### Software for Partial Fraction Expansion

Figure 6.3 illustrates the use of `residuez` (§J.5)
for performing a partial fraction expansion on the transfer function

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

we obtain the output of

`residued`(§J.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 -8Thus, 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 -8That 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 [].

#### Separating the Transfer Function Numerator and Denominator

From Eq.(6.5) we have that the transfer function of a recursive filter is a ratio of polynomials in :

where

By elementary properties of complex numbers, we have

These relations can be used to simplify calculations by hand, allowing the numerator and denominator of the transfer function to be handled separately.

**Next Section:**

Frequency Response in Matlab

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Summary of the Partial Fraction Expansion