Noise in oscillators: a review of state space decomposition approaches

Abstract

We review the state space decomposition techniques for the assessment of the noise properties of autonomous oscillators, a topic of great practical and theoretical importance for many applications in many different fields, from electronics, to optics, to biology. After presenting a rigorous definition of phase, given in terms of the autonomous system isochrons, we provide a generalized projection technique that allows to decompose the oscillator fluctuations in terms of phase and amplitude noise, pointing out that the very definition of phase (and orbital) deviations depends of the base chosen to define the aforementioned projection. After reviewing the most advanced theories for phase noise, based on the use of the Floquet basis and of the reduction of the projected model by neglecting the orbital fluctuations, we discuss the intricacies of the phase reduction process pointing out the presence of possible variations of the noisy oscillator frequency due to amplitude-related effects.

Appendix 1: Floquet theory basics

Floquet theory forms the basis for the most advanced oscillator noise theories. Details can be found in [23], while [37] deals with the extension of the classical Floquet theory of ODEs to DAEs.

Let us consider a LTV homogeneous ODE of size $n$

$$\begin{aligned} \frac{\displaystyle \text {d}\mathbf {y}}{\displaystyle \text {d}t}=\mathbf {A}(t)\mathbf {y}(t) \end{aligned}$$

(58)

where $\mathbf {A}(t)=\mathbf {A}(t+T)$ is a $T$-periodic matrix of size $n$. Given the initial condition $\mathbf {y}(0)=\mathbf {y}_0$, Floquet theorem states that the solution of (58) reads

$$\begin{aligned} \mathbf {y}(t)=\mathbf {S}(t,0)\mathbf {y}_0 \end{aligned}$$

(59)

where $\mathbf {S}(t,s)$, the state transition matrix of the LTV system, is expressed as

$$\begin{aligned} \mathbf {S}(t,s)=\mathbf {U}(t)\mathbf {D}(t-s)\mathbf {V}(s), \end{aligned}$$

(60)

where $\mathbf {U}$ and $\mathbf {V}$ are two $T$-periodic invertible square matrices of size $n$ such that $\mathbf {U}(t)=\mathbf {V}^{-1}(t)$, while matrix $\mathbf {D}(t)$ is a diagonal matrix:

$$\begin{aligned} \mathbf {D}(t)={\mathrm{\text {diag} }\left\{ \exp (\mu _1 t), \dots , \exp (\mu _n t) \right\} }. \end{aligned}$$

(61)

The $n$ complex numbers² $\mu _i$ are the Floquet exponents (FEs) of (58), while $\lambda _i=\exp (\mu _i T)$ are the corresponding Floquet multipliers (FMs). According to the FM definition, for each FM $\lambda _i$ an infinite set of FEs exists, namely $\mu _i+{\text {i}}k 2\pi$ where $k$ is an integer number: this splitting of the FEs is important when the exponents are calculated by means of frequency domain techniques, such as Harmonic Balance [38].

Denoting with $\mathbf {u}_i(t)$ (resp. $\mathbf {v}_i^{\text {T}}(t)$) the $i$th column (resp. row) of $\mathbf {U}(t)$ (resp. $\mathbf {V}(t)$), the two sets $\{\mathbf {u}_i(t)\}$ and $\{\mathbf {v}_i(t)\}$ both span the entire $\mathbb {R}^n$, and form a bi-orthogonal basis (see (19)). Furthermore:

• $\mathbf {u}_i(t)\exp (\mu _i t)$ is a solution of (58) with initial condition $\mathbf {u}_i(0)$. For this reason, $\mathbf {u}_i(t)$ is the direct Floquet eigenvector associated to the $\mu _i$ FE of (58);

• $\mathbf {v}_i(t)\exp (-\mu _i t)$ is a solution of the adjoint system associated to (58), i.e.

  $$\begin{aligned} \frac{\displaystyle \text {d}\mathbf {z}}{\displaystyle \text {d}t}=-\mathbf {A}^{\text {T}}(t)\mathbf {z}(t), \end{aligned}$$

  (62)

  with initial condition $\mathbf {v}_i(0)$. Correspondingly, $\mathbf {v}_i(t)$ is the adjoint Floquet eigenvector associated to $\mu _i$.

Considering the limit cycle $\mathbf {x}_{\text {S}}(t)$ solution of (1), and the LTV system defined by a linearization of (1), i.e.

$$\begin{aligned} \mathbf {A}(t)=\left. \nabla _\mathbf {x}\mathbf {f}(x)\right| _{\mathbf {x}_{\text {S}}(t)}, \end{aligned}$$

(63)

it follows

$$\begin{aligned} \frac{\displaystyle \text {d}}{\displaystyle \text {d}t} \frac{\displaystyle \text {d}\mathbf {x}_{\text {S}}}{\displaystyle \text {d}t}=\frac{\displaystyle \text {d}}{\displaystyle \text {d}t} \mathbf {f}(\mathbf {x}_{\text {S}}(t))=\mathbf {A}(t) \frac{\displaystyle \text {d}\mathbf {x}_{\text {S}}}{\displaystyle \text {d}t}. \end{aligned}$$

(64)

Therefore, for the LTV system associated to an autonomous circuit, $0$ is always a FE (or, equivalently, $+1$ is always a FM) associated to the direct Floquet eigenvector $\text {d}\mathbf {x}_s/\text {d}t$. With no loss of generality, we assume $\mu _1=0$ and $\mathbf {u}_1(t)=\text {d}\mathbf {x}_s/\text {d}t$.

Due to the exponential dependence on $\mu _i$ of the solution of (58), an oscillator has an asymptotically stable orbit if and only if all the FEs $\mu _i$ ($i=2,\dots ,n$) have negative real part, or equivalently all the FMs $\lambda _i$ ($i=2,\dots ,n$) have magnitude lower than 1.

The computation of the FEs and eigenvectors (direct and adjoint) is a fundamental step for Floquet-based oscillator noise analysis. Specifically, the adjoint Floquet eigenvector $\mathbf {v}_1(t)$, associated to $\mu _1=0$, is the so-called perturbation projection vector that plays the man role in the assessment of phase noise [16, 21, 22]. Due to their importance for oscillator noise and for the assessment of the stability of limit cycles [39], the Floquet quantities have been the object of research for several years. In most of the cases, the computation is perfomed in time domain [40–43]. However, efficient algorithms for the frequemcy domain evaluation, based on the harmonic balance technique, are proposed in [38, 42, 44–47].