τ /2π S, ρ < ρ <... generate Dirac distribution δ ( τ ). Spectral density

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1 IOM and ession of e Commission of Acousics, Bucares 5-6 May COMPARATIVE ANALYI OF THREE IMULATION METHOD FOR BAND LIMITED WHITE NOIE AMPLE Ovidiu OLOMON, Carles W. TAMMER, Tudor IRETEANU 3 *Corresponding auor: solomonovidius@yaoo.com In is paper ree simulaion meods for samples of Gaussian band limied wie noise random process are analyzed. Te relaionsips beween parameers a deermine e specral bandwid of simulaed random processes are obained by imposing e condiion o ave equal mean square values. Keywords: wie noise, Wiener process, socasic differenial equaion, Mone Carlo meod.. PRINCIPLE OF IMULATION METHOD Wie noise is a process widely used in modeling dynamic sysems subjeced o random eciaions. Tis process is no pysically realizable (i.e. energy is infinie). I is used as a useful approimaion of many random processes encounered in pracice, ofen being filered; i leads o imporan simplificaions in e sudy of socasic differenial equaions.. FILTERING METHOD We consider () a saionary random process, normal, zero mean and correlaion funcion []: τ R ( τ) = π / e,, > () ( ) For sufficienly large values of, e properies of socasic process () approimae ose of a saionary wie noise, because e correlaion funcion ( τ ) a epresses e saisical dependence beween () and ( + τ ) akes very small values, even for values very close o zero. Wen is an ineger, e series of funcions R ( ) n τ /π, < <... generae Dirac disribuion δ ( τ ). pecral densiy of random process wi correlaion funcion given above is: R iντ ( ) ( ) ν = R τ e dτ π = () + ( ν / ) Wen is obain ( ν ) =. By analogy wi wie lig a conains all frequency Z componens, is propery was given e name wie noise random process z (). If a random process specral densiy is approimaely consan unil sufficienly ig frequencies, we are dealing wi a broadband random process. In is case, e simulaed wie noise can be a useful approimaion. Romanian-American Universiy, Bvd. Epoziiei, B, RO- Universiy of Ba, Ba BA 7AY, England, 3 Insiue of olid Mecanics of e Romanian Academy, Cons. Mille 5, RO-4

2 355 Comparaive analysis of ree simulaion meods for band limied wie noise samples Te random process () belongs o e class of saionary random processes wi normal disribuion and raional specral densiy of e form []: were Pi ( υ) ( υ) = (3) Qi ( υ) p q p k q k k k k= k= Pi ( υ) = β ( iυ), Qi ( υ) = α ( iυ) β, k =,,.., p, α, k =,,.., q, p< q k k An imporan propery of ese processes is a ey can be modeled by Iô socasic differenial equaions. From a ecnical poin of view, is propery is equivalen o e fac a any saionary random process wi normal disribuion and raional specral densiy can be obained by filering saionary normal wie noise. Te ransfer funcion of e linear sysem corresponding o is filer and e socasic equaion are Pi ( υ) Hi ( υ) = (5) Qi ( υ) and wi e inpu z () and e oupu () p k d () d z() βk p k d q q k p αk = q k k= d k= (6). By applying (4), e parameers of equaion (6) are β =, α =, α =, p=, q=. In is case, e Iô socasic equaion becomes d () = () d + dw() (7) E dw() = π d. were ( ) (4). FORMAL DIFFERENTIATION OF WIENER PROCEE In e disribuion eory cone [3], saionary wie noise is normally defined as e derivaive of Wiener process, specifying is definiion by e noaion []: dw() = z() d (8) were dw() = w( + d) w() (9) represens e infiniesimal increases of brownian moion w(), aving e properies: Edw [ ( )] =, E ( dw() ) = cd, Edwdw [ ( ) ( )] =, () Te generalized Gaussian wie noise process is defined as e derivaive of Wiener generalized random processes, wi zero mean and auocorrelaion funcion equal wi cδ ( s), were c is a consan a can be aken by a convenien coice equal o [4]. For a fied posiive consan, is defined e random process z () as [5]:

