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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) $


$ 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:

$\displaystyle \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}]$ .

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``Impact of String Stiffness on Virtual Bowed Strings'', by Stefania Serafin<>, (From CCRMA DSP Seminar Presentation, Music 423).
Copyright © 2019-02-05 by Stefania Serafin<>
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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