## Frequency-Dependent Damping

In real vibrating strings, damping typically increases with frequency for a variety of physical reasons [73,77]. A simple modification [392] to Eq.(6.14) yielding frequency-dependent damping is*frequency-dependent rates*. This means the loss factors of the previous section should really be

*digital filters*having gains which decrease with frequency (and never exceed for stability of the loop). These filters

*commute*with delay elements because they are

*linear*and

*time invariant*(LTI) [449]. Thus, following the reasoning of the previous section, they can be lumped at a single point in the digital waveguide. Let denote the resulting

*string loop filter*(replacing in Fig.6.12

^{7.7}). We have the stability (passivity) constraint , and making the filter

*linear phase*(constant delay at all frequencies) will restrict consideration to

*symmetric*FIR filters only. Restriction to FIR filters yields the important advantage of keeping the approximation problem

*convex*in the weighted least-squares norm. Convexity of a norm means that gradient-based search techniques can be used to find a global miminizer of the error norm without exhaustive search [64],[428, Appendix A]. The linear-phase requirement halves the degrees of freedom in the filter coefficients. That is, given for , the coefficients are also determined. The loss-filter frequency response can be written in terms of its (impulse response) coefficients as

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The Stiff String

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The Damped Plucked String