Sampling of random data streams

INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 9, NO. 1,MARCH 2011 1 ©2011 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119 Sampling o Random Da a S eams Gus á CEPCIANSKY.1, Ladisla SCHWARTZ 1 1 Depa men o Telecommunica ions and Mul imedia, Uni e si y o Zilina, Uni e zi na 1, 01026 Zilina, Slo akia [email protected], schwa z@ el.uniza.sk Abs ac . Mode n elecommunica ion ne wo ks wo k on he ansmission me hod o common da a s eams in which da a bu s s consis ing o packe s ha u he consis o pa icula bi s a e mul iplexed om a ious a ic sou ces. The la ge amoun o da a is ansmi ed h ough a ansmission medium (op ical ib e), he mo e equen ly bu s s occu , and he lowe amoun o da a, he mo e a ely hey ollow. I i is equi ed o moni o how la ge amoun o da a is being ansmi ed in a ne wo k b anch in o de o ind ou , o which measu e o his b anch is occupied, i is no necessa y o ake each in o ma ion uni (each packe o e en each pa icula bi ). I will do i in o ma ion whe he a da a bu s occu s in he ansmission o does i no occu is aken in ce ain ime in e als – wi h a ce ain sampling equency. The pape deals wi h his sampling in e als. Keywo ds Sampling, a ic, da a bu s , passi e op ical ne wo k. 1. In oduc ion The e a e many ypes o da a bu s s on a ansmission ou e be ween wo ne wo k nodes om a ious a ic sou ces. The da a bu s leng h is conside ed as a andom a iable. Acco ding o Figu e 1, he bu s ain in a common da a s eam can be desc ibed as a andom p ocess X( ) consis ing o se ies o andom occu ing ec angula pulses wi h a andom ampli ude ha only gains 2 alues o x0 = 0 andx1 = 1 and wi h a andom ime leng h Q ha gains alues o u k,k = 1, 2, ... Fig. 1. Packe ain in he da a s eam The mean le el o he a e age alue o his andom p ocess å = ¥® =N k k TT y 1 1 lim J (1) also exp esses he a e age a ic use o a communica ion channel, e.g. wi h which pa o he o al ime T heo e ically in ini ely long he channel was occupied by he ansmi ed da a bu s s. In o he wo ds, he alue y also exp esses he p obabili y ha he communica ion channel will be in any momen o sampling ound as occupied. Fu he , le l be he coun o seizu es o he communica ion channel by he ansmi ed da a bu s s and m he coun o eleases o he communica ion channel du ing a ime uni . The a io m l =a(2) ep esen s he a ic o e ed o he communica ion channel. The in e se alue o m å ¥ = ¥® == 1 1 lim 1 k k NN J m J (3) is he a e age bu s leng h. 2. Theo y 2.1 Random P ocess Cha ac e is ics The occu ence o da a bu s s in he common da a s eam can be conside ed as he Ma ko andom p ocess because he ea u es o such a p ocess a e ul illed: ·a da a bu s occu ence is ully andom, ·bu s s las o a andom ime, · he p obabili y o a da a bu s occu ence o end o a bu s du ing a e y small ime in e al D will be p opo ional o he du a ion o his in e al. u 1 u 2 u 3 u k u N-1 u N T 2INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 9, NO. 1,MARCH 2011 ©2011 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119 These ea u es ep esen a hidden de e minis ic elemen in he andom p ocess. The consequence o his is ha i is possible o explici ly de i e basic cha ac e is ics – co ela ion and co- a ia ion unc ions, dispe sion and spec um o ha andom p ocess [1], [2], [3], [4]. 1) Solu ion o Random P ocess The andom p ocess gains only 2 s a es: S0 – communica ion channel is ee and S1 – communica ion channel is occupied. The p ocess s ays in he ze o s a e S0 i no da a bu s occu s du ing D , he p obabili y o which is: ( ) p D-=D+ l 1, 00 .