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Phase Noise in Oscillators: Daes and Colored Noise Sources Alper Demir Alpdemir @Research.Bell-Labs.Com Bell Laboratories Murray Hill, New Jersey, USA

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Phase Noise in Oscillators: Daes and Colored Noise Sources Alper Demir Alpdemir @Research.Bell-Labs.Com Bell Laboratories Murray Hill, New Jersey, USA Phase Noise in Oscillators: DAEs and Colored Noise Sources Alper Demir alpdemir @research.bell-labs.com Bell Laboratories Murray Hill, New Jersey, USA Abstract as 1= f noise has a significant impact on phase noise of practical oscil- Oscillators are key components of electronic systems. Undesired perturbations, lators. Understanding how colored noise sources affect the oscillator i.e. noise, in practical electronic systems adversely affect the spectral and tim- spectrum is crucial for low phase noise, low cost, integrated oscillator ing properties of oscillators resulting in phase noise, which is a key perfor- designs that can meet the stringent specifications of today’s RF com- mance limiting factor, being a major contributor to bit-error-rate (BER) of RF munications applications. communication systems, and creating synchronization problems in clocked and First, in Section 2, we establish the equivalent of Floquet theory sampled-data systems. In this paper, we first present a theory and numerical (and some related results) for periodically time-varying systems of lin- methods for nonlinear perturbation and noise analysis of oscillators described ear DAEs, which we then use in Section 3 to develop the theory and by a system of differential-algebraic equations (DAEs), which extends our re- numerical methods for nonlinear perturbation analysis of autonomous cent results on perturbation analysis of autonomous ordinary differential equa- tions (ODEs). In developing the above theory, we rely on novel results we es- DAEs. In Section 4, we analyze the case of colored noise perturbations tablish for linear periodically time-varying (LPTV) systems: Floquet theory for and obtain a stochastic characterization of the phase deviation. Models DAEs. We then use this nonlinear perturbation analysis to derive the stochastic for burst (popcorn) and 1 = f (flicker) noise, the most significant col- characterization, including the resulting oscillator spectrum, of phase noise in ored noise sources in IC devices, are discussed in Section 5. Then, oscillators due to colored (e.g., 1= f noise), as opposed to white, noise sources. in Section 6, we calculate the resulting oscillator spectrum with phase The case of white noise sources has already been treated by us in a recent publi- noise due to a colored noise source. Our treatment of phase noise due cation. The results of the theory developed in this work enabled us to implement to colored noise sources is general, i.e., it is not specific to a partic- a rigorous and effective analysis and design tool in a circuit simulator for low ular type of colored noise source. Hence, our results are applicable phase noise oscillator design. to the characterization of phase noise due to not only 1= f and burst 1 Introduction noise, but also other types of possibly colored noise, e.g., substrate or power supply noise. In Section 7, we consider the presence of white Oscillators are ubiquitous in physical systems, especially electronic and colored noise sources together, and derive the resulting oscilla- and optical ones. For example, in radio frequency (RF) communica- tor spectrum. Finally, in Section 8, we present simulation results in tion systems, they are used for frequency translation of information phase noise characterization of oscillators. All proofs are omitted due signals and for channel selection. Oscillators are also present in digital to space limitations. electronic systems which require a time reference, i.e., a clock signal, in order to synchronize operations. 2 Floquet theory for DAEs Noise is of major concern in oscillators, because introducing even We now consider the n-dimensional inhomogeneous linear system of small noise into an oscillator leads to dramatic changes in its frequency DAEs1 spectrum and timing properties. This phenomenon, peculiar to os- cillators, is known as phase noise or timing jitter. A perfect oscilla- d + + = C t x Gtxbt0(1) tor would have localized tones at discrete frequencies (i.e., harmon- dt ics), but any corrupting noise spreads these perfect tones, resulting in n n ! high power levels at neighboring frequencies. This effect is the ma- where the matrix C :IR IR is not necessarily full rank, but we jor contributor to undesired phenomena such as interchannel interfer- assume that its rank is a constant, m n, as a function of t. C t and ence, leading to increased bit-error-rates (BER) in RF communication G t are