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Physical Audio Signal Processing
    The Digital Waveguide Oscillator
       Digital Waveguide Resonator

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Digital Waveguide Resonator

Converting a second-order oscillator into a second-order filter requires merely introducing damping and defining the input and output signals. In Fig.J.3, damping is provided by the coefficient $ G(n)$, which we will take to be a constant

$\displaystyle G(n)\equiv\message{CHANGE eqv TO equiv IN SOURCE}g. $

When $ g<1$, the oscillator decays exponentially to zero from any initial conditions. The two delay elements constitute the state of the resonator. Let us denote by $ x_1(n)$ the output of the delay element on the left in Fig.J.3 and let $ x_2(n)$ be the delay-element output on the right. In general, an output signal $ y(n)$ may be formed as any linear combination of the state variables:

$\displaystyle y(n) = c_1 x_1(n) + c_2 x_2(n) $

Similarly, input signals $ u(n)$ may be summed into the state variables $ x_i(n)$ scaled by arbitrary gain factors $ b_i$.

The foregoing modifications to the digital waveguide oscillator result in the so-called digital waveguide resonator (DWR) [308]:

$\displaystyle \tilde{x}_1(n)$ $\displaystyle =$ $\displaystyle g x_1(n)\protect$ (J.1)
$\displaystyle v(n)$ $\displaystyle =$ $\displaystyle c[\tilde{x}_1(n) + x_2(n)]$ (J.2)