

Bandlimited Signals Interpretation

A more practically useful correspondence between physical and digital waveguide networks is obtained by assuming the inputs to the physical networks are bandlimited continuous-time Kirchoff variables. The signals propagating throughout the physical network are assumed to consist of frequencies less than $f_s/2$ Hz, where $f_s=1/T$ denotes the sampling rate. Therefore, by the Shannon sampling theorem, if we record a sample of the pressure wave, say, at each unit-delay element every $T$ seconds, the bandlimited continuous pressure fluctuation can be uniquely reconstructed throughout the waveguide network. Saying the pressure variation is frequency bandlimited to less than $f_s/2$ is equivalent to saying the pressure distribution is spatially bandlimited to less than $f_s/2c$, or, a one-sample section of waveguide is less than half a cycle of the shortest wavelength contained in a traveling wave. In summary, a DWN is equivalent to a physical waveguide network in which the input signals are bandlimited to $f_s/2$ Hz. This equivalence does not remain true for time-varying or non-linear DWNs.

In the case of time-varying wave impedances, the time-variation of the resultant scattering coefficients applies a continuous amplitude modulation to the continuous propagating signals, thereby generating sidebands. If the signals incident on a junction are bandlimited to $ f_1 $ Hz and the scattering coefficients are bandlimited to $ f_2 $ Hz, then the signals emerging from an interconnection of waveguides are bandlimited only to $ f_1 + f_2 $ Hz. If the network is nontrivial, a portion of the amplitude-modulated signals may return to the same time-varying junction, and the signal bandwidth expands to $ f_1 + 2
f_2$, and so on. A time-varying junction eventually expands the bandwidth of the signals contained in the network to infinity. With a fixed sampling rate $f_s$ in the DWN, we obtain aliased versions of the physical signals. The general result is that time-varying physical waveguide networks cannot be simulated by DWNs at a fixed sampling rate. In practice, however, we obtain good approximate digital simulations by working with wave variables and junction parameters bandlimited to much less than $f_s/2$, and by using sufficient damping (implemented using low-pass filtering) in the network so that the aliased signal energy is attenuated below significance.

Particular care must be used also when inserting nonlinear elements into waveguide networks, since they also extend the bandwidth, and frequency foldover will result. Nonlinear elements are almost always present in the excitation blocks of musical instruments (e.g., reeds, bows, and felt-covered hammers), sometimes also in the resonators (e.g., brasses, sitar strings and cymbals), and sometimes also in the output amplification/diffusion stage (e.g., a saturating amplifier or speaker simulation). Nonlinearities in the resonators (such as a sitar string) may be sufficiently weak so that inherent low-pass filtering can attenuate nonlinearly generated high frequencies to insignificance before they alias. Also, nonlinear excitations can be bandlimited using a lowpass filter if they are not strongly coupled to the resonator they excite (the ``source-filter'' decomposition). Similarly, nonlinearities in the output path can be implemented without aliasing in the absence of feedback to the resonator or excitation. In some practical cases, conditions for the passivity of nonlinearities can be determined [127,62,81]. However, preventing aliasing is much more difficult. One strategy is to use low-order polynomials to implement nonlinearities. A polynomial of order $n$ expands the bandwidth by only a factor of $n$. Therefore, bandlimiting the nonlinearity input to $f_s/(2n)$ and oversampling by a factor of $n$ will avoid aliasing.

A third reason (beyond time-variation and nonlinearity support) for adopting a sampling rate significantly higher than the audio frequency bandwidth is the necessity of using fractional delays. These are needed whenever a high spatial resolution is needed, or when the waveguide length must be ``continuously'' variable. In all these cases digital interpolators must be used, in the form of FIR or IIR filters, ideally allpass (see next subsection). To some extent, both FIR and IIR filter types introduce amplitude and/or phase distortion at high frequencies, so it is typically beneficial to increase the sampling rate and provide a ``guard band'' between $20$ kHz (for high fidelity audio) and $f_s/2$.


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