Colored Noise in Oscillators. Phase-Amplitude Analysis and a Method to Avoid the Ito-Stratonovichˆ Dilemma Michele Bonnin, Fabio L

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Colored noise in oscillators. Phase-amplitude analysis and a method to avoid the Ito-Stratonovichˆ dilemma Michele Bonnin, Fabio L. Traversa, Fabrizio Bonani, Senior Member, IEEE

Abstract—We investigate the effect of time-correlated noise on spectral purity, while time jitter describes the time domain the phase fluctuations of nonlinear oscillators. The analysis is accuracy of the oscillator waveforms. Phase noise spreads based on a methodology that transforms a system subject to the bandwidth of the fundamental frequency of oscillators colored noise, modeled as an OrnsteinUhlenbeck process, into an equivalent system subject to white Gaussian noise. A description and may produce interference with neighboring channels, in terms of phase and amplitude deviation is given for the thus degrading the whole system performances. Therefore, transformed system. Using stochastic averaging technique, the characterizing phase noise in oscillators is a major problem equations are reduced to a phase model that can be analyzed for practical applications. to characterize phase noise. We find that phase noise is a drift- Since the seminal work [1], linear time invariant models diffusion process, with a noise-induced frequency shift related to the variance and to the correlation time of colored noise. (LTI) have been applied to high-Q resonant and quartz- The proposed approach improves the accuracy of previous phase crystal oscillators. While of great practical importance, such a reduced models. technique is often too simplistic and fails to capture essential Index Terms—Oscillator noise, phase noise, colored noise, features such as spectral dispersion. Inclusion of linear time stochastic differential equations (SDEs), Fokker-Planck equation, variant effects (LTV) can yield more accurate results [2], [3]. stochastic averaging, phase models. The most rigorous treatment of phase noise in nonlinear oscillator perhaps dates back to [4], where the author de- composes the oscillator response into phase and magnitude I.INTRODUCTION components, and successfully derived a differential equation Oscillators and phase locked loops (PLLs) are fundamental for the phase deviation. In [5], the oscillator response is components of electronic and optical systems. For instance, decomposed into orthogonal components, and equations for in digital systems they are used to establish a reference time purely phase and amplitude deviations are derived. Unfortu- to synchronize operations. In communication systems, they nately, as shown in [6], using an orthogonal decomposition to are used for frequency coding and decoding, and for channel separate phase and amplitude deviations leads to inaccurate selection. results. The method proposed in [6], based on using Floquet Noise sources, both intrinsic and external, are a major vectors to project the response of the noisy oscillator onto nuisance plaguing oscillator and PLL performance. They can a shifted version of the unperturbed response, exploits a be classified as white (frequency independent) fluctuations, linear periodically time varying approximation of the oscillator such as thermal noise in electrical circuits with resistive behavior leading, ultimately, to a nonlinear phase equation. elements or shot noise in semiconductor devices, and time- The idea that Floquet vectors constitute the ideal basis to correlated (colored) noise sources. Among the latter, partic- decouple phase and amplitude dynamics was confirmed in [7], ularly relevant in oscillators based on bipolar transistors and [8]. Phase domain models based on the ideas introduced in [6], MOSFET devices used as radio frequency sources, we find have been extensively used to derive an analytical stochastic Lorentzian low-frequency noise and flicker noise. Flicker or characterization of oscillator phase noise due to both white 1/f fluctuations can be in several cases traced back to the and colored noise sources [9]–[16]. superposition of low-frequency Lorentzian noise sources, that In [4], [6], [9], simplified scalar stochastic differential therefore are of paramount importance in assessing oscillator equations for the phase variable were derived, neglecting all arXiv:1905.12994v1 [nlin.AO] 30 May 2019 random variations. contributions of noise and amplitude fluctuations, i.e small The performance and reliability of oscillators depend cru- deviations from the limit cycle, beyond the first order. This cially on noise sources, which deteriorate the oscillator re- assumption is well justified for most electronic oscillators, sponse and are responsible for phase noise and time jitter. that are typically subject to small noise and exhibit strongly Phase noise and time jitter are strictly related concepts, defin- stable limit cycle, thus leading to very accurate phase models. ing oscillator short term frequency instabilities. In particular, LTV techniques yield reliable and precise predictions for the phase noise is a frequency domain measure of the oscillator phase diffusion process, but they fail to capture the frequency shift phenomenon [11], [17], [18]. Such a frequency shift is M. Bonnin and F. Bonani are with the Dipartimento di Elettronica e usually negligibly small for strongly stable oscillators, such Telecomunicazioni, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. F.L. Traversa is with MemComputing Inc., San Diego, as those customarily used in electronic systems, but may play CA, 92093-0319, USA. a relevant role in autonomous systems from other fields, e.g. 2

