residues may be complex. The poles may be found by factoring the polynomial into first-order terms,7.4e.g., using the roots function in matlab. The residue corresponding to pole may be found analytically as

 (7.8)

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 .

Example

Consider the two-pole filter

The poles are and . The corresponding residues are then

We thus conclude that

As a check, we can add the two one-pole terms above to get

as expected.

Complex Example

To illustrate an example involving complex poles, consider the filter

where can be any real or complex value. (When is real, the filter as a whole is real also.) The poles are then and (or vice versa), and the factored form can be written as

Using Eq.(6.8), the residues are found to be

Thus,

A more elaborate example of a partial fraction expansion into complex one-pole sections is given in §3.12.1.

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 somewhere among the terms in the one-pole expansion. These two terms can be paired to form a real second-order section as follows:

Expressing the pole in polar form as , and the residue as , the last expression above can be rewritten as

The use of polar-form coefficients is discussed further in the section on two-pole filtersB.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:

 (7.9)

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.

Inverting the Z Transform

The partial fraction expansion (PFE) provides a simple means for inverting the z transform of rational transfer functions. The PFE provides a sum of first-order terms of the form

It is easily verified that such a term is the z transform of

Thus, the inverse z transform of is simply

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, .

In the improper case, discussed in the next section, we additionally obtain an FIR part in the z transform to be inverted:

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.

FIR Part of a PFE

When in Eq.(6.7), we may perform a step of long division of to produce an FIR part in parallel with a strictly proper IIR part:

 (7.10)

where

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).

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

 (7.11)

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.

We may compare these two PFE alternatives as follows: Let denote , , and . (I.e., we use a subscript to indicate polynomial order, and ' is omitted for notational simplicity.) Then for we have two cases:

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 [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.

The general second-order case with (the so-called biquad section) can be written when as

To perform a partial fraction expansion, we need to extract an order 0 (length 1) FIR part via long division. Let and rewrite as a ratio of polynomials in :

Then long division gives
yielding

or

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

Long division now looks like
giving

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.

Alternate PFE Methods

Another method for finding the pole residues is to write down the general form of the PFE, obtain a common denominator, expand the numerator terms to obtain a single polynomial, and equate like powers of . This gives a linear system of equations in unknowns , .

Yet another method for finding residues is by means of Taylor series expansions of the numerator and denominator about each pole , using l'Hôpital's rule..

Finally, one can alternatively construct a state space realization of a strictly proper transfer function (using, e.g., tf2ss in matlab) and then diagonalize it via a similarity transformation. (See Appendix G for an introduction to state-space models and diagonalizing them via similarity transformations.) The transfer function of the diagonalized state-space model is trivially obtained as a sum of one-pole terms--i.e., the PFE. In other words, diagonalizing a state-space filter realization implicitly performs a partial fraction expansion of the filter's transfer function. When the poles are distinct, the state-space model can be diagonalized; when there are repeated poles, it can be block-diagonalized instead, as discussed further in §G.10.

Repeated Poles

When poles are repeated, an interesting new phenomenon emerges. To see what's going on, let's consider two identical poles arranged in parallel and in series. In the parallel case, we have

In the series case, we get

Thus, two one-pole filters in parallel are equivalent to a new one-pole filter7.8 (when the poles are identical), while the same two filters in series give a two-pole filter with a repeated pole. To accommodate both possibilities, the general partial fraction expansion must include the terms

for a pole having multiplicity 2.

Dealing with Repeated Poles Analytically

A pole of multiplicity has residues associated with it. For example,

 (7.12)

and the three residues associated with the pole are 1, 2, and 4.

Let denote the th residue associated with the pole , . Successively differentiating times with respect to and setting isolates the residue :

or

Example

For the example of Eq.(6.12), we obtain

Impulse Response of Repeated Poles

In the time domain, repeated poles give rise to polynomial amplitude envelopes on the decaying exponentials corresponding to the (stable) poles. For example, in the case of a single pole repeated twice, we have

Proof: First note that

Therefore,
 (7.13)

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.

To study this phenomenon further, consider the convolution of two one-pole impulse-responses and :

 (7.14)

The finite limits on the summation result from the fact that both and are causal. Recall the closed-form sum of a truncated geometric series:

Applying this to Eq.(6.14) yields

Note that the result is symmetric in and . If , then becomes proportional to for large , while if , it becomes instead proportional to .

Going back to Eq.(6.14), we have

 (7.15)

Setting yields

 (7.16)

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.

We also see that the polynomial amplitude-envelopes fundamentally arise from iterated convolutions. This corresponds to the repeated poles being arranged in series, rather than in parallel. The simplest case is when the repeated pole is at , in which case its impulse response is a constant:

The convolution of a constant with itself is a ramp:

The convolution of a constant and a ramp is a quadratic, and so on:7.9

Alternate Stability Criterion

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.

