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). Then

= [    ;::: ;    ; ;::: ; ]

Dt s diag exp µ1 t s exp µm t s 0 0

{z }

| Φ ;  =         = Φ  ;  t 0 x0 U t D t V 0 C 0 x0 t 0 x0eff

k

 =    =Φ ; 

since C 0 x0 C 0 x0eff. Hence, x t t0x0is a solution of (2)

    

U t : n n and V t : n n are both T -periodic and nonsingular (for =

all t), and satisfy satisfying the effective initial condition x0 x0eff.   State-transition matrix for the adjoint system

Im0

;  Ω

    = V t C t U t (11) The state-transition matrix t s for the adjoint system (6) is not sim-

00 T

;  Φ ; =

ply given by Φ s t in terms of the state transition matrix t s

       UtDtsVsCsfor (2), as it would be the case for ODEs. In- µ are called the characteristic (Floquet) exponents of (2), and λ =

i i stead, it is given by 

exp µiT are called the (Floquet) characteristic multipliers. Note that

(2) has k = n m Floquet multipliers that are 0. Ω T T T ;  =        

T t s V t D s t U s C s

       Let ui t be the columns of U t ,andv t be the rows of V t : i m

T T

        

= ∑ exp µ t s v t u s C s

=[  ;::: ;  ;  ;::: ;  ]  i i U t u1tumtum+1tunt(12) i

T i=1

 = [  ;::: ;  ;  ;::: ;  ]

V t v1tvmtvm+1tvnt(13)

Monodromy matrix

f  ;::: ;  g   f  ;::: ;  g

u1 t umtis a basis for S t ,and um+1 t untis a ba- Φ ; 

T We define the monodromy matrix for (2) as T 0 , and it is given by

 f  ;::: ;  g   sis for N t . Similarly, v1 t vmtis a basis for S t ,and

T mm

f  ;::: ;  g      =

vm+1t vntis a basis for N t .For1 i m,xt T T

; =        =        

ΦT 0 ∑exp µiT ui T v 0 C 0 ∑exp µiT ui 0 v 0 C 0

    =  

uitexp µit is a solution of (2) with the initial condition x 0 ui0.i i =

i=1i1

  =    

Similarly, for 1  i m, y t vitexp µit is a solution of (6) with

=  

the initial condition y0 vi0.

 = ;::: ; ui 0 for i 1 mare the eigenvectors of the monodromy ma-

2 Φ ;      Authors in [3] derive a similar result for the state-transition matrix of a DAE system. trix T 0 with corresponding eigenvalues exp µiT ,andui 0 for

+ ;::: ; Φ  ; 

i = m 1 nare the eigenvectors of the monodromy matrix T 0 where

 corresponding to the k-fold eigenvalue 0. vi 0 are not the eigenvec-

T d

;  Φ 

 =   tors of the transposed monodromy matrix T 0 . Here, we must C t q x (26)

Ω ;   dx consider the monodromy matrix T 0 for the adjoint system (6), x =xs which is given by

d

 =   G t g x (27) m dx

T T x =xs

; =         Ω T 0 ∑exp µiT vi 0 ui 0 C 0 (21)

i=1 Φ ; 

Let t s be the state transition matrix of (25). Since (25) has a T- 

periodic solutionx ˙s t , we can choose, without loss of generality,

 = ;::: ; Now, vi 0 for i 1 mare the eigenvectors of the monodromy ma-

Ω ;     

trix T 0 with corresponding eigenvalues exp µiT ,andvi 0 for

=  

u1 t x˙st(28)

+ ;::: ; Ω ;  i = m 1 nare the eigenvectors of the monodromy matrix T 0 corresponding to the k-fold eigenvalue 0. and Numerical computation of the monodromy matrix

λ  = The eigenvalues of the monodromy matrix determine the stability of 1 = exp µ1T 1 (29)

(2) [4, 5]. Hence, one would like to calculate the monodromy ma- Φ ;  trix and its eigenvalues. Authors in [5] define a reduced monodromy in the representation of t s in (17). Hence, one of the Floquet (char- acteristic) multipliers of (25) is 1. One can show that if the remaining matrix for (2) as a nonsingular m  m matrix, as opposed to the mon-

m 1 Floquet multipliers have magnitudes less than 1, i.e., odromy matrix we defined which is n  n and has k eigenvalues equal

to 0. The monodromy matrix they define has the same eigenvalues

λ j = j  j < = ;::: ; j exp µ T 1 i 2 m(30) as the one we define, except for the k-fold eigenvalue 0. For the nu- i i

merical computation of the reduced monodromy matrix, as proposed  then x t is an asymptotically orbitally stable solution of (22).