system biology and neuroscience [19]. Uhlenbeck process with deterministic initial state η0 is char- This paper proposes a novel methodology to analyze phase acterized by the expectation value noise in nonlinear oscillators subject to time-correlated noise, t hη i = η exp − (3) modeled as an Ornstein-Uhlenebeck process. Making use of t 0 τ the method proposed in [20], [21] the system with colored noise is first transformed into an equivalent system subject and by the correlation function to white Gaussian noise. The advantage is twofold: first the D2 |t − s| hη η i = exp − (4) transformation allows to use the whole machinery of stochas- t s 2τ τ tic differential equations on a reduced dimensional system. Thus we consider equations (1) and (2), that we rewrite adopt- Second, and more important, the transformation avoids the ing the standard notation of stochastic differential equations Ito-Stratonovichˆ dilemma [22]. Phase and amplitude deviation (SDEs) equations are derived for the transformed system, and then reduced to a phase model, that describes the oscillator dynamics dxt = [a(xt) + B(xt)ηt] dt (5) in terms of the phase variable only. This description is the ideal τdηt = −ηt dt + D dWt (6) tool for phase noise analysis, since it gives an approximate yet very accurate description of the phase dynamics. We show that where Wt is a Wiener process, i.e. the integral of a white phase noise in nonlinear oscillators is a drift-diffusion process, noise. that is, noise sources not only induce a spread in the oscillator The SDEs (5)-(6) describe a diffusion process with un- spectrum, but also a shift in the oscillation mean frequency. modulated (additive) noise. Therefore the equations can be The frequency shift is related to the variance of the colored interpreted using any of the two main interpretation schemes, noise and to the noise correlation time. Some examples are namely Itoˆ or Stratonovich, obtaining the same solution. In presented to assess the validity of the approach. the full space (x, η) the system is Markovian, that is, future states are completely determined by the current state and the stochastic process shows no memory. Methods for analysis II.MODELING of Markovian systems, based on the Fokker-Planck or Kol- Consider the nonlinear system subject to random noise mogorov equations, are well developed. However, the practical source solution of the equations obtained using these approaches may become unbearable because of the large number of state x˙ t = a(xt) + B(xt) ηt (1) variables involved. n where xt : R 7→ R denotes the state of the system, A possible solution strategy amounts to study the system a : Rn 7→ Rn is a smooth vector field that defines the system dynamics in a reduced dimension space. However, if we internal dynamics, B : Rn 7→ Rn is a smooth vector valued consider the (x) space only the system is non Markovian function and ηt : R 7→ R is a scalar function describing due to the presence of multiplicative noise, as the increments random fluctuations, both internal and external. of the state variables depend on the past history of the Random fluctuations are often modeled as zero mean, noise process. Problems arise even in the simpler case of a Gaussian distributed white noise. A zero mean Gaussian quasi-white approximation: for τ → 0, (6) shows that the white noise ηt = W˙ t is characterized by hW˙ ti = 0 and external fluctuations reduce to a white noise ηtdt = DdWt. hW˙ tW˙ si = δ(t − s), where δ is the Dirac delta function, Substituting this approximation into (5) yields while t and s denote two time instants. White Gaussian noise dxt = a(xt)dt + D B(xt)dWt (7) is a reasonable approximation in the case where the typical time scales of the underlying deterministic dynamics are much The white noise is now modulated by the state dependent larger than the noise correlation time (quasi-white approxima- function B(x) (multiplicative noise), therefore the question tion). Unfortunately, white noise is rarely an accurate model arises whether (7) should be interpreted according to Itoˆ or to represent all of the noise sources that induce fluctuations in Stratonovich. real electronic devices and systems. As a consequence, a Dirac Applying the procedure presented in [20], [21], in the next delta correlation in time is more justified by mathematical section we shall derive a reduced description in the (x) space convenience than being physically plausible. Real processes of problem (5)-(6) where the SDE system is transformed into have typically finite correlation times, and often 1/f power an SDE (for the state vector x only) subject to white Gaussian spectra, e.g. flicker noise. noise. The main results are the following A more realistic description of noise in electronic systems • The reduced system holds for small, but not necessarily is given by an exponentially correlated process. Better known vanishing correlation time τ. as colored noise, it can be modeled as an Ornstein-Uhlenbeck • The reduced system resolves the Ito-Stratonovichˆ process [23]. In this case the noise source is modeled accord- dilemma, that is, we shall derive equivalent SDEs for the ing to two interpretations. By equivalent, we mean two different SDEs, interpreted following different rules, having the τη˙t = −ηt + DW˙ t (2) same solution. Because the solution is unique, it is just a where τ is a parameter proportional to the finite noise cor- matter of personal preference to choose one interpretation relation time, and D is the diffusion constant. The Ornstein- rather than the other. 3

