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Intro

My question is related to a previous one linked here. I am interested in non-linear phase FIR filters with a specific desired phase response.

After I tried the options in the linked question I also stumbled upon the idea to separately define weights for phase and magnitude responses developed in Matt's PhD thesis linked here (see section 2.1.3).

The thesis states that a discrete approximation variant can be similarly developed which is what I'm after. While Matt provides a MATLAB implementation, it follows the continuous approximation, and therefore the desired phase response is limited, and only allows for resampling data (fractional delay filter). On the other hand, I have measurements of the required phase response on a grid.

Question

When I try to formulate the problem on a discrete frequency grid I get to a problem. After applying the approximation, the contribution of passband to approximation error reads:

$$\tilde\epsilon_p = \sum_{i=0}^{L-1}\left(W_m(\omega_i)\{Re[\mathbf{e}^H(\omega_i)\mathbf{h}e^{-j\phi_d(\omega_i)}]-|D(e^{j\omega_i})|\}^2 + W_\phi(\omega_i)\{Im[\mathbf{e}^H(\omega_i)\mathbf{h}e^{-j\phi_d(\omega_i)}]\}^2\right) $$

However, I'm not sure how to proceed due to the separate treatment of the real and imaginary parts. How would I get the normal equation form or solve it another way?

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