by the authors in [5], one needs to calculate m linearly independent s

  + α   consistent initial conditions for (2). With our definition, we avoid hav- Lemma 3.1 If xs t is a solution of (22), then xs t t is a solution ing to compute m linearly independent consistent initial conditions for of

(2). Instead, we integrate (2) with an effective matrix (rank m) initial d



+  +    +α   + α  = condition X0eff :n n(columns of which are consistent initial condi- qx gxctCttu t t 0 (31)

dt 11 = tions for (2)) from t = 0tot Tto calculate the monodromy matrix

as follows: In numerical integration of (2), we set the initial condition

 α  

where the scalars c1 t and t satisfy = X 0 In, the n-dimensional identity matrix. From (5), we can write

3

+   I = X X where the columns of X lie in S 0 of (3) ,and

n 0eff kerC 0eff d α  =  

t c1 t

    = the columns of XkerC lie in N 0 (null space of C 0 ). X 0 X0eff is dt

effectively realized during numerical integration by

= <  =

c1 t 0for t 0 al pha 0 0

 =   +   =   C 0 In C 0 X0eff C 0 XkerC C 0 X0eff Now, consider a small, additive, state-dependent perturbation of the

Note that during the numerical integration of (2), one does not need n n  p p

   
 ! 
 !

form Bx b t to (22) (where B :IR IR and b :IR IR ):

 =   to calculate X0eff itself, but only C 0 X0eff C 0 In. The numerical

integration of (2) is started with an order 1 method (i.e., backward d

+  +    =

qx gxBxbt0 (32)

     Euler) at t=0 that requires only C 0 X 0 to compute the X t at the dt

first time step. If one would like to obtain the effective initial condition

   X0eff, a backwards step in time (with the same time step) can be taken Next, we decompose the (small) perturbation B x b t into its compo- after the first forward time step is computed.4 This is a much more nents using

efficient way of computing consistent initial conditions for (2) then the

 + α   + α ;::: ;  + α   + α ;

one used in [5]. fC t t u1 t t Cttum t t

 + α   + α  ::: ;  + α   + α g 3 Perturbation analysis for autonomous DAEs G t t um+1 t t Gttun t t

5

   Consider the system of autonomous DAEs: as the basis ,whereuit are the columns of U t of Section 2

d m

 +  =

  =  ;α ;   +α   + α  + q x gx0 (22) Bx b t ∑cxttCttu t t dt ii

i=1 (33)

We assume that (22) has an asymptotically orbitally stable periodic n

 ; α ;   + α   + α  

solution xs t with period T, i.e., a stable limit cycle in the solution ∑ ci x t t G t t ui t t +

space. Hence i=m 1

; α ; ;  

d where the coefficients ci x t t 1 i m are given by

+  = qxs gxs0 (23)

dt T

; α ; =  +α        ci x t t vittB x b t for 1 i m (34)

Let us take the derivative of both sides of (23) with respect to t: !

 which was obtained using the orthogonality/biorthogonality relation-

d d d ships in (14) and (16). We distinguish the component in (33) along

    =

q x x˙ + g x x˙ 0 (24)

+ α   + α  =  + α   + α 

s s 

dt dx dx C t t u1 t t C t t x´s t t from the rest: =

x=xs x xs

;  =  ; α ;   + α   + α 

b1 x t c1 x t t C t t u1 t t  Thus,x ˙ t is a T-periodic solution of the LPTV system of DAEs

s ˜

;  =      ;  b x t B x b t b1 x t

d

  +   = C t x Gtx0 (25)

dt 5 n  Recall that the columns of U t form a basis for IR for index-1 DAEs. It can

3

 

f  + α    + α  ;::: ;  + α   + α  ;  + α    +

They also span S 0 , since the columns of In are linearly independent. be shown that C t t u1 t t Cttum t t G t t um+1 t

4 n

 ::: ;  + α   + α  g The author would like to thank Hans Georg Brachtendorf for pointing this out. α t Gttun t t also forms a basis for IR for the index-1 DAE case.

   Theorem 3.1 5. Compute the periodic vector v1 t for 0 t T by numerically

integrating the adjoint system backwards in time

+ α   1. xs t t solves

T d T

   =

d d C t y G t y 0 (39)

+  +  ; =  +  + qx gxbxtq x gx dt

dt 1dt

=    

T using v1 0 v1Tas the initial condition. Note that v1 t is a

+α       + α   + α  = v1ttB x b t C t t u1 t t 0 periodic steady-state solution of (39) corresponding to the Flo-

where quet multiplier that is equal to 1. It is not numerically stable to  calculate v1 t by numerically integrating (39) forward in time.

