In §6.7 we discussed damping filters for vibrating string models, and in §6.9 we discussed dispersion filters. For vibrating strings, which are well described by a linear time-invariant (LTI) partial differential equation, damping and dispersion filtering are the only deviations possible from the ideal string discussed in §6.1.
The ideal damping filter is ``zero phase'' (or linear phase) ,7.10while the ideal dispersion filter is ``allpass'' (as described in §6.9.1). Since every desired frequency response can be decomposed into a zero-phase frequency-response in series with an allpass frequency-response, we may design a single loop filter whose amplitude response gives the desired damping as a function of frequency, and whose phase response gives the desired dispersion vs. frequency. The next subsection summarizes some methods based on this approach. The following two subsections discuss methods for the design of damping and dispersion filters separately.