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In many audio applications, digital all-pass filters are of central importance; a key property of such filters is energy (l<sub>2</sub> norm) preservation. In audio effect and sound synthesis algorithms, it is desirable to have filters that behave as all-passes with time-varying characteristics, but direct generalizations of time-invariant designs can lose the important norm-preserving property; for fast parameter variation, large gain increases are possible. We here call attention to some simple time-varying filter structures, based on wave digital filter designs, that do preserve signal energy and that reduce to simple first- and second-order all-pass filters in the time-invariant case.
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Time-varying Generalizations of Allpass Filters
Stefan Bilbao
Sonic Arts Research Centre, Queen’s University Belfast
United Kingdom, BT7 1NN
Tel: +44(0)2890974465, Fax: +44(0)2890274828, Email: s.bilbao@qub.ac.uk
Abstract— In many audio applications, digital allpass fil-
ters are of central importance; a key property of such fil-
ters is energy (l
2
norm) preservation. In audio effect and
sound synthesis algorithms, it is desirable to have filters
which behave as allpasses with time-varying characteristics,
but direct generalizations of time-invariant designs can lose
the important norm-preserving property; for fast parame-
ter variation, large gain increases are possible. We here
call attention to some simple time-varying filter structures,
based on wave digital filter designs, which do preserve signal
energy, and which reduce to simple first- and second-order
allpass filters in the time-invariant case.
Keywords allpass filters, time-varying filters, wave digital
filters, musical sound synthesis, digital audio effects
EDICS Category: 2-AUEA
I. Introduction
A
LLPASS digital filter designs [1] play a fundamental
role in almost all areas of audio signal processing.
The defining property of such a design is that it possesses
unity gain, at all frequencies. A corollary is that in the
time domain, the squared norm of a sequence is preserved
through the filtering operation; in other words, such a filter
is energy-preserving.
In several important applications, it is necessary to ex-
tend the definition of the allpass filter to the time-varying
coefficient case. Certain audio effects [2] rely on this, as
do many physics-based musical sound synthesis algorithms
(such as digital waveguides as applied to nonlinear strings
[3], [4], [5], and scattering representations of woodwind
toneholes [6], [7]). If the time variation of the filter co-
efficients is slow, then generally it is safe to treat such
a filter as a quasi-static system, though strictly speaking,
frequency domain analysis (and the use of terms such as
“unity gain,” etc.) is not generally meaningful. For faster
time variation (as may be the case for the physical models
mentioned above), such analysis can be misleading; large
gain variations are a possibility.
Scattering-type filter designs, such as digital waveguides
[8], [9] and, in particular, wave digital filters [10], [11] offer
a useful set of design tools for time-varying filters; though
it of course remains impossible to perform any meaningful
frequency-domain analysis, energy-based stability guaran-
tees are still within reach. They have appeared, in partic-
ular, in vocal tract modelling in the wave digital filtering
S. Bilbao is with the Sonic Arts Research Centre, Queen’s
University Belfast, Northern Ireland, United Kingdom, BT7
1NN. Tel: +44(0)2890974465, Fax: +44(0)28 90274828, Email:
s.bilbao@qub.ac.uk.
context [12], [13], and, earlier as lattice filter designs [14].
In this short paper, we point out a simple generalization
of the first- and second-order allpass filters to the time-
varying coefficient case, suitable for audio applications, us-
ing wave digital filters as a starting point. It is worth not-
ing that these algorithms are quite different in character
from algorithms developed by Mourjopolis [15], Zetterberg
and Zhang [16] and alim¨aki et al. [17] for time-varying
filters, for which the emphasis is on transient suppression
and minimizing distortion.
II. First-order Allpass Filters
For a given real input sequence x
n
, indexed by integer n,
a general first-order allpass filter is defined by the recursion
y
n+1
= β (ax
n+1
+ x
n
) ay
n
(1)
where y
n
is the output sequence, again indexed by integer
n. Here, a is a real filter coefficient, assumed constant, and
constrained to be of magnitude less than unity for stability.
β is a parameter taking on the value 1 or -1.
The familiar transfer function for this filter, obtained by
taking z-transforms, is
H(z
1
) =
Y (z
1
)
X(z
1
)
= β
a + z
1
1 + az
1
(2)
which possesses the well-known property
|H(z
1
)| = 1 |Y (z
1
)| = |X(z
1
)| (3)
on the unit circle (i.e., for z = e
jω
). Furthermore, through
Parseval’s relation, we then have that
ky
.
k = kx
.
k (4)
where
kf
.
k =
X
n=−∞
f
2
n
!
