Download Moog Ladder Filter Generalizations Based on State Variable Filters We propose a new style of continuous-time filter design composed
of a cascade of 2nd-order state variable filters (SVFs) and a global
feedback path. This family of filters is parameterized by the SVF
cutoff frequencies and resonances, as well as the global feedback
amount. For the case of two identical SVFs in cascade and a specific value of the SVF resonance, the proposed design reduces to
the well-known Moog ladder filter. For another resonance value,
it approximates the Octave CAT filter. The resonance parameter
can be used to create new filters as well. We study the pole loci
and transfer functions of the SVF building block and entire filter.
We focus in particular on the effect of the proposed parameterization on important aspects of the filter’s response, including the
passband gain and cutoff frequency error. We also present the first
in-depth study of the Octave CAT filter circuit.
Download Stability Analysis of Time-Varying Virtual Analog Filters Time-varying virtual analog filters used in digital audio effects and synthesizers are often implemented by discretizing continuous-time state-space systems using trapezoidal integration. When filter parameters such as cutoff frequency or resonance vary over time, as is the norm in musical applications, proving BIBO stability of the resulting time-varying discrete-time system becomes nontrivial. In this paper, we review the technique of common quadratic Lyapunov functions (CQLFs) from the control systems literature and show how a continuous-time CQLF is preserved through discretization. This allows us to prove stability of some time-varying virtual analog filters by working in the often simpler continuous-time domain. We apply this framework to several filters of musical interest, providing new proofs of stability for the state variable filter and the Sallen-Key filter, and new bounds on the stable time-varying parameter range for the Moog ladder and diode ladder filters.
Download Time-Varying Filter Stability and State Matrix Products We show a new sufficient criterion for time-varying digital filter stability: that the matrix norm of the product of state matrices over a certain finite number of time steps is bounded by 1. This extends Laroche’s Criterion 1, which only considered one time step, while hinting at extensions to two time steps. Further extending these results, we also show that there is no intrinsic requirement that filter coefficients be frozen over any time scale, and extend to any dimension a helpful theorem that allows us to avoid explicitly performing eigen- or singular value decompositions in studying the matrix norm. We give a number of case studies on filters known to be time-varying stable, that cannot be proven time-varying stable with the original criterion, where the new criterion succeeds.