# 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

• whoa! parallel, eh? uhm, you start with your transfer function that is a ratio of two polynomials a and b. then whatever polynomial is in the denominator (i guess it's a) is factored into roots that will include complex conjugate root pairs. then use Heaviside partial fraction expansion to get the parallel first-order sections. then combine each conjugate pole sections into a single second-order section. – robert bristow-johnson Dec 3 '19 at 7:02
• Sir thank you but can you help me how to write the matlab code which produces the first order sections? – Jason Dec 3 '19 at 8:01

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