3 Ovidiu OLOMON, Carles W. TAMMER, Tudor IRETEANU 356 () w ( + ) w ( ) z = () Te process defined above as a zero mean normal disribuion, e auocorrelaion funcion R z ( τ ) = ma, τ and specral densiy is: z sin( πν ) ( ν ) = πν Mean square of random process () is given by z () (3) σ = (4) Te random process defined by (), for small enoug, is a good approimaion of a Gaussian wie noise process [4], [5]..3 MONTE CARLO METHOD A discree ime isory (sample funcion) of Gaussian saionary limied band-noise zn+ = z( n Δ ), n =,,..., N (5) can be derived from a sequence of pseudo-random numbers u, n=,,..., N uniformly disribued on e n inerval [,] [6]. Mos digial compuers include pseudo-random number generaor (Mone Carlo simulaion) aving e following recursive form: = a + b(mod T), (, T], n=,,,... (6) n+ n n+ were b and T are relaively prime. For an ineger iniial value or seed, e relaion () generaes a sequence aking ineger values from o T-, e reminders wen e n+ = an + b are divided by T. Wen e coefficiens a, b and T are cosen appropriaely, e numbers: un = n / T (7) are uniformly disribued on e inerval [,]. If ( un, un), n=,,... represens a series of pairs of suc values, en e following series of values: zn = π/ Δ ln un cos πun, n=,,... (8) z = π / Δ ln u sin πu, n=,,... n n n are asympoically ( n ) independen Gaussian random variables, wi zero mean and mean square value equal o π / Δ. Te discree random process defined by: zδ () = zn, ( n ) Δ nδ (9) is mean square convergen o e wie noise process z () wen Δ..4. CALIBRATION OF AMPLING PARAMETER In order o compare e numerically simulaed samples roug e described meods, e relaions beween parameers, and Δ, are esablised by imposing e same mean square for all obained samples. Te mean square values of processes () and z () are given by

4 357 Comparaive analysis of ree simulaion meods for band limied wie noise samples and υ σ = d d,, d υ = = π = = dυ + υ + sinπυ sin σ z = dυ = d=, = π, d d πυ π υ = π υ Taking ino accoun e relaions () and () and e fac a e mean square value of random process zδ () is π / Δ, e following relaionsip( = ) is obained π = π = () Δ () (). NUMERICAL IMULATION REULT For e numerical simulaion of ese processes, was cosen e sampling rae oal ime inerval T = s. According o (), =, =, 59. In order o simulae e process (), is used e ieraive relaion [7]: Δ =. ( + Δ ) = ( ) μ + σ n (3) and e were μ e Δ =, σ = ( μ ) and n are independen normally disribued random numbers wi zero mean and mean square value equal o πδ. Figures -3 sows e ime isories wiin e ime inerval -s for ree samples obained by e described meods. 3 σ = () Fig.. ample of wie noise process ( ) 3 σ = () z Fig.. ample of wie noise process z ()

5 Ovidiu OLOMON, Carles W. TAMMER, Tudor IRETEANU σ = () z Δ Fig. 3. ample of wie noise process z ( Δ ) As one can see, e mean square values, obained for e ree numerically generaed samples, are almos equal if e sampling parameers fulfill e imposed condiions (). 3. CONCLUION Tree simulaion meods of samples of Gaussian band limied wie noise random process are analyzed, imposing e equaliy of mean square values for e numerically generaed samples. Te wellknown Mone Carlo meod is compared wi wo oer simulaion meods, based on socasic differenial equaions and on e formal derivaive of e Brownian process. Te resuls obained by e las wo meods can be viewed as a rigorous maemaical assessmen of e more pracical Mone Carlo meod. REFERENCE. Jazwinski A. ocasic Proccesses and Filering Teory, Acedemic Press, 97. ireeanu T., iseme mecanice oscilane neliniare supuse la eciaţii aleaoare, eză de docoara, Arnold L., ocasic differenial equaions: eory and applicaions, Wiley, Bălaş C., Conrolul semiaciv în procese socasice de ip Iô, Asab, 9 5. Kloeden P., Plaen E.,,Numerical oluion of ocasic Differenial Equaions, pringer-verlag, Abramowiz M., egun I., Handbook of Maemaical Funcions wi Formulas, Graps, and Maemaical Tables, Dover, Gillespie D.T., Eac numerical simulaion of e Ornsein Ulenbeck process and is inegral, Pys.Rev.E 54:84 9, 996