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ADAPTIVE PHASE DISTORTION SYNTHESIS
Victor Lazzarini, Joseph Timoney Jussi Pekonen, Vesa Välimäki
Sound and Music Technology Group
National University of Ireland, Maynooth
Ireland
Victor.Lazzarini@nuim.ie
JTimoney@cs.nuim.ie
ABSTRACT
This article discusses Phase Distortion synthesis and its application to arbitrary input signals. The main elements that compose
the technique are presented. Its similarities to Phase Modulation
are discussed and the equivalence between the two techniques is
explored. Two alternative methods of distorting the phase of an
arbitrary signal are presented. The first is based on the audio-rate
modulation of a first-order allpass filter coefficient. The other
method relies on a re-casting of the Phase Modulation equation,
which leads to a heterodyned form of waveshaping. The relationship of these implementations to the original technique is explored in detail. Complementing the article, a number of examples are discussed, demonstrating the application of the technique as an interesting digital audio effect.
1. INTRODUCTION
Phase Distortion (PD) [1] is a synthesis technique based on the
table lookup of a sinusoidal function using a non-linear mapping
of a modulo counter (also called a phasor). It was first introduced
in the Casio CZ-series of synthesisers [2], where it was used to
emulate a typical subtractive synthesis signal flow composed of
source-filter controls. This was actually a clever way of disguising what otherwise might have been a less intuitive method of
synthesis. In fact, as we will see, PD is effectively a subset of
Phase Modulation (PM) synthesis, the usual implementation
method of Frequency Modulation (FM) [3] in hardware synthesisers [4] such as the Yamaha DX-series [5]. FM was admitedly
a non-intuitive synthesis method for musicians, although it was
very powerful computationally.
PD has not been extensively explored in the signal processing literature. However, it remains possibly an interesting
method for many applications, including the design of Virtual
Analogue (VA) oscillators. In the present work, we will try to
explore its possibilities for adaptive signal processing, in the vein
of Adaptive FM (AdFM) [6] and Adaptive SpSB [7]. This paper
is organised as follows. We will first sketch out the basic elements of PD and its theoretical foundations. This will be followed by two proposed methods of phase distortion of arbitrary
signals, employing audio-rate coefficient-modulated allpass filters [8][9] and using a heterodyne arrangement [10]. The paper
concludes with a discussion of a number of examples using different inputs and parameters.
Dept. of Signal Processing and Acoustics
TKK Helsinki University of Technology
Espoo, Finland
Jussi.Pekonen@tkk.fi
Vesa.Valimaki@tkk.fi
2. PHASE DISTORTION SYNTHESIS
The principles of PD synthesis as described in its original formulation [1] can be formalised as follows. A sinusoidal function
with modulo phase φ(t) defined as
))(2cos()( ttx πφ−= (1)
can produce a complex harmonic spectrum if its phase is shaped
by a non-linear function f(x) as in
)))((2cos()( tftx φπ−
(2)
Depending on the shape of f(x), different spectra can be produced. If f(x) is linear, no distortion is effected and we have a
pure sinusoidal tone. Thus, PD can be seen as form of phase-
shaping, in analogy to the non-linear amplitude distortion
method of waveshaping [11]. In its original implementation, PD
synthesis used piecewise linear functions of two or more segments to implement the non-linear mapping. Fig. 1 shows one
such function, plotted against a linear phase increment. The resulting distorted waveshape is depicted in Fig. 2.
Figure 1. Phase distortion function (continuous line),
plotted against linear phase (dots), with d = 0.1.
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+
φ
x
Figure 2. Phase distortion waveform (continuous line)
plotted against an inverted cosine wave (dots).
