Chaos on Phase Noise of Van Der Pol Oscillator

Phase noise is the most important parameter in many oscillators. The proposed method in this paper is based on nonlinear stochastic differential equation for phase noise analysis approach. The influences of two different sources of noise in the Van Der Pol oscillator adopted this method are compared. The source of noise is a white noise process which is a genuinely stochastic process and the other is actually a deterministic system, which exhibits chaotic behavior in some regions. The behavior of the oscillator under different conditions is investigated numerically. It is shown that the phase noise of the oscillator is affected by a noise arising from chaos than a noise arising from the genuine stochastic process at the same noise intensity.


Introduction
Phase noise and short term frequency stability are two representations of the same physical phenomenon.Short term frequency stability is a time domain description.Phase noise is a frequency domain description which describes stochastic fluctuation of oscillator phase.Phase noise is defined as the ratio of the single side-band power at a frequency offset of from the carrier with a measurement bandwidth of 1Hz to the carrier power.Signal contained phase noise is whether an emission excitation signal or local oscillation signal, noise will likewise appear in receiving end with signal.Thereby, Signal-to-noise ratio will be descended and error rate will be worsened as a result of the unwanted carrier modulation.So, it becomes increasingly important to research phase noise of oscillators.
Although the topic of noise in oscillators has engaged classical investigations of a qualitative nature [1], [2], Leeson [3] was the first to propose a simple intuitive phenomenological model [4] relating the level of phase noise in a widely used class of resonator-based oscillators to voltage and current noise sources in the circuit elements.This model has been widely embraced, and serves well to predict oscillator phase noise induced by sources of white noise.However, while Leeson admits that device flicker (1/f) noise may determine the phase noise very close to the oscillation frequency, his model cannot explain why.Using a linear time variant (LTV) model for the oscillator, Hajimiri and Lee [5] has proposed a phase-noise analysis method, which explains this up-conversion phenomenon, but it cannot predict the phase noise at frequency offsets very close to carrier.Kaertner [6] and Demir [7], [8] use perspective of a state-space trajectory to analyze phase noise of oscillators.In their works, the noise causing perturbation is decomposed into two parts.One part causes a deviation in the state-space solution along the unperturbed trajectory, effectually altering the phase of the solution.The other part results in a deviation considered an orbital perturbation.The orbital perturbation can be shown to remain small given a small noise.And then analyze oscillator phase noise by a linearization of the oscillator equations around the noiseless periodic steadystate solution.
In fact, all of the previous approaches to oscillator phase noise analysis are based on some kind of explicit or implicit perturbation analysis, some linear and some nonlinear.However, oscillator system is essentially a nonlinear system.Any linearization method will change the essence of oscillator, so a new nonlinear stochastic analysis method is needed to analyze the phase noise without linearization operation.This paper directly describes oscillators by using nonlinear autonomous differential equation, and introduces noise signal as a term of nonlinear autonomous differential equation.Setting up nonlinear stochastic differential equation (NSDE) analyses phase noise of oscillators.And then adopt this method to compares the influence of two different sources of noise in the Van der Pol oscillator, one source of noise is a white noise process, which is a genuinely stochastic process; the other source of noise is actually a deterministic system, which exhibits chaotic behavior in some regions.The behavior of the oscillator under different conditions is investigated numerically.It is shown that the phase noise of the oscillator is affected more by noise arising from chaos than by noise arising from the genuine stochastic process at the same noise intensity.

Research Method
As any ideal oscillator has an output form as follows where x is output of oscillator, A 0 is amplitude of oscillator, f is output frequency of oscillator, φ 0 is initial phase, A 0 , f and φ 0 are all constant.Oscillators' output is a signal spectral line in frequency domain at this case.Carefully observe equation (1), it must satisfy differential equation as follows Considering the nonlinear essence of oscillator, introduce nonlinear term ε to describe nonlinear active device of oscillator.Without loss of generality, without noise, the oscillator is described by the scalar, ordinary differential equation where ε is a real number.The function f is nonlinear so ε is a parameter that controls the degree of nonlinearity of the system.Introduced noise term, to describe oscillator with noise can be gotten the equation where w(t) is the noise.

Results and Analysis
Using Van der Pol oscillator as example, although it may not be as completely applicable to modern transistor oscillators, it is often used as a example to illustrate features of nonlinear oscillators.Considering the general sense of nonlinear stochastic differential equation, it is not loss of generality to explain problem by using Van der Pol oscillator.Figure 1 shows a schematic of a Van der Pol oscillator.

