PRESERVING THE STRUCTURE OF THE MOOG VCF
IN THE DIGITAL DOMAIN
Federico Fontana
Universita' di Verona
Dipartimento di Informatica
15 Strada le Grazie, Verona 37134 - Italy
ABSTRACT
A discrete-time version of the Moog VCF structure is proposed based on the explicit computation of the delay-free
loop contained in that filter. Compared to previous solutions, the model presented here shows accurate frequency
responses when working in linear regimes meanwhile preserving good numerical stability and direct access to the
parameters of cutoff frequency and feedback gain. These
features make the proposed model a candidate for efficient implementation on fixed-point architectures as well
as in normal PC's, in the form of software plug-in. The
structural correspondence with the Moog VCF will allow
future direct transpositions of the analog system in the
discrete-time model.
1. INTRODUCTION
The Moog Voltage-Controlled Filter (VCF) has a recognized place in the history of electronic music [7]. By exploiting a clever interconnection of analog components it
realizes the filtering scheme depicted in Figure 1, in which
we note the existence of a loop containing a series of four
identical filtering stages.
Every stage in this loop has the same lowpass characteristic. In linear regimes this characteristic is expressed
with good approximation by the transfer function
Figure 1. Structure of the Moog VCF.
G(s)
ic
wc + s
After that work, other discrete-time models of the Moog
VCF have been proposed [6]. In the meantime some commercial products have been launched into the market of
virtual analog electronic musical instruments, especially
in the form of software plug-ins. Whatever their quality
and cost, their specification is obviously protected by nondisclosure terms.
The work presented here is mostly built on the research
made by Stilson and Smith. Rather than designing digital filters that preserve the acoustical and control features
of the Moog VCF, in this paper we will derive a discretetime version that preserves the structure of the analog system. In practice, the one-pole analog filters are mapped
into corresponding digital filter blocks. This design strategy can be pursued since a versatile formal tool capable of handling the delay-free loop, whose computation is
known to be impossible using traditional sequential procedures, has been developed in the meantime. We will
show that by means of this tool the digital system obtained
by bilinear transformation of the Moog VCF has a magnitude response that matches that of the analog filter with
good accuracy, once we is preliminary bilinearly antitransformed to compensate for the frequency warping introduced by this transformation. Furthermore we will show
that the resulting model preserves the direct access to the
driving parameters to a good extent. As a consequence, it
can be efficiently implemented in a fixed-point digital signal processing architecture at the cost of performing two
table lookups and few multiply-and-add operations at every temporal step.
Besides these results, the proposed discrete-time model
is numerically robust and prospectively versatile for future
accurate digital transpositions of the Moog VCF. In fact,
the one-by-one correspondence between the one-pole analog and digital filter blocks allows to transpose features of
the analog filters directly into the discrete-time blocks.
In the Moog VCF both the parameter we and the loopback gain k can be dynamically varied. The former sets
the cutoff frequency. The latter ranges in the interval O~4:
by increasingly feeding back the output y(t) to the input
u(t), the system's pure lowpass behavior (3 dB-cutoff at
we with k 0) progressively changes into that of a resonant filter. At the stability limit k = 4 the Moog VCF
oscillates at we rad/s.
An excellent analysis of the Moog VCF, concerning
many acoustically relevant aspects of this system, has been
presented in 1996 by Stilson and Smith [8]. On top of that
analysis, the authors propose some discrete-time models
aiming at preserving the response of the Moog VCF without sacrificing the accessibility to its driving parameters.
Controllability is in fact a crucial property that should be
maintained in a good digital realization.
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