Considering Noise Orbital Deviations on the ... - Semantic Scholar
438
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 6, JUNE 2006
Considering Noise Orbital Deviations on the Evaluation of Power Density Spectrum of Oscillators Adriano Carbone and Fabrizio Palma
Abstract—This brief presents a new application of the theory of noise in free running oscillators based on the Floquet eigenvector decomposition. In oscillators, all orbital deviations contribute to the power density spectrum (PDS) as much as the “phase” term, usually considered. Each orbital deviation component shows a time evolution depending on the related Floquet eigenvalue, which thus characterizes statistical properties related to that component. Orbital deviations are partially correlated, due to their common origin from noise sources, thus also correlation terms are considered in the evaluation of the PDS. In this brief, we introduce a simplified method of calculation of PDS and apply it to an example of RLC negative resistance oscillator. Results show the relevance of orbital deviations in PDS in presence of stationary noise, these contributions becomes particularly relevant when noise is cyclostationary.
problematic of contribution to PDS from orbital deviation components is depicted by an example RLC oscillator. Results obtained using our simplified approach illustrate degradation of PDS due to the orbital deviation, and their particular relevance in the case of cyclostationary noise. These results are verified by straightforward multistep integration of nonlinear differential equations.
Index Terms—Floquet eigenvector decomposition, orbital deviation, oscillator noise, oscillators.
(1)
I. INTRODUCTION
S
EVERAL efforts have been made in the past few years in order to characterize effects of noise in oscillators. Hajimiri and Lee [1] proposed a theory that acknowledges the periodically time-varying nature of the problem. It was also shown [2], [3] that noise in oscillators must be decomposed into its components along the Floquet eigenvectors of the system in order to obtain a correct description of the system evolution. In these models, displacements out of the orbit of the state of a stable oscillator (in the following orbital deviations) remain small if noise is small. Moreover, displacements along the orbit (phase deviations) grow unbounded. Demir et al. [4] presented the asymptotic statistical properties of the phase deviation process and related them to the power-density spectrum (PDS) of the oscillator output. Nevertheless, autocorrelation of the oscillator output is sensitive to short time behavior of both phase deviation and orbital deviations. For this reason, despite the unbounded growth of the phase, also the orbital deviations may significantly influence the spectrum of the oscillator [2]. Recently, Magierowski and Zukotynski [5] proposed a nonlinear stochastic analysis, including effects of orbital deviations but limiting the analysis to the case of quasi-sinusoidal oscillations. In this brief, we introduce a simplified approach to the determination of statistical properties of the orbital deviation components, basing on Floquet eigenvector decomposition. The
Manuscript received February 11, 2005; revised November 3, 2005. This paper was recommended by Associate Editor B. C. Levy. The authors are with the Dipartimento di Ingegneria Elettronica, Università di Roma “La Sapienza,” Rome 00184, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TCSII.2006.873527
II. MATHEMATICAL PRELIMINARIES The dynamic of any autonomous system without undesired perturbations can be described by a system of differential equations as
where is the -vector of state variables, is the -vector of their time-derivatives and is a -vector of functions of that satisfies the existence and uniqueness conditions for the initial value problem. Equation (1) describes an oscillator if and only if it has at least one periodic solution that describes a closed trajectory in the state space. In respect of this trajectory, the system has to be asymptotically orbitally stable in order to describe a physical oscillator. Note that a linear system never satisfies this last condition and so a physical oscillator is surely nonlinear. With small noise perturbations, the system is described by (2) where is a -vector of stochastic processes, each deis a scribing a different noise source and matrix that maps noise sources into the state variables. We assume that the autocorrelation matrix of is diagonal with elements along the diagonal equal to unitary amplitude Dirac distribution. In an electronic system, the noise induced by electronic devices may vary with time. We assume that the information about the cyclostationary behavior is included in the coefficients of . If noises are “small” [noise projections by ] matrix are small with respect to the time-derivatives of state variables) it is possible to study oscillator’s behavior around its stable trajectory using the system linearized along the orbit (3) where describes the deviation from the stable trajectory is the Jacobian of evaluated on the stable trajectory. and By definition, and are periodically time-variant matrices and thus the system in (3) is linear with periodically time-variant
