Digital Waveguide Resonator

Another attractive second-order filter structure, based on the ``digital waveguide oscillator'' [1], is the digital waveguide resonator (DWR):

$${x^\prime}(n) = g x(n)$$ (29)

$$v(n) = {c^\prime}[{x^\prime}(n) + y(n)]$$ (30)

$$x(n+1) = v(n) - y(n) + {\tilde b}_1 u(n)$$ (31)

$$y(n+1) = {x^\prime}(n) + v(n) + b_2 u(n)$$ (32)

where⁴

$$g = r_1^2$$ (33)

$${\tilde b}_1 = b_1 \sqrt{\frac{1-{c^\prime}}{1+{c^\prime}}}$$ (34)

$${c^\prime} = \sqrt{\frac{1}{1 + \frac{\tan^2(\theta)(1+g)^2+(1-g)^2}{4g}}}$$ (35)

$$\approx 1 - \frac{\tan^2(\theta)(1+g)^2 + (1-g)^2}{8g}.$$ (36)

Note that ${c^\prime}=\cos(\theta_1)$ when $g=1$ (undamped case). Like the MCF, the DWR produces steady sinusoidal oscillations indefinitely, even for short word lengths. This happens because the resonance tuning is controlled by only one coefficient, when the damping goes to zero. As would be the case for any oscillator structure which is controlled by one coefficient affecting tuning only, quantization of that coefficient can only affect tuning as well. Unlike the MCF and complex multiply, which require four multiplies per sample, the DWR requires only two multiplies per sample. Moreover, when the decay is set to $\tau=\infty$ ($r_1=1$), one of the multiplies in the DWR disappears, leaving only one multiply per sample for sinusoidal oscillation. As a result, the DWR appears to be best suited for VLSI implementation. As an added bonus, the $x$ and $y$ outputs of the DWR are in exact phase quadrature, like the complex-multiply case considered first above. Note, however, that the choice of input (to either the $x(n)$ or $y(n)$ state variables) results different amplitude scaling.

Figure 1 shows an overlay of initial impulse responses for the three resonators discussed above. The decay factor was set to $r_1=0.99$, and the output of each multiplication was quantized to 16 bits, as were all coefficients. The three waveforms sound and look identical. (There are small differences, however, which can be seen by plotting the differences of pairs of waveforms.)

$\parbox{3.2in}{\caption{{\it Overlay of... ...oupled form (MCF), and (3) second-order digital wave\-guide resonator (DWR).}}}$

Figure 2 shows the same impulse-response overlay but with $r_1=1$ and only 4 significant bits in the coefficients and signals. The complex multiply oscillator can be seen to decay toward zero due to coefficient quantization ( $x_1^2+y_1^2<1$). The MCF and DWR remain steady at their initial amplitude. All three suffer some amount of tuning perturbation.

$\parbox{3.2in}{\caption{{\it Overlay of... ...r_1=1$\ and quantization of coefficients and signals to 4 significant bits.} }}$