- The waveguide ``input impedance''
*h*(*x*_{0},*n*) for this system can be determined by calculating its response to a unit volume velocity impulse and transforming it to the frequency domain using the DFT. - The continuous-time pressure response to a volume velocity impulse is given by the inverse Fourier transform of the characteristic impedance
- This function is given by

(26) - It is necessary to determine a digital filter that appropriately models the response of
*h*^{+}(*x*,*t*) in the discrete-time domain. - The spherical-wave characteristic impedance is given in terms of its Laplace transform as

(27) *A*(*x*) is the spherical-wave surface area at*x*. - Using the bilinear transform, an equivalent discrete-time filter is given by

(28) *a*_{1}is equal to the allpass truncation filter coefficient given in Eq. (25). - The impulse response
*h*(*x*_{0},*n*) of a closed-open truncated cone is found from the digital waveguide structure of Fig. 7 as the pressure response at*x*=*x*_{0}to the impulse response*h*^{+}(*x*_{0},*n*). - Because the cone is excited at
*x*=*x*_{0}, output scale factors are appropriately weighted by*x*_{0}. - Figure 8 is a plot of
*h*(*x*_{0},*n*), calculated using a Levine and Schwinger model of the open-end radiation and neglecting viscothermal losses.**Figure 8:**Impulse response of conical bore closed at*x*=*x*_{0}and open at*x*=*L*. Open-end radiation is approximated by a 2nd-order discrete-time filter and viscothermal losses are ignored. - Salmon (1946) and Benade (1988) proposed a transmission-line conical waveguide model directly analogous to that described above.
- Shown in Fig. 9, Benade's model represents a conic section by a cylindrical waveguide, a pair of ``inertance'' terms, and a transformer whose ``turns ratio'' is equal to the ratio
*P*_{x0}/*P*_{L}. - The digital waveguide truncated cone model implements the cylindrical waveguide section using delay lines, as previously described, and the transformer operation wherever a physical pressure measurement is taken.

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