## Input Impedance Calculations

• The waveguide input impedance'' 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)

where is the Heaviside unit step function.

• 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)

where A(x) is the spherical-wave surface area at x.

• Using the bilinear transform, an equivalent discrete-time filter is given by
 (28)

where is the bilinear transform constant and a1 is equal to the allpass truncation filter coefficient given in Eq. (25).

• The impulse response h(x0,n) of a closed-open truncated cone is found from the digital waveguide structure of Fig. 7 as the pressure response at x=x0 to the impulse response h+(x0,n).

• Because the cone is excited at x=x0, output scale factors are appropriately weighted by x0.

• Figure 8 is a plot of h(x0,n), calculated using a Levine and Schwinger model of the open-end radiation and neglecting viscothermal losses.

• 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 Px0/PL.

• 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.