A new kind of locked circuit: the Quasi-Periodic Locked ...

A new kind of locked circuit: the Quasi-Periodic Locked LoopA new kind of locked circuit: the Quasi-Periodic Locked Loop(Q-PLL)(Q-PLL)This paper was downloaded from TechRxiv (https://www.techrxiv.org).

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16-12-2020 / 18-12-2020

CITATION

Gonzalez, Diego Luis; Grassi, Lorenzo; Maurizi, Alberto (2020): A new kind of locked circuit: the Quasi-Periodic Locked Loop (Q-PLL). TechRxiv. Preprint. https://doi.org/10.36227/techrxiv.13385591.v1

DOI

10.36227/techrxiv.13385591.v1

https://www.techrxiv.org

https://dx.doi.org/10.36227/techrxiv.13385591.v1

A new kind of locked circuit: the Quasi-Periodic Locked Loop (Q-PLL)

Journal: IEEE Transactions on Circuits and Systems I: Regular Papers

Manuscript ID TCAS-I-01682-2020

Manuscript Type: Regular Paper

Date Submitted by the Author: 15-Dec-2020

Complete List of Authors: Gonzalez, Diego; Istituto per la Microelettronica e Microsistemi Consiglio Nazionale delle Ricerche Unità di BolognaGrassi, Lorenzo; ASPER S.r.lMaurizi, Alberto; Istituto per la Microelettronica e Microsistemi Consiglio Nazionale delle Ricerche Unità di Bologna,

EDICS:

ACS400 - Circuits and systems for biomedical applications, life science and biology < ACS - Analog and Mixed Mode Circuits and Systems, COMM150A5 - Multicarrier communication < COMM150 - Multichannel and multicarrier systems < COMM - Circuits and Systems for Communications, ACS280 - Phase locked loops < ACS - Analog and Mixed Mode Circuits and Systems, ACS300 - Frequency synthesizers < ACS - Analog and Mixed Mode Circuits and Systems, CTRL150A0 - Control of nonlinear systems < CTRL150 - Applications of control < CTRL - Control Theory topics in Circuits and Systems, NOLIN100 - Analysis/modeling/simulation of nonlinear networks < NOLIN - Nonlinear Circuits and Systems, NOLIN170C5 - Syncronization of chaotic systems < NOLIN170 - Chaotic circuits and systems < NOLIN - Nonlinear Circuits and Systems

IEEE Transactions on Circuits and Systems I: Regular Papers

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS – I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. XX, NO. XX, MONTH YYYY 1

A new kind of locked circuit: the Quasi-PeriodicLocked Loop (Q-PLL)

Diego Luis Gonzalez, Lorenzo Grassi, and Alberto Maurizi

Abstract—A new nonlinear circuit with frequency locking ca-pability in the case of a generic quasi-periodic input, is presented.Due to this capability the circuit is called a Quasi-Periodic LockedLoop (Q-PLL). The locked frequency is parametrically selectedfrom among those prescribed by the theory of resonances indynamical systems. In particular, the locked frequency formsa three-frequency resonance with the frequencies of the quasi-periodic input. The circuit is able to lock also in case ofdeterministic perturbation (harmonics of the input frequencies)and stochastic perturbation (wide-band noise). The circuit isclosely related to the pitch perception of complex sound inhumans and, as such, can be considered a bio-inspired technology.From the point of view of applications, it may be considered asan extension of the Phase Locked Loop (PLL) with the additionalability of locking simultaneously to more than one frequency. Dueto the dynamical and structural robustness of the locked states,the Q-PLL represents a tangible advance for the development ofspecific applications, for example, in medicine (hearing aids, andcochlear implants), in robotics (artificial senses), and in industrialand consumer electronics (improvement of speech intelligibility,pitch-based processing, etc.).

Index Terms—Quasi-periodic signals, Frequency locking, Pitchperception

I. INTRODUCTION

PHASE-Locked Loop (PLL) circuits are widely used inelectronics. PLL circuits can be used as frequency mul-

tipliers or dividers, tracking generators, or clock recoverycircuits. Their main technological applications derive from acapacity to recover very weak periodic signals, which wouldotherwise be lost in background noise, and the detection and/orsynthesis of frequency and phase-controlled signals. Notwith-standing their ubiquitous presence in modern electronics, PLLshave a basic limitation in their locking capability: they can lockonly to one periodic signal at a time. PLLs fail to lock whenthe input is perturbed by additional deterministic frequenciesof competing amplitudes. However, theoretically, it is possibleto have a system locked simultaneously to two or moreperiodic signals which are usually mutually incommensurate.

In fact, the PLL in its locked regime should be consideredas a synchronized oscillator [1]–[4]. Usually the workingregime is with 1/1 synchronisation, in which forcing andresponse frequencies are the same. However, it is not unusual,depending on the kind of PLL and its operating regime, to have

Diego Luis Gonzalez and Alberto Maurizi are with the Istituto di Microelet-tronica e Microsistemi of the Consiglio Nazionale delle Ricerche, Bologna,Italy and Dipartimento di Scienze Statistiche “Paolo Fortunati” Universita diBologna, Italy. Emails: [email protected] and [email protected].

Lorenzo Grassi is with ASPER S.r.l., email: [email protected] author: Alberto Maurizi, CNR-IMM, via Gobetti 101, 40129

Bologna, Italy; email: [email protected].

locked states corresponding to other synchronised conditionssuch as, for example, 1/n, where the loop is locked to asub-harmonic of the forcing signal. The locked responses areindeed resonances of a system with two frequencies (two-frequency resonances).

Dynamical systems theory provides a framework for sys-tems with n-frequency resonances. It has been demonstrated[5] that locking in the case of 3-frequency resonances ispossible, i.e., the response synchronises with two independentforcing frequencies.

Moreover, it has also been shown that three-frequencyresonances are related to a relevant aspect of neural dynamics,namely the pitch perception of complex sounds [6], [7]. Thisaspect is relevant for applications because it opens up thepossibility of technological developments in various fieldsincluding medicine, artificial senses, and robotics.