Figure 1. Van der Pol oscillator
Considering the situation of no noise, described Figure 1 equation can be written Then, substituting the above expression into equation (5), it be obtained In order to gain numerical solution for equation (8), let LC=1，1/RC=0.1, the equation is had Rewriting equation (9) as a system of equations, it is gotten It exist only one equilibrium position (0,0), in this position, which has a pair of conjugate complex root with positive real part, so this equilibrium point is an unstable focal point.According to Lienard's limit cycle theory [9], stable limit cycle exist in equation (9).
Introduced noise term w(t), the equation for described Van der Pol oscillator with noise is

The output of Van der Pol oscillator without noise
Using four-order fix-step Runge-Kutta method to obtain numerical solution for equation ( 9), a stable periodic solution is gotten, whose phase diagram and solution is shown in Figure 2.

The output and phase noise of Van der Pol oscillator with chaos noise
In order to produce chaos noise, considering the situation of nonlinear differential equation actuated by periodic signal without loss of generality is Where δ is a damping coefficient, 0 1 ε < << , n is number of periodic signal, f i is amplitude of periodic signal, i ω frequency of periodic signal.
Rewriting equation (12) as a system of equations, it is gotten When ε =0, system (12) is Hamilton system, whose Hamilton quantity is Due to It is known that (1,0) and (-1,0) is saddle point of system ( 14), (0,0) is center of system (14).When const=0.25, there are two heteroclinic orbits to connect (1,0) with (-1,0), two parameter equation of heteroclinic orbits are x t th t y t h t The Melnikov function for equation ( 16) is where According to Melnikov theory, when , , It is selected n=3, (f i , ω i )={(35,1),(40,4),(10,13)}, according to the conditions mentioned above, produced chaos noise is shown in Figure 3. Adopting algorithm proposed by Wolf [10] for produced chaos noise by equation ( 12), it is gotten two Lyapunov exponents are λ 1 = 0.089254, λ 2 = -0.189278at t=500, whose time-evolvement curve is shown in Figure 4.By reason of maximal Lyapunov exponent greater than zero, it indicates that x's values are determinately chaos data in Figure 3.  Let w(t) is chaos noise in equation (11), using produced chaos noise by equation ( 12) and setting its variance is 0.01, obtained solutions by using stochastic Runge-Kutta method [11] are shown in Figure 5.Let w(t) is white noise in equation (11), setting its intensity is 0.01, obtained solutions by using stochastic Runge-Kutta method are shown in Fig. 6.

Conclusion
By setting up NSDE to describe oscillator, it can be conveniently used numerical methods to analyses phase noise of oscillator system.Analyzing Fig. 8, it can be discovered that phase noise produced by chaos noise has a larger value than that of by white noise and very close to phase noise by combination noise under the same intensity conditions.This is due to pseudo-random of chaos noise.Because chaos noise is produced by deterministic system, amongst chaos noise has a long-larger correlative degree than that of white noise, which results in a small stochastic averaging of NSDE, so produces a large phase noise output.Because of universality of the chaos phenomena, chaos noise determinately exists in oscillator system.Therefore, in order to reduce phase noise in output of oscillator, it should be used the chaos control method to reduce chaos noise as great as possible by the time it is designed the oscillator.It is also an important research direction to use analytic methods to obtain analytic solution of NSDE in some sense for guiding minimized phase noise in an oscillator design.

Figure 3 .
Figure 3. Phase diagram and solution of produced chaos noise by equation (12).

TELKOMNIKAFigure 5 .
Figure 5. Phase diagram, solution and phase noise of Eqn (11) whose w(t) is chaos noise.(a) is phase diagram, (b) is x's solution, (c) is phase noise

Figure 6 .
Figure 6.Phase diagram, solution and phase noise of equation (11) whose w(t) is white noise.(a) is phase diagram, (b) is x's solution, (c) is phase noise

Figure 7 .
Figure 7. Phase diagram, solution and phase noise of equation (11) whose w(t) is combination noise.(a) is phase diagram, (b) is x's solution, (c) is phase noise

Figure 8 .
Figure 8.Compared phase noise produce by white noise, chaos noise with combination noise in NSDE, (a) is compared phase noise within large offset frequency, (b) is compared phase noise within small offset frequency.