III.WHITENOISEAPPROXIMATION implies that ψ is the stationary distribution of the Ornstein- Dividing both sides of (6) by τ, substituting τ = ε2 and Uhlenbeck process (9), thus [25] r introducing ηt = yt/ε, equations (5)-(6) become 1 y2 ψ(y) = pst(y) = exp − (17) 1 πD2 D2 dxt = a(xt) + B(xt)yt dt (8) ε As a consequence, condition (bn, ψ) = 0 amounts to require 1 D that each term b averages to zero with respect to y dy = − y dt + dW (9) n t ε2 t ε t Z (bn, ψ) = hbniy = bn pst(y) dy = 0 (18) Usually, in electronic systems the correlation time τ is small R compared to the characteristic time constants for the deter- Taking into account that hyi = 0, it is straightforward to ministic part of the dynamics. Therefore, we can assume the y verify that hΛ u i = 0. Thus equation (15) is solvable, and correlation time τ small enough such that ε 1. Under this 1 0 y direct substitution shows that assumption, equations (8) and (9) show a time scale separation, ∂u0(x, t) −1 since the Ornstein-Uhlenbeck process yt is one order, in u (x, y, t) = y B(x) = −Λ Λ u (19) 1 ∂x 2 1 0 the parameter ε, faster than the state variables xt. Notice that a straightforward application of stochastic averaging [24] Similarly, equation (16) is solvable if would lead to inconsistent results. In fact, since asymptotically ∂u 0 − hΛ u i + Λ Λ−1Λ u = 0 (20) hyti = 0, the averaged equation would simply coincide with ∂t 0 0 y 1 2 1 0 y the deterministic (noiseless) system. y Now, let u(x, t) = E[f(xt, t)] denote the expected value for The three averages on the left hand side can be expressed as Z a generic, smooth enough function f(xt, t). The Kolmogorov ∂u0(x, t) ∂u0(x, t) backward equation corresponding to (8)-(9) takes the form pst(y) dy = (21) R ∂t ∂t [22] Z ∂u 1 1 ∂u0(x, t) = Λ0 + Λ1 + Λ2 u (10) Λ0u0 pst(y) dy = a(x) (22) ∂t ε ε2 R ∂x Z 2 where −1 D ∂u0(x, t) ∂B(x) Λ1Λ2 Λ1u0 pst(y) dy = − B(x) ∂u R 2 ∂x ∂x Λ0 u = a(x) (11) ∂x ∂2u (x, t) + BT (x) 0 B(x) (23) ∂u ∂x2 Λ u = y B(x) (12) 1 ∂x where ∂B(x)/∂x is the Jacobian matrix of the vector ∂u D2 ∂2u function B with respect to the x variables with elements Λ2 u = −y + 2 (13) 2 2 ∂y 2 ∂y (∂B(x)/∂x)ij = ∂Bi(x)/∂xj, while ∂ u0(x, t)/∂x is the Hessian matrix of the scalar function u with respect to x and ∂u/∂x is a row vector denoting the gradient of the scalar 0 with elements (∂2u (x, t)/∂x2) = ∂2u (x, t)/(∂x ∂x ). function u with respect to the vector x (thus ∂u/∂x a(x) is 0 ij 0 i j Furthermore, to solve the last integral we used the fact that, the scalar product of the two vectors). We look for a solution from (17), hy2i = D2/2. of (10) in the form of a power series expansion u = u + y 0 Substituting equations (21)-(23) into (20) yields the Kol- εu + ε2u + ... 1 2 mogorov backward equation Introducing this ansatz into (10) and equating the coefficients 2 of the same powers of ε yields the hierarchy of equations ∂u0(x, t) ∂u0(x, t) D ∂u0(x, t) ∂B(x) = a(x) + B(x) −2 ∂t ∂x 2 ∂x ∂x ε :Λ2 u0 = 0 (14) 2 T ∂ u0(x, t) ε−1 :Λ u = −Λ u + B (x) B(x) (24) 2 1 1 0 (15) ∂x2 0 ∂u0 The corresponding Stratonovich SDE is (we use the symbol ε :Λ2 u2 = − Λ0u0 − Λ1u1 (16) ∂t ◦ to denote Stratonovich stochastic integral) . . dxt = a(xt)dt + D B(xt) ◦ dWt (25) while the equivalent Itoˆ SDE is The first equation in the hierarchy implies that u0 does not 2 depend on y, so that u0 = u0(x, t). D ∂B(xt) dxt = a(xt) + B(xt) dt + D B(xt) dWt The other equations are of the type Λ2un = bn. According 2 ∂x to Fredholm alternative theorem, these equations are solvable (26) ∗ provided that a function ψ exists such that: (1) Λ2 ψ = 0, By equivalent we mean that the SDEs (25) and (26) are ∗ where Λ2 is the conjugate operator of Λ2, and (2) each bn interpreted according to different rules but they have the same satisfies (bn, ψ) = 0, where (·, ·) denotes the inner product solution. The Stratonovich SDE (25) can be transformed into 2 ∗ in the L Banach space. Taking into account (13), Λ2 ψ = 0 the Itoˆ SDE (26) (and vice versa) by addition (respectively, 4

