Appendix A Definitions and Solution Techniques of SDEs

Transcription

1 Appendix A Definitions and Solution Techniques of SDEs Here we summarize the mathematical machinery we needed in Chapter 5. This material is from [15]. 1. Mathematical Preliminaries DEFINITION A.1 lf0. is a given set, then au-algebra :F on 0, is a family :F of the subsets of0. with the following properties 1 0 E :F 2 F E :F =;. pc E :F, where pc = n \ F is the complement off in n 3 A1,A2,... E :F=:- A= U::1 Ai E :F The pair (fl, :F) is called a measurable space. A probability measure IP' on a measurable space ( n' :F) is a function IP' : :F --+ [0, 1 J such that 1 IP' [0] = 0, IP' [fl] = 1 2 if A1, A2,... E :F and {Ai}~ 1 is disjoint then The triple (fl, :F, IP') is called the probability space. The subsets F off! which belong to :F are called :F-measurable sets. In a probability context these sets are called events and we use the interpretation IP' [F] = "the probability that the event F occurs" In particular, if IP' [F] = I we say that F occurs with probability I, or almost surely (a.s.). Given any family U of subsets of n there is a smallest u-algebra 1-lu containing U namely 1-lu is called the a-algebra generated by U. 'H.u = n{'h. : 'H. u-algebra of!1, U E 'H.}

2 178 NOISE ANALYSIS OF RADIO FREQUENCY CIRCUITS For instance, ifu is the collection of all open subsets of a topological space 0 (e.g. 0 = lrn), then B = Hu is called the Borel a-algebra on n and the elements B E B are called Borel sets. B contains all open sets, all closed sets, all countable unions of closed sets, all countable intersections of such countable unions etc. If (0, F, IP') is a given probability space, then a function Y : 0 ~!Rn is called F-measurable if y- 1 (U) ={we 0: Y(w) E U} E F for all open sets U E!Rn (or, equivalently, for all Borel sets U C!Rn). If X : n ~!Rn is any function, then the a-algebra Hx generated by X is the smallest a-algebra on n containing all the sets It can be shown that Hx = {X- 1 (B): BE B} where B is the Borel a-algebra on!rn. Clearly X will then be Hx-measurable and Hx is the smallest a-algebra with this property. 2. Ito Integrals The following integral is defined as the Ito integral lst f(t,w)dbt(w) where Bt ( w) is a!-dimensional Brownian motion. We need the following definitions to describe the class of function f(t,w) for which the Ito integral will be defined. DEFINITION A.2 Let Bt(w) be ann-dimensional Brownian motion. Then Ft = Ft) is defined to be the a-algebra generated by the random variables B. ( ), s :::; t. In other words, Ft is the smallest a-algebra containing all sets of the form where k = 1, 2..., t; > 0 and F; C JRn are Borel sets. (All sets of measure zero are assumed to be included in Ft.) DEFINITION A.3 Let {Nt}t~o be a nondecreasingfamily of a-algebras of subsets ofil. A process g( t' w) : JR+ X n ~!Rn is called M -adapted if for each t 2: 0 the function ism-measurable. w ~ g(t,w) Ito integral can be defined for the following class of functions: DEFINITION A.4 Let V = V(S, T) be the class of functions such that j(t,w): JR+ X i1 ~ JR I (t,w) ~ f(t,w) is B x F-measurable where B denotes the Borel a-algebra on JR.+

3 APPENDIX A: Definitions and Solution Techniques ofsdes f(t,w) is :Ft adapted 1 IE [r; J(t,w) 2 ctt] < = DEFINITION A.5 A functions cjj E Vis called elementary if it has the form (A. I) where X denotes the characteristic (indicator) function. Since cjj E Veach function ej must be :Ft-measurable. DEFINITION A.6 (THE ITo INTEGRAL) Let f E V(S, t). Then the Ito integral of f (from S tot) is defined by LT j(t,w)db1 (w) = nl!!,! LT cpn(t,w)db1 (w) (limit in L 2 (P)) (A.2) where { n} is a sequence of elementary functions such that (A.3) An important property of the It6 integral is that it is a martingale. DEFINITION A. 7 A filtration (on (n, :F)) is a family {Mt}t2:0 of a-algebras Mt C :F such that 0 <::; s < t =;. Ms c Mt (i.e., {M,} is increasing). Ann-dimensional stochastic process { Mt}12:o on (n, F, IP') is called a martingale with respect to a filtration {Mt}t20 (and with respect to P 0 ) if Mt is Mt-measurableforall t 2 IE [IMtl] <=for all t 3 IE [MsiMt] = Mtforall s :C:: t Here the expectation and the conditional expectation are taken with respect to P 0 For continuous martingales we have the following important inequality due to Doob: THEOREM A.l (DooB'S MARTINGALE INEQUALITY) If M 1 is a martingale such that t ~ M 1 ( w) is continuous a.s., then for all p :C:: 1, T :C:: 0 and all A > 0 This inequality can be used to prove that the It6 integral can be chosen to depend continuously on t. 1t f(s,w)dbs THEOREM A.2 Let f E (0, 1'). Then there exists at-continuous version of,[ f(s,w)dbs(w) 0 'S t 'S T

