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Delay Line Interpolation

A deceptively simple problem that applies to nearly all digital waveguide models is that of delay-line interpolation. Integer delay lengths are not sufficient for musical tuning of digital waveguide models at commonly used sampling rates [31]. The simplest scheme which is typically tried first is linear interpolation. However, poor results are obtained in some cases (such electric guitars) because the pitch-dependent damping caused by interpolation can be much larger than the desired damping in the string model. In these cases, the interpolation filter becomes the dominant source of damping, so that when the pitch happens to fall on an integer delay-line length, the damping suddenly decreases, making the note stand out as ``buzzy.''

Allpass interpolation is a nice choice for the nearly lossless feedback loops commonly used in digital waveguide models, because it does not suffer any frequency-dependent damping [31]. However, allpass interpolation instead has the problem that instantly switching from one delay to another (as in a hammer-on or pull-off simulation in a string model) gives rise to a transient artifact due to the recursive nature of the allpass filter. Recently, Vesa Välimäki has developed a general transient elimination scheme for recursive digital filters of arbitrary order [64].

Another popular choice is Lagrange interpolation [33] which is a special case of FIR filter interpolation; while the switching problem does not arise since the interpolating filter is nonrecursive, there is still a time-varying amplitude distortion at high frequencies. In fact, first-order Lagrange interpolation is just linear interpolation, and higher orders can be shown to give a maximally smooth frequency response at DC (zero frequency), while the gain generally rolls off at high frequencies. Allpass interpolation can be seen as trading off this frequency-dependent amplitude distortion for additional frequency-dependent delay distortion [16]. A comprehensive review of Lagrange interpolation appears in [61].

Both allpass and FIR interpolation suffer from some delay distortion at high frequencies due to having a nonlinear phase response at non-integer desired delays. This distortion is normally inaudible, even in the first-order case, causing mistuning or phase modulation only in the highest partial overtones of a resonating string or tube.

Optimal interpolation can be approached via general-purpose bandlimited interpolation techniques [56]. However, the expense is generally considered too high for widespread usage at present. Both amplitude and delay distortions can be eliminated over the entire band of human hearing using higher order allpass or FIR interpolation filters in conjunction with some amount of oversampling. A comprehensive review of delay-line interpolation techniques is due to appear in [39].

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``Physical Modeling Synthesis Update'', by Julius O. Smith III, Computer Music Journal, vol. 20, no. 2 (summer), pp. 44-56, MIT Press, 1996.
Copyright © 2005-12-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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