Parallel form filter design I have a transfer function coefficients as following:
[b,a] = ellip(4,.2,40,[.41 .47]);

I'd like to factor this system into its PFE using [r,p,k] = residue(b,a). Then, I'd like to implement a filter which uses a parallel combination of second-order subsections. To do this, I need to combine each complex pole with its complex conjugate so that the overall second order subsections will have real valued coefficients. Hence; I selected four pairs of first order terms and recombined them into second order subsections using "residue" function. How can I store the coefficients of the resulting second-order subsections in the matrices c and d so that each row corresponds to one second order system?

I tried to do this but I don't know whether it's correct or not?

[c1 d1]=residue([r(1) r(2)],[p(1) p(2)],0);
[c2 d2]=residue([r(3) r(4)],[p(3) p(4)],0);
[c3 d3]=residue([r(5) r(6)],[p(5) p(6)],0);
[c4 d4]=residue([r(7) r(8)],[p(7) p(8)],k);

[c d]=[c1 c2 c3 c4 d1 d2 d3 d4]


Can anybody please give me an idea? Any help would be appreciated
A few points first


Since you are using digital filters, you should be using residuez() not residue()
Matlab's residuez() is not particularly well implemented: in case of higher order or steep filters, you can get a lot of numerically noise or results that are just totally wrong. Always check the result against your original target.


Back to the question: each residue comes as a conjugate complex pair (at least if you are lucky and don't have to deal with real or multiple poles). Just add these to form a single real second order section.

$$H(z) = \frac{r}{1-p \cdot z^{-1}} + \frac{r^{*}}{1-p^* \cdot z^{-1}} $$
$$= \frac{r \cdot (1-p^* \cdot z^{-1})+r^* \cdot (1-p^* \cdot z^{-1})}{(1-p \cdot z^{-1}) \cdot (1-p^* \cdot z^{-1})}$$

$$= \frac{(r + r^*) - (r \cdot p^* + r^* \cdot p  ) \cdot z^{-1} }{1 -(p+p^*)\cdot z^{-1}+ p \cdot p^* \cdot z^{-2}}$$

The last equation  gives you directly the real valued biquad coefficients (if I did the math correctly,  that is).