1057-7130/$20.00 © 2006 IEEE
CARBONE AND PALMA: NOISE ORBITAL DEVIATIONS ON THE EVALUATION OF PDS
coefficients (LPTV system). This particular kind of systems can be studied by means of the Floquet theory which generalizes the well known results of stationary linear system theory. In particular, the state transition matrix for LPTV systems and their forced response can be written, respectively, as [6]
439
The output of the oscillator is, in general, a linear combination of all the state variables. For the sake of simplicity, and with no lack of generality, we assume here that output is one of the state variables. of the noise components along the Floquet The vector by a suiteigenvectors, can thus be obtained by multiplying , delayed by able periodic matrix
(4) (5) is a nonsingular, T-periodic matrix; and is a diagonal matrix containing the (Floquet) eigenvalues . The columns of are the system’s right eigenvectors (Floquet eigenvectors), they indicate the directions of independent dynamics. It can be demonstrated that the linearized system’s response is the sum of the evolutions along the directions of the Floquet eigenvectors. Each of this evolution is forced by the component of perturbation on that direction. where
III. EFFECT OF ORBITAL DEVIATION Decomposition of orbital deviations in components along the directions of Floquet eigenvectors simplifies the description of the system response, since dynamic of each component, referred to in the following as modal component, is independent. Kaertner [2] pointed out that the matrixes and of the system in (3) have to be considered translated by the amount of the overall phase deviation . In a stable oscillator always exists one Floquet eigenvalue equal to zero, we name it (and the corresponding eigenvector) as the “first” one. All the other eigenvalues have negative real part. The first eigenvector is tangential to the orbit, thus noise on its direction only affects the phase of the oscillator [2]. Since the first eigenvalue is zero, the variance of phase displacement, increases unbounded with time, so that a nonlinear differential equation must be solved for the phase deviation process. It was shown that, for approaching infinite, becomes gaussian white, with variance linearly increasing, [2]. With this condition can be considered uniformly distributed along the variable the oscillation period [7]. In the PDS characterization, one can assume that the condition of uniformly distributed random phase has already been reached. Starting from this condition, we evaluate output autocorrelation for limited time intervals, short enough that further phase deviation remains limited, and the growth equation of can be considered linear. We note that investigating oscillator behavior for short time intervals, corresponds to evaluate PDS only at frequencies far enough from the carrier. Under these conditions, we may evaluate PDS assuming that all modal components undergo to a linear evolution. This allows calculation of statistical properties under small noise perturbations condition. are white, unWe assume that noise stochastic processes correlated. In general, noise components arising from one particular noise source are correlated. As a consequence, time evolutions of modal components show cross-correlations that have to be taken into account.
(6) as . In the following, we will refer to the rows of Noise projections are therefore cyclostationary white prois defined as cesses. The correlation matrix of (7) where superscript H indicates the hermitian operator. In order to , we note from (6) that terms are funcevaluate and , which are station of two stochastic processes, tistically independent. The statistical average operator in (7) can thus act separately on the two variables. For each term of matrix , we first calculate
(8) is referred to the stowhere the statistical average operator chastic variable is the Dirac distribution, and is the -dimension identity matrix. Hereafter, will be omitted. . Since on its own is a stochastic process, so is may vary along the orbit. As indicated in (8) From the knowledge of the terms , it is possible to obtain the statistical properties of the modal components. The terms of correlation matrix of the modal components are given by (9) and are the terms on the diagonal matrix of the impulse responses, , that describe dynamics on the different Floquet eigenvectors in (4), and do not depend on . We now introduce the following approximation: (10) In Appendix A, we show from a result of [2], how the approx. imation is valid for can be exOnce assumed to be constant with time pressed as
(11)
440
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 6, JUNE 2006
In short, each term of the correlation matrix of the modal components results to be the convolution of the averaged correlation function of the corresponding noise components, by the correlation function of the impulse responses on the directions of Floquet eigenvectors. Using results from (4) in (11) and Fourier transforming, we obtains for the generic terms
Fig. 1. (a) Oscillator. (b) Nonlinear generator transcharacteristic.