subtraction) of the Wong-Zakai drift correction term [22]. The The remaining n−1 vectors u2(t),..., un(t) can be chosen as solution of (25) and (26) is also a weak solution for the original the Floquet vectors (apart from the limit cycle tangent u1(t)) system with colored noise (5)-(6). Weak means that the two of the linearized variational equation [18], [26], [27] solutions for a specific realization of the the noise process dWt are different in details, but they converge in probability, i.e. dx(t) e = J (t) x(t) (29) they have the same probability density function and therefore dt a e the same statistical properties. Whether to use the Stratonovich SDE (25) or the equivalent where J a(t) = ∂a/∂x is the Jacobian matrix of the vector Itoˆ SDE (26) is at this point a matter of personal taste. As function a(x(t)) evaluated on the limit cycle xs(t). Thus a rule of thumb, Stratonovich interpretation may be better the vectors {u1(t),..., un(t)} are independent, although in suited for algebraic manipulations, since traditional calculus general they are not orthogonal. We construct the matrix rules apply. By contrast, Itoˆ interpretation requires a whole U(t) = [u1(t),..., un(t)], and we define the reciprocal T T new set of calculus rules, known as Itoˆ calculus [22], but it is vectors v1 (t),..., vn (t) to be the rows of the inverse matrix −1 n more suitable for numerical simulations and for the calculation V (t) = U (t). Thus {v1(t),..., vn(t)} also span R T T of expected quantities. and the bi-orthogonality condition vi uj = ui vj = δij for all t, holds. Finally we introduce matrices Y (t) = [u2(t),..., un(t)], Z(t) = [v2(t),..., vn(t)], and the mag- IV. PHASE-AMPLITUDEEQUATIONSFORNONLINEAR nitude (in the L2 norm) of the vector field evaluated on the OSCILLATORSWITHCOLOREDNOISE limit cycle, r(t) = |a(xs(t))|. Let us now consider the case where equations (5)-(6) Following [18], [26], [27], we decompose the solution of describe a nonlinear oscillator subject to colored noise. In the (26) into two components absence of random fluctuations, equation (5) reduces to the autonomous ordinary differential equation (ODE) xt = xs(θt) + Y (θt)Rt (30) dx = a(x) (27) The first component x (θ ) represents the projection of the dt s t stochastic process xt onto the limit cycle, evaluated at an We assume that (27) admits of an asymptotically stable T - unknown time instant θt. The second component Y (θt)Rt periodic solution xs(t), represented by a limit cycle in its represents the distance between the solution and the limit state space. cycle, measured along the directions spanned by the vectors We shall derive an equivalent description of system (25) v2,..., vn at the random time θt. Because xt is a stochastic (or (26)) in terms of phase and amplitude deviation variables, process, both θ : R 7→ R and R : R 7→ Rn−1 are analogous to the one derived in [18], [26], [27]. The phase stochastic processes as well. Itoˆ equations defining the time function used in our description coincides locally, in the evolution of these stochastic processes can be found following neighborhood of the limit cycle, with the asymptotic phase the procedure given in [18, Theorem 3.1] and [27, Theorem defined in [6], [13], [28]. As a second step, we shall derive 1], obtaining a phase reduced model, that describes the oscillator dynamics in terms of the phase variable alone. dθt = [1 + aθ(θt, Rt) +a ˆθ(θt, Rt) + bθ(θt, Rt)] dt For our purpose it is more convenient to work with the Itoˆ + Bθ(θt, Rt) dWt (31) SDE (26). The reason to prefer the Itoˆ over the Stratonovich interpretation is that Itoˆ integrals are adapted processes, i.e. state dRt = [L(θt)Rt + aR(θt, Rt)+ variables and the noise increment are independent. By contrast, in the Stratonovich interpretation state variables and noise +aˆR(θt, Rt) + bR(θt, Rt)] dt+ increments are correlated, a property known as “anticipating + BR(θt, Rt) dWt (32) nature” or “look in the future property” of the Stratonovich integral. The far reaching consequence is that when one tries to where (the 0 sign denotes the derivative with respect to θ) describe the dynamics by using only a subset of state variables, an additional piece of information is lost, represented by the T 0 correlation between eliminated variables and noise increments aθ(θ, R) =κ v1 a(xs + YR) − a(xs) − Y R (33) [17]. T 0 To make the paper self-contained, we introduce some aˆθ(θ, R) = − κ v1 Y BR(θ, R)Bθ(θ, R) notation. We consider a set of time dependent vectors Rn 1 2 00 00 {u1(t),..., un(t)}, forming a basis for , for all t. These + Bθ (θ, R)(xs + Y R) (34) basis vectors can be conveniently constructed as follows: the 2 vector u1(t) is chosen as the unit vector tangent to the limit D2 ∂B(x + YR) b (θ, R) = κ vT s B(x + YR) (35) cycle at any t θ 2 1 ∂x s a(xs(t)) T u1(t) = (28) Bθ(θ, R) =D κ v1 B(xs + YR) (36) |a(xs(t))| 5

T 0 L(θ) = − Z Y (37) frequency shift aθ becomes negligibly small, and all points rotate with a uniform angular frequency. In many of the T 0 aR(θ, R) =Z a(xs + YR) − Y R aθ(θ, R) (38) practical applications noise perturbations are small if compared to deterministic effects, that is, aˆθ(θ, R), bθ(θ, R) and T 0 0 aˆR(θ, R) = − Z Y R aˆθ(θ, R) + Y BR(θ, R)Bθ(θ, R) Bθ(θ, R) can be considered as perturbation terms. They either include some explicit small parameter, or the condition D 1 1 2 00 00 + Bθ (θ, R)(xs + Y R) (39) holds. As a consequence, the stationary distribution for the 2 angle is expected to remain close to the uniform distribution T 0 p (θ) = 1/(2π), that describes the phase diffusion process bR(θ, R) = − Z Y R bθ(θ, R) st in a nonlinear oscillator with uniform angular frequency [18], D2 ∂B(x + YR) + ZT s B(x + YR) (40) [27]. 2 ∂x s Similar considerations can be made for the amplitude devi- T 0 T BR(θ, R) = − Z Y R Bθ(θ, R) + D Z B(xs + YR) ation SDE (32). It can be shown that (see [27, Theorem 3]) (41) 2 and L(θ)R + aR(θ, R) = DR + O(R ) (44) T 0 −1 κ = r + v1 Y R (42) where D = diag[ν2, . . . , νn] is a diagonal matrix whose The SDEs (31)-(32) describe phase noise in nonlinear entries are the Floquet characteristic exponents, with the ex- oscillators with colored noise as a drift-diffusion process. ception of the structural one ν1 = 0. The amplitude deviation The responsibility of random fluctuations to phase diffusion dynamics is the balance of two competing forces: random does not come as a surprise, since, contrary to the amplitude, fluctuations drive the system out of the limit cycle, while the phase deviation does not have a self-limiting mechanism. asymptotic stability of the limit cycle implies that the system is Phase deviations are not damped, and may eventually grow continuously pushed toward the periodic orbit. Electronic sys- unbounded as time passes. However, noise is also responsible tems are usually strongly stable, meaning that Re{νi} 0 for for phase drift, that is, it produces a shift in the position of all i = 2, . . . , n, and as a consequence amplitude fluctuations the peaks of the oscillator frequency spectrum. Because of the remain confined to a small neighborhood of the limit cycle. nonlinear response of the oscillator, random forces applied at Thus we can linearize the amplitude deviation SDE around the a certain angle are amplified, while other are reduced. This noiseless solution R = 0, and after averaging with respect to results in a net, non null contribution to the expected angular the phase stationary distribution pst(θ) = 1/(2π) we obtain frequency. Noise induced frequency shift is also observed in nonlinear oscillators subject to white Gaussian noise [17], [18], dRt = (MRt + m) dt + n dWt (45) [27], but in presence of colored noise there is an additional shift contribution, represented by the term bθ, that can be where (as usual ∂aˆR/∂R, ∂bR/∂R and ∂BR/∂R are the ascribed to the finite correlation time of the noise source. Jacobian matrices with respect to R)