(4) The p ocess ansi s om he s a e S0 o he s a e S1 (communica ion channel will be seized) i a da a bu s occu s du ing D . The p obabili y o his is: ( ) p D=D+ l , 01 .(5) The p ocess s ays in he s a e S1 i a da a bu s does no inish du ing D , he p obabili y o which is: ( ) p D-=D+ m 1, 11 .(6) The p ocess ansi s om he s a e S1 o he s a e S0 (communica ion channel will be eleased) i a da a bu s inishes du ing D . The p obabili y o ha is: ( ) p D=D+ m , 10 .(7) The p obabili y ha mo e han 1 change occu s in he s a e o he channel du ing D is equal o ze o. The p obabili ies p00,p01,p10,p11 a e a anged in o he ma ix o ansi ion p obabili ies: = ÷ ÷ ø ö ç ç è æ D+D+ D+D+ =D+ ),(),( ),(),( ),( 1110 0100 p p p p P = ÷ ÷ ø ö ç ç è æ D-D DD- =)1) )1 mm ll AJ ×+=D ÷ ÷ ø ö ç ç è æ - - + ÷ ÷ ø ö ç ç è æ =. 10 01 mm ll .(8) The ma ix J is he uni one. The ma ix A is he ma ix o in ensi ies o ansi ion p obabili ies. In gene al, he elemen s o he ma ix Acan be ime dependen . Bu i i is supposed he e godici y o he andom p ocess, e.g. he Ma ko p ocess is homogenous, he ime dependence does no exis . By he solu ion o he basic equa ion o he Ma ko andom p ocess A p p ×= )( )( d d,(9) i can be calcula ed, wi h which p obabili y p0( ) he communica ion channel will be ee and wi h which p obabili y p1( ) he communica ion channel will be occupied by a da a bu s in a ime . The equa ion is sol ed by means o Laplace ans o m: () ò ¥- == 0 ).()}({ psF p deL s,(10) })({ )( A p p ×= þ ý ü î í ìL d d L.(11) Hence, when he ans o m is ca ied ou ( ) ( ) [ ] 1 0- -××= AJspsF .(12) The sough ec o p( ) will be ob ained a e he in e se Laplace ans o m ( ) ( ) { } ( ) [ ] { } 1 11 0- -- -××== AJspLsFL p . (13) Fi s , le s.J – A be w i en in: ÷ ÷ ø ö ç ç è æ +- -+ = = ÷ ÷ ø ö ç ç è æ - - - ÷ ÷ ø ö ç ç è æ =-× mm ll mm ll s s sAJs 10 01 (14) The in e se ma ix: [ ] = ÷ ÷ ø ö ç ç è æ + + ++× =-× - lm lm ml s s ss AJs . )( 1 1 ÷ ÷ ÷ ÷ ø ö ç ç ç ç è æ ++× + ++× ++×++× + = )()( )()( ml l ml mml l ml m ss s ss ssss s .(15) To ca y ou he in e se Laplace ans o m, he elemen s o he in e se ma ix shall be decomposed in o he sum o pa icle ac ions. Then he basic sys em o equa ions (13) acqui es he o m: () () + ê ê ê ê ë é ÷ ÷ ÷ ÷ ø ö ç ç ç ç è æ ++ ++ = ml l ml mml l ml m s psF 1 .0 ú ú ú ú û ù ÷ ÷ ÷ ÷ ø ö ç ç ç ç è æ ++ - + - + ++ + ml m ml mml l ml l ml s 1.(16) The esul will be ob ained by he in e se Laplace ans o m: () (){ } () + ê ê ê ê ë é ÷ ÷ ÷ ÷ ø ö ç ç ç ç è æ ++ ++ ==- ml l ml mml l ml m 0L 1psF p INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 9, NO. 1,MARCH 2011 3 ©2011 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119 ú ú ú ú û ù ÷ ÷ ÷ ÷ ø ö ç ç ç ç è æ ++ - + - + ++- ml m ml mml l ml l ml . )( e.(17) Hence [ ] [ ] ml ml mm ml ml )0()0( )0()0( 1 )( 10 )( 100 pp pp p - + + ++ + = +- e,(18) [ ] [ ] ml ml ll ml ml )0()0( )0()0( 1 )( 10 )( 101 pp pp p +- + + ++ + = +- e.(19) These equa ions desc ibe he Ma ko andom p ocess wi h 1 channel in ha manne ha hey gi e he p obabili y wi h which he andom p ocess will s ay in one o bo h s a es in an eligible u u e ime . 2) Co ela ion and Co- a ia ion Func ion The alue o he andom p ocess X( ) acqui es in a ime he andom a iable X ha can acqui e only 2 alues: x0 = 0 i a da a bu s is no p esen o x1 = 1 i a da a bu s is p esen . Thus, he e a e 4 a ious combina ions o p oduc s o de e mine he co ela ion unc ion R( ): () ( ){ } () ( ){ } () ( ){ } () ( ){ } () ( ){ } 1111 })()({.. )().