d α T =  +α    + α    α = t v1ttB xs t t b t 0 0 (35) In implementing the above algorithm, one can increase the efficiency

dt by saving LU factored matrices that need to be calculated in Step 2

+ α   +  

2. xs t t z t solves and reuse them in Step 5. One can also avoid calculating the full

Ω ;  n  n monodromy matrix T 0 explicitly, and use iterative meth-

d Ω ; 

ods (which require only the computation of products of  T 0 with

+  +    =

qx gxBxbt0 (36) Ω ;  dt some vectors) at Step 4 to calculate the eigenvector of  T 0 that

corresponds to the eigenvalue 1. 

where zt does not grow without bound and it indeed stays

  = ; >

small (within a factor of bt ). If b t 0ttcfor some 4 Stochastic characterization of the phase deviation

 ! !∞  + α   tc > 0,thenzt 0 as t , and xs t tc solves (36) with colored noise sources ∞ for t ! . We now find the probabilistic characterisation of the phase deviation

We considered the perturbed system of DAEs in (32), and obtained the α 

t (which satisfies the differential equation (35)) as a stochastic pro- 

following results: The unperturbed oscillator’s periodic response x t 

s cess when the perturbation bt is a (one-dimensional) stationary, zero-

+ α     α 

is modified to xs t t + z t by the perturbation, where t is a 7

[  ] = τ mean ( E b t 0), Gaussian colored stochastic process .LetR

changing time shift, or phase deviation, in the periodic output of the N 

be the autocovariance function, and SN  f be the power spectral den- 

unperturbed oscillator, zt is an additive component, which we term 

α sity, of the stationary Gaussian stochastic process bt : 

the orbital deviation, to the phase-shifted oscillator waveform. t

τ  = E [  + τ =   τ = ]

 α  and z t are such that: t will, in general, keep increasing with time RN b t 2 b t 2 (40)

Z ∞ 

even if the perturbation bt is always small, and if the perturbation

 = F f τ g = τ   π τ  τ

α S  f R R exp j2 f d (41) 

is removed, t will settle to a constant value. The orbital deviation N N N

∞ 

zt , on the other hand, will always remain small (within a bounded

τ τ  =

   factor of bt ), and if the perturbation is removed, z t will decay to Note that RN is a real and even function of .Letvt

T

   : : zero. Furthermore, we derived a nonlinear differential equation (35) vtBxst, which is a scalar (both v1 and B are vectors) that is

α 1  for the phase deviation t . periodic in t with period T . Hence, (35) becomes

Numerical methods α 

d t

 + α   ; α = = v t t b t 0 0 (42) For perturbation analysis and phase noise/timing jitter characteriza- dt tions of an oscillator, one needs to calculate the steady-state periodic

In this section, we will follow the below procedure to find an adequate

  

solution xs t and the periodic vector v1 t that appears in (35). Below, α 

probabilistic characterization of the phase deviation t due to the 

we describe the numerical computation of v1 t . 

colored noise source bt for our purposes:  1. Compute the large-signal periodic steady-state solution x t for

s 1. We first derive a partial integro-differential equation for  0  t T by numerically integrating (22), possibly using a tech-

nique such as the shooting method [6]. the time-varying marginal probability density function (PDF)

η;  α   pα  t of t defined as

Ω 6 ; 

2. Compute the monodromy matrix  T 0 in (21) by numeri-

∂Pα η

cally integrating t

η ; =  pα  t ∂η t 0 (43)

T d T

   =  =

C t Y G t Y 0 Y 0 I(37)

: dt nwhere P denotes the probability measure.

2. We then show that the PDF of a Gaussian random variable,

    from 0 to T, backwards in time, where C t and G t are de-

Ω “asymptotically” with t, solves this partial integro-differential ; =   fined in (26). Note that  T 0 YT. Note that it is not

Ω equation. A Gaussian PDF is completely characterised by the ; 

numerically stable to calculate T 0 by integrating (37) for- α 

mean and the variance. We show that t becomes (under

  =

ward in time. Since C t is not full rank, in general, Y 0 In

τ    some conditions on R  or S f ), for “large” (to be con- can not be realized in solving (37). Please see the discussion at N N cretized) t, a Gaussian random variable with a constant mean the end of Section 2.

and a variance that is given by

   =   Z 3. Compute u 0 using u 0 x˙0. Z

1 1 s t t

  Ω ; 

α   = β   4. v1 0 is an eigenvector of T 0 corresponding to the eigen- var  t RN t1 t2 dt1 dt2 (44)

0 0  value 1. To compute v1 0 , first compute an eigenvector of

Ω  ;  

T 0 corresponding to the eigenvalue 1, then scale this Theorem 4.1 If bt is a stationary, zero-mean, Gaussian stochastic