A transfer function with no pole-zero cancellations is said to be irreducible. For example, is irreducible, while is reducible, since there is the common factor of in the numerator and denominator. Using this terminology, we may state the following stability 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.

Summary of the Partial Fraction Expansion

In summary, the partial fraction expansion can be used to expand any rational z transform

as a sum of first-order terms

 (7.17)

for , and

 (7.18)

for , where the term is optional, but often preferred. For real filters, the complex one-pole terms may be paired up to obtain second-order terms with real coefficients. The PFE procedure occurs in two or three steps:
1. When , perform a step of long division to obtain an FIR part and a strictly proper IIR part .
2. Find the poles , (roots of ).
3. If the poles are distinct, find the residues , from

4. If there are repeated poles, find the additional residues via the method of §6.8.5, and the general form of the PFE is

 (7.19)

where denotes the number of distinct poles, and denotes the multiplicity of the th pole.

In step 2, the poles are typically found by factoring the denominator polynomial . This is a dangerous step numerically which may fail when there are many poles, especially when many poles are clustered close together in the plane.

The following matlab code illustrates factoring to obtain the three roots, , :

A = [1 0 0 -1];  % Filter denominator polynomial
poles = roots(A) % Filter poles


See Chapter 9 for additional discussion regarding digital filters implemented as parallel sections (especially §9.2.2).

Software for Partial Fraction Expansion

Figure 6.3 illustrates the use of residuezJ.5) for performing a partial fraction expansion on the transfer function

The complex-conjugate terms can be combined to obtain two real second-order sections, giving a total of one real first-order section in parallel with two real second-order sections, as discussed and depicted in §3.12.

 B = [1 0 0 0.125]; A = [1 0 0 0 0 0.9^5]; [r,p,f] = residuez(B,A) % r = % 0.16571 % 0.22774 - 0.02016i % 0.22774 + 0.02016i % 0.18940 + 0.03262i % 0.18940 - 0.03262i % % p = % -0.90000 % -0.27812 - 0.85595i % -0.27812 + 0.85595i % 0.72812 - 0.52901i % 0.72812 + 0.52901i % % f = [](0x0) 

Example 2

For the filter

 (7.20) (7.21)

we obtain the output of residuedJ.6) shown in Fig.6.4. In contrast to residuez, residued delays the IIR part until after the FIR part. In contrast to this result, residuez returns r=[-24;16] and f=[10;2], corresponding to the PFE

 (7.22)

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.

 B=[2 6 6 2]; A=[1 -2 1]; [r,p,f,m] = residued(B,A) % r = % 8 % 16 % % p = % 1 % 1 % % f = % 2 10 % % m = % 1 % 2 

Polynomial Multiplication in Matlab

The matlab function conv (convolution) can be used to perform polynomial multiplication. For example:

B1 = [1 1];   % 1st row of Pascal's triangle
B2 = [1 2 1]; % 2nd row of Pascal's triangle
B3 = conv(B1,B2) % 3rd row
% B3 = 1  3  3  1
B4 = conv(B1,B3) % 4th row
% B4 = 1  4  6  4  1
% ...

The matlab conv(B1,B2) is identical to filter(B1,1,B2), except that conv returns the complete convolution of its two input vectors, while filter truncates the result to the length of the input signal'' B2.7.10 Thus, if B2 is zero-padded with length(B1)-1 zeros, it will return the complete convolution:
B1 = [1 2 3];
B2 = [4 5 6 7];
conv(B1,B2)
% ans = 4  13  28  34  32  21
filter(B1,1,B2)
% ans = 4  13  28  34
filter(B1,1,[B2,zeros(1,length(B1)-1)])
% ans = 4  13  28  34  32  21


Polynomial Division in Matlab

The matlab function deconv (deconvolution) can be used to perform polynomial long division in order to split an improper transfer function into its FIR and strictly proper parts:

B = [ 2 6 6 2]; % 2*(1+1/z)^3
A = [ 1 -2 1];  % (1-1/z)^2
[firpart,remainder] = deconv(B,A)
% firpart =
%   2  10
% remainder =
%    0    0   24   -8

Thus, this example finds that is as written in Eq.(6.21). This result can be checked by obtaining a common denominator in order to recalculate the direct-form numerator:
Bh = remainder + conv(firpart,A)
%  = 2 6 6 2


The operation deconv(B,A) can be implemented using filter in a manner analogous to the polynomial multiplication case (see §6.8.8 above):

firpart = filter(B,A,[1,zeros(1,length(B)-length(A))])
%       = 2 10
remainder = B - conv(firpart,A)
%         =  0 0 24 -8
`
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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