Digital Waveguide Interpolation

A more compact simulation diagram which stands for either sampled or continuous waveguide simulation is shown in Fig.C.4. The figure emphasizes that the ideal, lossless waveguide is simulated by a bidirectional delay line, and that bandlimited spatial interpolation may be used to construct a displacement output for an arbitrary $x$ not a multiple of $cT$ , as suggested by the output drawn in Fig.C.4. Similarly, bandlimited interpolation across time serves to evaluate the waveform at an arbitrary time not an integer multiple of $T$ (§4.4).

Figure C.4: Simplified picture of ideal waveguide simulation.

Ideally, bandlimited interpolation is carried out by convolving a continuous ``sinc function'' sinc$(x)\isdeftext \sin(\pi x)/\pi x$ with the signal samples. Specifically, convolving a sampled signal $x(t_n)$ with sinc$[(t_n-t_0)/T)]$ ``evaluates'' the signal at an arbitrary continuous time $t_0$ . The sinc function is the impulse response of the ideal lowpass filter which cuts off at half the sampling rate.

In practice, the interpolating sinc function must be windowed to a finite duration. This means the associated lowpass filter must be granted a ``transition band'' in which its frequency response is allowed to ``roll off'' to zero at half the sampling rate. The interpolation quality in the ``pass band'' can always be made perfect to within the resolution of human hearing by choosing a sufficiently large product of window-length times transition-bandwidth. Given ``audibly perfect'' quality in the pass band, increasing the transition bandwidth reduces the computational expense of the interpolation. In fact, they are approximately inversely proportional. This is one reason why oversampling at rates higher than twice the highest audio frequency is helpful. For example, at a $44.1$ kHz sampling rate, the transition bandwidth above the nominal audio upper limit of $20$ kHz is only $2.1$ kHz, while at a $48$ kHz sampling rate (used in DAT machines) the guard band is $4$ kHz wide--nearly double. Since the required window length (impulse response duration) varies inversely with the provided transition bandwidth, we see that increasing the sampling rate by less than ten percent reduces the filter expense by almost fifty percent. Windowed-sinc interpolation is described further in §4.4. Many more techniques for digital resampling and delay-line interpolation are reviewed in [269].