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.C.40, damping is provided by the coefficient , which we will take to be a constant

When , the oscillator decays exponentially to zero from any initial conditions. The two delay elements constitute the state of the resonator. Let us denote by the output of the delay element on the left in Fig.C.40 and let be the delay-element output on the right. In general, an output signal may be formed as any linear combination of the state variables:

Similarly, input signals may be summed into the state variables scaled by arbitrary gain factors .

The foregoing modifications to the digital waveguide oscillator result in the so-called digital waveguide resonator (DWR) [304]:

 (C.127) (C.128) (C.129) (C.130) (C.131)

where, as derived in the next section, the coefficients are given by
 (C.132) (C.133) (C.134) (C.135)

where denotes one desired pole (the other being at ). Note that when (undamped case). The DWR requires only two multiplies per sample. As seen earlier, when the decay time is set to (), one of the multiplies disappears, leaving only one multiply per sample for sinusoidal oscillation.

Figure C.41 shows an overlay of initial impulse responses for the three resonators discussed above. The decay factor was set to , 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.)

Figure C.42 shows the same impulse-response overlay but with 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 ( ). The MCF and DWR remain steady at their initial amplitude. All three suffer some amount of tuning perturbation.

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Application to FM Synthesis