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˜aoPaulo Maynooth - Co. Kildare - Ireland S˜aoPaulo - Brazil {joseph.timoney, victor.lazzarini}@nuim.ie {ag, mqz}@ime.usp.br

Abstract Linear Time-Varying (PLTV) systems. With In this survey we explore implementations of re- these new approaches we operate by modulat- 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. These techniques are based on Periodic 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- a desired spectrum. In order to illustrate this survey and encourage exploration of these new techniques propriate tools for properly understanding and we present examples in the Csound language. using these new synthesis processes. These sys- tems are intended to operate differently than Keywords audio effects such as time-varying delay lines allpass filter coefficient modulation, feedback ampli- used in flanging [Zolzer, 2011] and allpass filters 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 systems where users manually change (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) systems theory [Oppenheim and context, such as the ones that will be presented. 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 varying tools were indeed considered a long time [Lazzarini, 2009]. Waveshaping [LeBrun, 1979], ago [Layzer, 1971] [Risset, 1969] for audio appli- phaseshaping [Ishibashi, 1987], and summation cations within our context, but only recently are 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 us new kinds of synthesis/effects techniques and code. More recent or unusual approaches like different ways for implementing established ones asymmetrical FM [Palamin et al., 1988], Phase (which can bring nice variations or reduce com- Aligned Formants [Puckette, 1995], and Modi- putational costs). fied FM synthesis [Lazzarini and Timoney, 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 1 family, namely techniques based on Periodic http://www.ime.usp.br/~ag/dl/lac15-code.zip 2 Allpass coefficient modulation In order to get an approximation of a sawtooth An allpass filter contains poles and zeros at re- from a cosine, we must read the rising part of ciprocal distances from the origin, so their ef- the cosine in a shorter time, and take more time fects on all frequencies are balanced [Moore, to read the decaying portion. The function we 1990]. A typical 1st order invariant allpass must add to the phase generator, in this case, would have a fixed coefficient, but a time- is drawn on the upper panel of Figure 1 and is varying one has the transfer function given by [Lazzarini et al., 2009b] ( −m(t) + z−1 ( 1 − d) x , x < d g(x) = 2 d (6) H(z, t) = −1 , (1) 1 1−x 1 − m(t)z ( 2 − d) 1−d , x ≥ 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-  −m(t) sin (ω)  φ(ω, t) = −ω+2 tan−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 2 sin (ω) − (φ(ω, t) + ω) cos (ω) of the sawtooths generated with both the origi- The allpass filtering described can be imple- nal technique and the modulated allpass based mented with the difference equation [Lazzarini one. Notice that the missing components of the et al., 2009b] classic phase distortion technique are actually present in the spectrum of the modulated all- pass output. We can also see and hear that y(n) = x(n − 1) − m(n)(x(n) − y(n − 1)). (4) the latter spectrum is richer. The code for im- 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) dio effect, distorting the sound of a pre-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) this technique the emulation of the classic phase distortion [Ishibashi, 1987]. The phase distor- 1 2 tion algorithm consists of reading a cosine table 3 in an unusual way. Instead of getting indexes 4 −o dac from a regular phase generator (which goes from 5 0 to 1 in a period related to the chosen funda- 6 7 mental frequency) multiplied by the table size, 8 we add another function to the phase genera- 9 0 dbfs=1 tor values and then read the cosine wavetable. 