4 180 NOISE ANALYSIS OF RADIO FREQUENCY CIRCUITS i.e., there exists at-continuous stochastic process lt on (!1, :F, IP') such that IP' [Jt =lot f(s,w)db.(w)] = 1 Vt 0 ::=; t ::=; T (A.4) COROLLARY A.3 Let f(s,w) E V(O, T)forall T. Then Mt(w) =lot f(s,w)db. is a martingale with respect to :Ft and The Ito integral J fdb can be defined for a larger class of integrands f than V. First, the measurability condition (2) of Definition A.4 can be relaxed to the following: 2': There exists an increasing family of a-algebras Ht t 2 0 such that 1 Bt is a martingale with respect to Ht 2 ft is Ht-adapted Condition 1 implies that :Ft c Ht. The essence of this extension is that we can allow ft to depend on more than :Ft as long as the Bt remains a martingale with respect to the "history" of Is s :::; t. If the above conditions hold then le [B. - Bt IHt] = 0 for all s > t and this is sufficient to carry out the construction of Ito integral. Condition (3) of Definition A.4 can also be weakened to 3' IP'[fotf(s,w) 2 ds<=forallt20] =1 DEFINITION A.8 W denotes a class of stochastic processes satisfying 1 ofdefinitiona.4 and 2' and 3' above. 3. Stochastic Differential Equations THEOREM A.4 (EXISTENCE AND UNIQUENESS THEOREM) LetT> Oand b(, ): [0, T] X Rn-> Rn, a(, ): [0, T] X Rn-> Rnxm be measurable functions satisfying lb(t, x)l + la(t, x)l ::=; C(1 +!xi) x ERn t E [0, T] (A.6) for some constant C, where ial 2 = L iaii 1 2 and such that lb(t,x)- b(t,y)i + ia(t,x)- a(t,y)i:::; Dlx- Yi x,y ERn t E [O,Tj (A.7) for some constant D. Let Z be a random variation which is independent of the a -algebra :F 00 generated by B.( ), s 2 0 and such that Then the stochastic differential equation (A.8) dxt = b(t, Xt)dt + a(t, Xt)dBt 0:::; t:::; T Xo = Z (A.9) has a unique t-continuous solution Xt(w) each component of which belongs to V[O, T].

5 APPENDIX A: Definitions and Solution Techniques of SDEs 181 REMARK A.l Conditions (A.6) and (A.7) are natural in view of the following two simple examples from deterministic differential equations (i.e.. O" = 0): 1 The equation dxt _ xz X 1 dt - t 0 = (A.lO) corresponding to b(x) = x 2 (which does not satisfy (A.6)) has the (unique) solution 1 Xt = -- 0 < t < 1 1- t - Thus it is impossible to find a global solution (defined for all t) in this case. More generally, condition (A.6) ensures that the solution Xt(w) does not explode, i.e., that IX(w)l does not tend to oo in a finite time. 2 The equation dxt = 3X;13 X(O) = 0 dt has more than one solution. In fact for any a > 0 the function t~a t >a (A.ll) solves (A.ll). in this case b(x) = 3x does not satisfy the Lipschitz condition (A.7) at X= 0. Thus condition (A.7) guarantees that equation (A.9) has a unique solution. Here uniqueness means that if X 1 (t, w) and X2(t, w) are two t-continuous processes in V[O, T] satisfying (A.9) then X1(t,w) = Xz(t,w)forallt 2:: Ta.s. (A.l2)

