Delay Loop Expansion

When a subset of the resonating modes is nearly harmonically tuned, it can be much more computationally efficient to use a filtered delay loop (see §2.6.5) to generate an entire quasi-harmonic series of modes rather than using a biquad for each modal peak [443]. In this case, the resonator model becomes

$$H(z) \eqsp \sum_{k=1}^N \frac{a_k}{1 - H_k(z) z^{-N_k}},$$

where $N_k$ is the length of the delay line in the $k$ th comb filter, and $H_k(z)$ is a low-order filter which can be used to adjust finely the amplitudes and frequencies of the resonances of the $k$ th comb filter [432]. Recall (Chapter 6) that a single filtered delay loop efficiently models a distributed 1D propagation medium such as a vibrating string or acoustic tube. More abstractly, a superposition of such quasi-harmonic mode series can provide a computationally efficient psychoacoustic equivalent approximation to arbitrary collections of modes in the range of human hearing.

Note that when $H_k(z)$ is close to $-1$ instead of $+1$ , primarily only odd harmonic resonances are produced, as has been used in modeling the clarinet [435].