τ  α  

eigenvector so that process with autocovariance function RN  , and if t satisfies (42),

η;  α

T then the time-varying marginal PDF pα  t of satisfies

    =

v 0 C 0 u 0 1 (38)

1 1

Z t

η;  ∂ ∂  + η

∂pα  t v t

τ + η  τ  η;  τ + is satisfied. For some oscillators (encountered quite often in = v RN t pα t d

Ω ∂t ∂η ∂η ; 

practice), there can be “many” other eigenvalues of  T 0 0

with magnitudes very close to 1, and they may not be numer- ∂2 Z t

+ η τ + η  τ  η;  τ vt v R t pα t d ically distinguishable from the eigenvalue that is theoretically ∂η2 N equal to 1. In this case, to choose the correct eigenvector of 0

Ω (45) ;     T 0 as v1 0 , calculate the inner products of these eigen-

7

 

   vectors with C 0 x˙s 0 and choose the vector which has the The extension to the case when b t is a vector of uncorrelated white and colored noise largest inner product. Theoretically, the inner products of the sources is discussed in Section 7. Noise sources in electronic devices usually have inde-

pendent physical origin, and hence they are modeled as uncorrelated stochastic processes.

   “wrong” eigenvectors with C 0 x˙s 0 are 0. Hence, we consider uncorrelated noise sources. However, the generalization of our results 6Note the minus sign in front of T. to correlated noise sources is trivial. with the initial/boundary condition for t = 0 This is satisfied when the bandwidth of the colored noise source is

much less than the oscillation frequency ω0, or equivalently, the cor-

η; =δη

pα  0 (46) relation width of the colored noise source in time is much larger than

 

π=ω

2 the oscillation period T = 2 0. We will further comment on this

[α ] = E α   = i.e., E 0 0and 0 0. condition in Section 5, where we discuss the models for burst and 1 = f

The partial integro-differential equation (45) for the time-varying noise. With (54), (52) and (53) become

η;  α   marginal PDF pα  t of t is a generalization of a partial differ-

σ2 Z t   

ential equation known as the Fokker-Planck equation [7, 8] derived for dµt d t 2

= j j  τ  τ

α = 0 2 V0 RN t d (55)    the PDF of t satisfying (42) when b t is a white noise process, dt dt 0 which is given below

From (55)

∂ ∂ η;  ∂  + η

pα  t v t

Z Z

λ  + η η;  + = v t pα t tt2

∂ ∂η ∂η 2 2

= j j   t σ t 2V0RNt2t1dt1dt2(56)

(47) 00

∂2

1 2

+ η η; 

v t pα t τ 2 ∂η2 follows trivially. Since, the autocovariance RN  is an even function

of τ, (56) can be rewritten as

 λ  Z

where 0 1 depends on the definition of the stochastic integral [7] Z tt 

used to interpret the stochastic differential equation in (42) with bt σ2 2 =j j  

t V0RNt2t1dt1dt2(57)

 α 

as a white noise process. If bt is a white noise process, then t is 00

 α  a Markov process. However, when bt is colored, t , in general, is not Markovian. (47) is valid for any initial/boundary condition. On Thus, we obtained (44).

2

 α   the other hand, (45) is valid only for initial/boundary conditions of the Lemma 4.2 The variance σ t of t in (56) can be rewritten with type a single integral as follows:

Z t

η ; =δηα

pα  0 0(48) σ2 2 = j j  τ τ τ t 2V0tRNd(58) 0

for some α0.

η ; 

We would like to solve (45) for pα  t . We do this by first solving It can also be expressed in terms of the spectral density of the colored

ω;  α  

for the characteristic function F  t of t , which is defined by  noise source bt as follows:

Z ∞

Z ∞

 π 

1exp j2 ft

ω; = E [  ωα ] =  ωη  η ;  η

F  t exp j t exp j pα t d 2 2

= j j   ∞ σ t 2V0SNfdf (59) ∞π2 2

4 f

   v t is T-periodic, hence we can expand v t into its Fourier series:

5 Models for burst (popcorn) and 1= f (flicker) noise ∞

2π Burst noise

=  ω ; ω = vt ∑Vexp ji t

i0 0 T The source of burst noise is not fully understood, although it has been

∞ i= shown to be related to the presence of heavy-metal ion contamina-

α  ω;  Lemma 4.1 The characteristic function of t ,F t , satisfies tion [9]. For practical purposes, burst noise is usually modeled with a

colored stochastic process with Lorentzian spectrum, i.e., the spectral

∞ ∞

ω;  ∂F  t density of a burst noise source is given by

 2

 ω  ω ω ω = ∑ ∑ V V exp j it i

∂t i k 0 0

∞

= a

∞

i k = (49) I

Z

 =

t Sburst f K (60)

f2

 τ   ω τ  ω  +ω;  τ RN t exp j 0k F 0 i k td 1+ 0 fc

where  denotes complex conjugation. where K is a constant for a particular device, I is the current through