1/2
(5)
defines the l
2
norm for square-summable sequences f
n
(the .
refers to summation variable n). In other words, the allpass
filter is norm-preserving. (It is of course also possible to
derive (4) directly from (1) without using frequency domain
concepts.)
A. Time-varying Parameters
The most straightforward approach to extending the all-
pass to the time-varying case is to simply make the coeffi-
cients a variable, as per, e.g., [15]. One way of doing this
is to use the recursion
y
n+1
= β (a
n+1
x
n+1
+ x
n
) a
n+1
y
n
(6)
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A full analysis of this filter is, of course, difficult, but it is
easy enough to show that a simple condition such as
|a
n+1
| < 1 (7)
is not sufficient to ensure energy preservation. Consider
the case where x
n
is simply an impulse, i.e., x
0
= 1 and
takes on the value zero otherwise. Then, we have
y
0
= βa
0
y
1
= β(1 a
0
a
1
)
.
.
.
y
n
= β(1 a
0
a
1
)
n
Y
j=2
(a
j
)
in which case
ky
.
k
2
= (a
0
)
2
+ (1 a
0
a
1
)
2
1 +
X
n=2
n
Y
j=2
a
2
j
Then, from (7), we have that
ky
.
k
2
2
+ (1 +
2
)
2
X
n=0
2n
=
1 + 3
2
1
2
This bound is tight, and can be met with equality for filter
coefficients a
n
= (1)
n
. For this choice of coefficients,
the output energy is greater than the input energy for any
choice of < 1, and becomes unbounded as approaches
1.
Other obvious attempts to generalize the allpass also fail;
for instance, instead of (6), we could use
y
n+1
= β (a
n+1
x
n+1
+ x
n
) a
n
y
n
Such a choice can lead to similar energy growth, under
condition (7). (Considering, for example, a unit impulse
input, and filter coefficients a
0
= 0, a
n
= (1)
n
, for n 1,
we again get unbounded growth of ky
.
k in the limit as
approaches 1.)
B. A Wave Digital One-Port
A full introduction to wave digital filtering principles is
beyond the scope of this paper—we refer the reader to [11]
for an overview and introduction.
Consider the wave digital one-port shown in Figure 1,
which consists of a single two-port adaptor (either series
or parallel) terminated on a delay, with or without sign
inversion (β takes on the value 1 or -1), corresponding to
a wave digital capacitor or inductor, respectively. At the
free port, the input is a sequence x
n
, and the output is
the sequence y
n
. At the terminated port, the input to the
adaptor is the sequence w
n
, and the output is v
n
.
The adaptor, in either the series or parallel case, is de-
fined by values at its two ports, M at the free port, and M
n
at the port connected to the delay element. We assume M
to be a constant, and M
n
to be a time-varying sequence.
y
n
x
n
w
n
v
n
β
M M
n
T
Parallel
Series
or
Fig. 1. Wave digital one-port, corresponding to a generalization of
a first-order allpass section.
Both M and M
n
are constrained to be strictly positive. We
note that we have used the neutral letter M in this case,
to indicate that these port values may be taken as port
resistances (in the case of a series adaptor) or as conduc-
tances (in the case of a parallel adaptor). Instantaneous
scattering at the adaptor can be described by the following
matrix equation:
y
n
v
n
= α
γ
n
p
1 γ
2
n
p
1 γ
2
n
γ
n
x
n
w
n
(8)
where γ
n
, the time-varying reflection coefficient, is defined
by
γ
n
=
M
n
M
M
n
+ M
(9)
which, due to the positivity condition on M and M
n
, must
satisfy
|γ
n
| < 1 (10)
for all n. The parameter α is set to 1 for a series junction
or -1 for a parallel junction. Note that we have assumed
power-normalized scattering here [11]. The reactive one-
port is defined simply by
w
n+1
= βv
n
(11)
The scattering operation, when viewed as a matrix trans-
formation, is easily shown to be orthogonal at each time
step n, and thus we must have
y
2
n
+ v
2
n
= x
2
n
+ w
2
n
(12)
It then follows immediately, summing over n, that
ky
.
k
2
+ kv
.
k
2
= kx
.
k
2
+ kw
.
k
2
(13)
From the definition (11) of the reactive one-port, we also
have that
kv
.
k
2
= kw
.
k
2
(14)
from which it then follows that
ky
.
k
2
= kx
.
k
2
(15)
Thus, this wave digital one-port has the same norm-
preserving property as the allpass filter, but now in the
time-varying case. It is important to note that power-
normalization is crucial here, in that otherwise (using, say,
the more standard voltage wave scattering), the scattering
matrix is not orthogonal. The distinction is identical to
that between normalized and non-normalized lattice digi-
tal filter sections [14].