Our main objective here is to develop, on the basis of abiologically oriented paradigm, a loop circuit able to locksimultaneously to two independent frequencies. Such a circuitwould extend the capabilities of the well-known phase-lockedloop (PLL) and overcome the structural stability limitation ofpreviously studied dynamical systems (coupled nonlinear os-cillators, coupled phase-locked loops and coupled circle maps[5], [8], [9]). This new non-linear circuit is termed a Quasi-Periodic Locked Loop (Q-PLL, patent pending [10], [11]) dueto its main property, that is, its ability to stably lock on aquasi-periodic input given by two independent frequencies.The circuit is interesting for applications envisaging real timeresponses of the auditory system, and also shows promisefor other electronic applications representing a significant stepforward with respect to standard PLL circuits.

First, the properties of three-frequency resonances from adynamical systems point of view will be recalled and theirimplications for pitch perception outlined. The structural sta-bility problems of known implementations will be discussed.Then, the circuit architecture will be presented and the strategyof the implementation for obtaining robust three-frequencyresonant responses will be explained. After this, the responseof the Q-PLL circuit in parameter space will be studied and itsdeterministic (signal perturbations) and structural (parameterperturbations) stability properties analysed. Finally, severalimplementations and possible future developments and appli-cations will be discussed.

II. THREE-FREQUENCY RESONANCES AND PITCHPERCEPTION

Synchronization was discovered by Christiaan Huygens in1665 [12]. He found that the pendulums of two clocks, fixed

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on the same mounting, after some time swung synchronously.Moreover, Huygens found that the phases of clock oscillationsshowed an asymptotic tendency towards a certain fixed valueof the phase difference. Because of this tendency, synchro-nization is also known as phase locking.

Synchronization is a well-known universal phenomenonoccurring widely in nature, from heartbeats to the movementof celestial bodies [13], [14]. The most common case, welldescribed by the Huygens clocks, is when the forcing andresponse frequencies are equal. This is called 1/1 synchro-nization and corresponds, for example, to the case of orbitaland rotational synchronised frequencies characterizing themovement of our moon and responsible of the fact that wesee always the same face.

Synchronization can appear in dynamical systems with morethan one external periodic forcing. Consider the general caseof a nonlinear system forced by n independent external forces.For n = 1, a periodic response corresponds to a synchronized,or locked, state. As said above, this resonance can differfrom simple 1/1 synchronization. In fact, when the quotientbetween the proper frequency f0 of the unperturbed systemand the frequency f1 of the external forcing, approach arational number, i.e., f0/f1 ' p/q with p, q ∈ N, a solutioncharacterized by a sub-harmonic of the forcing frequency, i.e.,f1/q can arise. When the response locks exactly to the q sub-harmonic, then f0/f1 = p/q and thus pf0 − qf1 = 0. Forp = q = 1, it reduces to 1/1 synchronization which impliesf0 = f1. This last relationship is a special case of the moregeneral case represented by

N−1∑k=0

mkfk = 0 (1)

where mk are integers. A system which satisfies Equation (1)non-trivially, i.e., mk 6= 0 for all values of k, is said to beN -resonant (the frequencies form an N -resonance).

Analog phase-locked loop circuits represent an electronicrealization of a 2-resonant system showing responses locked tothe forcing frequency, i.e., the case with N = 2 and |mk| = 1in Equation 1.

The next case in increasing dimensionality (N = 3)corresponds to quasi-periodic forcing, i.e., a system forcedwith two independent external frequencies f1 and f2. Quasi-periodically forced systems exhibit a rich variety of dynamicalbehaviour [15]. These range from periodic behaviour, to 2- and3-frequency quasi-periodic responses, to strange non-chaotic,and chaotic attractors. The Ruelle-Takens-Newhouse theoremestablishes that quasi-periodic attractors with 3 incommen-surate frequencies are on the border of chaos. These quasi-periodic attractors imply that Equation (1) with N = 3 hasonly the trivial solution (mk = 0 for all k values). The theoremimplies that in the immediate neighborhood of these responses,chaotic attractors must be present. As a consequence, quasi-periodic attractors with three-frequencies would be unstable inparameter space.

However, when Equation (1) has non-trivial solutions forN = 3, the three frequencies satisfy a resonant condition,i.e., are not independent and, therefore, the Ruelle-Takens-

P = ω

P = ω

P = ω + P

a)

b)

c)

P = 2- ω/(p+q)

ω0

ω

1

23

45 6

7

8

9

10

45 6

7

8

9

10

d)

ω0

ω0

Fig. 1. Schematic representation of the relationship between tones and pitchfor different stimuli. Musical sounds are complex tones consisting of a lowestfrequency, called fundamental (ω0 here) and integer multiples of it, calledovertones or harmonics. Panel a) reports the complex tone time series andpower spectrum of the first 10 harmonics of a violin whose pitch concideswith the fundamental. In panel b) the fundamental and the first few highercomponents of the spectrum are removed. The pitch does not change withrespect to a): it remains that of the fundamental (missing fundamental). Panelc) displays the effect of uniformly shifting the harmonics of panel b) by aquantity ∆ω. Although the difference combination tones remain unchanged,the pitch shifts by a quantity ∆P . Panel d) reports the relationship betweenthe frequency shift and the perceived pitch according to [6] superimposed ondata from the pitch shift experiment in [16].

Newhouse theorem does not apply. Thus, the system is ex-pected to have resonant responses with stability regions ofnon-zero size in parameter space. This has already been shownto hold in several nonlinear quasi-periodically forced systems[8].

Although in the study of nonlinear dynamical systems thecharacterization of deterministic chaos has attracted a greatdeal of attention, more regular dynamics are found to beimportant for modelling many real processes. In particular, thestudy of 3-frequency resonances was found to be particularlyimportant for modelling the auditory perception of complexsounds [6], [7].