However, we can demonstrate that PD is no more than a dis-
guised way of implementing PM, albeit only a subset of it. We
can decompose the phaseshaping function f(x) into a linear phase
and a modulation term:
)()( xxgxf
= (3)
Now eq. 2 becomes the more familiar PM expression:
π−= (4)
))])(()([2cos()( tgttx φ+
So what is the phase modulation function g(x)? This can be
found by subtracting the linear phase term from the phase distortion function. The piecewise linear function of fig.1 is defined
as:
x
1
⎧
⎪
⎪
d
2
xgx
=+
)(
⎨
1
⎪
1[
⎪
2
⎩
,
dx
−
)(
+
d
−
)1(
dx
<
dx
>=
],
x
<≤
1 0h wit
(5)
where d is the point at which the two pieces of the function join
together. We can then extract g(x) using eq. 3, which yields:
⎧
1
x
⎞
⎛
−
d
⎟
⎜
⎪
⎪
2
d
⎠
⎝
)(
=
xg
⎨
1
⎞
⎛
⎪
⎪
⎩
[
−
d
⎟
⎜
2
⎠
⎝
,
<
dx
)1(
−
x
],
)1(
−
d
>=
dx
1 0h wit
<≤
(6)
i.e. a sawtooth wave inflected at d with an amplitude of (0.5 – d).
PD is then characterised as a form of phase-synchronous
complex PM. The ratio between carrier and modulator fundamental frequency is always integral (in this particular case 1) and
the modulation index, controlling spectral energy is 2π(0.5 – d),
reaching a maximum of
π
in the limit of d → 0 (100% sawtooth).
Figure 3. PD spectrum, using sawtooth-shape distortion
of fig.1 (d = 0.1).
Moving the point d (to the left) not only increases the modu-
lation index, but also increases the number of significant components in the modulating wave. At the limit, this type of Complex
PM can be expanded as [12]:
∞
1
)([2cos()(
tts
∞
1
π
+=
)(2sin(
t
πφ
=
∑
k
k
∞
∑
2
n
=
1
n
∞
⎛
⎜
∏
⎜
n
⎝
1
π
J
k
n
=
n
1
1
++−=
∑
414
n
=
1
n
)))(2sin(
tn
φπ
⎞
⎟
⎟
⎠
πφ
+
As it can be seen, there is scope for producing a very wideband
spectrum with the technique, possibly with aliasing issues at high
fundamentals. However, in a straightforward implementation of
PD, d typically will not be too small. The resulting spectrum for
d = 0.1 is plotted on fig.3, up to the fiftieth harmonic (this corresponds to the waveform and distortion functions of figs.2 and 1,
respectively). Of course, when implementing PD in terms of PM,
we can use a sawtooth wave liberally, and raise the modulation
index above the π threshold, if needed.
Other distortion shapes can be used. For instance, a function
with two inflections, instead of one, can produce spectra without
even harmonics, similar to a square wave. This is equivalent to
implementing PM with a carrier to modulator fundamental frequency ratio of 2, since the two inflections are equivalent to
halving the period of the modulating function. Other distortion
shapes can also be produced by concatenating simpler functions.
Typically, PD can be implemented by using a simple flowcontrol logic to shape the phase increment of a sinusoidal table
lookup oscillator. It is also possible to implement the non-linear
phase function by using a table lookup, in manner similar to
waveshaping. However, possibly the most efficient method is to
use the PM equivalence discussed above. In order to generate
time-varying spectra, we can, depending on the implementation
method, change the position of the inflection point or points, in-
]))(2sin(
tn
φπφπ
(7)
∞
])1)[(2(sin)2(...
nkt
n
∑∑
=
1
n
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(
()(
()()(
)
terpolate between a linearly and non-linearly shaped phase or
vary the modulation index.
In any case, all of these methods require the use of a precalculated lookup table, which will hold the signal whose phase
is distorted (generally a single wave period). In order to apply
phase distortion to arbitrary input signals, in analogous fashion to
AdFM and related techniques, we will need to find alternative
methods of implementation.
3. GENERAL-PURPOSE PHASE DISTORTION
METHODS
We will now investigate two methods for distorting the phase of
arbitrary signals. The first of these employs a first-order allpass
filter, which is a well-known method for imparting small delays
and phase corrections to signals. The second takes advantage of
the PD-PM equivalence demonstrated above and a rearrangement of the PM equation. Both methods will be shown to be
comparable to the original PD formulation, whilst allowing formore general-purpose applications.