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It is simple enough to arrive at a recursion corresponding
to this one-port which relates y
n
to x
n
directly:
y
n+1
= αγ
n+1
x
n+1
β
s
1 γ
2
n+1
1 γ
2
n
(αγ
n
y
n
x
n
) (16)
Setting a
n
= βαγ
n
gives, finally,
y
n+1
= β (a
n+1
x
n+1
+ φ
n
x
n
) φ
n
a
n
y
n
(17)
where
φ
n
=
s
1 a
2
n+1
1 a
2
n
(18)
and reduces to (1) when a
n
= a, a constant.
C. A Numerical Example
It is useful to make a comparison of the behavior of the
two designs (6) and (17). Consider a simple time-variation
of the parameter a
n
, which takes on the constant value
0.999 for n 540, and which decreases progressively to a
value of 0.999 at n = 597; it remains at 0.999 thereafter
(see Figure 2, at top). The input is a sinusoid at 1000 Hz,
of amplitude 1; the sample rate is taken to be 44 100 Hz.
The output of the filters defined by (6) and (17) are plot-
ted in the middle and bottom panels, respectively, of Figure
2. Notice that in the case of the time-invariant design, a
large transient offset results; it occurs even though the pa-
rameter undergoes a gradual transition over a number of
samples. It is important to add that for filter coefficients
a
n
varying near 1 or -1, the factor φ
n
used in the wave dig-
ital design (17) can also exhibit strong variation, leading,
in this case, to the ”kink” observed in the output at bot-
tom in Figure 2. Further examination of the effect of the
smoothness of the coefficient transition on output smooth-
ness is necessary, but cannot be discussed in detail in this
short article; neither is there space to discuss the difference
in terms of audibility between the two responses, but we
do reiterate that stability is the goal here, not necessarily
transient suppression.
III. Second-order Allpass Filters
A second-order allpass filter is defined by
y
n+1
+ ay
n
+ by
n1
= α (bx
n+1
+ ax
n
+ x
n1
) (19)
If the coefficients a and b satisfy
|a| 1 < b < 1 (20)
then the filter is stable.
Keeping in mind the discussion in the previous section,
it should be clear that an appropriate generalized structure
for the second-order allpass will have the form of a wave
digital one-port, as shown in Figure 3. Here, we again read
the input and output sequences x
n
and y
n
from the free
port of a three-port adaptor (again, either series or par-
allel). The other two ports of the adaptor are terminated
on a wave digital inductor, and a wave digital capacitor.
100 200 300 400 500 600 700 800 900 1000
−2
0
2
100 200 300 400 500 600 700 800 900 1000
−2
0
2
100 200 300 400 500 600 700 800 900 1000
−2
0
2
n
n
n
y
n
y
n
a
n
Fig. 2. Top, parameter a
n
, in a first-order design, plotted against
n, the time index. Middle, output y
n
of filter defined by (6), and
bottom, of filter defined by (17).
y
n
x
n
w
2,n
v
2,n
w
1,n
v
1,n
1
M
M
1,n
M
2,n
T
T
Parallel
Series
or
Fig. 3. Wave digital one-port, corresponding to a generalization of
a second-order allpass section.
Again, the sequences M
1,n
and M
2,n
, always strictly posi-
tive, are to be interpreted as port resistances (for a series
adaptor) and as port conductances (for a parallel adaptor).
The input and output waves at the two ports are as indi-
cated in the figure. In this case, the scattering equations
can be written as
y
n
v
1,n
v
2,n
= α
I q
n
q
T
n
x
n
w
1,n
w
2,n
(21)
where α = 1 for a series connection, and α = 1 for a
parallel connection, and where the vector q
n
is defined by
q
n
= [
2M,
p
2M
1,n
,
p
2M
2,n
]
T
/
p
M + M
1,n
+ M
2,n
In addition, we have
w
1,n
= v
1,n1
w
2,n
= v
2,n1
(22)
From orthogonality of the scattering operation (21), and
using (22), it is again possible to show that the one-port is
norm-preserving, i.e.,
ky
.
k = kx
.
