Designing allpass filters for dispersion simulation

We choose a numerical filter made of a delay line delay line $q^{\tau-\tau_{0}}$ , and a n-order stable all-pass filter:

$H(q)=q^{n}P(q^{-1})/P(q)$

where

$P(q)=p_{0}+\ldots +p_{n-1}q^{n-1}+q^{n}$

and $\tau$ and $n$ are appropiately chosen.

We minimize the infinity-norm of a particular frequency weighting of the error between the internal loop phase and its approximation by the filter cascade:

$$\delta_{D}=\min_{p_{1}, \ldots, p_{m}} \Vert W_{D}(\Omega) [\varphi_{d}(\Omega)-( \varphi_{D}(\Omega)+\tau \Omega) ] \Vert _{\infty}$$

where $\varphi_{D}(\Omega)$ =phase of $H(e^{j\Omega})$ ,
$W_{D}(\Omega)$ = frequency weigthing ( $W_{D}(\Omega)$ is zero outside the frequency range), i.e. $[\Omega_{c}, \Omega_{N}]$ .