∂aˆ ∂b V. PHASE EQUATION M =D + R + R (46) ∂R ∂R The phase and amplitude deviation SDEs (31)-(32) have θ θ the same solution as the Stratonovich SDE (25) or the Itoˆ m = haˆRiθ + hbRiθ (47) SDE (26), that in turn are characterized by the same statistical n = hB i properties of the solution of the SDEs (5)-(6). Since (31)- R θ (48) (32) are exact, they are not easier to solve than the white noise approximated SDE (26). However, they can be used and to derive a phase reduced model [26], [29], [30] that in Z 2π 1 Z 2π turn can form the basis to find useful, albeit approximate, hf(θ)i = f(θ) p (θ) dθ = f(θ) dθ (49) θ st 2π results. The main advantage of a phase reduced model is 0 0 that methods for Markovian systems, e.g. Fokker-Planck and In general, the solution of the linear SDE (45) is not a Gaussian Kolmogorov equations, can be (comparatively) easily applied, process, but the vector of the expected values µ(t) = hRti and the obtained equations can be more easily solved, being T and the matrix of second moments P (t) = hRt Rt i can be one dimensional (single variable). found solving the linear ODE [31] To derive a simplified phase equation, we exploit a stochastic averaging technique. First we observe that if the Floquet dµ =Mµ + m (50) basis is used as vectors ui, vj, then dt 2 dP aθ(θ, R) = O(R ) (43) =MP + PM T + mµT + µmT + nnT (51) dt where O(R2) denotes terms quadratic in the amplitude deviation components, see [27, Theorem 3] for a proof. Then Finally, the first and second moment are used to obtain a in the neighborhood of the limit cycle the “deterministic” phase reduced equation. Expanding the terms of the phase 6