autocorrelation can thus be calculated first with an average with respect to , which leads to (12) (15) indicates the unit step function. We note that In (12), some terms are complex, but their sum is always real (as it must be), in fact
where and are elements of matrix , periodic. Applying the statistic average operator with respect to on (15), and being uniformly distributed along the period, we obtain
(16)
since
(13)
The integral in (16) represents the correlation of -components of the ith and jth eigenvectors. The Fourier transform of this term shows the same harmonic frequencies of the oscillator. The resulting output PDS of the noisy oscillator is the Fourier transform of the expression (16) as established by the Wiener–Kintchine theorem. Since the deviation are assumed small it results to be the sum of single term contributions translated around the harmonic frequencies of the oscillator.
(14)
IV. SIMULATION RESULTS
, we have
which is real. We note that, in this approximation, variance of , the first as predicted modal component, grows uniformly with rate in [2], while the variances of the others modal components are constant. The output noise can be obtained by projection of the orbital , the eigenvector madeviations onto the state variables. trix, is the projection matrix. Eigenvectors rotate in the period together with the state along the stable trajectory of the oscillator, the modal components thus represents a modulation of the oscillator output. These modulations have three fundamental characteristics. 1) They are small index modulations. 2) The autocorrelation matrix is constant along the orbit. 3) They are partially correlated because of their origin from the same physical phenomenon. Due to the assumed independence of on the phase deviation (property 2), the projected orbital deviation, , can be treated as a function of two stochastic variables, respectively and . Its
We apply the former theoretical results to the simple oscillator reported in Fig. 1(a), constituted by an LC parallel resonator, loaded with a conductance G. Resonator is connected to a nonlinear current generator controlled by the voltage drop . In Fig. 1(b), the piecewise-linear I–V at the two terminals, characteristic of the generator is reported. In the central zone, and , generator behaves as a negative conbetween and below the generator current is ductance. Above constant. In order to simplify the interpretation of results the was chosen for the transconductance, particular value of in the negso that the total conductance of the resonator is as the ative conductance region, and outside. We choose output variable. Floquet eigenvalues, and the eigenvectors at a given must be calculated respectively as eigenvalues, and eigenvectors of the Monodromy matrix. Monodromy matrix can be defined as the Jacobian of the nonlinear state transition function through one period of oscillation, starting at time . In a piecewise-linear system state variables evolve linearly within each zone. During one cycle the state passes through five
CARBONE AND PALMA: NOISE ORBITAL DEVIATIONS ON THE EVALUATION OF PDS
441
TABLE I
Fig. 3. PDS simulation with periodic noise source (continuous lines) compared with the previous case of stationary noise (dashed lines), for two values of . Noises variance showed in the inset, and compared to the oscillation amplitude.
Q
Fig. 2. PDS (continuous), I eigenvector contribution (dotted), II eigenvector contribution (dashed), crosscorrelation terms contribution (dotted-dashed) for = 5 and = 100.