The pitch of a complex sound is a psychoacoustic quantity,i.e., a subjective placement of the sound along an innertemplate that goes from bass to high tones, as in a musicalscale. If a correspondence between pitch and the frequencyspectrum of the stimulus is sought, for the majority of musical

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sounds, i.e., those composed of a fundamental and a series ofharmonics, the task is relatively easy: the pitch can be welldescribed by the fundamental component of the sound (seeFigure 1a). However in some cases, the pitch is not triviallyconnected to the lowest frequency of the stimulus. In fact,even when a few of the lowest components of a musical soundare removed, the perceived pitch still corresponds to that ofthe (removed) fundamental (see Figure 1b). This phenomenonis called missing fundamental or residue. In order to explainthis behaviour, a static nonlinear theory was formulated byHelmholtz [17]: the difference tone (one of the possiblecombination tones), produced within the ear, plays the role ofthe missing fundamental: the difference between two adjacentpartials equals the fundamental component. As an example, inFigure 1b, f5−f4 ≡ 5f0−4f0 = f0 . Nevertheless, pitch shiftexperiments, consisting of the observation that the pitch shiftswhen all the frequency components are shifted by the sameamount (see Figure 1c where it is shown that the pitch is alsoshifted by the amount ∆P ) [16], demonstrated the fallacy ofthis view (Figure 1c). This fact was intepreted historically asa failure of the nonlinear approach [18]–[20]. However, morerecently, a nonlinear dynamical theory [6], [7] (as opposed tothe static view of Helmholtz) was successful in explaining thekey feature of pitch shift (Figure 1d). This new theory is basedon the concept of dynamical resonances. The main results ofthe theory and its application to pitch perception modelling arebriefly outlined here below (see [6], [7] for a more detaileddescription).

When a generic dynamical system characterized by anintrinsic frequency f0 is forced by two external independentfrequencies f1 and f2, a web of three-frequency resonancesis generated in its parameter space. The general procedure forfinding the possible resonances starts with the continued frac-tion development of the quotient of the external frequencies,r = f1/f2 (see, e.g., [21]):

r = [a0; a1, ..., an, ...] . (2)

The development is finite if the quotient is rational and infiniteotherwise. Successive truncations up to a given finite order ofthe development give the so-called approximants of the ratio,

rn =pnqn

= [a0; a1, ..., an] . (3)

For any approximant a main three-frequency resonance existswhich is given by:

f0n =f1 + f2pn + qn

(4)

The hierarchy of the frequencies f0n characterizing the web ofthree-frequency resonances is described by a generalized Fareysum operation. The usual Farey sum operation on rationalnumbers, (p/q) ⊕ (r/s) = (p + r)/(q + s), describes thehierarchy of the web of synchronized responses in periodicallyforced oscillators (two-frequency resonances). Such hierarchi-cal organization is known as the Devil’s staircase.

Analogously, for three-frequency resonances, there is ahierarchical organization given by the generalized Farey sumoperation [8] defined as follows: for a given convergent

Fig. 2. Sketch of the first few levels of the generalized Farey sum hierarchy.

rn = pn/qn ≡ (pn, qn), define an interval of frequencies[f1/pn; f2/qn]. Then, the first step of the generalized Fareysum is defined as

f1pn⊕ f2qn

=f1 + f2pn + qn

. (5)

The operation can be iterated between adjacents, taking intoaccount that two frequencies fx and fy are considered asadjacents if sfx − rfy = qf1 − pf2. At variance withthe usual Farey sum, the generalized Farey sum acts onquotients between (mixed) real and integer numbers. In thisway, a hierarchical organization of resonances (similar to thatcorresponding to periodically forced oscillators) is formed ingeneric quasi-periodically forced systems [22]. The first stepsof this hierarchy, based on the generalized Farey sum, aresketched in Figure 2. The fundamental fact here, is that thepitch of complex sounds, including the phenomenon of miss-ing fundamental, can be described in terms of three-frequencyresonances. In particular, the pitch elicited by a quasi-periodicstimulus corresponds to a main three-frequency resonance. Inthe simplest case of a two frequencies stimulus, the pitch isrepresented by a three-frequency resonance obtained as thegeneralized Farey sum between the subharmonics f1/p andf2/q where p/q is an approximant of the stimulus frequencyratio. Such a description is in excellent agreement with experi-mental psycho-acoustical data explaining, with high precision,the phenomenon of pitch shift (see Figure 1c and d).

Having a robust dynamical model for the prediction ofthe pitch of complex sounds may be very useful in someapplications. However, from the practical point of view, amajor drawback appears: the stability intervals correspondingto the main three-frequency resonances appear to be verynarrow in all the dynamical systems studied so far: 1) nonlin-ear coupled oscillators with piecewise analytical solutions, 2)coupled PLLs [9], 3) quasi-periodically forced circle-maps [5],[8]. This is due to the fact that the web of resonances existingin the narrow interval determined by f1/p andf2/q is verycomplex (in principle, infinite resonances, which correspondto the successive levels of the generalized Farey sum operation,can coexist). This is similar to the case of the infinite steps ofthe Devil’s staircase in periodically forced systems.

In order to tackle this stability problem, a system whichcan stably produce these main three-frequency resonances overwide intervals of parameter space is required. This implies that


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F'(t)

x'(t)

VCO

S/H

1s

F(t)

x(t)

fQ G

f(t)

Fig. 3. Implementation of Q-PLL in Simulink®. The input signal F (t) (seeEquation (6) for symbols) enters the S/H module whose output F ′(t) ismultiplied by F (t) itself and then integrated over time (low-pass filtered)to give x′(t) which, in turn, determines the VCO frequency. The outputof the system x(t) is the locked signal which stems from the VCO. Theinstantaneous frequency f(t) of the output signal is also given, obtained usingthe same formula implemented in the VCO. The sign function between theVCO output and the S-H trigger input, is used to transform a sine wave toan impulse in order to trigger the S/H.

some kind of dynamic control must be implemented in orderto enhance the existence intervals of the main resonances. Thefollowing Section describes an electronic circuit, based on asampling locked loop control displaying very large stabilityintervals for the main three-frequency resonances, and whichcan lock not only on periodic but also on quasi-periodic inputs.