SDE (31) in Taylor series around R = 0, and averaging with Fully neglecting the amplitude fluctuations may be a reason- respect to the amplitude deviation, yields able approximation for strongly stable oscillators, however a more detailed analysis shows that in some cases the amplitude X ∂aˆθ ∂bθ dθ = 1 +a ˆ + b + + µ noise impacts on the cycle frequency inducing a non-negligible θ θ ∂R ∂R i i i i shift [11], [17], [18], especially for some autonomous systems 2 2 2 1 X ∂ aθ ∂ aˆθ ∂ bθ exploited in computational biology and neuroscience [19]. + + + P dt 2 ∂R ∂R ∂R ∂R ∂R ∂R ij i,j i j i j i j VI.EXAMPLES 2 X ∂Bθ 1 X ∂ Bθ + B + µ + P dW (52) A. Stuart-Landau oscillator with colored noise θ ∂R i 2 ∂R ∂R ij t i i i,j i j As a first example we consider a Stuart-Landau oscillator with colored noise. The reason to choose such a simple system where the functions a , aˆ , b ,B and their derivatives are θ θ θ θ is twofold. First, most of the analysis can be made analyti- evaluated at (θ, 0), and R denotes the i-th component of i cally, making the example useful to illustrate the theory and vector R. techniques described in the previous sections. Second, because We compare the phase equation (52) with the analogous many of the equations admit of an exact solution, the example equations obtained for a nonlinear oscillator subject to white permits to assess the accuracy of exploited approximations. Gaussian noise [18], [27]. Apart from 1 that represents the The state equations are the following oscillator’s normalized angular frequency, the terms in the first 2 two rows describe a frequency shift, whereas the terms in the dφ = α − βρ + ρ ηt dt last row describe a diffusion. aˆθ and its derivatives resolve 3 2 dρ = (ρ − ρ + ρ ηt)dt (56) the correlation between the phase and noise increments. These terms were already discussed for a nonlinear oscillator subject τdηt = −ηt dt + D dWt to white Gaussian noise [18], [27]. By contrast, bθ and its where α and β are real parameters that define the oscillator derivatives describe the different action that colored noise free running frequency. exerts on the phase with respect to white noise only, due to Applying the methodology described in section III we the non null noise correlation time. obtain the following SDE with modulated white Gaussian It is worth noticing that in the weak noise limit, if higher or- noise der contributions of amplitude fluctuations and the correlation 2 D 2 resolving term are neglected (implying µ = 0,P = 0 and dφ = α + − β ρ dt + D ρ dWt i ij 2 (57) aˆ (θ, R) = 0, respectively), then the simplified phase equation θ 2 3 2 is obtained dρ = ρ + D − 1 ρ dt + D ρ dWt We can now take advantage of the particularly simple structure dθ = [1 + b (θ, 0)] dt + B (θ, 0)dW (53) θ θ t of the Stuart-Landau system. Because the SDE for the ampli- Eq. (53) is the equivalent of the phase equations derived in tude is independent on the phase, the Fokker-Planck equation [4], [6] for the case of colored noise, where bθ(θ, 0) is a zero for the amplitude is single variable order approximation of the frequency shift effect produced by ∂p ∂ D2 ∂2 = − ρ + D2 − 1 ρ3 p + ρ4p (58) the finite noise correlation time. ∂t ∂ρ 2 ∂ρ2 In order to compare our results against previous literature on phase noise in oscillators subject to colored noise sources, The stationary distribution can be found analytically we consider here the approach developed in [9] where higher −2(1+ 1 ) 1 p (ρ) = N ρ D2 exp − (59) order contributions of amplitude fluctuations are neglected. st D2ρ2 For a strongly stable limit cycle, fluctuations are expected where N is a constant determined through a normalization to keep the trajectory in a small neighborhood of the limit +∞ condition R p (ρ)dρ = 1. It is worth noticing that the cycle, so that amplitude noise plays no influence on the phase 0 st same result cannot be obtained if the original problem is dynamics. The noisy solution is then approximated as a time considered, because in the SDE (56) the amplitude equation shifted version of the noiseless limit cycle x = x (θ ), and t S t and the Ornstein-Uhlenbeck process are coupled. a phase equation is readily derived (see [9], eq. (3)1). For our The theoretical prediction (59) is compared to the amplitude system (5), (6) the phase model in [9] reads stationary distribution obtained through numerical integration vT (θ) B(x (θ)) of (56) in figure 1. Milstein numerical integration scheme has dθ = 1 + 1 S η dt (54) r(θ) t been used in the simulation, and the probability to find the amplitude in the interval ρ + dρ has been evaluated as the τdηt = − ηt dt + D dWt (55) fraction of time spent in that interval divided by the total simulation length. As expected, the accuracy of the white Notice that this phase equation is further approximated in [9] noise approximation increases as the noise correlation time to derive the noise spectra. τ decreases. 1The division by r(θ), absent in [9], is a normalization required to guarantee In absence of noise the Stuart-Landau oscillator admits of T that the oscillator’s free running frequency is equal to one. an asymptotically stable limit cycle xs(t) = [(α − β)t, 1] . 7

2.5 2 0.18 0.18 Reference (56) Reference (56) 2 Analytical (59) Analytical (59) 0.17 0.17 1.5 0.16 0.16 1.5 1 0.15 0.15 1 0.14 0.14 0.5 0.5 0.13 0.13

0 0 0.12 0.12 0 1 2 3 0 1 2 3 0 2 4 6 0 2 4 6

Figure 1. Stationary distribution for the amplitude of a Stuart-Landau Figure 3. Stationary distribution for the phase of a Stuart-Landau oscillator, oscillator, for different values of the correlation time τ. Blue bars are obtained for two different values of D. Left: D = 0.2. Right D = 0.5. Parameters from a numerical solution of the oscillator subject to colored noise (56). The are α = 4, β = 2. red line is the theoretical stationary distribution (59) obtained with the white noise approximation. On the left the results for τ = 0.25, on the right for τ = 0.1 α = 4 β = 2 D = 0.5 . Other parameters are , , . Figure 2 shows the comparison between the amplitude deviation stationary distribution for the full system (59), and 5 its counterpart for the linearized system (63), for different values of D. As expected, the distribution of the linearized 4 system approximates well the full distribution for small values of D. The stationary distribution for the phase, obtained using 3 numerical integration, is shown in figure 3 for two different D 2 values of . It confirms that the uniform distribution is a good approximation even for fairly large values of D. 1 Solving equations (50), (51), we find the first two moments D2 0 hRi = (64) -0.5 0 0.5 1 2 − 3D2 2 2 2 2D hRi + D Figure 2. Stationary distribution for the amplitude deviation of a Stuart- hR i = 2 (65) Landau oscillator, for different values of D. Solid and dashed blue lines are 4 − 6D the stationary distributions (59) and (63), respectively, for D = 0.25. Solid Finally, taking stochastic expectation on both sides of the first and dashed red lines are the stationary distributions (59) and (63), respectively, 3 for D = 0.5. of (60) and neglecting O(R ) terms we obtain the expected angular frequency 2 T T dθ 1 D 2 The Floquet vectors are u1(t) = [1, 0] , u2(t) = [β, 1] , =1 + (1 − 2β) + D (1 − 3β)hRi T T dt α − β 2 while the co-vectors are v1(t) = [1, −β] , v2(t) = [0, 1] . It is straightforward to derive the phase and amplitude deviation D2 + (1 − 3β) + 2β hR2i (66) equations 2 1 D2 dθ = 1 + − βR + − β (1 + R)2 A comparison with [9] can be made making explicit (54) α − β 2 1 − β dθ = 1 + η dt (67) α − β t − β D2 − 1 (1 + R)3 dt τdηt = − ηt dt + D dWt (68) D and taking the stochastic expectation on both sides of (67). + (1 + R)[1 − β(1 + R)]dW (60) α − β t Using (3) we find the expected angular frequency dθ − t 2 3 2 = 1 + η0 e τ (69) dR = 1 + R + D − 1 (1 + R) dt + D(1 + R) dWt dt (61) Therefore, according to the model in [9], noise has no influence at all on the asymptotic expected angular frequency. The linearized SDE for the amplitude is Figure 4 shows the expected normalized angular frequency 2 2 for the Stuart-Landau oscillator with colored noise (56), the dR = −2 + 3D R + D dt + D dWt (62) equivalent system with white noise (57), and the theoretical and the stationary distribution obtained solving the associated predictions (69) and (66), as functions of D. Under the Fokker-Planck equation is hypothesis that the system is ergodic, the normalized expected frequencies for systems (56) and (57) have been obtained 3D2 − 1 pˆ (R) = Nˆ exp R2 + 2R (63) through numerical integration, using the time average st D2 dθ 1 φ(t2) − φ(t1) where Nˆ is the normalization constant. = (70) dt α − β t2 − t1 8

