The Exponential, Gaussian and Uniform Truncated Discrete Density Functions for Discrete Time Systems Analysis

Transcription

1 tvan Brbr Th Exponntial, Gauian an Uniform Truncat Dicrt Dnity Function for Dicrt Tim ytm nalyi TEVN BERBER Elctrical an Computr Enginring Dpartmnt Th Univrity of ucklan Princ trt, 00 ucklan Cntral NEW btract: - Continuou nity function an thir truncat vrion ar wily u in nginring practic. Howvr, limit work wa icat to th thortical analyi an prntation in clo form of truncat nity function of icrt ranom variabl. Th rivation of xponntial, uniform an Gauian icrt an truncat nity function an rlat momnt, a wll a thir application in th thory of icrt tim tochatic proc an for th molling of communication ytm, i prnt in thi papr. om imprci olution an common mitak in th xiting book rlat to icrt tim tochatic ignal analyi ar prnt an rigorou mathmatical olution ar offr. Ky-Wor: - truncat niti, Gauian truncat icrt nity, xponntial truncat nity, momnt of truncat niti. Introuction. Motivation Digital tchnology i wily u in th vlopmnt an ign of lctronic vic in communication ytm. Thrfor, th thortical analyi of th ytm aum rprntation of ignal in icrt tim omain. Conquntly, th ranom variabl in th ytm n to b rprnt in icrt form having th valu in a limit intrval. Th problm motivat u to work on truncat icrt nity function rivation. Hr, w will tart with prnting thr ca whr thi kin of analyi i ncary. FIRT CE: In th analyi of icrt tim tochatic proc, w ar uually intrt in calculating thir man, varianc an autocorrlation function. For th calculation, w n to u th probability nity function of a ranom variabl involv. In oing thi, a common mitak i that th nity function i fin a a continuou function of th rlat ranom variabl, which impli that th ranom variabl i of a continuou typ vn though it i not. Thrfor, in orr to prrv mathmatical xactn, w n to fin an u nity function of icrt ranom variabl. Th milaing an mathmatically incorrct procur in uing continuou probability nity function in th thory of icrt tim tochatic proc can b foun in publih papr an book. For xampl, in book [], pag 78, xampl 3.3. an book [], pag 7 an 7, xampl..3, th man an autocorrlation function of a icrt tim harmonic proc, which i fin a X ( m) in( m ) with uniformly itribut pha in th intrval (-π, π) having continuou nity function f ( ) =/π, ar calculat a ( m) E{ X ( m)} in( m ) f ( ) an in( m ) 0 R ( m, l) E{ X ( m) X ( l)} X in( m )in( l ) f ( ) co ( m l) in( ( m l) ) 4 co ( ml) () () E-IN: Volum 6, 07

