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Use of a bilinear conformal map to achieve a frequency warping nearly identical to that of the Bark frequency scale is described. Because the map takes the unit circle to itself, its form is that of the transfer function of a first-order allpass filter. Since it is a first-order map, it preserves the model order of rational systems, making it a valuable frequency warping technique for use in audio filter design. A closed-form weighted-equation-error method is derived that computes the optimal mapping coefficient as a function of sampling rate, and the solution is shown to be generally indistinguishable from the optimal least-squares solution. The optimal Chebyshev mapping is also found to be essentially identical to the optimal least-squares solution. The expression 0:8517 [arctan(0:06583fs)] 1=2 00:916 is shown to accurately approximate the optimal allpass coefficient as a function of sampling rate fs in kHz for sampling rates greater than 1 kHz. A filter design example is included that illustrates improvements due to carrying out the design over a Bark scale. Corresponding results are also given and compared for approximating the related "equivalent rectangular bandwidth (ERB) scale" of Moore and Glasberg using a first-order allpass transformation. Due to the higher frequency resolution called for by the ERB scale, particularly at low frequencies, the first-order conformal map is less able to follow the desired mapping, and the error is two to three times greater than the Bark-scale case, depending on the sampling rate.

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Basic principles of digital waveguide modeling of musical instruments are presented in a tutorial introduction intended for graduate students in electrical engineering with a solid background in signal processing and acoustics. The vibrating string is taken as the principal illustrative example, but the formulation is unified with that for acoustic tubes. Modeling lossy stiff strings using delay lines and relatively low-order digital filters is described. Various choices of wave variables are discussed, including velocity waves, force waves, and root-power waves. Signal scattering at an impedance discontinuity is derived for an arbitrary number of waveguides intersecting at a junction. Various computational forms are discussed, including the Kelly-Lochbaum, one-multiply, and normalized scattering junctions. A relatively new three-multiply normalized scattering junction is derived using a two-multiply transformer to normalize a one-multiply scattering junction. Conditions for strict passivity of the model are discussed. Use of commutativity of linear, time-invariant elements to greatly reduce computational cost is described. Applications are summarized, and models of the clarinet and bowed-string are described in some detail. The reed-bore and bow-string interactions are modeled as nonlinear scattering junctions attached to the bore/string acoustic waveguide.

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