Decomposing the numerator and the denominator polynomials into their even and odd parts

Question

Here there is a continues time transfer function (G(s)) in form of:

G(s) = N(s)/D(s);
G(s) = (s^3+4s^2-s+1)/(s^5+2s^4+32s^3+14s^2-4s+50)     (1)

and (s = j*w) where w = frequency symbol.

Now, how is it possible to decompose the numerator and the denominator polynomials of Eq. (1) into their even and odd parts and get the G(jw) as (Using Matlab) :

Answer 1

You could probably take the real and imaginary parts after the substitution with s=j*w. However, you can actually select the even and odd parts of your polynomials:

% G(s) = N(s)/D(s);

syms s;
N = s^3+4*s^2-s+1;

p = sym2poly(N);

%// do this in fewer lines:
%{
/*
if mod(length(p),2)==0  %// then first index is odd
    imin_o = 1;  %// for odd part
    imin_e = 2;  %// for even part
else
    imin_o = 2;  %// for odd part
    imin_e = 1;  %// for even part
end
*/
%} 
imin_o = mod(length(p),2) + 1;
imin_e = 2 - mod(length(p),2);

% odd part of numerator
p_o = zeros(size(p));
p_o(imin_o:2:end) = p(imin_o:2:end);
% even part of numerator
p_e = zeros(size(p));
p_e(imin_e:2:end) = p(imin_e:2:end);

% restore
N_o = poly2sym(p_o,s);
N_e = poly2sym(p_e,s);

and the same for the denominator.