2 tvan Brbr Howvr, u to th icrt natur of th tochatic proc X(m) an it raliation, which ar icrt tim function x(m) of argumnt m, th prnt calculation ar not mathmatically corrct. In thi papr, th xprion will b rigorouly calculat aftr riving th propr nity function f ( ). Namly, th ranom pha θ ha to b of a icrt typ ini th intrval (-π, π) having a icrt nity function. On th othr han, ba on xiting thory, th nity function of th icrt pha n to b prnt a a ma function. Howvr, th ma function cannot b u to irctly olv th intgral. Thrfor, in thi papr, w riv an u th nity function of th icrt ranom variabl xpr in trm of Dirac lta function, which implifi th olution of rlat intgral, a will b hown in ction 5. ECOND CE: Th gnral xprion for th probability of bit rror of a irct qunc pra-pctrum (D) ytm in a wiban channl (WBC) with whit Gauian noi, auming xitnc of on primary an a t of conary channl, wa riv in [3]. It i aum that th ignal in conary channl ar tranmitt with ranom lay τ in rpct to th rlativ zro-lay in th primary channl. Th probability of rror can b xpr in thi form [3] / ( ) / E b P ( ) rfc / / 4 / N 0 (3) whra ψ i th qunc factor, i th numbr of intrpolat ampl in a chip, β i th praing factor an E b /N 0 i th ignal-to-noi ratio in th channl. Th probability function (3) i a ranom function conition on th lay τ a a ranom variabl. Thrfor, th probability of rror i th man valu of thi function that can b calculat uing thi intgral P P ( ) f ( ). (4) t Th probability nity function of th ranom lay τ i gnrally rprnt by th continuou xponntial nity function, f τ (τ). Howvr, in th abov ca th intrval of τ valu i limit an contain poibl icrt valu. Thrfor, th lay ha to b xpr uing a truncat icrt nity function f t (τ). Furthrmor, thi function ha to b xpr in a form uitabl to calculat intgral (4). To olv all th problm w vlop th xprion of a truncat icrt xponntial function in clo form an montrat it application for thi ca. THIRD CE: In irct qunc prapctrum (D) an co iviion multipl acc (CDM) ytm with imprfct qunc ynchroniation, th probability of rror conitional on th lay τ btwn tranmittr an th rcivr praing qunc, for a ingl-ur ytm with intrlavr, can b xpr a 4( / 4) 4 E b P ( ) rfc X ( ) ( X ( )) b N 0 / (5) whra X( ) /( ), ψ i th qunc factor, i th numbr of intrpolat ampl in a chip, b i th man quar valu of faing cofficint, β i th praing factor an E b /N 0 i th ignal-tonoi ratio in th channl. Th imilar phnomnon occur for wllinvtigat carrir ynchronization, whr a ranom pha iffrnc xit btwn th rciv ignal an th locally gnrat carrir. Thi ranom pha wa uually rprnt by Gauian an Tikhonov nity function, a can b n in Lo an Lam [4], Eng an Miltain [5] an Richar papr [6]. ha bn point out by Richar, Tikhonov itribution can b xpr a Gauian or uniform nity function a it pcial ca. Th niti ar u to fin th man valu of th probability of rror that wa conition on th pha ranom variabl, a may b n from papr of Polprart an Ritcy [7], ong at al. [8] an Chanra at al. [9]. In all th papr, th nity function ar aum continuou an th intrval of thir valu omtim wa byon th intrval of poibl valu of th pha rror a xplicitly not in Richar papr [6]. Thrfor, in thortical analyi an practic, th nity function of icrt ranom variabl τ ha to b xpr uing a truncat icrt nity function. Furthrmor, thi function ha to b xpr in a form uitabl to olv th intgral (4). For th raon, in thi papr w riv th xprion of truncat icrt xponntial, Gauian an uniform nity function an xpr thm in trm of Dirac lta function an montrat thir application.. Backgroun Th thory of continuou nity function an thir truncat vrion, xpr a conitional nity E-IN: Volum 6, 07

3 tvan Brbr function, i wll known. Th problm of icrtiation of Gauian nity function an th rivation of rlat icrt nity function i analy in tail in Roy papr [0] pointing out th importanc of normal itribution that mak ky rol in tochatic ytm molling, apotrophiing that th appli icrtiation mtho approximat th valu of probability of particular vnt fatr than in th ca of imulation. Thi papr wa a goo gui for our vlopmnt in pit of that th notion of th icrtiation intrval i not prrv an ymmtry of th obtain nity function aroun th man valu i lot. imilar analy for gomtric an hyprgomtric probability function wa prnt in Xkalaki work []. Roy an Dagupta [] uggt icrt approximation procur for th valuation of th rliability of complx ytm. Roy [3] invtigat th icrt Rayligh itribution function with rpct to two maur of failur rat, an u thi itribution for valuation rliability of complx ytm. In Ho an Chng papr [4], th xprion for th probability ma of a truncat xponntial nity function wa riv. Howvr, thy i not prnt th rlat nity function in clo form an i not riv th momnt of th nity function. hanullah [5] prnt hi analyi of xponntial itribution. om obrvation on th xponntial half logitic nity function an rlat itribution wr prnt in o an Kang papr [6]. Rachk [7] point out th importanc of applying th truncat xponntial function in molling th amplitu of an arthquak in imology an uggt th u of th gnraliz truncat xponntial itribution. In thi papr, w will u our uniqu approach to riv th xprion for icrt probability nity function, an thn xtn thm for th rivation of rlat truncat nity function that ar mor uitabl in practic whn th icrt ranom variabl xit in a limit intrval of it poibl valu. pcifically, th icrt truncat nity function of our intrt n to fulfil th conition: ) Dicrtiation of th rlat continuou nity function. Th icrt nity function houl b obtain by aigning probability valu a th wight of Dirac lta function. ) Prrvation of th valu of icrtiation intrval T that allow u to rcontruct th ampling intrval an rlat it to th ral valu in practical application. For xampl, in th ca of fining a lay in communication ytm th ampling intrval will b xpr in appropriat tim unit. 3) Exprion of nity function in clo form: By uing Dirac lta function th obtain nity function of icrt ranom variabl can b u to calculat th man valu of a ranom function accoring to th intgral (4). Th Dicrt Truncat Exponntial Dnity Function an It Momnt Thi ction contain baic rivativ of th nity function an rlat momnt. Th obtain rult can b u to fin man valu of th probability of rror xpr a ()..Th icrt xponntial nity function Th procur of a continuou xponntial nity function icrtiation i prnt in Fig.. Th icrtiation i prform at uniformly pac icrt intrval T. Th problm i how to fin th xprion of thi icrt nity function an how to fin it man an varianc. In our approach, in th proc of icrtiation th probability of ach intrval n to b calculat an aign to th icrt tim intant nt. Thr ar two poibiliti in thi ca: Firtly, th probabiliti ar aign to th lft i of th intrval tarting with icrt valu τ=0. conly, th probability ar aign to th right i of th intrval tarting with icrt valu τ=. f (τ) P{0 T } 0 T T (n-)t nt Figur. Dicrtiation of a continuou xponntial nity function. Th two poibiliti will hav om conqunc to th xprion for nity function, man an varianc of th truncat icrt ranom variabl an particular car houl b takn about thi iu. Firtly, w will riv th icrt nity an itribution function in clo form an rlat momnt. Propoition: Th icrt nity function, having th valu at th uniformly pac intant T, i xpr a τ E-IN: Volum 6, 07