Q
Q
linear zones so that the Monodromy matrix can be defined as the product of the state transition matrices of each zone
(17) are related to the stawhere the state transition matrixes . Each tionary linear system in th time interval time interval has duration . For simplicity, the only noise source considered is the thermal noise due to the conductance . In Table I we report the calculated two Floquet eigenvalues, obtained normalizing the coefficients and the coefficients in (11) by noise variance. One can note a growth trend for and , while the quality factor the crosscorrelation terms decreases. This is due to the deviation from the quadrature condition of the two eigenvector contributions to the output, which occurs when the oscillator present larger distortion. Fig. 2 reports the simulated PDS (continuous lines) for, reand , reporting also the different conspectively, tributions. We note that in the region of frequencies far from the dB/decade carrier, where all the terms contribute with their term zone, the power density is twice the contribution of the term remains predominant in ( times in the general case). the nearer the carrier region, while it can be found a transition region. Also note that in the example the cut frequency of the second modal component moves toward the carrier when the quality factor increases. This can be justified observing that a higher implies lower losses with oscillator recovering the stable trajectory more slowly. Slower recovering dynamic is related to a second eigenvalue closer to zero. As the second eigenvalue becomes closer to the value zero, the cut frequency in expressions (12)
reduces, becoming closer to the carrier, while the low frequency value increases. The overall effect is an increase of the relative relevance of the orbital noise at frequency closer to the carrier. An ideal LC oscillator without any loss (infinite ) does not recover the original trajectory and phase noise and orbital noise give exactly the same contribution at any frequency. The relevance of orbital deviations can be even higher when the noise process is cyclostationary. In Fig. 3 are reported PDS results from a simulation with periodic noise source (continuous lines) compared with the previous case of stationary noise (dashed lines), for two values of . Noise variance is nonnull is around its absolute peak values, as shown in only when the figure inset, where noise pulses are reported together with oscillation amplitude, and the stationary noise variance. In this case average of noise component onto the second eigenvector is much higher than the average projection onto the first eigenvector. This reflects on a reduction of PDS at low frequencies. The increased component onto the second eigenvector becomes relevant only at high offset frequencies, cut-off depending on the associated eigenvalue (thus on the ). This effect was also proven by nonlinear simulations, using a multi-step integration method. This approach required simulation times three order of magnitude longer than the proposed method. Moreover, in order to be able to neglect errors due to numerical integration, we choose to increase noise power by four order of magnitude, thus achieving only a qualitative comparison. . Shaping of PDS Results are reported in Fig. 4, for in presence of cyclostationary noise is confirmed. V. CONCLUSION We presented an analysis of the effects of orbital deviation driven by noise in oscillators, using a decomposition of this deviation onto Floquet eigenvectors of the system. Even if the phase grows unbounded, the components which remain limited may influence the overall PDS of oscillators.
442
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 6, JUNE 2006
is always negative, while the integral terms in (18) is always positive. The following inequality thus holds:
(20)
Constance of can be used to state the time independence of autocorrelation since
(21)
Q
Fig. 4. PDS simulation by multi-step method for = 10. Case of periodic noise source (continuous) compared with the stationary noise case (dashed). The PDS of oscillator without any noise source is also reported (dotted) in order to evaluate errors due to numerical calculation.
REFERENCES The expression of all contributes were calculated, showing the dependence on the Floquet eigenvalues and eigenvectors. Example pointed out the particular relevance of orbital deviation in presence of cyclostationary noise, and the influence of the quality factor of the resonant circuit. APPENDIX A Reference [2] calculates the value of variances of orbital de, for a periodic system as in (3). This result can be viations, used to evaluate percentage variation of the variances shown in . Since is (18) at the bottom of the page, with definite positive and eigenvalues real parts are non positive we can assume (19)
[1] A. Hajimiri and T. H. Lee, “A general theory of phase noise in electrical oscillators,” IEEE J. Solid-State Circuits, vol. 33, no. 2, pp. 179–194, Feb. 1998. [2] F. X. Kaertner, “Analysis of white and noise in oscillators,” Int. J. Circuit Theory Appl., vol. 18, pp. 485–519, 1990. [3] G. J. Coram, “A simple 2-D oscillator to determine the correct decomposition of perturbations into amplitude and phase noise,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 48, no. 7, pp. 896–898, Jul. 2002. [4] A. Demir, A. Mehrotra, and J. Roychowdhury, “Phase noise in oscillators: A unifying theory and numerical methods for characterization,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 47, no. 5, pp. 655–674, May 2000. [5] S. K. Magierowski and S. Zukotynski, “CMOS LC-oscillator phase noise analysis using nonlinear models,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 51, no. 4, pp. 664–677, Apr. 2004. [6] R. Grimshaw, Nonlinear Ordinary Differential Equation. New York: Blackwell Scientific, 1990. [7] A. Papoulis, Probability, Random Variables and Stochastic Processes. New York: McGraw Hill, 1991.