III. CIRCUIT DESCRIPTION

The Q-PLL circuit robustly implements three-frequencyresonator dynamics. The main idea underlying the circuitdesign is to use a sub-Nyquist approach taking into accountthat the desired three-frequency resonances are below theNyquist frequency of the forcing. Within this framework, afeedback system is developed which locks stably to one ofthe possible three-frequency resonances.

In Figure 3 a possible realization of the circuit in termsof a Simulink® model is shown. It also represents well thegeneral functional scheme of the system. The quasi-periodicinput signal F (t) is sent to a triggered sampler (Sample andHold, S/H). The sampling frequency of the S/H is determinedby the output frequency of a Voltage Controlled Oscillator(VCO). The sampled signal F ′(t) is then multiplied (×) by thenon-sampled one and, subsequently sent to a time integrator(1/s). Then, the integrated signal x′(t) is sent to the VCO todetermine the VCO frequency as fVCO = Gx′(t) + fQ, whereG and fQ are the sensitivity and quiescent frequency of theVCO, respectively. The output of the VCO, once transformedinto a square wave, is sent to the trigger port of the S/H. Theresponse of the system is the output signal of the VCO x(t)with instantaneous frequency f(t).

-150

-100

-50

0

50

0 100 200 300 400 500

Po

we

r (d

Bm

)

frequency (Hz)

inputoutput

Fig. 4. Example of locking for quasi-periodic input: f1 = 200 Hz,f1/f2 = φ, the golden ratio, and for VCO parameters: fQ = 100 Hz andG = 2000 V/Hz (NB: the result is independent of the initial phase ϕ0).The only surviving frequency in the output signal is the “residue” frequencycorresponding to f0 ' 104.72 Hz, according to three-frequency resonancetheory as outlined in Section II.

The equation that describes the dynamics of the Q-PLLshown in Figure 3, can be written as:

x(t) = A0 cos (2π[ fQ +G

∫ t

0

F (τ)F ′(τ) d τ ] t+ ϕ0) (6)

where ϕ0 is the initial phase of the VCO, F (t) is theinput (forcing) and F ′(t), the sampled signal, is formallyexpressed by F ′(t) = F (ti) for ti ≤ t < ti+1 withti = {t|dtH(x(t)) > 0} where H is the Heaviside functionand dt is the distributional derivative. The sampled signalfunction F ′(t) makes this equation, analytically speaking,rather intractable. Features of the model will be studied usingnumerical simulations.

Figure 4 reports, as an example, the power spectrum of theQ-PLL output (continuous line) along with that of the inputsignal (dashed line) for an input f1 = 200 Hz and f2 = φf1where φ = (1 +

√5)/2 is the golden ratio. This irrational

number ensures that the frequency ratio is as far as possiblefrom a rational ratio (quasi-periodic forcing).

In the general case, the sampling frequency fs ≡ fVCO,including its initial value fQ, is below the Nyquist frequencyof the input.

It is well known from sampling theory (see, e.g., [23]), thatsampling a sinusoidal signal of frequency f1 with a samplingfrequency fs < 2f1 produces a spectrum with aliasing compo-nents. The sequence fak

of aliases (on the positive frequencyaxis) is based on the quantity ∆1 ≡ |f1− k1fs| (k1 being thenearest integer to f1/fs.):

fak= {∆1, ∓∆1 + kfs; k ∈ N}. (7)

As a direct consequence, for k = k1, it results that either∆1 + k1fs = f1 or −∆1 + k1fs = f1, in other words, theresulting signal contains the input frequency.

If the signal to be sampled (input signal) is a superpositionof two sinusoids of frequencies f1 and f2 satisfying thecondition fs < 2 min{f1, f2}, the power spectrum of the


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f1

f2

fs

S/H

F'(t)

Fx

Fig. 5. Scheme of the sampling system implemented in Q-PLL. Two sinesignals of frequencies f1 and f2 are added and constitute the input signal.Sampling is triggered in S/H when a sine wave of independent frequencyfs crosses the zero from negative values. The signals to be analysed forthe Q-PLL locking mechanism are the sampling output F ′(t) and the signalresulting from the product of the input and the sampled signal F×.

sampled signal, for linearity, is the superposition of two powerspectra containing aliasing components based on ∆1 and∆2 = |f2 − k2fs|.

Thus, according to the sub-Nyquist sampling properties justrecalled, the sampled signal contains components at frequen-cies f1 and f2 coinciding with the input components. Takingthe product (×) of the two signals (input and sampled) pro-duces a signal F× with a non-null zero-frequency componentF(0) of the power spectrum. This Direct Current (DC) term isthe sum of the contribution resulting from the zero-frequencycomponents, F1(0) and F2(0), of the two input signals.

Furthermore, along with the DC component, the two lowestalias frequencies ∆1 and ∆2 constitute the low frequencypart of the spectrum which survives the subsequent low-passfiltering produced by the integrator. The analysis of thesequantities allows a deeper insight into the locking mechanismas shown below.

Consider the open submodel, i.e., without feedback, shownin Figure 5, where the triggering signal (input to the S/Hmodule) is replaced by a controlled frequency fs. The Fouriertransform of the product signal F× is computed and the zero-frequency component F(0) is considered. Calculation of thisquantity is performed for different values of the samplingsignal frequency fs over a range covering the convergencefrequency interval, while its phase ϕs is varied over the fullrange [0 : 2π]. Results are reported in Figures 6–8.