k
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It is also possible, through tedious algebraic manipulation,
to derive a two-step recursion involving only x
n
and y
n
;
this form is rather complex. In terms of the scattering
parameters
γ
1,n
=
r
M
1,n
M
γ
2,n
=
r
M
2,n
M
(23)
and defining the auxiliary parameters
d
1,n
=
(1 + γ
2
1,n
+ γ
2
2,n
)γ
1,n1
2α(γ
2,n
γ
1,n1
+ γ
2,n1
γ
1,n
)
d
2,n
=
(1 + γ
2
1,n1
+ γ
2
2,n1
)γ
1,n
2(γ
2,n
γ
1,n1
+ γ
2,n1
γ
1,n
)
c
1,n
=
(1 γ
2
1,n
γ
2
2,n
)γ
1,n1
2(γ
2,n
γ
1,n1
+ γ
2,n1
γ
1,n
)
c
2,n
=
(1 γ
2
1,n1
γ
2
2,n1
)γ
1,n
2α(γ
2,n
γ
1,n1
+ γ
2,n1
γ
1,n
)
the recursion may be written as
y
n+1
+
d
2,n+1
+ αd
1,n
γ
2,n
d
1,n+1
y
n
+ α
d
2,n
d
1,n+1
y
n1
=
c
1,n+1
d
1,n+1
x
n+1
c
2,n+1
+ αc
1,n
+ αγ
2,n
d
1,n+1
x
n
α
c
2,n
d
1,n+1
x
n1
If M
1,n
and M
2,n
are constant, it reduces to the form
(19), with
a = 2α
M
1
M
2
M + M
1
+ M
2
b =
M + M
1
+ M
2
M + M
1
+ M
2
a and b, defined as above in terms of the positive quantities
M, M
1
and M
2
automatically satisfy (20), as expected.
IV. Conclusions
We have discussed here some simple generalizations of
digital allpass filters to the time-varying case, in particular
first- and second-order sections. While in many situations
(i.e., when time-variation of parameters is slow), simple de-
signs such as (6) and (19) may be used directly, we have
shown here a means of extending these designs such that
the important energy-preserving property is retained. (We
make no claims, however, about the effects on the output in
terms of transients and other audible distortion.) Though
these designs are wave digital in origin, they may be imple-
mented as recursions just as standard filters are, though,
for simplicity, a wave digital realization may be desirable,
and does not lead to a significant increase in memory or
computational requirements. These designs would appear
to be of general applicability throughout all areas of audio
signal processing.
Acknowledgment
The author would like to thank Jyri Pakarinen, Matti
Karjalainen, Cumhur Erkut and Vesa alim¨aki of the
Acoustics Laboratory at the Helsinki University of Tech-
nology for many useful discussions on this topic.
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Author Information Stefan Bilbao (B.A.,
Harvard, 1992, MSc., Stanford, 1996, PhD.,
Stanford, 2001) was born in Montreal, Canada
in 1969. He is currently a lecturer at the Sonic
Arts Research Centre, and the Department of
Music, at the Queen’s University, Belfast.
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Wave Digital Filters were developed to discretize linear time invariant lumped systems, particularly electronic circuits. The time-invariant assumption is baked into the underlying theory and becomes problematic when simulating audio circuits that are by nature time-varying. We present extensions to WDF theory that incorporate proper numerical schemes, allowing for the accurate simulation of time-varying systems. We present generalized continuous-time models of reactive components that encapsulate the time-varying lossless models presented by Fettweis, the circuit-theoretic time-varying models, as well as traditional LTI models as special cases. Models of time-varying reactive components are valuable tools to have when mod-eling circuits containing variable capacitors or inductors or electrical devices such as condenser microphones. A power metric is derived and the model is discretized using the alpha-transform numerical scheme and parametric wave definition. Case studies of circuits containing time-varying resistance and capacitance are presented and help to validate the proposed generalized continuous-time model and discretization.
... It is well-known that systems with constant parameters can become unstable when those parameters are made time varying. In [8], it is shown that if the first-order allpass filter (1) is made time-varying, ...
... which is greater than the input for any choice of < 1 [8], and thus contradicting the all-pass filter assumption that all frequencies pass with equal (unity) gain. A solution is suggested in [8] in terms of a wave digital one port and a corresponding orthogonal matrix formulation that ensures energy preservation-a solution that is almost equivalent to the power preserving rotation matrix to be discussed in Section 4. ...
... which is greater than the input for any choice of < 1 [8], and thus contradicting the all-pass filter assumption that all frequencies pass with equal (unity) gain. A solution is suggested in [8] in terms of a wave digital one port and a corresponding orthogonal matrix formulation that ensures energy preservation-a solution that is almost equivalent to the power preserving rotation matrix to be discussed in Section 4. ...