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 6, JUNE 2006
Considering Noise Orbital Deviations on the Evaluation of Power Density Spectrum of Oscillators Adriano Carbone and Fabrizio Palma
Abstract—This brief presents a new application of the theory of noise in free running oscillators based on the Floquet eigenvector decomposition. In oscillators, all orbital deviations contribute to the power density spectrum (PDS) as much as the “phase” term, usually considered. Each orbital deviation component shows a time evolution depending on the related Floquet eigenvalue, which thus characterizes statistical properties related to that component. Orbital deviations are partially correlated, due to their common origin from noise sources, thus also correlation terms are considered in the evaluation of the PDS. In this brief, we introduce a simplified method of calculation of PDS and apply it to an example of RLC negative resistance oscillator. Results show the relevance of orbital deviations in PDS in presence of stationary noise, these contributions becomes particularly relevant when noise is cyclostationary.
problematic of contribution to PDS from orbital deviation components is depicted by an example RLC oscillator. Results obtained using our simplified approach illustrate degradation of PDS due to the orbital deviation, and their particular relevance in the case of cyclostationary noise. These results are verified by straightforward multistep integration of nonlinear differential equations.
Index Terms—Floquet eigenvector decomposition, orbital deviation, oscillator noise, oscillators.
(1)
I. INTRODUCTION
S
EVERAL efforts have been made in the past few years in order to characterize effects of noise in oscillators. Hajimiri and Lee [1] proposed a theory that acknowledges the periodically time-varying nature of the problem. It was also shown [2], [3] that noise in oscillators must be decomposed into its components along the Floquet eigenvectors of the system in order to obtain a correct description of the system evolution. In these models, displacements out of the orbit of the state of a stable oscillator (in the following orbital deviations) remain small if noise is small. Moreover, displacements along the orbit (phase deviations) grow unbounded. Demir et al. [4] presented the asymptotic statistical properties of the phase deviation process and related them to the power-density spectrum (PDS) of the oscillator output. Nevertheless, autocorrelation of the oscillator output is sensitive to short time behavior of both phase deviation and orbital deviations. For this reason, despite the unbounded growth of the phase, also the orbital deviations may significantly influence the spectrum of the oscillator [2]. Recently, Magierowski and Zukotynski [5] proposed a nonlinear stochastic analysis, including effects of orbital deviations but limiting the analysis to the case of quasi-sinusoidal oscillations. In this brief, we introduce a simplified approach to the determination of statistical properties of the orbital deviation components, basing on Floquet eigenvector decomposition. The
Manuscript received February 11, 2005; revised November 3, 2005. This paper was recommended by Associate Editor B. C. Levy. The authors are with the Dipartimento di Ingegneria Elettronica, Università di Roma “La Sapienza,” Rome 00184, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TCSII.2006.873527
II. MATHEMATICAL PRELIMINARIES The dynamic of any autonomous system without undesired perturbations can be described by a system of differential equations as
where is the -vector of state variables, is the -vector of their time-derivatives and is a -vector of functions of that satisfies the existence and uniqueness conditions for the initial value problem. Equation (1) describes an oscillator if and only if it has at least one periodic solution that describes a closed trajectory in the state space. In respect of this trajectory, the system has to be asymptotically orbitally stable in order to describe a physical oscillator. Note that a linear system never satisfies this last condition and so a physical oscillator is surely nonlinear. With small noise perturbations, the system is described by (2) where is a -vector of stochastic processes, each deis a scribing a different noise source and matrix that maps noise sources into the state variables. We assume that the autocorrelation matrix of is diagonal with elements along the diagonal equal to unitary amplitude Dirac distribution. In an electronic system, the noise induced by electronic devices may vary with time. We assume that the information about the cyclostationary behavior is included in the coefficients of . If noises are “small” [noise projections by ] matrix are small with respect to the time-derivatives of state variables) it is possible to study oscillator’s behavior around its stable trajectory using the system linearized along the orbit (3) where describes the deviation from the stable trajectory is the Jacobian of evaluated on the stable trajectory. and By definition, and are periodically time-variant matrices and thus the system in (3) is linear with periodically time-variant