1.02 2

1.5 1

1 0.98

0.5 0.96

0 0.94 -10 -5 0 5

0 0.1 0.2 0.3 0.4 Figure 5. PDF p(θ − t, t) at three different time instants. Solid lines: PDF with hRi and hR2i given by (64) and (65), respectively. Dashed lines: PDF Figure 4. Expected normalized angular frequency for a Stuart-Landau obtained neglecting amplitude ﬂuctuations, that is imposing hRi = hR2i = 0. oscillator versus D. Blue lines: Numerical result for the system with colored Other parameters are α = 4, β = 2, D = 0.4. noise (56). Solid: τ = 0.5. Dashed: τ = 0.1. Red line: numerical result for the equivalent system with white Gaussian noise (57). Black dotted line: Asymptotic theoretical prediction (69). Black solid line: Theoretical prediction 15 (66). Parameters are α = 4, β = 2. 10

5 for t2 t1. An analysis of the phase equation (60) provides further 0

information. Averaging over the amplitude, substituting (64) (dB/Hz) PSD and (65), and neglecting O(R3) terms, proves that the angle -5 variable is well approximated by a Brownian motion with -10 2 drift θ(t) = µt + σWt, where 0.2 0.25 0.3 0.35 0.4 1 D2 µ =1 + (1 − 2β) + D2(1 − 3β)hRi (71) α − β 2 Figure 6. Power spectral density for the orbital noise component xs of the Stuart–Landau oscillator computed using the Welch’s method. Blue line: PSD D2 for the full system with colored noise (56), noise intensity D = 0.4 and noise + (1 − 3β) + 2β hR2i (72) correlation time τ = 0.5. Black solid line: PSD for the full system without 2 noise. Red line: PSD obtained from the reduced phase equation. Black dashed D dθ E line: expected angular frequency dt given by (66). Parameters are α = 4 D and β = 2. σ = 1 − β + (1 − 2β)hRi − βhR2i (73) α − β

For the sake of simplicity we assume a perfectly localized show excellent agreement around the carrier frequency, while initial condition θ(0) = 0, and that θ ∈ (−∞, +∞), with the system with colored noise shows a significantly reduced boundary conditions p(±∞, t) = 0. Then the phase has power content at high frequency (not shown in this figure). 2 a normal distribution p(θ, t) ∼ N(µt, σ t), and the auto- The black solid line is the PSD for the system without noise, correlation is here shown to put in evidence the frequency shift induced s 4 min(t, s) by noise. PSDs were obtained considering 10 oscillations, R(θ(t), θ(s)) = (74) divided into 228 points, corresponding to a sampling rate of max(t, s) 5369 samples per second for the discrete Fourier transform Figure 5 shows the PDF p(θ − t, t) for the phase deviation calculation. Finally, the black dotted line marks the theoretical (i.e. the difference between θ(t) and the phase in absence of prediction for the expected angular frequency hdθ/dti as given noise) at three different time instants. by (66). The power spectral density (PSD) for the orbital noise component, calculated using Welch’s method together with B. van der Pol oscillator with colored noise an average over 200 realizations of the corresponding time- domain process, is shown in figure 6. The blue line is the As a second example we consider the nonlinear oscillator PSD for the orbital noise xs = ρ(t) cos φ(t) in presence of the (van der Pol) shown in figure 7. The nonlinear resistor NR colored noise source as defined in (56). The noise correlation is assumed to be noiseless, with characteristic iG = g(v) = time is τ = 0.5 and the noise intensity is D = 0.4. The v3/3−v. The random source on the right models environment red line is the PSD for the orbital noise component xs = and internal noise. Using Kirchhoff current law it is straight- (1 + µ) cos[(α − β)θ(t) + βµ], where θ is the solution of the forward to derive the state equations reduced phase equation (52), again for D = 0.4. The two PSDs dv =u dt0 (75) 2This process is sometime referred to simply as Brownian motion, whereas 1 1 0 1 0 0 the case µ = 0 is called Standard Brownian motion, in accordance with the du = − v − g (v)u − sn(t ) dt (76) corresponding PDFs. LC C C 9

iG i L + iC iN NR L v C −

Figure 7. Second order nonlinear oscillator.

