Delay Loop Expansion

When a subset of the resonating modes are nearly harmonically tuned, it can be more efficient to use a filtered delay loop (see §1.6.5) to generate an entire quasi-harmonic series of modes rather than using a biquad for each modal peak [444]. In this case, the resonator model becomes

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

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]. Note that a single filtered delay loop has been used extensively to efficiently model distributed media such as vibrating strings, wind instrument bores, horns, and pipes [239,211,231,437,442], as will be elaborated later.

Note that when $H_k(z)$ is close to $-1$, primarily only odd harmonic resonances are produced, as has been used in modeling the clarinet [435]. We call this case the odd-harmonic filtered delay loop (FDL).