Conference Paper
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This work builds upon existing work in which second-order allpass filters are used in a feedback network, with parameters made time-varying to enable effects such as phase distortion in a generative audio system; the term " audio " is used here to distinguish from generative " music " systems, emphasizing the strong coupling between processes governing the production of high-level music and lower-level audio. The previous system was subject to issues of instability that can arise when time-invariant filter parameters are allowed to vary over time. These instabilities are examined herein, along with the adoption of a power-preserving rotation matrix formulation of the allpass filter to ensure stability and ultimately an improved synthesis for a generative audio system.
... Schroeder allpass filters are also "nested" in cascade with delay lines inside of Feedback Delay Networks (FDNs) [5][6][7][8][9][10] or another Schroeder allpass [2,[11][12][13][14][15][16]. Time-varying first-order allpass filters (Schroeder allpasses with length-one delay lines) have also been explored in digital audio effect and synthesizer design [10,[17][18][19][20][21][22][23][24][25]. ...
... Fig. 8(g)/(h) has been studied for its scaling properties [54]. Fig. 8(q) was proposed as an energy-preserving time-varying first-order allpass filter [17] and inspired the current work [31]. ...
... In this model, the length of a DWG is varied over time, using a fractional delay filter to produce a smoothly varying pitch. To account for the non-physical energy variations that are caused by on-line string length adjustment, the authors use a method of energy compensation [7,8]. In [5] the need to address such issues is avoided by introducing a time-varying scattering junction at the slide-string contact point based on a balanced perturbation method proposed in [9], and in addition methods are proposed for modelling slidestring interaction in the horizontal and longitudinal transversal polarisations. ...
... Using (36), (8) and (65), F n o can also be written as: ...
Conference Paper
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Slide-string instruments allow continuous control of pitch by articulation with a slide object whose position of contact with the string is time-varying. This paper presents a method for simulation of such articulation. Taking into account sensing and musical practice considerations, an appropriate physical model configuration is determined, which is then formulated in numerical form using a finite difference approach. The model simulates the attachment and detachment phases of slide articulation which generally involve rattling, while finger damping is modelled in a more phenomenological manner as a regionally induced time-varying damping. A stability bound for the numerical model is provided via energy analysis, which also reveals the articulatory source terms. The approach is exemplified with simulations of slide articulatory gestures that involve glissando, vibrato and finger damping.
... A direct implementation of the time-varying allpass gains may become unstable under strong variation [22], whereas the stability of the time-varying feedback matrix was shown for arbitrary variation [16]. More advanced allpass structures can ensure stability under timevariation [23], but also increases the complexity of the design. Further, allpass filters introduce a frequency dependent delay which decreases the accuracy of the reverberation time specification. ...
... In [97] it is shown that power wave adaptors inherently remain stable with timevarying port resistance values, but [67] claims that they lead to slightly higher computational costs and worse numerical properties. Finally, it is worth noticing that the time-varying generalization of allpass filters presented in [98] is based on power-normalized WDFs. ...
Thesis
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Recent advances in semiconductor technology eventually allowed for affordable and pragmatic implementations of sound processing algorithms based on physical laws, leading to considerable interest towards research in this area and vast amounts of literature being published in the last two decades. As of today, despite the efforts invested by the academic community and the music technology industry, new or better mathematical and computational tools are called for to efficiently cope with a relatively large subset of the investigated problem domain. This is especially true of those analog devices that inherently need to be studied by lumped nonlinear models. This research is, in this sense, directed towards both general techniques and specific problems. The first part of this thesis presents a generalization of the wave digital filter (WDF) theory to enable interconnections among subnetworks using different polarity and sign conventions. It proposes two new non-energic two-port WDF adaptors, as well as an extension to the definitions of absorbed instantaneous and steady-state pseudopower. This technique eventually removes the need to remodel subcircuits exhibiting asymmetrical behavior. Its correctness is also verified in a case study. Furthermore, a novel, general, and non-iterative delay-free loop implementation method for nonlinear filters is presented that preserves their linear response around a chosen operating point and that requires minimal topology modifications and no transformation of nonlinearities. In the second part of this work, five nonlinear analog devices are analyzed in depth, namely the common-cathode triode stage, two guitar distortion circuits, the Buchla lowpass gate, and a generalized version of the Moog ladder filter. For each of them, new real-time simulators are defined that accurately reproduce their behavior in the digital domain. The first three devices are modeled by means of WDFs with a special emphasis on faithful emulation of their distortion characteristics, while the last two are described by novelly-derived systems in Kirchhoff variables with focus on retaining the linear response of the circuits. The entirety of the proposed algorithms is suitable for real-time execution on computers, mobile electronic devices, and embedded DSP systems.
... • suitability for time-varying filters [29], [34]- [36], and • low computational complexity and real-time capability. ...
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