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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!

Paul
on 2 Jan 2021

Edited: Paul
on 2 Jan 2021

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

Paul
on 3 Jan 2021

Walter may be able to work some symbolic magic; he's really good at that. The best I can do is get the numerical answer:

sq_denom(w)=simplify(pw*conj(pw));

gain = polynomial(end);

f(w) = sq_denom(w) - 2*gain^2; % this is the equation to solve for w

fplot(f(w)); % looks like there are two real roots, which makes sense from the form of f(w)

w0 = vpasolve(f,w);

w0(imag(w0)==0)

ans =

-2.1139176749042158430196891286339

2.1139176749042158430196891286339

Walter Roberson
on 3 Jan 2021

solve(f, 'maxdegree', 4) I suspect

If so then the solutions are not likely to clarify anything, as they will be more than 1000 characters each.

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