3.1. Allpass filter-based phase distortion
A first-order allpass with the correct properties for a phase distortion application can be implemented by any one of the following expressions [9]:
)]1()([)1()( −−−−= nynxanxny
(8)
or
() () ( )
⎧
⎨
() () ( )
⎩
or
() () ()
⎧
⎨
() () ( )
⎩
where a is the filter coefficient and its transfer function is
()
zH
=
1
−
For our present purpose, we will be using the implementation
of eq.8, as this will be shown to provide better results when the
coefficient a is time-varying. The output phase delay for this allpass filter can be defined as a function of an input frequency
[13]:
⎛
−
1
()
ωωφ
+−=
We can vary the coefficient a at an audio-range rate with a
modulating function m(t). This will yield a time-varying phase
shift, which can now be put in terms of two variables (frequency
and time):
⎜
tan2
⎜
⎝
−+=
1
nawnxnw
(9)
−+−=
1
nwnawny
+=
naynxnw
(10)
−+−=
1nwnaxny
1
−
za
+−
(11)
1
−
az
⎞
()
ω
sin
a
−
a
−
cos1
⎟
(12)
⎟
()
ω
⎠
ω
as
⎛
() ( )
tm
−
−
1
()
One of the notable aspects of eq.13 is that it shows that the
relationship between the time-varying phase and the modulation
function is non-linear. In addition, it has been shown that the
modulating signal needs to be placed in the range of zero to one
to avoid dispersive effects associated with allpass filters, as well
as to keep the filter stable [8].
As we are interested in phase distortion, we need to find a
suitable modulation function for the allpass filter coefficient that
will induce desired phase deviations in its input. To determine an
expression for the modulation m(t) from the phase devia-
φ(ω
,t) it is possible to rearrange eq.11 and use a simplifying
tion
approximation for the tan(x) function (for small x) [14]:
Experiments have indicated that the approximation in eq.14
does not lead to significant differences in the output signal. This
will now provide an expression for the time-varying modulation
function given a time-varying phase shift
Using eq.15, it is then possible to emulate the Phase Distortion technique by modulating an allpass filter coefficient. To apply the modulation function g(x) defined in eq.6 to eq.15 some
preliminary processing must be first carried out. The modulation
function, when applied in eq.4, lies between 0 and 2π(0.5 − d),
while the phase deviation for the allpass filter lies between −ω
and −π. We will shift and scale it to bring it to the appropriate
range:
Figure 4 shows the coefficient modulation function obtained
using eq.16, with d = 0.1. The figure also shows that there are a
number of significant dips in the modulation waveform. These
correspond to the points of maximum phase distortion. In [9] an
expression was used to smooth these dips as they caused undesired spikes in the output signal. However, by using a different
filter implementation, namely that of eq.8, we are able to avoid
these spikes almost completely. In fig.5, we plot the PD waveform generated from a sinusoidal input to coefficient-modulated
allpass filters implemented using eqs.9 and 10. The latter is identified as the one used in [8] and [9].
A plot of the output of the allpass filter PD implementation
using eq. 8 is given on fig.6, which shows a close approximation
of the original PD waveform (shown in a dashed line). It will not
match it exactly because the instantaneous frequency of the allpass filter output must be dynamic unlike that of PD signal, as
explained in [9]. As it can be clearly noted, the spikes of fig.5 are
completely removed.
ωωφ
t (13)
+−=
,
) ()()
⎛
tan
⎜
⎜
2
⎝
()
=
tm (15)
()
ωφ
= d
t 21
,
⎜
tan2,
⎜
() ( )
tm
−
⎝
+ tt
⎞
⎟
⎟
⎠
() ( )()()
()
−
,
=
2
ωωφ
,+−+−
t
t
ωπ
−−
dtg
21
π
d
21
⎞
ω
sin
⎟
⎟
ω
cos1
⎠
ωωφωωφ
+
(14)
)
ωωωφω
cos,sin2
()
(16)
ωπ
−−−
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φφπ
−
=
Figure 4. Coefficient modulation function for PD.
Figure 5. Plots of PD waveforms corresponding to filters
implemented with eqs. 9 (solid line) and 10 (dashed
line).
The actual shape of the phase distorted waveform at the output of the allpass filter depends on the phase of the input signal.
In fig.6, in order to match the shape of a PD waveform based on
a cosine wave, we adjust the input phase by (0.5 – d)2π. The
main reason for the difference between the original PD and allpass PD appears to be related to transient effects related to the
use of time-varying coefficients, which are not accounted in the
fixed-coefficient transfer function of eq.11. Moreover, these effects clearly depend on the implementation used (eqs. 8, 9 or 10).
In fig.7, a spectral plot of the smoother allpass PD waveform
from fig.6 is compared to the one generated by the original PD
method with the same parameters (d = 0.1). It can be seen that
the output of the allpass filter is quite close to the original PD
method up to harmonic 11. However, it has a richer highfrequency content in contrast to the spectrum of the original PD
waveform which is much sparser in the higher frequencies, with
some missing components, as previously noted in [1]. In the following section, in place of the cosine input we will be employing
input signals with richer spectra, such as instrumental tones of
various instruments, as will be discussed later in this article.
Figure 6. Allpass PD implemented according to eq.8
(solid line), plotted against the original PD waveform
(dashed line).
Figure 7. Spectral plot of allpass PD (star markers) and
original PD (dot markers).
3.2. Heterodyne phase distortion
A second alternative method for general-purpose phase distortion
is provided by a heterodyne formula for PM or FM, as already
noted in [7] and [11]. Starting with the PD-equivalent PM formula of eq.4, we can use the relevant trigonometric identity to
re-cast it into a sum of ring-modulated signals, as in:
))])(()([2cos()(
tgttx
+
))((2sin())(2sin(
tgt
=
−
The advantage of this formulation is that we can separate the
carrier signal cleanly from the modulator, thus allowing us to
substitute the former by any arbitrary input. PD becomes a heterodyned form of waveshaping (using sinusoidal transfer functions). In order to achieve the correct result, it is essential that
both carrier and modulator in the two terms have the correct
φππφ
(17)
))((2cos())(2cos(
tgt
φππφ
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Figure 8. Heterodyne PD signal.
Figure 9. General-purpose PD using the heterodyne
method.
phase offsets (relative to each other). With that in place, we can
reproduce the original PD waveforms faithfully. Fig. 8
output of eq. 17 using the same parameters employed to produce
eq. 2.
Now, turning our attention to substituting the carrier signal
with an arbitrary input, we will only need to preserve the 90 degree offset between the carriers in the two terms of eq. 17. As
these actually make up an analytic signal, we can perform a Hilbert Transform [15] in an input signal to place it in quadrature. A
flowchart for the complete algorithm is provided in fig.9.
It is clear from the present discussion that the heterodyne method
presents a faithful realisation of the original PD algorithm,
whereas the allpass implementation of the previous section does
not. This also indicates that the latter method will produce a
spectrally richer output than the former for the same parameters.
Thus both techniques offer advantages and disadvantages, as
well as different timbral characteristics that can be harnessed for
the designed digital audio effects.
plots the
Figure 10. Bassoon C2 tone, steady-state spectrum.
4. ADAPTIVE PHASE DISTORTION
The methods present in section 3 above are the central element in
our technique of Adaptive Phase Distortion. In addition to them,
we will control the modulation rate by tracking the pitch of the
input signal. This ensures that PD is applied correctly according
to the principles outlined above. The present technique uses the
principles already outlined in other forms of adaptive distortion
synthesis [6][7][10]. We will presently discuss some applications
of Adaptive PD to musical signals.
4.1. Allpass filtering method
The allpass PD method can be used to enrich the spectra of instrumental sounds. We will examine three different inputs and
the effect of this technique on them. Fig. 10 shows the steadystate spectrum of a bassoon C2 tone, which is dominated by a
strong formant around 800 Hz. Applying PD to this input, using
a distorting function shaped as in fig.1 (sawtooth-like), with the
inflection point at d = 0.1, we obtain the spectra plotted in fig.11.
Upon close examination, it is clear to see that the method
adds several low-energy partials to the mid-high spectral range.
The overall shape of the spectrum is preserved, but the secondary
peak seen in the original tone at around 2 kHz is now blended
into the overall spectral decay. The perceptual result is that of a
sharper, raspier, tone, as a result of the added harmonics.
The effect can be used to generate dynamic spectra, by timevarying the amount of modulation from 0 to the maximum. This
is shown on fig.12, which shows the spectrogram of a flute C4
tone processed by allpass PD. Here we vary the modulation
amount, from 0 at the start to the maximum at 1 sec. It can be
seen that as the distortion increases, extra partials are added at
the higher end of the spectrum. Further modulating this parameter can generate interesting sweeping effects.
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Figure 11. Allpass phase-distorted Bassoon tone.
Figure 12. Allpass PD flute tone, with varying modula-
tion amount (amplitudes in dB).
Finally, we can use a more pronounced phase distortion for
an increased effect. We can do this by moving the inflection position d in the phase distortion function. The next example demonstrates this for d = 0.1 and d = 0.05. The first plot in fig.13
shows the steady-state spectrum of the original sound, a clarinet
C3 tone. In fig.14, we plot the result of allpass PD with d = 0.1,
which only imparts small changes to the spectrum. The most significant of these are in the increase of lower-order even harmonic
energy.
To make more substantial modifications to the spectrum, we
set d = 0.05, which as discussed previously, will not only increase the effective modulation index, but also add energy to
higher harmonics of the modulating function. This result is plotted on fig.15. A side-effect of this is the increased possibility of
noticeable aliasing, which, in this particular case, is not significant.
Figure 13. Clarinet C3 tone, steady-state spectrum.
Figure 14. Allpass PD of clarinet tone, with moderate
distortion.
4.2. Heterodyne method
As discussed above, the heterodyne PD method reproduces the
original technique more faithfully. As a result, it will impart less
distortion if compared to the allpass implementation with similar
parameters.
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Figure 15. Allpass PD of clarinet tone, with increased
distortion.
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Proc. of the 12th Int. Conference on Digital Audio Effects (DAFx-09), Como, Italy, September 1-4, 2009
Figure 16. Spectrum of heterodyne PD of bassoon tone,
d = 0.1.
Figure 17. Spectrum of heterodyne PD of bassoon tone
with a higher equivalent modulation index.
Fig.16 illustrates this point, where we plot the resulting spec-
trum of heterodyne PD applied to the same bassoon tone of
fig.10, using d = 0.1. By comparing both plots, it can be seen
that not too much change has been effected by the technique.
There are two ways of improving the effect to make it more
noticeable. The first one was already discussed in the previous
section, namely, moving the inflection point d in the distortion
function to the left (i.e. decreasing it). The other is basically to
take advantage of the fact that we are actually implementing PM,
so the limit on the distortion amount, ie. the modulation index, is
removed. The next example, shown in fig.17 demonstrates the
result of multiplying the modulation function by a factor of 5 (so
that the equivalent modulation index is now 4π). In this particular case, not only extra partials are added to the spectrum, but the
secondary formant region in the original sound is enhanced.
Figure 18. Double-inflection PD function.
Figure 19. Steady-state PD spectrum using a flute tone as
input (single-inflection).
Complementing this discussion, we would like to examine
the use of a different distortion function. In this case, we will
select a function such as shown in fig.18. This generates, in the
original method with a sinusoidal input, a signal with odd harmonics only. The effect it has on a complex input such as flute
tone is shown on figs.19 and 20. The former shows the resulting
spectrum PD using the single-inflection distortion function (d =
0.2) and while the latter uses the double-inflection one (d1 = 0.1
and d2 = 0.6).
The richer spectrum of fig.20 can be explained by the fact
that the double inflection distortion function is equivalent to single inflection modulation at twice the frequency. Hence the inflection points d1 and d2 are relatively smaller (i.e. towards the
left of the function), resulting in more distortion. In addition, the
spectrum is also different because of the use of an equivalent c:m
ratio of 2. In fact, as an extension of both of the general-purpose
PD techniques discussed here, we can possibly set this ratio to
other values, something that is not directly possible with the
original method as described in [1].
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H. Massey, A. Noyes and D. Shklair. A Synthesist’s Guide
[2]
to Acoustic Instruments. Amsco publications, New York,
USA, 1987.
J. Chowning, “The synthesis of complex audio spectra by
[3]
means of frequency modulations”, Journal of the Audio En-
gineering Society, vol. 21, no. 7, Sept. 1973, pp. 525-534.
R. Pinkston, “FM synthesis in Csound”. The Csound Book,
[4]
R. Boulanger (ed.), MIT Press, Cambridge, Mass, 2000.
J. Chowning, “Method of synthesizing a sound”, United
[5]
States Patent 4018121, 1977.
V. Lazzarini, J. Timoney and T. Lysaght. “The Generation
[6]
of Natural-Synthetic Spectra by Means of Adaptive Frequency Modulation”, Computer Music Journal, 32(2), 2008,
pp. 12-22.
V. Lazzarini, J. Timoney and T. Lysaght, “Non-linear dis-
[7]
tortion synthesis using the split sideband method with appli-
Figure 20. Steady-state PD spectrum of flute-tone (dou-
ble-inflection).
5. CONCLUSION
In this article we have explored the technique of PD synthesis,
including two alternative implementations for it. We have shown
the equivalence of PD and PM, and discussed the specific characteristics of the original technique. The two novel implementations were shown to be general-purpose and with possible applications in adaptive digital audio effects. We have then presented
several examples of PD as applied to arbitrary input signals, discussing the qualities of the resulting tones. It is our belief that the
technique of Adaptive PD is a useful addition to its sister methods of AdFM and Adaptive SpSB.
6. ACKNOWLEDGMENTS
We would like to acknowledge the support of An Foras Feasa.,
who partially funded the research leading to this article. The
work of J. Pekonen and V. Välimäki has been financed by the
Academy of Finland (projects no. 122815 and no. 126310).
7. REFERENCES
[1] M. Ishibashi, “Electronic musical instrument”, United States
Patent 4658691, 1985.
cations to adaptive signal processing”, Journal of the Audio
Engineering Society, vol. 56, no. 9, Sept. 2008.
J. Pekonen, “Coefficient Modulated first-order allpass filter
[8]
as a distortion effect,” Proceedings of the 11th Conference
on Digital Audio Effects (DAFx), Espoo, Finland, 2008, pp.
893-87.
J. Timoney, V. Lazzarini, J. Pekonen and V. Välimäki,
[9]
“Spectrally rich phase distortion sound synthesis using an
allpass filter”, Proceedings of IEEE ICASSP 2009, Taipei,
Taiwan, 2009, pp. 293-296.
V. Lazzarini, J. Timoney and T. Lysaght, “Asymmetric-
[10]
spectra methods for adaptive FM synthesis”, Proceedings of
the 11th Conference on Digital Audio Effects (DAFx),
Espoo, Finland, 2008.
M. Le Brun, “Digital Waveshaping Synthesis”, Journal of
[11]
the Audio Engineering Society, 27(4), 1979, pp. 250-266.
M. Le Brun, “A Derivation of the Spectrum of FM with a
[12]
Complex Modulating Wave”, Computer Music Journal,
1(4), 1977, pp. 51-52.
T. I. Laakso, V. Välimäki, M. Karjalainen, and U. Laine,
[13]
”Splitting the unit delay—Tools for fractional delay filter
design,” IEEE Signal Processing Magazine, vol. 13, no. 1,
pp. 30–60, Jan. 1996.
K. Steiglitz, A Digital Signal Processing Primer, Prentice
[14]
Hall, USA, Jan. 1996.
Wardle, S. “A Hilbert Transform Frequency Shifter for Au-
[15]
dio”. Proceedings of the International Conference on Digi-
tal Audio Effects, Barcelona, Spain, pp. 25-29, 1999.
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