1057-7130/$20.00 © 2006 IEEE
CARBONE AND PALMA: NOISE ORBITAL DEVIATIONS ON THE EVALUATION OF PDS
coefficients (LPTV system). This particular kind of systems can be studied by means of the Floquet theory which generalizes the well known results of stationary linear system theory. In particular, the state transition matrix for LPTV systems and their forced response can be written, respectively, as [6]
439
The output of the oscillator is, in general, a linear combination of all the state variables. For the sake of simplicity, and with no lack of generality, we assume here that output is one of the state variables. of the noise components along the Floquet The vector by a suiteigenvectors, can thus be obtained by multiplying , delayed by able periodic matrix
(4) (5) is a nonsingular, T-periodic matrix; and is a diagonal matrix containing the (Floquet) eigenvalues . The columns of are the system’s right eigenvectors (Floquet eigenvectors), they indicate the directions of independent dynamics. It can be demonstrated that the linearized system’s response is the sum of the evolutions along the directions of the Floquet eigenvectors. Each of this evolution is forced by the component of perturbation on that direction. where
III. EFFECT OF ORBITAL DEVIATION Decomposition of orbital deviations in components along the directions of Floquet eigenvectors simplifies the description of the system response, since dynamic of each component, referred to in the following as modal component, is independent. Kaertner [2] pointed out that the matrixes and of the system in (3) have to be considered translated by the amount of the overall phase deviation . In a stable oscillator always exists one Floquet eigenvalue equal to zero, we name it (and the corresponding eigenvector) as the “first” one. All the other eigenvalues have negative real part. The first eigenvector is tangential to the orbit, thus noise on its direction only affects the phase of the oscillator [2]. Since the first eigenvalue is zero, the variance of phase displacement, increases unbounded with time, so that a nonlinear differential equation must be solved for the phase deviation process. It was shown that, for approaching infinite, becomes gaussian white, with variance linearly increasing, [2]. With this condition can be considered uniformly distributed along the variable the oscillation period [7]. In the PDS characterization, one can assume that the condition of uniformly distributed random phase has already been reached. Starting from this condition, we evaluate output autocorrelation for limited time intervals, short enough that further phase deviation remains limited, and the growth equation of can be considered linear. We note that investigating oscillator behavior for short time intervals, corresponds to evaluate PDS only at frequencies far enough from the carrier. Under these conditions, we may evaluate PDS assuming that all modal components undergo to a linear evolution. This allows calculation of statistical properties under small noise perturbations condition. are white, unWe assume that noise stochastic processes correlated. In general, noise components arising from one particular noise source are correlated. As a consequence, time evolutions of modal components show cross-correlations that have to be taken into account.
(6) as . In the following, we will refer to the rows of Noise projections are therefore cyclostationary white prois defined as cesses. The correlation matrix of (7) where superscript H indicates the hermitian operator. In order to , we note from (6) that terms are funcevaluate and , which are station of two stochastic processes, tistically independent. The statistical average operator in (7) can thus act separately on the two variables. For each term of matrix , we first calculate
(8) is referred to the stowhere the statistical average operator chastic variable is the Dirac distribution, and is the -dimension identity matrix. Hereafter, will be omitted. . Since on its own is a stochastic process, so is may vary along the orbit. As indicated in (8) From the knowledge of the terms , it is possible to obtain the statistical properties of the modal components. The terms of correlation matrix of the modal components are given by (9) and are the terms on the diagonal matrix of the impulse responses, , that describe dynamics on the different Floquet eigenvectors in (4), and do not depend on . We now introduce the following approximation: (10) In Appendix A, we show from a result of [2], how the approx. imation is valid for can be exOnce assumed to be constant with time pressed as
(11)
440
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 6, JUNE 2006
In short, each term of the correlation matrix of the modal components results to be the convolution of the averaged correlation function of the corresponding noise components, by the correlation function of the impulse responses on the directions of Floquet eigenvectors. Using results from (4) in (11) and Fourier transforming, we obtains for the generic terms
Fig. 1. (a) Oscillator. (b) Nonlinear generator transcharacteristic.
