Finding derivative of a transfer function in Matlab Symbolic

Hi,

I've had a look through the answers here but haven't found anything that really helped me. This is a problem I've been able to solve in the past, but I can't seem to figure out how to do it now and I'm wasting a lot of time taking various approaches and not getting the result I'm looking for.

I have several transfer functions, which are defined in the form:

    syms z b w;
    H(z) = (z - exp(-b)*cos(w) - 1i* exp(-b)*sin(w)) * (z - exp(-b)*cos(w) + 1i * exp(-b)*sin(w));

(this is the simplest example - some are rational)

I'm trying to find the extremum of the magnitude frequency response. So the approach I want to take is something like this:

    H_ps = abs(H)^2;
    H_ps_prime = diff(H_ps, z);

with z restricted to the unit circle. I've tried doing this by:

    syms r;
    z = exp(1i* r);
    % in this case, r essentially replaces z as the dependent variable in H

    %I've also tried this, but it doesn't seem to work properly
    assume(abs(z) == 1);

Once I've got it, I want to solve to H_ps_prime with respect to r - i.e. get the value of r for which the H_ps_prime is zero. I would also like to find the value of w for which this is the case, as the whole purpose of this exercise is to get a function that I can use to "correct" w so that it coincides with the magnitude extremum.

I've tried this by:

    solve(H_ps_prime == 0, r);

.. but this does not give me the right result.

In the case of the two zero filter I've used as an example here, I already know what the result should be as I've obtained it previously and have it written down. The magnitude extremum is at:

    r_ext = 2*atan(sqrt( - (exp(b)^2 * cos(w) - 2 * exp(b) + cos(w))/(exp(b)^2 * cos(w) + 2 * exp(b) + cos(w))));

and the solution for w is:

    w_c = acos((2 * exp(b) * cos(r_ext)) / (exp( 2 * b) + 1));

whereas I'm getting the following:

    syms r b w;
    H(r) = (exp(1i*r) - exp(-b)*cos(w) - 1i* exp(-b)*sin(w)) * (exp(1i*r) - exp(-b)*cos(w) + 1i* exp(-b)*sin(w));
    H_ps = abs(H)^2;
    H_ps_prime = diff(H_ps, r);
    solve(H_ps_prime == 0 , r);

    ans = 
        -log(exp(-b)(cos(w) - sin(w) *1i)) * 1i;
        -log(exp(-b)(cos(w) + sin(w) *1i)) * 1i;
        -log(exp(-b)*cos(w))*1i;

Can anybody tell me where I'm going wrong? I feel like I must be doing something stupid.

edit:

To add some further details: I have set assumptions on w, b and r that all should be real. r should also be restricted to the range [-pi, pi] and w and b should be in the range [0, pi]. Despite this, when solving for r (or w for that matter), matlab gives me a complex number. I have also been getting a "maximum recursion" error when calling solve on occasion (not sure what triggers it, but it throws and exits the call).

Answer 1

[Walter Roberson]

Once you have done the solve() that gives you the log() results, use rewrite() to express in terms of atan.

rewrite(ans, 'atan')

There are equivalences between log and atan

[David Browne]

The problem isn't the form of the equation, it's that the equation does not provide the result I'm after.

[Walter Roberson]

I cross-checked with another package; it agrees with the answers MATLAB produces, and confirms that the answer you expect does not match any of the three answers MATLAB produces.

Question: can we place constraints on any of the symbols? For example are w, b, and r all real-valued?