Interpolated Digital Waveguides

Integer delay lengths are not sufficient for musical tuning of digital waveguide models at commonly used sampling rates [40]. The simplest scheme which is typically tried first is linear interpolation. However, poor results are obtained in some cases (such electric guitar models) due to the pitch-dependent damping caused by delay-line interpolation. As is well known [88], linear interpolation is equivalent to a time-varying FIR filter of the form $y(n) = \alpha(n)x(n) + [1-\alpha(n)]x(n-1)$, and for $\alpha(n)=0.5$ (for interpolation half way between available samples), the gain of the interpolation filter at $f_s/2$ drops to zero. In some cases, such as for steel string simulation, the interpolation filter becomes the dominant source of damping, so that when the pitch happens to fall on an integer delay-line length ($\alpha=0$), the damping suddenly decreases, making the note stand out as ``buzzy.'' In such cases, something better than linear interpolation is required.

Allpass interpolation is a nice choice for the nearly lossless feedback loops commonly used in digital waveguide models [40,104], because it does not suffer any frequency-dependent damping. Its phase distortion normally manifests as a slight mistuning of high-frequency resonances, which is usually inaudible and arguably even desirable in most cases. 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. Transient artifacts can be reduced or eliminated by ``warming up'' a second instance of the filter using the new coefficients in advance of the transition (ideally several time-constants in advance) and, at the desired transition time, switching out the old filter and switching in the new [120]. In this way, the new filter is switched in with state consistent with the new coefficients. Another approach is to cross-fade from the old filter to the new filter, allowing consistent state to develop in the new filter before its output fades in.

Another popular choice is Lagrange interpolation [46,115,1,88] which is a special case of FIR filter interpolation; while the transient artifact problem is minimal 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 [13]. A comprehensive review of Lagrange interpolation appears in [115].

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. As mentioned above for the allpass case alone, 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 [105,70]. 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 IIR (e.g., allpass) or FIR interpolation filters in conjunction with some amount of oversampling. A comprehensive review of delay-line interpolation techniques is given in [53].