MATLAB: How to obtain Real and Imaginary parts of symbolic polynomial

imaginaryMATLABrealsymbolicSymbolic Math Toolbox

I have the below code. I cannot figure out how to obtain the real and imaginary parts of the symbolic polynomial (I have already solved the complete problem in standard Matlab, trying to learn the Symbolic Toolbox). My code is:
polynomial = [1 10 45 105 105]; % This polynomial is the Bessel Filter 
% polynomial computed above, and changes with the order of the filter.
p = poly2sym(polynomial, s)
pw = subs(p, 1*j*w)
% attempt from Walter Roberson's comment at: 
% https://www.mathworks.com/matlabcentral/answers/
% 465739-real-and-imaginary-part-of-this-function-please
real_pw = simplify( rewrite(real(pw), 'exp') )
imag_pw = simplify( rewrite(imag(pw), 'exp') )
% NOT WORKING FOR ME
% Assuming I can get the above done, my next steps would be to convert the
% below (from standard Matlab) to symbolic.  Any help with the below would
% also be appreciated.
sq_real = conv(real_pw, real_pw);
sq_imag = conv(imag_pw, imag_pw);
sq_denom = sq_real + sq_imag;
The above code gives me:
real_pw =
10*imag(w^3) – 45*real(w^2) + real(w^4) – 105*imag(w) + 105
imag_pw =
imag(w^4) – 45*imag(w^2) – 10*real(w^3) + 105*real(w)
Instead, I am expecting something like (obtained from my working plain Matlab code):
real_poly = 1 0 -45 0 105
imag_poly = 0 -10 0 105 0
Any help you can provide is appreciated!

Best Answer

From the context of the code, it appear that w is a real number. Assuming it being so will help:
>> syms s
syms w real
polynomial = [1 10 45 105 105]; % This polynomial is the Bessel Filter 
% polynomial computed above, and changes with the order of the filter.
p = poly2sym(polynomial, s);
pw = subs(p, 1j*w);  % note use of 1j, not 1*j
real_pw = simplify( rewrite(real(pw), 'exp') )
imag_pw = simplify( rewrite(imag(pw), 'exp') )
% but this is simpler
real_pw = real(pw)
imag_pw = imag(pw)
 
real_pw =
 
w^4 - 45*w^2 + 105
 
 
imag_pw =
 
- 10*w^3 + 105*w
 
 
real_pw =
 
w^4 - 45*w^2 + 105
 
 
imag_pw =
 
- 10*w^3 + 105*w
I don't believe that conv is supported for symbolic polynomials (version 2019a)
>> conv(real_pw,real_pw)
Error using conv2
Invalid data type. First and second arguments must be numeric or logical.
Error in conv (line 43)
c = conv2(a(:),b(:),shape);
But conv is the same as polynomial multiplication, so maybe this is what you want:
>> sq_real = real_pw*real_pw
 
sq_real =
 
(w^4 - 45*w^2 + 105)^2
 
>> sq_imag = imag_pw*imag_pw
 
sq_imag =
 
(- 10*w^3 + 105*w)^2
 
>> sq_denom = sq_real + sq_imag
 
sq_denom =
 
(- 10*w^3 + 105*w)^2 + (w^4 - 45*w^2 + 105)^2
 
>> expand(sq_denom)
 
ans =
 
w^8 + 10*w^6 + 135*w^4 + 1575*w^2 + 11025
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