### 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*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:

*digital waveguide resonator*(DWR) [304]:

where, as derived in the next section, the coefficients are given by

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.)

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Application to FM Synthesis