To gain an idea of how the locking mechanism acts, it isimportant to observe that a necessary condition for the outputof the Q-PLL circuit in Figure 3 to lock to a fixed value, is thatthe input to the VCO be constant. Therefore, the input signalto the integrator must vanish. A necessary condition for thisis that the DC component of F× vanishes. Figure 6 displaysthe DC component, F(0), over a wide range of fs that coverthe first three approximants of the frequency input ratio. Theshaded areas enclose the intervals defined by [f1/pk, f2/qk],for (pk, qk) = (1, 2), (2, 3), (3, 5), with pk/qk being the k-

-10

-5

0

5

10

100

F(0)

fs (Hz)

f0=174.54 Hz

f1/1f2/2

f0=104.72 Hz

f1/2 f2/3

f0=65.45 Hz

f1/3f2/5

Fig. 6. DC component F(0) (in arbitrary units) and convergence intervalsfor the first three approximants. It can be observed that the value of F(0)alternates its sign moving between adjacent convergents. The shaded regionsrepresent the area bounded by f1/p and f2/q. It is worth noting that theorder of f1/p and f2/q alternates between adjacent convergents. Frequencyof the first mediant f0 = f1+f2

p+qis reported for each convergence interval.

-5

-4

-3

-2

-1

0

1

2

3

4

98 100 102 104 106 108 110

sign(F1)=-sign(F2)

F (

0)

F1,F2>0F1=-F2

F1,F2<0

f1

fs (Hz)

0

q

Fig. 7. DC component F(0) (in arbitrary units) of the product of input andsampled (F×) signals over a range of frequencies around f0 for convergent(pk, qk) = (2, 3). Input frequencies are f1 = 200Hz and f2 = φf1. Shadedarea shows the range between f1/p and f2/q). Vertical line at fs = f0 isthe variation of F due to the change in phase ϕs which occurs only at thatfrequency. The dependence on this second variable is shown in the figureinsert which displays, with the same units, the variations in F with ϕs. Itcan be observed that for the rest of the plot, changes in F depend only uponfs. Labels indicate the relationship between F1 and F2 for different rangesof fs.

th convergent of f1/f2, and k = 1, 2, 3. The vertical linesrepresent the variations in F(0), caused by sampling signalphase change, which the system produces at exactly fs = f0k(and only for those values). To closely analyse this behaviour,Figure 7 displays a zoom of Figure 6 around the convergenceregion of the second approximant (pk, qk) = (2, 3). Considerthe values of the two components of F(0), namely F1 andF2: they are in some relationship with fs. Outside the shadedregion in Figure 7, for fs < f1/p they are both positive whilethey are both negative for fs > f2/q. Inside the shaded region,


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0

0 0.2 0.4 0.6 0.8 1

0

π

F(0)

Φ

ϕ/(2π)

F(0)Φ

Fig. 8. Phase response for fs = f0 for convergent (p, q) = (3, 5) in caseof input frequencies f1 = 200 Hz and f2 = φf1. Thick line is the DCcomponent (in arbitrary units) while the thin dashed line is the phase Φ ofthe response. It is important to observe that the number of cycles is exactlyp + q (compare to the insert in Figure 7). Moreover, the output phase onlyplays the role of the sign function, as it jumps from 0 to π when the DCcomponent changes its sign.

their sign is opposite and, for a given fs = f , they equal eachother in magnitude making F(0) vanish. However, Figure 7shows that f does not correspond to the main resonance and,therefore, the locking mechanism must rely on some additionalfeatures of the system.

In fact, F(0) is, in general, a function of both the frequencyfs and the phase ϕs of the sampling signal, and Figure 7should be viewed as a projection along the ϕs axis. However,it can be observed that over the range represented in Figure 7,except for fs = f0, F(0) does not depend on ϕs. Nevertheless,a strong dependence appears for fs = f0 as shown by the thickvertical line in Figure 7. It is known from Section II that in thiscase ∆1 = ∆2 and the two alias frequency sequences coincide.The plot inserted in Figure 7 represents the section of F(0)along the ϕs axis at fixed fs = f0 and displays explicitly thisdependence. It is important to observe that the variations ofF(0) due to ϕs are such that F(0) crosses the zero value,thus providing a mechanism of convergence towards the mainresonance frequency f0.

It is remarkable that this very special phase dependencehappens exactly at fs = f0, which underlines the strong con-nection between three-frequency resonances and the propertiesof the signal F× and, in turn, with the sub-Nyquist–sampledsignal F ′. Another remarkable feature is that the periodicityof F(0) as a function of ϕs is exactly p + q. The insertin Figure 7 corresponds to the convergent (pk, qk) = (2, 3)and the function presents 5 cycles. As another example ofthis feature, Figure 8 shows the case corresponding to theconvergent (pk, qk) = (3, 5) for which the periodicity of thephase response results to be 8.

Consider now the closed circuit (Figure 3). At the beginningof the locking cycle, if fQ < min(fk, f0k), the resulting DCis positive and forces the system to increase the samplingfrequency fs (in the opposite case, if fQ > max(fk, f0k) fsdecreases). When fs crosses f0k , a phase-locking mechanism

is activated and the system adjusts the phase of the samplingsignal to find one of the zeroes of the DC component (Fig-ures 7 and 8). This corresponds to a relative phase whichcancels the two low frequency components (∆1 = ∆2) ofthe aliases, ensuring a zero input to the integrator. Whenfar from the resonance, the two low frequency components(∆1 6= ∆2) provide a perturbation which leads the systemto cross fs = f0k and then to switch to the phase-lockingmechanism described above. The parameter acting as anamplifier of the perturbation is the VCO sensitivity G which, iftoo small does not allow the system to move far enough fromthe DC = 0 solution and cross the resonance value f0. On theother hand, if G is too large the circuit becomes unstable.

IV. EXPERIMENTAL SETUP AND RESULTS

The circuit features were studied by means of numericalexperiments. The implementation of the circuit in terms ofSimulink® blocks shown in Figure 3 is used here.