10 11 /∗ audio−rate coeff allpass ∗/ 12 opcode Allpass , a , aa 13 adel i n i t 0 14 setksmps 1 15 audio , a c o e f xin 16 aw = audio + a c o e f ∗ adel 17 ay = −aw∗ a c o e f + adel 18 adel = aw 19 xout ay 20 endop 21 22 23 /∗ PD f u n c t i o n ∗/ 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 i n s t r 1 : 32 PD s y n t h e s i s 33 using s i n e wave input 34 ∗/ 35 36 i n s t r 1 37 i f r = p5 38 iamp = p4 39 40 /∗ regular phase generator ∗/ 41 aph phasor i f r 42 /∗ phase distortion signal ∗/ 43 apd tablei aph, 1, 1, 0, 1 44 45 /∗ scaling of pd signal (eq.7) ∗/ 46 iw = 2∗$M PI∗ i f r / s r /∗omega value ∗/ 47 i a = (1 − 2∗ gip ) ∗$M PI 48 apd = apd∗ i a Figure 2: Waveforms generated with the classic 49 apd = apd ∗( i a − iw ) / i a − i a − iw technique (solid line) and the modulated allpass 50 (dashed line). Source: [Timoney et al., 2014] 51 /∗ coefficient modulation function 52 obtained from phase distortion 53 s i g n a l ( equation 3) ∗/ 54 /∗ envelope for modulation ∗/ 55 kmod l i n s e g 0 , 1 , 1 , p3 −2 ,1 ,1 ,0 56 amod = −kmod∗(apd + iw)/(2∗ s i n ( iw ) − ( apd+iw ) ∗ cos ( iw ) ) 57 58 /∗ s i n e input ∗/ 59 a s i n t a b l e i aph , −1 ,1 ,0 ,1 60 61 /∗ a l l p a s s ∗/ 62 asig Allpass asin,amod 63 64 /∗ envelope ∗/ 65 aout l i n e n r a s i g ∗ iamp, 0.01, 0.1, 0.01 66 67 outs aout , aout 68 endin Figure 3: Spectra obtained with the classic tech- 69 nique (upper panel) and with modulated allpass 70 /∗ (lower panel). Source: [Timoney et al., 2014] 71 i n s t r 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 i n s t r 2 soid and thus get a more musically interesting 77 iamp = p4 78 /∗ input signal ∗/ timbre. The expression we used to create our 79 afl diskin2 ”flutec3.wav”,1,0,1 example is given by 80 81 /∗ pitch tracking ∗/ 82 kfr,kamp ptrack afl,2048  π  y(n) = 0.4 cos (f ) + 0.4 cos 2f − 83 k f r port kfr , 0 . 0 1 0 0 3 84   85 /∗ master phase signal ∗/  π  4π + 0.35 cos 3f0 + + 0.3 cos 4f0 + 86 aph phasor k f r 7 3 87 /∗ phase distortion signal ∗/ (8) 88 apd tablei aph,1,1,0,1 89 90 /∗ scaling of pd signal ∗/ In order to shift the output of Equation 8 to the 91 kw = 2∗$M PI∗ k f r / s r appropriate range we use the scaling 92 i a = (1 − 2∗ gip ) ∗$M PI 93 π (y(n) + 1) y (n) = − . (9) 94 apd = apd∗ i a s 2 2 95 apd = apd ∗( i a − kw) / i a − i a − kw 96 Listing 2 presents Csound code for implement- 97 /∗ envelope for modulation ∗/ ing this new example. The upper panel of 98 kmod l i n s e g 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∗ s i n (kw) − ( apd+kw) ∗ cos (kw) ) pass modulation function resulting after apply- 101 ing Equation 3. Figure 5 shows the waveform 102 a s i g A l l p a s s a f l , amod and spectrum obtained. Albeit this was a naive 103 aout l i n e n r a s i g ∗ iamp, 0.01, 0.1, example, it demonstrates in an intuitive way 0.01 how a phase function could be easily generated 104 105 outs aout , aout using additive synthesis and then used to drive 106 endin the allpass filter coefficient after applying Equa- 107 tions 9 and then 3, highlighting another musical 108 possibility for using the allpass filter. 109 110 111 112 113 /∗ uncomment lines to 114 run instruments ∗/ 115 i 1 0 10 0 . 5 440 116 ; i 2 0 10 0 . 5 117 118 119 120 121 122

2.2 Finding a distortion function Now we present a new example with the allpass coefficient modulation technique to bring more insight. In this example, instead of choosing a Figure 4: Phase distortion (upper panel) and desired waveform and then finding out how to resultant modulation (lower panel) functions. 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 a priori can be considered for the process. 2 So we embarked on an experiment that in- 3 45 46 outs aout , aout 47 endin 48 49 50 51 52 i 1 0 10 0.25 440 53 54 55

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- ploiting its musical possibilities [Lazzarini et al., 4 −o dac 5 2009a], [Kleimola et al., 2011], [Lazzarini et al., 6 2011]. 7 First of all we will review its 1st order case. 8 0 dbfs=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 i n i t 0 y(n) = cos (ω0n)[1 + y(n − 1)], (10) 13 setksmps 1 14 audio , a c o e f xin with ω0 = 2πf0 and the initial condition y(n) = 15 aw = audio + a c o e f ∗ adel 0 for n ≤ 0. 16 ay = −aw∗ a c o e f + adel A first analysis [Kleimola et al., 2011] is made 17 adel = aw expanding Equation 10 as 18 xout ay 19 endop 20 21 /∗ PD f u n c t i o n ∗/ y(n) = cos (ω0n)+ 22 ipdfun ftgen 1, 0, 16384, 9, 1, 0.4, cos (ω n) cos (ω [n − 1])+ −90, 2 , 0 . 4 , −150, 3, 0.35, 0 0 180/7 −90 , 4 , 0 . 