Theorem 4.2 (49) has a solution that becomes (with time) the char- the device, a is a constant in the range 0:5to2,and fc is the 3 dB acteristic function of a Gaussian random variable: bandwidth of the Lorentzian spectrum [9]. Burst noise often occurs with multiple time constants, i.e., the spectral density is the summa-

2 2 

ω σ t tion of several Lorentzian spectra as given by (60) with different 3 dB

ω; =  ω    F  t exp j µ t (50) 2 bandwidths. A stationary colored stochastic process with spectral density solves (49) for t large enough such that

γ2

 = S  f (61)

1 2 2 2 1 i = k Nburst 22

  σ    ω

γ+ π

exp 0 i k t (51) 2f = 2 0 i 6 k has the autocorrelation function where γ

∞ Z

τ= γ jτ j

t 

  R exp (62)

dµ t 2 Nburst

ω j j  τ   ω  τ  τ = ∑ j 0i Vi RN t exp j 0i t d (52) 2

dt 0

∞ i = If the 3 dB bandwidth γ of (61) is much less than the oscillation fre- 2 ∞ Z t ω γ

σ = 

d t 2 quency 0, or equivalently, the correlation width 1 of (62) is much

j j  τ   ω  τ  τ

= ∑ 2 Vi RN t exp j 0i t d (53)

π=ω larger than the oscillation period T = 2 , then (54) is satisfied.

dt 0 0

∞ i =

1= f noise

Assumption 4.1 1= f noise is ubiquitous in all physical systems (as a matter of fact,

R =

 t in all kinds of systems). The origins of 1 f noise is varied. In IC

Z

τ τ =

t 0 RN t d i 0 devices, it is believed to be caused mainly by traps associated with

τ   ω  τ  τ  RN t exp j 0i t d (54) contamination and crystal defects, which capture and release charge

0 = 0 i 6 0 carriers in a random fashion, and the time constants associated with this process give rise to a noise signal with energy concentrated at low The following simple Lemma establishes the basic form of the autoco- frequencies. For practical purposes it is modeled with a “stationary” variance: and colored stochastic process with a spectral density given by Lemma 6.1

Ia ∞∞



 =

; τ =   ω   ω τ S f K(63)  = R t ∑∑XXexp j i k t exp jk

1 f fV ik0 0

=∞

∞

ik= (70)

[  ω  α   α  + τ ] where K is a constant for a particular device, I is the current through E exp j 0 i t k t

the device, and a is a constant in the range 0:5 to 2. There is a lot of

E [  ω  α   α  + τ ] controversy both about the origins and modeling of 1 = f noise. The The expectation in (70), i.e., exp j 0 i t k t is the

α  α + τ  spectral density in (63) is not a well-defined spectral density for a sta- characteristic function of i t k t . This expectation is inde-

tionary stochastic process: It blows up at f = 0. Keshner in [10] argues pendent of t for large time as established by the following theorem: that 1= f noise is really a nonstationary process, and when one tries to Theorem 6.1 If t is large enough such that

model it as a stationary process, this nonphysical artifact arises. We



are not going to dwell into this further, which would fill up pages and 1 ω2 2 2 1 i = k   σ   

exp 0 i k t (71) = would not be too useful other than creating a lot of confusion. In- 2 0 i 6 k stead, we will “postulate” a well-defined stationary stochastic process

= α  α + τ model for 1 f noise: We will introduce a cut-off frequency in (63), then i t k t is a Gaussian random variable and its charac-

below which the spectrum deviates from 1= f and attains a finite value teristic function, which is independent of t, is given by =

at f 0. To do this, we use the following integral representation [11]  =

0 i 6 k

Z

∞

[  ω  α   α  + τ ] 

1 1 E exp j 0 i t k t 1 2 2 2

ω σ jτ j = γ exp i i k = 0

4 2d(64) 2 j

j 2 π f 0 γ +2f (72)

We introduce the cut-off frequency γc in (64), and use 2  where σ t is as in (57), (58) or (59). Note that the condition (71) is

Z ∞ 1 same as the condition (51) in Theorem 4.2.