As already shown in the example reported in Figure 4, thesystem is able to lock to a quasi-periodic input composedof two incommensurate sinusoidal components whose ratio,notably, is the ”most irrational” number, i.e., the golden ratioφ. The ability to lock to different convergents of the f1/f2ratio over the entire range of the parameter fQ, is investigatedhere below.

Before proceeding further, some considerations on the de-pendence of the system on G are in order. G provides therate of change of the VCO frequency as a function of theintegrated signal. Roughly speaking, it must depend on theexpected frequency response which, in turn, depends on fQ.This can be observed through examining the intervals of theDC component around the locking frequencies in Figure 6. Asa rule of thumb, the smaller the interval extent, the smaller Gmust be in order to avoid jumps outside the given convergencebasin and to maximize the existence regions of the mainresonances. On this basis, appropriate values for G ≡ G(pk,qk)

are selected for each of the k-th approximant (pk, qk).Thus, by means of a trial-and-error procedure several values

of G(pk,qk) were selected: G(1,2) = 8000 Hz/V, G(2,3) =2000 Hz/V, G(3,5) = 1000 Hz/V, G(5,8) = 500 Hz/V,G(8,13) = 250 Hz/V. Small variations of G around the optimalvalues defined above have no significant effect and, therefore,this parameter is not investigated further.

A. Q-PLL features for quasi-periodic input

Figure 9 reports the response frequency fR of the Q-PLLas a function of fQ. It can be observed that fR locks exactlyto f0k , the main three-frequency resonance corresponding tothe k-th convergent. The extent of the intervals over whichthe solutions are stable are very large, much larger than the[f1/pk, f2/qk] intervals reported as continuous thin lines inFigure 9 (see also the shaded regions in Figure 6). It isworth noting that all other possible solutions correspondingto generalized mediants in the Devil’s staircase-like portrait(see Figure 2) expected to exist between f1/pk and f2/qk, aresuppressed. This is at variance with what has been found forsome systems previously investigated: coupled phase-locked


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Fig. 9. Response diagram as a function of fQ for optimal values of G(p,q).Input frequencies are: f1 = 200 Hz and f2 = f1φ. The frequency valuecorresponding to the first mediant (f1 + f2)/(p + q) (dotted lines) of thebounding values f1/p and f2/q for the Devil’s staircase (continuous thinlines) are shown (the latter not shown from (p, q) ≥ (5, 8)). Solutions (thicklines) converge to the first mediant. The convergence intervals width increasesfrom high to low p+q. Regions of fQ for which fR is not given, are regionsof instability of the solution.

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Fig. 10. Example of locking in response to a signal perturbed by noise. Inputis as in Figure 4 with additive white Gaussian noise of -45dB relative to thepeaks.

loops, coupled nonlinear oscillators with piece-wise analyticsolution and a quasi-periodically forced circle map [5], [8],[9]. Within the convergence intervals the attractors are stableand, therefore, evolve to asymptotic orbits independently ofthe initial conditions, mainly the initial phase of the VCO.

The highest reported convergent is (p6, q6) = (8, 13). Con-vergence intervals corresponding to higher convergents are tooshort to be represented fairly on the same scale. At the frontierof the reported stable solutions there are small regions in whichthe system is unstable where the responses may depend onthe initial conditions. In these regions, however, other stablesolutions may also occasionally appear (not reported here)which correspond to generalized mediants of the neighbouring(parent) solutions (see Figure 2).

Experiments to study the stability of the main resonances

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Fig. 11. Example of locking for one frequency input only (PLL-likebehaviour) for f1 = 200 Hz and fQ = 90 Hz. Output is the input frequencyf1 divided by n = 2. Q-PLL also produces harmonics of the responsefrequency of much lower amplitude.

facing various perturbations of the input signals were alsoperformed. First, the signal of Figure 4 was perturbed byan additive white Gaussian noise with an intensity of about-30 dBm relative to the input (0 dBm). For the presentsystem implementation and reported parameters, the systemresponse remains stable for noise figures below or equal tothis level. This can be appreciated as a peak corresponding tothe expected resonance at 104.72 Hz in Figure 10. A thoroughinvestigation about the stability to noise perturbation is beyondthe scope of this paper but would undoubtedly be interestingfor the design of electronics applications.

B. PLL-like behaviour

If the Q-PLL is forced with a signal containing a frequencyf1 = 200 Hz only, selecting fQ from a suitable range aroundfQ = 200 Hz the system locks at fR = 200 Hz as would anormal PLL.

If now the quiescent frequency is set, for example, to fQ =90 Hz, the system locks to fR = 100 Hz (Figure 11). Thisfrequency is the first sub-harmonic of the forcing frequency,fR = f1/2. In general, the system is able to lock to any sub-harmonic fR = f1/n depending on the value of fQ. Figure 12shows the response diagram as a function of fQ. Solutions arevery stable and do not depend on the VCO sensitivity G (inFigure 12, G = 1000 Hz/V is used). In this situation the Q-PLL behaves, in the limit, as a PLL with frequency-dividercapability such as, for example, those in the sampling class ofPLLs.

C. Q-PLL as a pitch detector

As the focus of this work is on the ability of the circuit tomodel the pitch of complex sounds, its ability to lock in thecase of the sum of two sine waves with harmonic frequencyratio f1/f2 = 2/3 is analysed. In musical terms, this particularratio corresponds to a perfect fifth interval. Such a ratio is usedas a paradigmatic example being considered one of the most


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Fig. 12. Response diagram for one input frequency only (PLL-like behaviour)as a function of fQ, for f1 = 200 Hz. Depending on the VCO quiescentfrequency various sub-harmonics of f1 can be selected.

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Fig. 13. Example of locking for harmonic input: f1 = 200 Hz, f2 = 300 Hzand fQ = 90 Hz. Harmonics of the response frequency are also producedwith lower amplitude.

consonant musical intervals. It should be borne in mind thatthis interval represents a stimulus with a missing fundamentalat f1/2 ≡ f2/3.