autocorrelation can thus be calculated first with an average with respect to , which leads to (12) (15) indicates the unit step function. We note that In (12), some terms are complex, but their sum is always real (as it must be), in fact
where and are elements of matrix , periodic. Applying the statistic average operator with respect to on (15), and being uniformly distributed along the period, we obtain
(16)
since
(13)
The integral in (16) represents the correlation of -components of the ith and jth eigenvectors. The Fourier transform of this term shows the same harmonic frequencies of the oscillator. The resulting output PDS of the noisy oscillator is the Fourier transform of the expression (16) as established by the Wiener–Kintchine theorem. Since the deviation are assumed small it results to be the sum of single term contributions translated around the harmonic frequencies of the oscillator.
(14)
IV. SIMULATION RESULTS
, we have
which is real. We note that, in this approximation, variance of , the first as predicted modal component, grows uniformly with rate in [2], while the variances of the others modal components are constant. The output noise can be obtained by projection of the orbital , the eigenvector madeviations onto the state variables. trix, is the projection matrix. Eigenvectors rotate in the period together with the state along the stable trajectory of the oscillator, the modal components thus represents a modulation of the oscillator output. These modulations have three fundamental characteristics. 1) They are small index modulations. 2) The autocorrelation matrix is constant along the orbit. 3) They are partially correlated because of their origin from the same physical phenomenon. Due to the assumed independence of on the phase deviation (property 2), the projected orbital deviation, , can be treated as a function of two stochastic variables, respectively and . Its
We apply the former theoretical results to the simple oscillator reported in Fig. 1(a), constituted by an LC parallel resonator, loaded with a conductance G. Resonator is connected to a nonlinear current generator controlled by the voltage drop . In Fig. 1(b), the piecewise-linear I–V at the two terminals, characteristic of the generator is reported. In the central zone, and , generator behaves as a negative conbetween and below the generator current is ductance. Above constant. In order to simplify the interpretation of results the was chosen for the transconductance, particular value of in the negso that the total conductance of the resonator is as the ative conductance region, and outside. We choose output variable. Floquet eigenvalues, and the eigenvectors at a given must be calculated respectively as eigenvalues, and eigenvectors of the Monodromy matrix. Monodromy matrix can be defined as the Jacobian of the nonlinear state transition function through one period of oscillation, starting at time . In a piecewise-linear system state variables evolve linearly within each zone. During one cycle the state passes through five
CARBONE AND PALMA: NOISE ORBITAL DEVIATIONS ON THE EVALUATION OF PDS
441
TABLE I
Fig. 3. PDS simulation with periodic noise source (continuous lines) compared with the previous case of stationary noise (dashed lines), for two values of . Noises variance showed in the inset, and compared to the oscillation amplitude.
Q
Fig. 2. PDS (continuous), I eigenvector contribution (dotted), II eigenvector contribution (dashed), crosscorrelation terms contribution (dotted-dashed) for = 5 and = 100.