()( 1111 0101 1010 0000 1 0 1 0 =+Ç=××= ==+Ç=××+ +=+Ç=××+ +=+Ç=××+ +=+Ç=××= ==+Ç== =+= åå == u X XP x Xx XPxx x Xx XPxx x Xx XPxx x Xx XPxx x Xx XPxx X XR ij jiji (20) I x0 = 0, only he las om 4 a icles has a non-ze o alue. As he andom p ocess is also s able and e godic, i does no depend on he alue , so i can be = 0, oo. Then: ( ) ( ) ( ) ( ) ( ) () (){ } 110 0 =Ç= =×=+×= u XXP XXX XR .(21) The p obabili y P{X(0) = 1 ÇX( ) = 1} can be ge om he o mula o he condi ional p obabili y: ( ) ( ) { } (){ } () (){ } 10/110 110 ==×= = = Ç = XXPXP XXP .(22) Supposing s abili y o he andom p ocess wi h p obabili y P{X(0) = 1}, ha he communica ion channel will be occupied a he beginning (in he ime = 0). The condi ional p obabili y is equal o P{X( ) = 1/X(0) = 1}, i he communica ion channel was occupied a he beginning, and i i will be also occupied a e a ime , i can be calcula ed om he equa ion (19), in which p0(0) = 0 and p1 = 1: ú ú û ù ê ê ë é+ + = = ú ú ú û ù ê ê ê ë é + ÷ ÷ ø ö ç ç è æ+ = = + + + == = = = = +- ÷ ÷ ø ö ç ç è æ+- +- J m m l ml l m m l m l ml m ml l ).1( ..1 )( 1 . 1 1. 1 .1. 1 .)( }1)0(/1)({}.1)0({)( a aa a p XXPXPR e e e (23) ú ú û ù ê ê ë é+ + =+- J ).1( . 1 1. 1 a aa ae.(24) he eby he equa ions (2) and (3) we e used, oo. Then he co ela ion unc ion R( ) using (22) and (24) will be: ú ú û ù ê ê ë é + + =+- J ).1( . 1 1. 1 .)( a aa a yR e.(25) The ollowing ela ionship holds be ween a ic a o e ed o a communica ion channel and hei a ic use y: ( ) apy ×-= 1 1.(26) whe e p1 is he p obabili y ha a channel is occupied in he s eady s a e: 1 1 1lim 1 )( )1( 11 + = ú ú û ù ê ê ë é + + =¥= +- ¥® a a aa a pp a J e. (27) Subs i u ing in o (24) we ob ain: . 1 1 1 1 1. 1 .).1()( )1( 2 )1( 1 ú ú û ù ê ê ë é+ ÷ ø ö ç è æ + = = ú ú û ù ê ê ë é+ + -= +- +- J J a a aa a aa a apR e e (28) Co- a ia ion unc ion is: ( ) ( ) 2 mRK -= .(29) whe e m is he mean alue (mean le el) o he andom p ocess. Then 4INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 9, NO. 1,MARCH 2011 ©2011 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119 [ ] 2 2 2 1 2 1 . 1 1).1(. ÷ ø ö ç è æ + = ú û ù ê ë é÷ ø ö ç è æ + -=-== a a a a a apyym .(30) So he co- a ia ion unc ion will be: . )1( 1 1 1 1 )( )1( 2 2 )1( 2 J J +- +- + = = ÷ ø ö ç è æ + - ú ú û ù ê ê ë é + ÷ ø ö ç è æ + = a a a a a a aa a K e e (31) I a will be exp essed by means o y om (26) and (27), we ob ain: y y a- =10 < y , and hen J J . ).1( 1 1 1 2).1.( 1 1 1 )( y y y yy y y y y K- - ÷ ÷ ø ö ç ç è æ+ - --= ÷ ÷ ø ö ç ç è æ+ - - =ee .(32) Dispe sion is: ( ) ( ) yyK -== 10 2 s .(33) 3) Spec um The spec um o a andom p ocess can be calcula ed by means o he Wiene -Khin chin ans o m which is he Fou ie ans o m o a co- a ia ion unc ion: ò ¥ ¥- - = w w de j ).()( KS.(34) Le in he equa ion (32) J a ).1( 1 y- =.(35) and s 2 is gi en by (33). Then == ò ¥ ¥- - - sw w a dee j ..)( 2 S = ú ú û ù ê ê ë é+= ò ò ¥ ¥ +--- 0 0 )()(2 s wa wa dede jj 22 22 211 wa a s wawa s + = ÷ ÷ ø ö ç ç è æ + + - =jj . (36) 0 0 R( ) a 2 S( w ) 0 w a) b) c) X ( ) Fig. 2. Shape, co- a ia ion unc ion and spec um o he andom p ocess The da a bu s s ain, he shape o he co- a ia ion unc ion and he spec um belonging o i a e d awn on Figu e 2 o he compa ison. The spec um on Figu e 2 c) has a shape o a p obabili y dis ibu ion which can ad an ageously be used o de e mina ion o he sampling equency. 2.2 Sampling F equency I he spec um S( w ) is o be conside ed o a p obabili y dis ibu ion, i mus be ul illed: ò ¥ ¥- =1)(. ww dSc,(37) hence 2 2 2 22 22 1 1 2 1 2 1 ps sw wa a s = + = + = òò ¥ ¥- ¥ ¥- x x cd d . (38) So he dis ibu ion will ha e he shape: 22 1 .)