T-periodic. The homogeneous system corresponding to (1) is systems. Another manifestation of the same phenomenon, timing jit- given by ter, is important in clocked and sampled-data systems: uncertainties d + = in switching instants caused by noise lead to synchronization prob- C t x Gtx0(2) lems. Characterizing how noise affects oscillators is therefore crucial dt for practical applications. The problem is challenging, since oscilla- Solution and related subspaces tors constitute a special class among noisy physical systems: their au- When C t is rank deficient, (2) does not have solutions for all initial tonomous nature makes them unique in their response to perturbations. n = 2 conditions x 0 x0IR . We assume that the DAEs we are dealing In a recent publication [1] (which has a brief review of previous with are index-1 [2]. Then, the solutions of the homogeneous system work on phase noise), we presented a theory and numerical methods (2) lie in an m-dimensional subspace defined by [2] for practical characterization of phase noise in oscillators described by asystemofordinary differential equations (ODEs) with white noise n ˙ = 2 + 2 S t zIR : G t CtzimC t (3) sources. In this paper, we extend our results to oscillators described by a system of differential-algebraic equations (DAEs) with colored 2 Also, every xt S t is a solution of (2) [2]. Let N t be the null as well as white noise sources. The extension to DAEs and colored space of C t : noise sources is crucial for implementing an effective analysis and de- = sign tool for phase noise in a a circuit simulator. Almost all of the N t ker C t (4) circuit simulators use the MNA (Modified Nodal Analysis) formula- = tion, which is basically a system of DAEs. Colored noise sources such which is an n m k-dimensional subspace. For index-1 DAEs, we \ =f g have [2, 3] S t N t 0and n = S t N t IR (5) 1 Note that the time derivative operates on the product C t x, not on x only. It will 0 become clear in Section 3 why we use this form. where denotes direct sum decomposition. For our purposes, Remark 2.1 The following orthogonality/biorthogonality conditions it suffices to know that (5) is equivalent to the following: If hold: = f ; ;::: ; g = Z t z1tz2tzmtis a basis for S t ,andWt T =δ= ;::: ; = ;::: ; v t C t u t i 1 mj 1 m(14) ; ;::: ; g [ fw1tw2twktis a basis for N t ,thenZ t W t is a basis j i ij n for IR . T = = ;::: ; = + ;::: ; v j t C t ui t 0i1mj m 1 n(15) Adjoint system and related subspaces T = = + ;::: ; = ;::: ; Before we discuss the forms of the solutions of (2) and (1), we would v j t G t ui t 0im1nj 1 m(16) like to introduce the adjoint or dual system corresponding to (2), and the related subspaces. The adjoint system corresponding to (2) is given (14) and (15) follow from (11). (16) follows from the fact that ; = ;::: ; ; = + by vi t exp µit j 1 m is a solution of (6), and ui t i m 1;::: ;n is in the null space N t of C t . ; T d T The state transition matrix Φt s in (10) can be rewritten as = C t y G t y 0(6) dt m T ; = Φ Note that the time derivative operates on y only, not on the product t s ∑exp µi t s ui t vi s C s (17) T = C t y, in contrast with (2). It will become clear shortly why (6) is i1 the form for the adjoint of (2). If y t is a solution of (6) and x t is a Theorem 2.2 The solution φ of (1) satisfying the initial condition solution of (2), then we have = 2 = x0 x0S0(for b 0 0) is given by d T d T Td Z t = + y t C t x t y t C t x t ytC t x t φ ; = Φ ; + Ψ ; +Γ dt dt dt t x0 t0x0tsbsdstbt(18) 0 T T = y t G t x t y t G t x t where = 0 ; = Ψ T T t s UtDtsVs(19) = Thus y t C t x t y0C0x0for all t 0. Let Γ T n T T and t : n n is a T-periodic matrix of rank k which satisfies = f 2 2 g S t zIR : G t z imC t (7) [ ;::: ; ] = Γ t C t u t ut0 (20) and 1 m Γ f ;::: ; g T T i.e., the null space of t is spanned by C t u t Ctut. = N t ker C t (8) 1 m From (10) and (19), the solutions of the homogeneous system (2) and T T T T n \ =f g = Then S t N t 0and S t N t IR . the inhomogeneous system (1) are respectively given by State-transition matrix and the solution m T = Having introduced the adjoint system for (2), we now consider the xH t ∑exp µit ui t vi 0 C 0 x 0 state-transition matrix and the solutions for (2) and (1). i=1 Theorem 2.1 The solution φ of (2) satisfying the initial condition Z = 2 x0 x0S0is given by mt T = + + Γ φ xIH t xHt∑uitexp µi t s vi s b s ds t b t ; = Φ ; t x0 t0x0(9) 0 i=1 2 ; Φ = where the “state transition matrix” t s is given by If the initial condition x0 x0is not in S 0 , i.e., if it is not a consis- =Φ ; tent initial condition for (2), then xt t0x0is still a solution of ; = Φ t s UtDtsVsCs(10) n = 2 (2), but it does not satisfy x 0 x0.Anyx0 IR can be written as + 2 2 where x0 = x0eff x0N where x0eff S 0 and x0N N 0 , which follows from (5).
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