The impulse response h(t) of a wind-instrument air column, defined as the inverse Fourier transform of its input impedance, implies an acoustic feedback loop because of the inherent closure of the bore entryway after the introduction of a volume velocity unit impulse.
The infinite termination impedance of the closed end results in a non-inverting reflection of pressure, so that the waveguide model of a closed-open cylindrical bore is as shown in Fig. 9.
Digital waveguide model of a closed-open cylindrical bore.
The waveguide ``input impedance'' is found by calculating the system impulse response and transforming it to the frequency domain using the discrete Fourier transform (DFT).
is approximated for low-frequency sound waves by a simple inversion, the impulse response of the feedback system of Fig. 9 will be periodic and of infinite duration.
The Fourier transform of the resulting infinite length periodic impulse train will then simplify to .
In comparison to traditional time-domain bore simulation techniques, the digital waveguide bore implementation offers high efficiency without sacrificing accuracy.
The lumped open-end reflectance filter
is, in general, adequately implemented using a first- or second-order digital filter. Thus, just a single short convolution/filtering operation is necessary in this model.