A first-order IIR filter that, when cascaded with a delay line,
simulates wave propagation in a lossy resonator of length , can
take the form

At the Nyquist frequency, and for , such filter gains and we have to use to have the correct attenuation at high frequency. Fig. 5 shows the magnitude and phase delay (in seconds) obtained with the first-order filter for three values of its parameter .

By comparison of the curves of Fig. 4 with the responses of Fig. 5, we see how the latter can be used to represent the losses in a section of one-dimensional waveguide section. Therefore, the simulation scheme turns out to be that of Fig. 6. Of course, better approximations of the curves of Fig. 4 can be obtained by increasing the filter order or, at least, by controlling the zero position of a first-order filter. However, the form (50) is particularly attractive because its low-frequency behavior is controlled by the single parameter .

As far as the wave impedance is concerned, in the discrete-time
domain, it can be represented by a digital filter obtained
from (49) by bilinear transformation, which leads to

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