Figure 13 displays the input and response frequencies for theperfect fifth case, f1 = 200 Hz and f2 = 300 Hz with fQ =90 Hz. The system locks stably to the missing fundamentalvalue fR = 100 Hz.

Next, the more complex case corresponding to a pitch-shift is analysed. Figure 14, reports the result for a non-harmonic signal obtained by applying a pitch shift operatorto the harmonic signal used in the perfect fifth example. Asmall shift ∆f = 21 Hz was selected to keep the ratio 2/3 asthe first approximant of the input frequency ratio. In fact, thecontinued fractions development of the ratio 221/321 is:

[1, 2, 4, 1, 3, 5] (8)

and the corresponding convergents: 1, 3/2, 13/9, ...Figure 14 shows the power spectrum density of the input and

output signals. The system locks exactly (within the numerical

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Fig. 14. Example of locking for inharmonic input (“pitch shift” of ∆f =21 Hz with respect to Figure 13): f1 = 221 Hz, f2 = 321 Hz and fQ =90 Hz. The only surviving frequency in the output signal is the “residue”frequency corresponding to f0 = (f1/2 + f2/3) = 108.4 Hz, according toEquation (4)

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Fig. 15. Example of locking in response to a signal comprised of seven shiftedharmonics with amplitude decay as a function of the harmonic number.

precision) to the first mediant of f1/2 and f2/3, i.e., f0 =108 Hz. Another remarkable feature is that, in virtue of thelocking mechanism described in Section III, only the residuefrequency survives.

Another interesting variation of the input signal, related tothe possibility of using the Q-PLL as a model for auditorypitch recognition, is represented by a signal which is aharmonic complex with two-to-several components under pitchshift [7]. This stimulus is the basis of several psychoacousticsexperiments (see, e.g., [16], [24]–[27]). Although the argumentis very complex and stimuli can be very different, to providean example: an input signal comprised of the first 7 harmonicsof a sine wave of frequency 100 Hz, shifted by the sameamount used in previuos pitch-shift examples (∆f = 21 hz),is fed into the Q-PLL. Figure 15 displays the spectrum of theinput and the response signals. The response remains stable tof0 = (f1+f2)/(2+3) = 108.4 Hz under significant variationsin the different magnitudes of the input components, provided


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that the amplitudes decay towards high harmonic numbers(typical of musical sounds).

The resulting spectrum is more complex than that reportedin Figure 14, displaying not only the locking peak, but alsoa number of lower power components. This happens becausein this case the low frequency components resulting from thesub-Nyquist sampling (see Section III) do not mutually cancelas in the case of a stimulus comprising only two componentsof the same amplitude.

V. CONCLUSION

A new nonlinear circuit which can maintain a locked statewhen forced by two incommensurate frequencies has beenpresented. The circuit was developed in a biologically inspiredparadigm within the framework of dynamical systems theory.Locking states are characterized by the generation of a thirdfrequency which forms a so-called three-frequency resonancetogether with the forcing frequencies. Moreover, it has beendemonstrated that these resonances characterize, in turn, thepitch of complex sounds when the fundamental componentis absent. Thanks to this property, the circuit can emulateone of the main psycho-acoustical parameters of auditoryperception. Thus, it opens the way for the development ofreal-time auditory processors which can be utilized in adiversity of applications, from medicine (hearing aids, arti-ficial cochlea), to robotics (artificial senses), to industrial andconsumer electronics (by including auditory-like processingin audio applications, as a natural extension of the univer-sal PLL). This new circuit, being orbitally and structurallystable, thanks to a control strategy based on sample-and-hold feedback, supersedes former attempts (such as coupledPLLs) and prompts further study regarding optimization of itsfeatures and performance in the light of future technologicalapplications.

ACKNOWLEDGEMENTS

The Italian [10] and International [11] Patent Applicationswere financially supported by CNR. The authors would liketo thank Oreste Piro and Julyan Cartwright for illuminatingdiscussions and Giuliano Gavina for the electronic implemen-tation of an analog version of the circuit.

REFERENCES

[1] H. de Bellescise, “La reception synchrone,” Onde Electrique, vol. 11,pp. 230–240, 1932.

[2] R. Urtel, “Entrainment and synchronization of oscillations,” Z. Techn.Phys., vol. 11, pp. 460–465, 1938.

[3] N. Margaris and P. Mastorocostas, “On the nonlinear behavior of theanalog phase-locked loop: Synchronization,” IEEE Trans. Ind. Electron.,vol. 43, no. 6, pp. 621–629, 1997.

[4] W. Mathis and J. Bremen, “Modelling and design concepts for electronicoscillators and its synchronization,” The Open Cybernetics & SystemicsJournal, vol. 3, pp. 47–60, 2009.

[5] O. Calvo, J. H. E. Cartwright, D. L. Gonzalez, O. Piro, and O. Rosso,“Three-frequency resonances in dynamical systems,” Int. J. Bifurcation& Chaos, vol. 9, no. 11, pp. 2181–2187, 1999.

[6] J. H. E. Cartwright, D. L. Gonzalez, and O. Piro, “Nonlinear dynamicsof the perceived pitch of complex sounds,” Phys. Rev. Lett., vol. 82, pp.5389–5392, 1999.

[7] ——, “Pitch perception: A dynamical-systems perspective,” Proc. NatlAcad. Sci. USA, vol. 98, pp. 4855–4859, 2001.

[8] ——, “Universality in three-frequency resonances,” Phys. Rev. E,vol. 59, pp. 2902–2906, 1999.

[9] O. Calvo, J. H. E. Cartwright, D. L. Gonzalez, O. Piro, and F. Sportolari,“Three-frequency resonances in coupled phase-locked loops,” IEEETrans. Circuits Syst., vol. 47, pp. 491–497, 2000.

[10] D. L. Gonzalez and A. Maurizi, “Circuit and method for modelingand emulating the pitch perception of complex sounds,” Italian patentapplication IT102 019 000 021 111, Nov. 13, 2019.