( wa p a w + = ,(39) which is he Cauchy dis ibu ion and i exp esses he p obabili y densi y o occu ence o ampli udes o pa icula spec al componen s. Then he elemen a y ec angle ( w ). .D w in Figu e 3 gi es he p obabili y wi h which he componen s in he in e al om w o w + D w will occu in he spec um. All spec al componen s in he ange om -¥ o +¥ occu wi h he p obabili y P = 1 because he in eg al om ( w ) in his ange equals o 1. Tha means i we wan o ca ch all changes in he cou se o he andom p ocess wi h he o al ce ain y, i mus be sampled wi h he in ini e high equency. The e o e, le he sampling equency is limi ed by he equency w and hen he p obabili y P wi h which each change will be caugh in he andom p ocess a his sampling equency will be calcula ed: INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 9, NO. 1,MARCH 2011 5 ©2011 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119 ( w ) 0 Dw w + w - w Fig. 3. Dis ibu ion o ampli udes o spec al componen s in he andom p ocess a w p wa w p aw w Pa c g 2 22 = + =ò - d.(40) The sampling equency o he sampling pe iod can be calcula ed om his equa ion o he gi en p obabili y P: 2 g )1(2 1 2 g. 2 1P y P p Jp p a p - == ,(41) 2 co g.)1(2 P yT p Jp -= .(42) 3. Expe imen al 3.1 Analysis o Resul s The equa ions (41) and (42) gi e he sampling equency o sampling in e al, espec i ely in he case da a bu s s occu o ally andomly and hei leng h is o ally andom, oo. I is also he case when he limi si ua ion can occu when da a bu s s ollow consecu i ely each o he and he gap be ween hem is in ini ely small (y ® 1), o he da a bu s s leng h is in ini ely sho ( u ® 0), o bo h cases occu simul aneously. 3.2 P ac ical Applica ion An example whe e his heo y could be applied is he gigabi passi e op ical ne wo k (GPON) [5], [6], [7], [8], [9]. The GPON is an access ne wo k whe e mo e subsc ibe s (up o 128) sha e he same op ical ib e. The ups eam communica ion be ween an op ical ne wo k e mina ion (ONT) on he subsc ibe side and he b oadband node – op ical line e mina ion (OLT) on he ne wo k side goes h ough op ical pulses bu s s called ansmission con aine s (T-CONTs) ca ying da a s eams. I is necessa y o hinde o e lapping T-CONTs o simul aneously communica ing ONTs. The e o e a ime window is ma ched o each T-CONT om OLT. This window de ines he T-CONT leng h and he ime ins an s in which an ONT may send T-CONTs wi h da a. To de e mine hese p ope ly, The OLT ei he ecei es and decodes dynamic bandwid h assignmen (DBA) epo s coming om ONT, o moni o s he a ic load on each b anch in he op ical ne wo k. The i s me hod is called he s a us DBA and he second he non-s a us DBA. The s a us DBA equi es exchange o a ious con ol in o ma ion be ween an OLT and an ONT in heade s o da a ba ches. The non-s a us DBA does no equi e any in o ma ion exchange bu i is necessa y o know how o en he a ic s a us shall be sampled on a ne wo k b anch. He e he o mula (41) o (42) can be help ul. The GPON bi a e is 1 244,16 Mbi /s in he ups eam. The sho es da a uni is one 125 ms long ame which is pe o med by 19 440 by es. The longes da a uni in he ups eam is 216 = 65 536 by es which las s 421,4 ms. Le ´s ake he mean alue o hese 2 igu es o he a e age bu s leng h: s2,273 2 4,421125 mJ = + =.(43) Conside ing P = 0,95, we ob ain o sampling in e al om (42): )1.