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Coefficient Interpolation for the Max Mathews Phasor Filter

Max Mathews described what he named the “phasor filter,” which is a flexible building block for computer music, with many desirable properties. It can be used as an oscillator or a filter, or a hybrid of both. There exist analysis methods to derive synthesis parameters for filter banks based on the phasor filter, for percussive sounds. The phasor filter can be viewed as a complex multiply, or as a rotation and scaling of a 2-element vector, or as a real valued MIMO (multiple-input, multiple-output) 2nd order filter with excellent numeric properties (low noise gain). In addition, it has been proven that the phasor filter is unconditionally stable under time varying parameter modifications, which is not true of many common filter topologies. A disadvantage of the phasor filter is the cost of calculating the coefficients, which requires a sine and cosine in the general case. If pre-calculated coefficients are interpolated using linear interpolation, then the poles follow a trajectory that causes the filter to lose resonance. A method is described to interpolate coefficients using a complex multiplication that preserves the filter resonance.

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