Can the Unwrapped Phase Function of a Fourier Transform be Derived?

[Cyrus]

I am stuck trying to derive the unwrapped phase function of a Fourier Transform. Here is the gist of the derivation.

We can express the Fourier Transform in polar form: X(e ^jω) = |X(e^jω)| e^jω

We can take the ln of both sides, resulting in:

ln X(ejω) = ln | X(ejω) | + j θ(ω)

Taking the derivative w.r.t. ω:

d ln X(ejω) / dω = d | X(ejω)| / dω + j dθ/dω

But, if we express X(ejω) = X_re(ejω) + j X_im(ejω) then we can also find the derivative to be:

d ln X(ejω) / dω = 1/ X(ejω) [ d X(ejω)/dω] = 1/ X(ejω) [dX_re(ejω)/dω + jdX_im(ejω)/dω]

Here is where I cannot get to: The author then states: " Therefore, the derivative of θ(ω) with respect to ω is given by the imaginary part of the right hand side of the second equation I wrote from the top. Somehow he is finding the equation below when equating/combining the two definitions of the derivatives of the ln X(ejw),"

dθ/dω = 1/ | X(ejω)|² [ X_re(ejω) d X_im (ejω) / dω - X_im(ejω)dX_re(ejω)/dω]

 

[Cyrus]

I was able to solve the problem, feel free to delete.