 = γ S  f 4 d(65) N1 = f

γ 2 2 We now obtain the stationary autocovariance function: π

c γ +2f

Corollary 6.1

γc

arctan π ∞

1 2 f 1



= 4 (66) 2 2 2

τ=  ω τ σ jτ j

R  ∑XXexp ji exp ω i (73)

j π

j f 2 f V ii0 2 0

∞ i=

for the spectral density of a stationary stochastic process that models

+ α 

To obtain the spectral density of x t t , we calculate the Fourier = 1= f noise. The spectral density in (65) has a finite value at f 0: s transform of (73):

4 ∞ = S 0 (67) N1 = f

γ 

=  + 

c SV  f ∑XiXiSifif0 (74)

∞ i= The autocorrelation function that corresponds to the spectral density in ω π

(65) is given by where 0 = 2 f0 and

τ = γjτj

 = F f τ g R 2E(68) 

N1 = f 1c Si f Ri (75)

 

1 2 2 2

 

= F σ jτ j where the exponential integral E1 z is defined as exp ω i (76) 2 0

Z ∞



exp  tz = E1 t dz The Fourier transform in (76) does not have a simple closed form. z 1 Mullen and Middleton in [12] calculate various limiting forms for this

The power in a 1= f noise source modeled with a stochastic process Fourier transform through approximating series expansions. We are with the spectral density (65) is concentrated at low frequencies, fre- going to use some of their methods to calculate limiting forms for (76) quencies much less than the oscillation frequency for practical oscilla- for different frequency ranges of interest, but before that, we would

2

= σ  τ

tors. Hence, (54) is satisfied for 1 f noise sources. like to establish some basic, general properties for t , Ri and  6 Spectrum of an oscillator with phase noise due to Si  f . colored noise sources Lemma 6.2

2

α 

σ  

Having obtained the stochastic characterization of t due to a col- t 2

j j   ored noise source in Section 4, we now compute the spectral density lim∞ = V0 SN 0 (77)

t ! t

+ α 

of the oscillator output, i.e., xs t t . We first obtain an expression

; τ   + α   for the non-stationary autocovariance function R t of xs t t . Corollary 6.2

Next, we demonstrate that the autocovariance is independent of t for 

6=

∞S00

+ α   N

“large” time. Finally, we calculate the spectral density of xs t t 2

σ  = by taking the Fourier transform of the stationary autocovariance func- lim∞ t (78)

t ! σ2 ∞  =

+ α 

tion for xs t t . SN00

+ α  We start by calculating the autocovariance function of x t t ,

s 2

∞ < ∞ given by where σ  is a finite nonnegative value.

For the models of burst and 1 = f noise discussed in Section 5, we have



; τ = E [  +α   + τ + α + τ]

RV t xsttxs t t (69) =

SNburst 0 1  Definition 6.1 Define Xi to be the Fourier coefficients of xs t :

∞ from (61), and

=  ω 

xs t ∑Xiexp ji 0t 4

 =

∞ = S 0 i N1 = f γc

2

 6= σ  =∞  ∞ case when both white and colored noise sources are present and unify from (67). Thus SN 0 0, and hence, limt ! t are satisfied for both. our results. Let there be p white noise sources and M colored noise

σ2 = Since 0 0wehave sources:

M =

Ri0 1 (79) d

+  +    +     qx gxBxbt∑Bx b t (84) dt wwcm cm

and hence m=1 Z

∞ n n  p n n

: ! 
 ! ; = ;::: ; 


where Bw  :IR IR , B cm :IR IR m 1 M,bw:

 = S  f d f 1 (80) p

i IR !IR is a vector of (uncorrelated) stationary, white Gaussian noise

∞

 ! ; = ;::: ;

processes, and bcm  :IR IR m 1 Mare zero-mean, Gaussian, 

for any colored noise source. The total power XiXi in the ith har- stationary colored stochastic processes (uncorrelated with each other, 

monic of the spectrum is preserved. The distribution of the power in and with bw t ) with autocorrelation function/spectral density pairs 

frequency is given by Si  f .If

= Ff τg = ;::: ; SNm f RNm m 1 M

2

σ  =∞ lim∞ t (81) where

t !

 6= = ;::: ;

SNm0 0 m 1 M  then Si 0 is nonnegative and finite, which is the case when the noise

source spectrum extends to DC. On the other hand, when the noise

τ = ;::: ; and RNm m 1 Mare assumed to satisfy (54). In this case,

source is bandpass, i.e., its spectrum does not extend to DC, then (81) α 

the phase error t satisfies the nonlinear differential equation

 δ will not be satisfied, and Si  f will have a function component at

α  

f = 0. Now, we concentrate on the case when (81) is satisfied, i.e., d t T

 + α  = v t t

when the spectrum takes a finite value at the carrier frequency (and its dt 1

"  harmonics). Next, we proceed as Mullen and Middleton in [12, 13] M

and calculate limiting forms to the Fourier transform in (76) through

  + α   +   + α    α = B x t t b t ∑Bx t t b t ; 0 0 approximating series: w s w cm s cm

m=1 Theorem 6.2 Let (81) be true. For f away from 0, (76) can be ap- (85)

proximated with  where v1 t is the periodically time-varying Floquet vector of Sec- 2