where iN is the integral of the stochastic process sn. With the change of variables √ 1 0 t = √ t x1 = v x2 = LC u LC and assuming that the random source sn is a colored noise Figure 8. Probability density function for the van der Pol oscillator with modulated by the current through the capacitor colored noise. Parameters are α = 0.5, D = 0.5. Noise correlation time is r τ = 0.5. C s (t) = u η n L t we obtain the state equations

dx1 = x2 dt 2 dx2 = −x1 + α 1 − x1 x2 + x2ηt dt (77)

τdηt = −ηt dt + D dWt where α = pL/C. Transformation to the equivalent system with white noise yields

dx1 = x2 dt D2 dx = −x + α 1 − x2 x + x dt + D x dW 2 1 1 2 2 2 2 t (78) Figure 9. Probability density function for the van der Pol oscillator with Figures 8 and 9 show the PDF p(x1, x2, t), at the same colored noise. Parameters are α = 0.5, D = 0.5. Noise correlation time is time instant, for the van der Pol oscillator with colored noise τ = 0.1. (77) and noise correlation time τ = 0.5 and τ = 0.1, respectively. Figure 10 show the PDF for the equivalent system • The functions M, m, N and n given by (46)-(48) are with white Gaussian noise (78). The PDF has been computed calculated, using the uniform distribution pst(θ) = 1/2π from numerical simulations, the same initial condition has for averaging. Equations (50) and (51) are solved to ﬁnd been used in all cases3. hRii and hRiRji for all i, j. Because the limit cycle and the Floquet vectors cannot be • The reduced phase equation is written and can be an- found analytically for the van der Pol oscillator, we resort alyzed to determine the expected normalized angular to semi analytical techniques and numerical methods for the analysis. In particular, we have developed a methodology based on the following steps: • First, the limit cycle of the noiseless system is determined in a semi analytical form using the Harmonic Balance technique [32], [33]. • The semi analytical expression of the limit cycle is used to determine the Floquet vectors and co-vectors. Because the example under investigation is a second order system, Floquet vectors an co-vectors are computed using the formulas given in [34]. For higher order systems, they can be found exploiting efﬁcient numerical techniques [33], [35]–[37]. • The limit cycle, and the Floquet vectors and co-vectors are used to determine the functions in equations (33)-(42).

3Parameter values are conveniently chosen to simplify calculations and to Figure 10. Probability density function for the van der Pol oscillator with highlight higher order contributions of noise. white Gaussian noise. Parameters are α = 0.5, D = 0.5. 10

1.01 30 30

1 20 20

10 10 PSD(dB/Hz) 0.99 PSD (dB/Hz)

0 0 0.98 0.14 0.15 0.16 0.17 0.14 0.15 0.16 0.17

0.97 Figure 12. Power Spectral Density for numerical solution of a van der Pol oscillator computed using Welch’s algorithm. Left: PSD for the system with 0.96 colored noise (77), τ = 0.5. Right: PSD for the equivalent system with white 0 0.1 0.2 0.3 0.4 Gaussian noise (78). The black dashed line identifies the expected frequency obtained using our theoretical model. Parameter α = 0.5. Figure 11. Expected normalized frequency for a van der Pol oscillator versus D. Blue lines: Numerical result for the system with colored noise (77). Solid: τ = 0.5. Dotted: τ = 0.1. Red line: numerical result for the leads to two equivalent stochastic differential equations for equivalent system with white Gaussian noise (78). Black dotted line: expected frequency obtained through numerical integration of (54), (55). Black solid the Itoˆ and the Stratonovich interpretations. Since the two line: Theoretical prediction using the proposed method. Parameter α = 0.5. equations are equivalent, they have the same solution and therefore which one should be used is just a matter of personal preference. An alternative description in terms of stochastic frequency, frequency shift, diffusion constant and so on. differential equations for the phase and the amplitude deviation Figure 11 shows the expected normalized angular frequency is derived for the transformed system. The phase variable used for the van der Pol system with colored noise (77), as a coincide locally, in the neighborhood of the unperturbed limit function of D. The expected frequency exhibits little depen- cycle, with the asymptotic phase defined using the concept of dence on the noise correlation time, in fact the blue curves isochrons. Using stochastic averaging, a reduced phase model (the solid curve corresponds to τ = 0.5, while the dashed is derived, where the system dynamics is described only in one to τ = 0.1) are very close. The red line represents terms of the phase variable. the expected normalized angular frequency for the equivalent The reduced phase model describes phase noise as a drift- system with white Gaussian noise (78). The curve has been diffusion process. The shift in the expected frequency is related determined through numerical integration of the full phase- to the variance of the colored noise and noise correlation time. amplitude deviation SDEs. Finally the black curves represents A numerical procedure is presented for the solution of the the theoretical predictions. The dotted line is the expected phase equation, that provides more accurate results than other angular frequency determined through numerical integration previously proposed models. of (54), (55) [9]. The solid line is the theoretical prediction given by our phase reduced model, obtained using the methods described above. REFERENCES As a further confirmation, we have computed the power [1] D. B. 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