6 Index a-algebra, 94, 177 MATLAB, 78 SPICE, 78 additive noise, 91 amplitude deviation, 33, 37 amplitude noise, 53, 91, 100 asymptotic phase property, 22 Backward Euler method, 103 basin of attraction, ISO bit error rate, 5 block Toeplitz matrix, 80 blocker signal, 5, 117 Boltzmann's constant, 6 Borel a-algebra, 177 Borel set, 177 Brownian motion, 60, 92, 97, 118, 119, 148, 178 characteristic multiplier, 46 charge-pump PLL, 157 clock generation, 147 cyclostationary process, 93, 98, 99 deviation techniques, ISO downconversion, 3 elementary function, 179 emitter-coupled multivibrator, 90 existence and uniqueness theorem Picard-LindelOf, 21 first exit time, ISO flicker noise, 7 Floquet eigenvector, 28 Floquet theory, 24 Fokker-Planck equation, 55, 61,93 forced oscillator circuit, 92 Fourier transform, 57 frequency synthesizer, 147 Gaussian process, 55, 70 Gaussian random variable, 55, 63 harmonic balance, 80 conversion matrix, 80, I 09 harmonic impulse response, 79, 108 harmonic power spectral density, 79, 107 image frequency, 117 image reject mixer, 117 image-reject filter, 3 indicator function, 179 infrared system, I inverse Fourier transform, 57 Ito formula, 95 Ito integral, 178, 180 martingale property of, 60, 179 Ito process, 94, 95 jammer signal, 117 Kramers-Moyal expansion, 55 coefficient, 56 Lindstedt-Poincare method, 38 linear analysis time invariant, 72 time-varying, 72 Lipschitz condition, 181 Lorentzian spectrum, 69, 72, 85, 101, 121 low noise amplifier, 3 LPTV system of equations, 24, 91, 93, 103 characteristic multipliers, 26 Floquet exponents, 26, 78 fundamental matrix of, 25 principal, 25 state transition matrix, 25 LPTV transfer function, 79

7 184 NOISE ANALYSIS OF RADIO FREQUENCY CIRCUITS martingale, 94, 179 measurable space, 177 memoryless nonlinearity, 118, 119 ~xed differential-algebraic equation, 21,92 modified nodal analysis, 92 Monte Carlo noise simulation, 87 multitone excitation, 118 noise, 2 coupling, 5 intrinsic, 6 noise analysis, 2 noise source figure of merit, 77 nonautonomous circuit, 91, 92, 118 orbital deviation, 33 orbital stability, 22 asymptotic, 22, 35, 46, 47, 81 Omstein-Uhlenbeck process, 148, 152 oscillator Colpitt's, 85 generic, 83 LC tank, 83, 85 relaxation, 83, 90 ring, 83, 87 van der Pol, 38 forced, 41, 44 voltage controlled, 90 oscillator perturbation analysis linear, 21, 23 nonlinear, 32 oscillator phase noise analysis frequency domain, 79 time-domain, 77 spectrum, 69 dbc/hz, 75 dbm/hz, 75 perfect time reference, 73 periodic steady-state solution, 102 phase deviation, 32 phase lock loop, 3 phase noise, 53, 91 phase noise/timing jitter sensitivity, 77 phase-locked loop, 147 phase-locked loop frequency division, 148 loop filter, 150 noise analysis, 147 Picard-Lindelof Theorem, 24 PLLnoise autocorrelation function, 153 power spectral density, 154 power amplifier, 3 probability density function, 54 characteristic function, 56, 61 moments, 55 probability measure, 54, 177 probability space, 94, 177 quasi-periodic, 117 radio frequency system, I RF front end, 3 Riemann-Stieljes, 60 shooting method, 77, 78 shot noise, 6 signal to noise ratio, 5 singular matrix, 82 null space of, 82 state transition matrix, 27, 77 stochastic differential equation, 14, 54, 93, 119, 132 existence & uniqueness theorem, 180 stochastic integral, 60 Ito's interpretation of, 60, 66, 68 Stratonovich's interpretation of, 60 stochastic process, 6, 54 autocorrelation, 6, 120 cyclostationary component, 71 stationary component, 71 autocorrelation function, 69 characteristic function, 71 cumulant generating function, 69 cumulants of, 68 cyclostationary, 73 ensemble, 54 finite-dimensional distribution, 54 power spectral density, 6 sample path, 54 stationary power spectral density, 69, 75 wide-sense stationary, 6 substrate coupling, 5 thermal noise, 6 timing jitter, 76 cycle-to-cycle, 76 Tow-Thomas filter, 83 transient simulation, 77 VCO control node, 149 white noise, 6, 58 wide-sense stationary process, 71, 98, 99 Wiener process, 63 zero-crossing, 76