2 2 f0 tion 3. Let

  j j    S  f i V S f f 0 (82) i 0 2 N Z T

f 1 T T

τ  τ  τ τ τ cw = v1 Bw xs Bw xs v1 d (86) π T 0

where ω0 = 2 f0. For f around 0, (76) can be approximated with

  and 1

2 2 2 Z

  F j j  jτ j + S  f exp ω i V S 0 T

i 2 0 0 N 1 T

τ   τ τ = ;::: ;

V0m = v1 Bcm xs d m 1 M(87)

  T 0

1 ω2 2 2 F j j  jτ j

exp 0i V0 SN 0 ~ α 

2 Lemma 7.1 t that satisfies (85) becomes a Gaussian random vari-

  Z

Z able with constant mean, and variance given by

j ∞ τj

2 2 2

ω j j τj   +   j F

0i V0 RN z dz zRN z dz M Z t τ j

j 0 2 2

 = + j j  τ  τ τ σ 

∑ t cw t 2 V0m t RNm d 2 2 0

22 22 m =1

j   j j   f0i jV0SN0 f0i V0SN0

Z ∞

+ ~

 M

 π 

2 2 2 1 exp j2 ft

+ j j  

244 2 2 244 2 2 = cwt ∑ 2 V0m SNm f df

π j j   + π j j   + fi V0SN0 f fi V0SN0 f ∞ π2 2 0 0 4 f

m=1

 

Z Z

∞ jτ j

22 2 for t large enough such that

ω j j jτj   +   

F 0iV0 RNzdz zRN z dz f 0

 τj j 0

(83) 1 ω2 2 2 1 i = k   σ   

exp 0 i k t (88) = 2 0 i 6 k

where ~ denotes convolution. The first term in (83) is a Lorentzian 

with corner frequency Theorem 7.1 With αt characterized as above, the oscillator output

+ α 

x t t is a stationary process, and its spectral density is given by

s

π 2 2 2 j  

f0 i jV0 SN 0 ∞



=  + 

SV  f ∑XiXiSifif0 (89)

=∞ and can be used as an approximation for (76) around f = 0 by ignoring i

the higher order second term. (83) contains the first two terms of a  series expansion for (76). where Xi are the Fourier series coefficients of xs t , and

8 22M2

+ j j  

  ∑ ficVS0

From (82), we observe that the frequency dependence of Si f is as 0wm=10mNm

>  > 2 f 0

2 < 2 4 4 M 2 2

=  

π + j j   + ∑  1 f multiplied with the spectral density S f of the noise source for f i c V S 0 f

N 0 w m=1 0m Nm =

offset frequencies away from the carrier. This result matches with mea- Si  f (90)

>

> 2 =

surement results for phase noise spectrum due to 1 f noise sources. : 2 f0 M 2

j j    i c + ∑ V S f f 0

2 w = 0m Nm 7 Phase noise and spectrum of an oscillator due to f m 1 white and colored noise sources The full spectrum of the oscillator with white and colored noise sources

has the shape of a Lorentzian around the carrier, and away from the car-  In [1], we considered the case where the perturbation bt is a vector of 2 (uncorrelated) stationary, white Gaussian noise processes and obtained rier, the white noise sources contribute a term that has a 1= f frequency

α  a stochastic characterization of the phase deviation t and derived dependence, and the colored noise sources contribute terms that have a 2 the resulting oscillator output spectrum, as we did it here for a col- frequency dependence as 1= f multiplied with the spectral density of ored noise source in Section 4 and Section 6. Now, we consider the the colored noise source. 8 Examples L C R b(t) We have derived an analytical expression, given by (89) and (90), for the spectrum of the oscillator output with phase noise due to white i=f(v) and colored noise sources. The analytical expression in (89) and (90) contains some parameters to be computed:

 Xi : The Fourier series coefficients of the large-signal noiseless (a) Simple oscillator  periodic waveform x t of the oscillator output. s v1(t) for Capacitor Voltage 1.5

1 R T T T

= τ  τ  τ τ τ  cw T 0 v1 Bw xs Bw xs v1 d : Scalar that char- acterizes the contributions of the white noise sources. 1

1 R T T

= τ   τ τ = ;::: ;  V v B x d m 1 M: Scalars that 0m T 0 1 cm s 0.5 characterize the contributions of the colored noise sources.