(89365 2 95,0 co g.2,273).1(2 yyT -=-= p p [ms], (44) ( ) yT -×» 166 [ms]. (45) The longes sampling in e al will be when he e is no a ic load (y = 0): 66 max, = Tms. (46) In o de o a oid collision o 2 consecu i e T- CONTs, he gua d in e al consis ing o 32 bi s is in oduced in he ups eam. They pe o m he gua d ime 32/1 244 160 000 = = 25,72 ns. Du ing his ime, ONT emi s no ene gy in o he op ical ib e. Tha means he e a e 4-by e gaps in he 65 536-by es bu s s. Tha ep esen s he a ic load o 99994,0 53665 453665 = - =y.(47) and so he sho es sampling in e al will be: ( ) smsT m 4004,09994,0166 min, ==-×= . (48) I is necessa y o ake in o conside a ion ha he maximum and minimum sampling in e als T ,max and T ,min a e calcula ed on condi ion ha he da a bu s s occu andomly on he op ical ibe and hei leng h is also andom in he ange om 125 o 421,4 ms. E en he minimum sampling in e al o 4 ms is 5000- imes longe han i each pa icula bi should be sampled which would gi e he sampling in e al o 1/1 244 160 000 = 0,8 ns. 6INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 9, NO. 1,MARCH 2011 ©2011 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119 4. Conclusion Applica ion o his heo y on da a s eams con ol may sa e many wo king ope a ions o digi al mic ocon olle s and hus enhance hei pe o mance. Acknowledgmen s The au ho s g a e ully acknowledge suppo om he VEGA p ojec No. 1/0375/08 “The Analy ic Model Home Access Ne wo k New Gene a ion”. Re e ences [1] LEVIN, B., L.:Teó ia náhodných p oceso a jej aplikácie ádio echnike. SNTL P aha, 1965. [2] SVEŠNIKOV, A. A.: Sbí ka úloh z eo ie p a děpodobnos i, ma ema ické s a is iky a eo ie náhodných unkcí. SNTL P aha, 1971. [3] VENTCEĽOVÁ, J., S.: Teó ia p a depodobnos i. Al a-SNTL B a isla a, 1973. [4] PRCHAL, J.: Teo ie p a děpodobnos i e sdělo ací echnice. NADAS P aha, 1975. [5] A b oadband op ical access sys em wi h inc eased se ice capabili y using dynamic bandwid h assignmen . ITU-T Recommenda ion G.983.4, 2001. [6] Gigabi -capable Passi e Op ical Ne wo ks (G-PON): Physical Media Dependen (PMD) laye speci ica ion. ITU-T Recommenda ion G.984.2, 2003. [7] Gigabi -capable Passi e Op ical Ne wo ks (G-PON): T ansmission con e gence laye speci ica ion. ITU-T Recommenda ion G.984.3, 2004. [8] Gigabi -capable Passi e Op ical Ne wo ks (G-PON): ONT managemen and con ol in e ace speci ica ion. ITU-T Recommenda ion G. 984.4, 2004. [9] Gigabi -capable Passi e Op ical Ne wo ks (G-PON): Enhancemen band. ITU-T Recommenda ion G. 984.5, 2007. Abou Au ho s ... Gus a CEPCIANSKY was bo n in 1947. 1971 – Mas e o sciences (MSc) in elecommunica ions on Uni e si y o Žilina. 1971 – 1976 – employed a Slo ak Telecom as a echnical de elopmen specialis . 1976 – 2002 – employed a Slo ak Telecom as he head o he ope a ional esea ch cen e. 2002 – 2009 – employed a T-Com as a echnical assis ance cen e specialis and a he same ime as a lec u e on he Telecommunica ion Depa men o he Elec ical Facul y, Uni e si y o Žilina. Nex o ma ion: 1984 – he 1s doc o deg ee (CSc). 1995 – he s udy s ay a Technical Uni e si y Supélec in Pa is. 1998 – he 2nd doc o deg ee (PhD). 2006 – he associa ed p o esso o Elec ical Facul y, Uni e si y o Žilina. Ladisla SCHWARTZ was bo n on 1950 in Žilina. In 1974 g adua ed om he Uni e si y o Žilina, Depa men o Telecommunica ions wi h he Mas e ’s deg ee (MSc). F om 1974 o 1991 he wo ked in he Resea ch Ins i u e o Compu e Technology in Žilina as he head o he Depa men o Da a Communica ions. In 1986 he was awa ded he i le PhD and in 1991 he became a ull ime eache and esea ch wo ke o he Uni e si y o Žilina. In 1999 he was appoin ed an associa e p o esso . His main ocuses a e unc ions, eliabili y, secu i y elecommunica ion and compu e ne wo ks and da a communica ions.