3 , 4∗180/3−90 cos (ω0n) cos (ω0[n − 1]) cos (ω0[n − 2])+ 23 ... (11) 24 i n s t r 1 25 26 i f r = p5 27 iamp = p4 ∞ k 28 X Y 29 iw = 2∗$M PI∗ ifr/sr ;omega y(n) = cos (ω0[n − m]), (12) 30 k=0 m=0 31 aph phasor ifr ;regular phase 32 /∗ phase distortion signal ∗/ showing that the resultant signal is composed by 33 apd tablei aph, 1, 1, 0, 1 harmonics of the fundamental f0. A more inter- 34 esting case is when we can control the amount 35 /∗ coef mod signal (eq.3) ∗/ 36 apd = −0.5∗$M PI ∗(apd+1)/2 ; scaling of feedback in the system, so we introduce the 37 amod = −1∗(apd + iw)/(2∗ s i n ( iw ) − ( feedback parameter β and Equation 10 becomes apd+iw ) ∗ cos ( iw ) ) 38 39 /∗ s i n e input ∗/ y(n) = cos (ω0n)[1 + βy(n − 1)]. (13) 40 a s i n t a b l e i aph , −1 ,1 ,0 ,1 41 /∗ a l l p a s s ∗/ 42 asig Allpass asin,amod The feedback parameter can be interpreted to 43 /∗ envelope ∗/ be similar to modulation index of conventional 44 aout l i n e n r a s i g ∗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 2 3 4 −o dac 5 6 7 8 9 0 dbfs= 1 10 11 opcode FBAM, a , kkki 12 ; set vector size to 1 sample 13 setksmps 1 14 ay i n i t 0 ; y [ 0 ] = 0 15 ka , kf , kb , i f n xin 16 /∗ current sample + 17 weighted previous sample ∗/ 18 ay o s c i l i ka + kb∗ay , kf , i f n 19 xout ay 20 endop 21 Figure 6: β influence on FBAM spectrum. 22 i n s t r 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 29 30 y(n) = x(n) + a(n)y(n − 1), (14) 31 f 1 0 16384 10 1 with 32 i 1 0 5 33 34 x(n) = a(n) = cos (ω0n), (15) 35 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 (ω0n)[1 + β1y(n − 1) + β2y(n − 2)]. by Cherniakov (2003) a condition for stability (18) of this system was derived as It was shown [Lazzarini et al., 2011] that with N this system we can get a narrower pulse, and Y β cos (ω m) < 1. (16) thus a richer spectral output for the FBAM sys- 0 tem, as shown in Figure 7, borrowed from [Laz- 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 βmax ≈ 1.9986 − 0.00003532(f0 − 27.5), (17) 2 3 and they also showed that as the value of β is in- 4 −o dac creased, the presence of significant components 5 in the output due to aliasing can be observed 6 7 before the system becomes unstable. Listing 3 8 ksmps = 10 presents code for the 1st order FBAM imple- 9 0 dbfs = 1 mentation. 10 nchnls = 2 modulation signals as independent streams, and treat the whole system as the combination of two first order units to reduce the difficulty of the analysis. Despite the initial results about the 2nd order FBAM that were already reported, we plan to proceed with more thorough investigations es- pecially regarding its stability and to determine any consequent restrictions it might have on the coefficient modulation waveform. 4 Conclusions In this survey we presented some techniques Figure 7: Comparison between 1st (in dots) and for sound synthesis derived from the concept 2nd order (solid line) FBAM for 500 Hz funda- of phase distortion. Results can be used to mental. Source: [Lazzarini et al., 2011] generate approximations of classic sawtooth os- 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 14 asigm1 i n i t 0 among our community and audio tools develop- 15 asigm2 i n i t 0 ers. 16 ain , kb1 , kb2 xin The application of time-varying systems is 17 a s i g = 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 19 asigm1 = a s i g ing explored for time-varying digital audio filter 20 xout a s i g 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 i n s t r 1 24 kb1 = 0 . 7 to investigation, in particular with regards to 25 kb2 = 0 . 7 2nd and higher order systems. 26 ; s i n u s o i d a l input We hope that this brief survey invites more 27 ain o s c i l i 1 , 440 , −1, 0.25 musicians and technicians to explore the nice 28 asig Fbam2 ain,kb1,kb2 retro motivated sounds obtained with these sys- 29 asig balance asig, ain 30 outs asig , a s i g tems, maintaining the interest in distortion syn- 31 endin thesis/effects techniques. We hope to encourage 32 the sharing of ideas and principles around these 33 techniques, such as, for instance, nice distortion 34 35 functions that could be used either in synthesis 36 i 1 0 5 or acoustic instruments processing. 37 38 5 Acknowledgements 39 The research leading to this paper was partially supported by CAPES (proc. number 8868-14-0) 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 References Mickhail Cherniakov. 2003. An introduction to parametric digital filters and oscillators. y(n) = x(n)+β a (n)y(n−1)+β a (n)y(n−2), 1 1 2 2 John Wiley & Sons Ltd, England. (19) with x(n) = a1(n) = a2(n) = cos (ω0n). This is Charles Dodge and Thomas Jerse. 1997. the simplest example but the analysis equations Computer Music: Synthesis, composition and are complicated for this second order system. performance. 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