[11] ——, “Circuit and method for modeling and emulating the pitchperception of complex sounds,” International patent applicationPCT/EP2020/081 140, Nov. 5, 2020.

[12] C. Huygens, Œuvres Completes de Christiaan Huygens. SocieteHollandaise des Sciences, 1888–1950, vol. 17, p. 185.

[13] B. Escribano, J. Vanyo, I. Tuval, J. H. E. Cartwright, D. L. Gonzalez,O. Piro, and T. Tel, “Dynamics of tidal synchronization and orbitcircularization of celestial bodies,” Phys. Rev. E, vol. 78, p. 036216,2008.

[14] B. Escribano, J. Vanyo, I. Tuval, J. H. Cartwright, D. L. Gonzalez,O. Piro, and T. Tel, “A minimal dynamical model for tidal synchro-nization and orbit circularization,” Celestial Mechanics and DynamicalAstronomy, vol. 209, pp. 181–200, 2010.

[15] T. Kapitaniak and J. Wojewoda, Attractors of Quasiperiodically ForcedSystems. World Scientific, 1994.

[16] J. F. Schouten, R. J. Ritsma, and B. L. Cardozo, “Pitch of the residue,”J. Acoust. Soc. Am., vol. 34, pp. 1418–1424, 1962.

[17] H. Helmholtz, Die Lehre von den Tonempfindungen als physiologischeGrundlage fur die Theorie der Musik. Vieweg , Braunschweig, 1863.

[18] F. L. Wightman, “The pattern transformation model of pitch,” J. Acoust.Soc. Am., vol. 54, pp. 407–416, 1973.

[19] E. Terhardt, “Pitch, consonance, and harmony,” J. Acoust. Soc. Am.,vol. 55, pp. 1061–1069, 1974.

[20] A. de Chevigne, “Pitch perception models,” in Pitch - Neural Coding andPerception, ser. Springer handbook of auditory research, C. J. Plack, A. J.Oxenham, R. R. Fay, and A. N. Popper, Eds. Springer ScienceBusinessMedia, Inc., 2005, vol. 24, ch. 6.

[21] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers.Oxford: Clarendon Press, 1938.

[22] J. H. E. Cartwright, J. Douthett, D. L. Gonzalez, R. Krantz, and O. Piro,“Two musical paths to the Farey series and devil’s staircase,” J. Math.Music, vol. 4, no. 1, pp. 57–74, 2010.

[23] W. M. Hartmann, Signals, Sound, and Sensation, ser. Modern Acousticsand Signal Processing. Springer, 1998.

[24] R. J. Ritsma and F. L. Engel, “Pitch of frequency-modulated signals,”J. Acoust. Soc. Am., vol. i36, no. 9, pp. 1637–1644, 1964.

[25] G. F. Smorenburg, “Pitch perception of two-frequency stimuli,” J.Acoust. Soc. Am., vol. 48, no. 4, pp. 924–942, 1970.

[26] R. D. Patterson, “The effect of relative phase and the number ofcomponents on residue pitch,” J. Acoust. Soc. Am., vol. 53, no. 6, pp.1565–1572, 1973.

[27] M. Greenhough and W. R. P., “Experiments on the pitch of signalsderived from three-tone complexes,” Journal of Sound and Vibration,vol. 41, no. 4, pp. 451–461, 1975.


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Diego Luis Gonzalez Diego L. Gonzalez was bornin Buenos Aires, Argentina, in 1951. He qualifiedas Professor of pianoforte in 1971. He receivedhis degree in physics from the University of LaPlata in Argentina in 1981 and PhD in TheoreticalPhysics from the same University in 1987. HisPhD focused on the study of synchronisation andchaos in nonlinear oscillators. He worked from 1988in the CNR LAMEL Institute (now IMM) in thefield of microelectronics and from 1999 to 2008he collaborated with the Acoustical Laboratory of

the St. George School Foundation and the CNR in Venice. At presenthe is an Associate Researcher at the CNR Institute for Microsystems andMicroelectronics (IMM), also working at the Statistical Sciences Departmentof Bologna University. His main research interests are the theory of low-dimensional nonlinear systems and its application to modelling complexdynamical systems and, in particular, biological ones. In this latter field, hehas formulated and developed an original theory of the pitch perception ofcomplex sounds based on multi-frequency resonances and a mathematicalmodel of the genetic code with consequences on its putative origin andevolution.

Lorenzo Grassi Studied at the University of Flo-rence, Italy, where he earned his Laurea (Italianequivalent MSc) in Physics in 1994. 1995-97 EECHCM Fellowship at the Max Planck Institut furMeteorologie (MPIfM), Hamburg, Germany. Spentthe next four years at the Institute of AtmosphericSciences and Climate (ISAC) of Italy’s NationalResearch Council (CNR). His research focused onoptical atmospheric remote sensing and, in particularon LIDAR (LIght Detecting And Ranging), DIAL(DIfferential Absorption Lidar), DOAS (Differential

Optical Absorption Spectroscopy) and the optical design of apparatus for insitu air quality monitoring. Worked as a consultant in optics and in softwaredevelopment for industrial automation. Since 2004, shareholder and projectmanager of Asper S.r.l., in charge of industrial and research projects, andsince 2005 he is the sole director of the company.

Alberto Maurizi Alberto Maurizi obtained his Lau-rea (Italian equivalent of MSc), cum laude, inphysics from the University of Bologna, Italy, in1992. His education continued with the support ofthe Italian National Research Council (CNR) atthe Institute of Atmospheric Sciences and Climate(CNR-ISAC) and subsequently supported by Post-Doc EC grants (EC HCM Programme). Promotedto a permanent position in 2001 at CNR-ISAC, heworked on complex geophysical problems (mainlyturbulence in geophysical flows). More recently, he

moved to the CNR-IMM (Institute of Microelectronics and Microsystems)to follow his growing interest in dynamical systems and their application tosound perception modelling.


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A new kind of locked circuit: the Quasi-Periodic Locked ...

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