0 

Once the periodic steady-state xs t of the oscillator and the scalars cw

= ;::: ; and V0m ; m 1 Mare computed, we have an analytical expres- sion that gives us the spectrum of the oscillator at any frequency f . −0.5 The computation of the spectrum is not performed separately for every frequency of interest. We compute the whole spectrum as a function −1 of frequency at once, which makes our technique very efficient. Using −1.5 0 1 2 3 4 5 6 7 the results of the theory developed in this work, an analysis and de- time

sign tool for low phase noise oscillator design was implemented in an  in-house circuit simulator. (b) v1 t for the capacitor voltage

−10 T T Oscillator with parallel RLC and a nonlinear current source x 10 v1 (t) BB v1(t) for the noise source 2

We now present simulation results in the phase noise characteriza- 1.8 tion of a simple oscillator in Figure 1(a). The resistor is assumed to be noiseless, but we insert a stationary external current noise source 1.6

1.4  across the capacitor. The Floquet vector v1 t is a two-dimensional vector, since the oscillator has two state variables, namely the capac- 1.2 itor voltage and the inductor current. Figure 1(b) shows the entry of 1

0.8  v1 t corresponding to the capacitor voltage. Now, let us assume that we have two current noise sources across 0.6 the capacitor, one of them a white stationary noise source, and the 0.4 other a colored stationary noise source with bandwidth much smaller 0.2 than the oscillation frequency. To calculate the spectrum of the capac- 0 0 1 2 3 4 5 6 7 itor voltage given by (89) and (90), we need to compute cw in (86) time for the white noise source, and V0 in (87) for the colored noise source.

T T

   

   For stationary noise sources, Bw t in (86) and Bc t in (87) are con- (c) v1 t BwBw v1 t for the noise source

T T

   stant functions of time t. Figure 1(c) shows v1 t BwBwv1 t which is a periodic function of time t. Note that cw is the time-average of this Figure 1: Simple oscillator with parallel RLC-nonlinearity quantity.

T  From (87), we observe that V0 is the time-average of v1 t Bc.The [6] R. Telichevesky, K.S. Kundert, and J. White. Efficient steady-state analysis based on

matrix-free krylov-subspace methods. In Proc. Design Automation Conference, June  time-average of v1 t in Figure 1(b) for the capacitor voltage is 0!

1995.  Thus, we conclude that any stationary (with the modulation Bc t a constant function of time) colored noise source (with bandwidth much [7] C.W. Gardiner. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer-Verlag, 1983. smaller than the oscillation frequency) connected across the capacitor [8] H. Risken. The Fokker-Planck Equation. Springer-Verlag, 1989. has no contribution to the oscillator spectrum due to phase noise, be- [9] P.R. Gray and R.G. Meyer. Analysis and Design of Analog Integrated Circuits. John cause V0 = 0 for this noise source. Wiley & Sons, second edition, 1984.

[10] M.S. Keshner. 1= f noise. Proceedings of the IEEE, 70(3):212, March 1982. CMOS ring-oscillator α [11] F. X. Kaertner. Analysis of white and f noise in oscillators. International Journal A ring-oscillator with CMOS inverter delay cells was simulated for its of Circuit Theory and Applications, 18:485–519, 1990.

phase noise spectrum. It oscillates at 2:86 GHz and burns 150 mW. [12] J.A. Mullen and D. Middleton. Limiting forms of fm noise spectra. Proceedings of

  Figure 2 shows S1  f in (90) using the expression for f 0. Contri- the IRE, 45(6):874–877, June 1957. [13] D. Middleton. An Introduction to Statistical Communication Theory. IEEE Press, butions of the thermal (white) and 1= f (colored) noise sources, as well as the total spectrum, are plotted. 1996. Acknowledgments Single SideBand Phase Noise We would like to thank Jaijeet Roychowdhury, Hans Georg Brachten- −20 dorf, David Lee, Amit Mehrotra, Peter Feldmann and Bob Melville for −40 useful discussions. References −60 −80 [1] A. Demir, A. Mehrotra, and J. Roychowdhury. Phase noise in oscillators: A unifying theory and numerical methods for characterisation. In ACM/IEEE Design Automa- −100 tion Conference, June 1998. dBc/Hz [2] R. M¨arz. On linear differential-algebraic equations and linearizations. Applied Nu- −120

merical Mathematics, 18, September 1995. −140 [3]R.Lamour,R.M¨arz, and R. Winkler. How floquet theory applies to differential- algebraic equations. preprint, 1997. −160

−180 [4] M. Farkas. Periodic Motions. Springer-Verlag, 1994. 3 4 5 6 7 10 10 10 10 10 [5] P.A. Selting and Q. Zheng. Numerical stability analysis of oscillating integrated Offset frequency circuits. Journal of Computational and Applied Mathematics, 82(1,2):367–378, Figure 2: CMOS ring-oscillator September 1997.