The Recursive Allpass as a Resonance Filter

In [Stilson96], Stilson and Smith explain the Moog four-pole lowpass VCF (Voltage Controlled Filter) and explore methods of creating a digital version of the filter. The digital transformations they employ attempt to render a digital filter faithful to Moog’s analog filter design. A distinct feature of Moog’s VCF is a resonant peak at the lowpass cutoff frequency. This peak arises because of the phase response of a cascade of four matched first-order lowpasses enclosed in a negativefeedback loop. Bilinearly transforming the first-order lowpasses will match the phase profile that creates the resonant peak, but creates a delay-free loop, making the filter unrealizable. Figure 1 shows the frequency responses of the bilinearly transformed Moog VCF, which serve as the digital ideal. This paper attempts to implement the resonant peak of the Moog VCF using an allpass filter instead of a lowpass cascade. Using allpasses as embedded subfilters is nothing new. For example, reference [Smith82] embeds an allpass filter to create dynamic spectral notches. This design scheme takes the bilinearly transformed lowpass filters (tuned by the single parameter a) outside of the loop and cascades them with an allpass filter (denoted by A(z)) in a negative-feedback loop (with feedback gain k) for the resonance: