LAC 2015 - Paper: Sound Synthesis with Periodic Linear Time-Varying Filters

2015·0.57 MB·English

by  Antonio Goulart, Marcelo Queiroz, Joseph Timoney, Victor Lazzarini

Sound Synthesis with Periodic Linear Time-Varying Filters
Antonio GOULART Joseph TIMONEY
Marcelo QUEIROZ Victor LAZZARINI
Computer Science Department Sound and Digital Music Technology Group
Sonology Research Center National University of Ireland - Maynooth
University of S˜ao Paulo Maynooth - Co. Kildare - Ireland
Sa˜o Paulo - Brazil {joseph.timoney, victor.lazzarini}@nuim.ie
{ag, mqz}@ime.usp.br
Abstract Linear Time-Varying (PLTV) systems. With
these new approaches we operate by modulat-
In this survey we explore implementations of re-
cently proposed distortion synthesis techniques, ing the coefficients of a filter, and it was shown
namely the Feedback Amplitude Modulation (1st [Pekonen, 2008] that it results in a kind of
and 2nd order cases) and Allpass filter coefficient dynamic version of phase distortion [Ishibashi,
modulation. ThesetechniquesarebasedonPeriodic 1987]. A thorough study of time-varying sys-
Linear Time-Varying systems, in which we operate
tems applied to these techniques [Cherniakov,
by finding a suitable modulation function to obtain
2003] [Timoney et al., 2014] discloses the ap-
adesiredspectrum. Inordertoillustratethissurvey
propriate tools for properly understanding and
and encourage exploration of these new techniques
using these new synthesis processes. These sys-
we present examples in the Csound language.
tems are intended to operate differently than
Keywords audio effects such as time-varying delay lines
usedinflanging[Zolzer, 2011]andallpassfilters
allpass filtercoefficientmodulation, feedback ampli-
tude modulation, periodic linear time-varying sys- used to obtain variable fractional delays [Peko-
tems, distortion synthesis nen et al., 2010]. Additionally, they do not re-
place another type of variation within musical
1 Introduction systemswhereusersmanuallychange(e.g. with
Knowledge of a good set of synthesis techniques a knob) parameter values to achieve a sonic ef-
is a must for electronic / electroacoustic / com- fect, such as mentioned in [Wishnick, 2014].
puter musicians, sound designers and synthe- Our motivation for presenting this survey
siser builders. Audio effects plugins developers comes from the fact that most audio program-
also benefit from dealing with synthesis tech- mers’ work is supported by classic Linear Time-
niques that can also be used in an audio effect Invariant(LTI)systemstheory[Oppenheimand
context, suchastheonesthatwillbepresented. Schafer, 1975], but Linear Time Varying (LTV)
In a previous survey, classic and recent dis- systems theory is less well covered in the liter-
tortion synthesis techniques along with their ature [Huang and Aggarwal, 1982]. Also, time-
implementations were presented at the LAC varyingtoolswereindeedconsideredalongtime
[Lazzarini, 2009]. Waveshaping [LeBrun, 1979], ago[Layzer,1971][Risset,1969]foraudioappli-
phaseshaping [Ishibashi, 1987], and summation cationswithinourcontext,butonlyrecentlyare
formulae [Moorer, 1976], some of the classic ap- being tackled in a more comprehensive fashion.
proaches to distortion synthesis, were analysed A good understanding of LTV theory can bring
mathematically and demonstrated in Csound usnewkindsofsynthesis/effectstechniquesand
code. More recent or unusual approaches like differentwaysforimplementingestablishedones
asymmetrical FM [Palamin et al., 1988], Phase (which can bring nice variations or reduce com-
Aligned Formants [Puckette, 1995], and Modi- putational costs).
fiedFMsynthesis[LazzariniandTimoney,2010] We will dedicate each of the next sections to
were also exposed. Other, more common dis- one technique and then conclude the text. The
tortion techniques, are also widely presented in aim of this paper is not to delve into the the-
computer music textbooks [Dodge and Jerse, ory of LTV systems; our intention, instead, is
1997] [Moore, 1990] [Puckette, 2007]. to present another look at some techniques and
In this paper, we address some recent tech- also provide code1 for their implementation.
niques which extend the distortion synthesis
family, namely techniques based on Periodic 1http://www.ime.usp.br/~ag/dl/lac15-code.zip
2 Allpass coefficient modulation In order to get an approximation of a sawtooth
from a cosine, we must read the rising part of
An allpass filter contains poles and zeros at re-
thecosineinashortertime, andtakemoretime
ciprocal distances from the origin, so their ef-
to read the decaying portion. The function we
fects on all frequencies are balanced [Moore,
must add to the phase generator, in this case,
1990]. A typical 1st order invariant allpass
is drawn on the upper panel of Figure 1 and is
would have a fixed coefficient, but a time-
given by [Lazzarini et al., 2009b]
varying one has the transfer function
(cid:40)
−m(t)+z−1 (1 −d)x, x < d
H(z,t) = , (1) g(x) = 2 d (6)
1−m(t)z−1 (1 −d)1−x, x ≥ d,
2 1−d
with m(t) as the modulating function which where d is how long it takes to read from the
drives the coefficient. Using this filter, a time- start of the cosine table up to its maximum,
varying phase distortion given by [Timoney et and the smaller it is the more abrupt the rising
al., 2009] [Laakso et al., 1996] ramp will be and the more distortion we will
obtain.
If we want to implement g(x) using a mod-
(cid:18) (cid:19)
−m(t)sin(ω)
φ(ω,t) = −ω+2tan−1 (2) ulated allpass filter we will get better results if
1−m(t)cos(ω)
φ(ω,t) goes from −ω to −π [Lazzarini et al.,
2009b], so we make
is introduced in the signal. If we know how
we want the phase to be distorted – in other
words, if we know φ(ω,t) – we can use the ap- g(t)((1−2d)π−ω)
φ(ω,t) = −(1−2d)π−ω.
proximation tan(x) ≈ x [Timoney et al., 2009] (1−2d)π
and determine the modulating signal as (7)
The resulting modulation function is shown on
the lower panel of Figure 1. On Figures 2
−(φ(ω,t)+ω)
m(t) = . (3) and 3 we can see the waveforms and spectra
2sin(ω)−(φ(ω,t)+ω)cos(ω)
of the sawtooths generated with both the origi-
nal technique and the modulated allpass based
The allpass filtering described can be imple-
one. Notice that the missing components of the
mented with the difference equation [Lazzarini
classic phase distortion technique are actually
et al., 2009b]
present in the spectrum of the modulated all-
pass output. We can also see and hear that
the latter spectrum is richer. The code for im-
y(n) = x(n−1)−m(n)(x(n)−y(n−1)). (4)
plementing this example is given in Listing 1,
The condition |m(n)| < 1,∀n, assures stability where we can see two instruments; the first one
[Cherniakov, 2003] and a DC offset works as a synthesis instrument, distorting the
sinusoid and producing richer spectra; the sec-
1−m(n) ond one applies the same technique as an au-
DC(n) = (5)
1+m(n) dioeffect,distortingthesoundofapre-recorded
flute (any sound file can be used, or even a mi-
is introduced in the signal [Pekonen, 2008].
crophone input).
2.1 Classic phase distortion emulation
Listing 1: Classic phase distortion emulation as
In [Pekonen, 2008] [Lazzarini et al., 2009b] [Ti-
synthesis (instrument 1) and technique used as
money et al., 2014], we find as an example for
an audio effect (instrument 2)
thistechniquetheemulationoftheclassicphase
distortion [Ishibashi, 1987]. The phase distor- 1 <CsoundSynthesizer>
2
tion algorithm consists of reading a cosine table
3 <CsOptions>
in an unusual way. Instead of getting indexes 4 −o dac
fromaregularphasegenerator(whichgoesfrom 5 </CsOptions>
0 to 1 in a period related to the chosen funda- 6
7 <CsInstruments>
mental frequency) multiplied by the table size,
8
we add another function to the phase genera-
9 0dbfs=1
tor values and then read the cosine wavetable. 10
11 /∗ audio−rate coeff allpass ∗/
12 opcode Allpass ,a,aa
13 adel init 0
14 setksmps 1
15 audio ,acoef xin
16 aw = audio + acoef∗adel
17 ay = −aw∗acoef + adel
18 adel = aw
19 xout ay
20 endop
21
22
23 /∗ PD function ∗/
24 /∗ inflection point ∗/
25 gip = 0.1
26
27 /∗ ftgen producing line segments ∗/
Figure 1: Phase distortion function (up- 28 ipdfun ftgen 1,0,16384,7,0,16384∗gip
per panel) and coefficient modulation function ,1,16384∗(1−gip),0
(lower panel). Source: [Timoney et al., 2014] 29
30 /∗
31 instr 1:
32 PD synthesis
33 using sine wave input
34 ∗/
35
36 instr 1
37 ifr = p5
38 iamp = p4
39
40 /∗ regular phase generator ∗/
41 aph phasor ifr
42 /∗ phase distortion signal ∗/
43 apd tablei aph, 1, 1, 0, 1
44
45 /∗ scaling of pd signal (eq.7)∗/
46 iw = 2∗$M PI∗ifr/sr /∗omega value∗/
47 ia = (1 − 2∗gip)∗$M PI
48 apd = apd∗ia
Figure 2: Waveforms generated with the classic
49 apd = apd∗(ia − iw)/ia − ia − iw
technique(solidline)andthemodulatedallpass
50
(dashed line). Source: [Timoney et al., 2014] 51 /∗ coefficient modulation function
52 obtained from phase distortion
53 signal (equation 3) ∗/
54 /∗ envelope for modulation ∗/
55 kmod linseg 0,1,1,p3−2,1,1,0
56 amod = −kmod∗(apd + iw)/(2∗sin(iw)
− (apd+iw)∗cos(iw))
57
58 /∗ sine input ∗/
59 asin tablei aph,−1,1,0,1
60
61 /∗ allpass ∗/
62 asig Allpass asin ,amod
63
64 /∗ envelope ∗/
65 aout linenr asig ∗ iamp, 0.01, 0.1,
0.01
66
67 outs aout, aout
68 endin
Figure3: Spectraobtainedwiththeclassictech-
69
nique(upperpanel)andwithmodulatedallpass
70 /∗
(lower panel). Source: [Timoney et al., 2014] 71 instr 2:
72 PD adaptive synthesis
73 using a monophonic instr input
74 ∗/ volved the summing of partials in order to find
75 a function to distort the phase of an input sinu-
76 instr 2
soid and thus get a more musically interesting
77 iamp = p4
timbre. The expression we used to create our
78 /∗ input signal ∗/
79 afl diskin2 ”flutec3.wav”,1,0,1 example is given by
80
81 /∗ pitch tracking ∗/
82 kfr ,kamp ptrack afl ,2048 (cid:16) π(cid:17)
y(n) = 0.4cos(f )+0.4cos 2f −
83 kfr port kfr ,0.01 0 0 3
84 (cid:18) (cid:19)
(cid:16) π(cid:17) 4π
85 /∗ master phase signal ∗/ +0.35cos 3f + +0.3cos 4f +
0 0
86 aph phasor kfr 7 3
87 /∗ phase distortion signal ∗/ (8)
88 apd tablei aph,1,1,0,1
89
InordertoshifttheoutputofEquation8tothe
90 /∗ scaling of pd signal ∗/
appropriate range we use the scaling
91 kw = 2∗$M PI∗kfr/sr
92 ia = (1 − 2∗gip)∗$M PI
93 π(y(n)+1)
y (n) = − . (9)
94 apd = apd∗ia s 2 2
95 apd = apd∗(ia − kw)/ia − ia − kw
Listing 2 presents Csound code for implement-
96
97 /∗ envelope for modulation ∗/ ing this new example. The upper panel of
98 kmod linseg 0,1,1,p3−2,1,1,0 Figure 4 shows the distortion function for this
99 example, while the lower panel shows the all-
100 amod = −kmod∗(apd + kw)/(2∗sin(kw)
pass modulation function resulting after apply-
− (apd+kw)∗cos(kw))
ing Equation 3. Figure 5 shows the waveform
101
102 asig Allpass afl ,amod and spectrum obtained. Albeit this was a naive
103 aout linenr asig ∗ iamp, 0.01, 0.1, example, it demonstrates in an intuitive way
0.01
how a phase function could be easily generated
104
using additive synthesis and then used to drive
105 outs aout, aout
106 endin theallpassfiltercoefficientafterapplyingEqua-
107 tions9andthen3, highlightinganothermusical
108 </CsInstruments> possibility for using the allpass filter.
109
110
111 <CsScore>
112
113 /∗ uncomment lines to
114 run instruments ∗/
115 i 1 0 10 0.5 440
116 ;i 2 0 10 0.5
117
118
119 </CsScore>
120
121
122 </CsoundSynthesizer>
2.2 Finding a distortion function
Now we present a new example with the allpass
coefficient modulation technique to bring more
Figure 4: Phase distortion (upper panel) and
insight. In this example, instead of choosing a
resultant modulation (lower panel) functions.
desired waveform and then finding out how to
modulate the allpass coefficient, we choose an
arbitrary signal to distort the phase. Keeping Listing 2: Implementation of an arbitrary func-
in mind that we should generate a signal within tion as a phase distorter.
the appropriate range, that is [-1,1], any signal
1 <CsoundSynthesizer>
a priori can be considered for the process.
2
So we embarked on an experiment that in- 3 <CsOptions>
45
46 outs aout, aout
47 endin
48
49 </CsInstruments>
50
51 <CsScore>
52 i 1 0 10 0.25 440
53 </CsScore>
54
55 </CsoundSynthesizer>
3 Feedback amplitude modulation
The Feedback Amplitude Modulation was men-
tioned by Layzer (1971) and described and im-
plemented by Risset (1969) in his catalogue ex-
Figure 5: Waveform and spectrum obtained
ample #510, but a rigorous mathematical anal-
with arbitrary function
ysis of the technique was lacking. Since 2009
there was a renewed research interest in ex-
ploitingitsmusicalpossibilities[Lazzarinietal.,
4 −o dac
2009a], [Kleimola et al., 2011], [Lazzarini et al.,
5 </CsOptions>
2011].
6
7 <CsInstruments> First of all we will review its 1st order case.
8 0dbfs=1 The basic idea is to modulate the amplitude of
9 an oscillator using its previous output, as in
10 /∗ audio−rate coeff allpass ∗/
11 opcode Allpass ,a,aa
12 adel init 0 y(n) = cos(ω0n)[1+y(n−1)], (10)
13 setksmps 1
with ω = 2πf and the initial condition y(n) =
14 audio ,acoef xin 0 0
0 for n ≤ 0.
15 aw = audio + acoef∗adel
16 ay = −aw∗acoef + adel Afirstanalysis[Kleimolaetal., 2011]ismade
17 adel = aw expanding Equation 10 as
18 xout ay
19 endop
20
21 /∗ PD function ∗/ y(n) = cos(ω0n)+
22 ipdfun ftgen 1, 0, 16384, 9, 1, 0.4,
cos(ω n)cos(ω [n−1])+
0 0
−90, 2, 0.4, −150, 3, 0.35,
180/7−90, 4, 0.3, 4∗180/3−90 cos(ω0n)cos(ω0[n−1])cos(ω0[n−2])+
23
... (11)
24 instr 1
25
26 ifr = p5
27 iamp = p4
∞ k
28 (cid:88) (cid:89)
y(n) = cos(ω [n−m]), (12)
29 iw = 2∗$M PI∗ifr/sr ;omega 0
30 k=0m=0
31 aph phasor ifr ;regular phase
32 /∗ phase distortion signal ∗/ showingthattheresultantsignaliscomposedby
33 apd tablei aph, 1, 1, 0, 1 harmonicsofthefundamentalf . Amoreinter-
0
34
esting case is when we can control the amount
35 /∗ coef mod signal (eq.3)∗/
of feedback in the system, so we introduce the
36 apd = −0.5∗$M PI∗(apd+1)/2 ;scaling
37 amod = −1∗(apd + iw)/(2∗sin(iw) − ( feedbackparameterβ andEquation10becomes
apd+iw)∗cos(iw))
38
39 /∗ sine input ∗/ y(n) = cos(ω n)[1+βy(n−1)]. (13)
0
40 asin tablei aph,−1,1,0,1
41 /∗ allpass ∗/
Thefeedbackparametercanbeinterpretedto
42 asig Allpass asin ,amod
be similar to modulation index of conventional
43 /∗ envelope ∗/
44 aout linenr asig∗iamp,0.01,0.1,0.01 FM synthesis, and its influence on FBAM’s
spectral evolution is shown in Figure 6, bor- Listing 3: 1st order FBAM
rowed from [Kleimola et al., 2011].
1 <CsoundSynthesizer>
2
3 <CsOptions>
4 −o dac
5 </CsOptions>
6
7 <CsInstruments>
8
9 0dbfs= 1
10
11 opcode FBAM, a, kkki
12 ; set vector size to 1 sample
13 setksmps 1
14 ay init 0 ; y[0] = 0
15 ka, kf , kb, ifn xin
16 /∗ current sample +
17 weighted previous sample ∗/
18 ay oscili ka + kb∗ay, kf , ifn
19 xout ay
20 endop
21
Figure 6: β influence on FBAM spectrum. 22 instr 1
23 ; amp, freq , beta
Source: [Kleimola et al., 2011]
24 a1 FBAM 0.5, 440, 0.7, 1
25 out a1
The interpretation of the system as a time-
26 endin
varying filter enables a better analysis. The 27
structure for the 1st order case is 28 </CsInstruments>
29
30 <CsScore>
y(n) = x(n)+a(n)y(n−1), (14)
31 f1 0 16384 10 1
with 32 i1 0 5
33 </CsScore>
34
x(n) = a(n) = cos(ω n), (15)
0 35 </CsoundSynthesizer>
so we have a coefficient modulated IIR filter.
The 2nd order FBAM is obtained using two
As in the previous section, the modulation will
previous outputs in the modulation, each with
create the dynamic phase distortion, generating
its own feedback parameter. The system equa-
new partials, and in this case the filter is not
tion is then given by
allpass, so it has a time-varying non-flat magni-
tude response.
Applying the equations for stability analysis
y(n) = cos(ω n)[1+β y(n−1)+β y(n−2)].
0 1 2
by Cherniakov (2003) a condition for stability
(18)
of this system was derived as
Itwasshown[Lazzarinietal.,2011]thatwith
(cid:12) (cid:12) this system we can get a narrower pulse, and
N
(cid:12) (cid:89) (cid:12) thusaricherspectraloutputfortheFBAMsys-
(cid:12)β cos(ω m)(cid:12) < 1. (16)
(cid:12) 0 (cid:12) tem, as shown in Figure 7, borrowed from [Laz-
(cid:12) (cid:12)
m=1
zarini et al., 2011]. Code for implementing the
An expression for the maximum value of beta
2nd order FBAM is presented in Listing 4.
was proposed by Kleimola et al. (2011) as
Listing 4: 2nd order FBAM
1 <CsoundSynthesizer>
β ≈ 1.9986−0.00003532(f −27.5), (17)
max 0 2
3 <CsOptions>
andtheyalsoshowedthatasthevalueofβ isin- 4 −o dac
creased, the presence of significant components 5 </CsOptions>
in the output due to aliasing can be observed 6
7 <CsInstruments>
before the system becomes unstable. Listing 3
8 ksmps = 10
presents code for the 1st order FBAM imple-
9 0dbfs = 1
mentation. 10 nchnls = 2
modulationsignalsasindependentstreams,and
treat the whole system as the combination of
two first order units to reduce the difficulty of
the analysis.
Despitetheinitialresultsaboutthe2ndorder
FBAM that were already reported, we plan to
proceed with more thorough investigations es-
pecially regarding its stability and to determine
anyconsequentrestrictionsitmighthaveonthe
coefficient modulation waveform.
4 Conclusions
In this survey we presented some techniques
for sound synthesis derived from the concept
Figure7: Comparisonbetween1st(indots)and
of phase distortion. Results can be used to
2nd order (solid line) FBAM for 500 Hz funda-
generate approximations of classic sawtooth os-
mental. Source: [Lazzarini et al., 2011]
cillators and more timbrally-involved FM-like
spectra. Implementations in Csound, which
11 are easily translated to other languages, were
12 opcode Fbam2,a,akk presented in order to promote the techniques
13 setksmps 1
among our community and audio tools develop-
14 asigm1 init 0
ers.
15 asigm2 init 0
16 ain ,kb1,kb2 xin The application of time-varying systems is
17 asig = ain∗(1 + kb1∗asigm1 + kb2∗ not new in general signal processing, but only
asigm2)
recently is the theory behind these systems be-
18 asigm2 = asigm1
ing explored for time-varying digital audio filter
19 asigm1 = asig
20 xout asig systems. This paper shows there is a signifi-
21 endop cant potential for a number of novel techniques
22 based on PLTV systems. The field is still open
23 instr 1
to investigation, in particular with regards to
24 kb1 = 0.7
2nd and higher order systems.
25 kb2 = 0.7
26 ; sinusoidal input We hope that this brief survey invites more
27 ain oscili 1, 440, −1, 0.25 musicians and technicians to explore the nice
28 asig Fbam2 ain ,kb1,kb2
retromotivatedsoundsobtainedwiththesesys-
29 asig balance asig , ain
tems,maintainingtheinterestindistortionsyn-
30 outs asig ,asig
31 endin thesis/effectstechniques. Wehopetoencourage
32 the sharing of ideas and principles around these
33 </CsInstruments> techniques, such as, for instance, nice distortion
34
functions that could be used either in synthesis
35 <CsScore>
or acoustic instruments processing.
36 i 1 0 5
37 </CsScore>
38 5 Acknowledgements
39 </CsoundSynthesizer>
The research leading to this paper was partially
supportedbyCAPES(proc. number8868-14-0)
For a deeper analysis we can understand the
and Science Foundation Ireland (ISCA-Brazil).
system as a 2nd order PLTV system by rewrit-
ing the system equation as
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