Q
Q
linear zones so that the Monodromy matrix can be defined as the product of the state transition matrices of each zone
(17) are related to the stawhere the state transition matrixes . Each tionary linear system in th time interval time interval has duration . For simplicity, the only noise source considered is the thermal noise due to the conductance . In Table I we report the calculated two Floquet eigenvalues, obtained normalizing the coefficients and the coefficients in (11) by noise variance. One can note a growth trend for and , while the quality factor the crosscorrelation terms decreases. This is due to the deviation from the quadrature condition of the two eigenvector contributions to the output, which occurs when the oscillator present larger distortion. Fig. 2 reports the simulated PDS (continuous lines) for, reand , reporting also the different conspectively, tributions. We note that in the region of frequencies far from the dB/decade carrier, where all the terms contribute with their term zone, the power density is twice the contribution of the term remains predominant in ( times in the general case). the nearer the carrier region, while it can be found a transition region. Also note that in the example the cut frequency of the second modal component moves toward the carrier when the quality factor increases. This can be justified observing that a higher implies lower losses with oscillator recovering the stable trajectory more slowly. Slower recovering dynamic is related to a second eigenvalue closer to zero. As the second eigenvalue becomes closer to the value zero, the cut frequency in expressions (12)
reduces, becoming closer to the carrier, while the low frequency value increases. The overall effect is an increase of the relative relevance of the orbital noise at frequency closer to the carrier. An ideal LC oscillator without any loss (infinite ) does not recover the original trajectory and phase noise and orbital noise give exactly the same contribution at any frequency. The relevance of orbital deviations can be even higher when the noise process is cyclostationary. In Fig. 3 are reported PDS results from a simulation with periodic noise source (continuous lines) compared with the previous case of stationary noise (dashed lines), for two values of . Noise variance is nonnull is around its absolute peak values, as shown in only when the figure inset, where noise pulses are reported together with oscillation amplitude, and the stationary noise variance. In this case average of noise component onto the second eigenvector is much higher than the average projection onto the first eigenvector. This reflects on a reduction of PDS at low frequencies. The increased component onto the second eigenvector becomes relevant only at high offset frequencies, cut-off depending on the associated eigenvalue (thus on the ). This effect was also proven by nonlinear simulations, using a multi-step integration method. This approach required simulation times three order of magnitude longer than the proposed method. Moreover, in order to be able to neglect errors due to numerical integration, we choose to increase noise power by four order of magnitude, thus achieving only a qualitative comparison. . Shaping of PDS Results are reported in Fig. 4, for in presence of cyclostationary noise is confirmed. V. CONCLUSION We presented an analysis of the effects of orbital deviation driven by noise in oscillators, using a decomposition of this deviation onto Floquet eigenvectors of the system. Even if the phase grows unbounded, the components which remain limited may influence the overall PDS of oscillators.
442
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 53, NO. 6, JUNE 2006
is always negative, while the integral terms in (18) is always positive. The following inequality thus holds:
(20)
Constance of can be used to state the time independence of autocorrelation since
(21)
Q
Fig. 4. PDS simulation by multi-step method for = 10. Case of periodic noise source (continuous) compared with the stationary noise case (dashed). The PDS of oscillator without any noise source is also reported (dotted) in order to evaluate errors due to numerical calculation.
REFERENCES The expression of all contributes were calculated, showing the dependence on the Floquet eigenvalues and eigenvectors. Example pointed out the particular relevance of orbital deviation in presence of cyclostationary noise, and the influence of the quality factor of the resonant circuit. APPENDIX A Reference [2] calculates the value of variances of orbital de, for a periodic system as in (3). This result can be viations, used to evaluate percentage variation of the variances shown in . Since is (18) at the bottom of the page, with definite positive and eigenvalues real parts are non positive we can assume (19)
[1] A. Hajimiri and T. H. Lee, “A general theory of phase noise in electrical oscillators,” IEEE J. Solid-State Circuits, vol. 33, no. 2, pp. 179–194, Feb. 1998. [2] F. X. Kaertner, “Analysis of white and noise in oscillators,” Int. J. Circuit Theory Appl., vol. 18, pp. 485–519, 1990. [3] G. J. Coram, “A simple 2-D oscillator to determine the correct decomposition of perturbations into amplitude and phase noise,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 48, no. 7, pp. 896–898, Jul. 2002. [4] A. Demir, A. Mehrotra, and J. Roychowdhury, “Phase noise in oscillators: A unifying theory and numerical methods for characterization,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 47, no. 5, pp. 655–674, May 2000. [5] S. K. Magierowski and S. Zukotynski, “CMOS LC-oscillator phase noise analysis using nonlinear models,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 51, no. 4, pp. 664–677, Apr. 2004. [6] R. Grimshaw, Nonlinear Ordinary Differential Equation. New York: Blackwell Scientific, 1990. [7] A. Papoulis, Probability, Random Variables and Stochastic Processes. New York: McGraw Hill, 1991.
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(18)
