The Second-Order Waveguide Filter

The first step is to make a second-order digital filter with zero damping by abutting two unit-sample sections of waveguide medium, and terminating on the left and right with perfect reflections, as shown in Fig.J.1. The wave impedance in section is given by , where is air density, is the cross-sectional area of tube section , and is sound speed. The reflection coefficient is determined by the impedance discontinuity via . It turns out that to obtain sinusoidal oscillation, one of the terminations must provide an inverting reflection while the other is non-inverting.

Figure J.1: The second-order, lossless, digital waveguide oscillator, built using two acoustic tube sections.

At the junction between sections and , the signal is partially transmitted and partially reflected such that energy is conserved, i.e., we have lossless scattering. The formula for the