Multiconvolution in waveguides with arbitrarily spaced discontinuities.

Abstract

The multiconvolution algorithm [Martínez etal., J. Acoust. Soc. Am. 84, 1620–1627 (1988)] used to calculate the impulse response or reflection function of a musical instrument air column has proven to be useful, but it has the limitation that the spacing between discontinuities is constrained to be some multiple of c Δt (for phase velocity c and time step Δt). Two approaches have been devised to remove this limitation. Each time-domain approach calculates the response of an air column, modeled as an arbitrary one-dimensional acoustic waveguide constructed using cylindrical or conical bore segments with viscothermal damping and tone-hole discontinuities. The band-limited discrete-time multiconvolution (BDTM) specifies band-limited discrete-time reflection and transmission functions at each discontinuity. The continuous-time interpolated multiconvolution (CTIM) uses continuous-time convolutions between analytical reflection and transmission functions and discrete-time pressure signals. The arbitrary spacing between discontinuities is accounted for in the BDTM method by multirate discrete-time processing of the response functions, and in the CTIM method by interpolation of the discrete-time pressure signals.