4 tvan Brbr T nt f ( ) ( ) ( nt ), (6) n whr δ(.) ar Dirac lta function an τ i a continuou variabl. Th valu of th nity function ar icrt an fin by th poition an wight of lta function. Proof: If thi nity i uniformly icrtiz in rpct to τ, with th intrval of icrtiation of T, th probability valu in any intrval fin by n i ftr calculating n ( ) n0, th xprion for can b foun an u in (0) to fin th man a xpr by (8). imilarly, th man quar valu can b foun a nt P{( n ) T nt} f( ). ( n) T ( ) ( n) T nt T nt To xpr th nity in a clo form, th probabiliti can b u a wight of Dirac lta function rprnting th icrt xponntial nity function at tim intant τ = nt, which rult in thi xprion f ( ) P{( n ) T nt} ( nt ) n, (7) T nt ( ) ( nt ) n which complt our proof. Propoition: Th man an varianc riv for any T, an for a unit intrval T =, can b xpr in th form T f( ) T (8) T T T T an T E{ }. (9) T ( ) T ( ) Proof: Th proof for th unit icrtiation intrval T = will b prnt. In thi ca th man of icrt nity function i n f ( ) ( ) ( n) n n 0 n n ( ) ( n) T n ( ) n ( ) (0) { } ( ) E f n ( ) n ( ) n () whr th can b calculat a a function of a. By inrting into (), w can gt th varianc a tat in (9). Propoition: Th man valu for th icrt ranom variabl tarting with n = 0 an with n = ar iffrnt a wa point out bfor. Thir rlationhip i xpr a 0. () Thi rlation n to b takn into account, pcially in th ca whn th ampling intrval T i larg.. Th truncat icrt xponntial nity function w point out in Introuction, our motivation in oing thi rarch i to vlop thortical xprion for th icrt truncat nity function that ar appropriat for icrt tim tochatic ytm molling. For xampl, th lay in icrt tim communication ytm ar taking valu in a limit intrval of, ay, poibl icrt valu. Thrfor, th function that crib th lay itribution i truncat to intrval. Propoition: Th nity an itribution function of a truncat icrt xponntial ranom variabl ar givn in clo form a an ( ) n ft( ) ( n) n E-IN: Volum 6, 07

5 tvan Brbr ( ) n Ft( ) U( n) (3) n Proof: In orr to fin thi truncat function, th whol omain of poibl τ valu from 0 to infinity, for th alray analy icrt nity function, will b ivi into non-ovrlapping intrval containing valu. ll corrponing valu in th intrval, tarting with t, (+)th, (+)th, tc. trm, will b a to obtain th truncat nity function valu for n =,,.,. Uing xprion for th nity function with T =, th firt valu of th nity function in th t, n an m-th intrval of valu will b f ( ) ( ), t f ( ) ( ) t f ( m) ( ) t ( ) m Th um of th valu, whn m tn to infinity, will giv th firt truncat valu fin for unit lay τ =, i.., f t ( ) ( ) an th valu for nity function for any lay n will b ( ) n ft( n). (4) If w aign th valu to th wight of Dirac lta function in th whol intrval of poibl truncat valu, w can gt th truncat nity function in thi form ( ) n ft( ) ( n), (5) n which complt our proof. In th proc of nity function icrtiation, th icrt probability valu wr calculat at tim intant nt, tarting with n =. Thi coul b alo on tarting with th valu n = 0. Howvr, thi will not b xactly th am nity function, bcau it will rult in iffrnt man valu of th lay an imprci imulation of th icrt lay, a w will how. For th ca whn th icrt valu of th nity function tart at n = 0, th man valu i ( nt ) f (( n ) T ). (6) t0 t n0 0 f ( T ) T f ( T )... T f ( T ) t t t Whn th firt nity valu i fin for n =, th man valu will b ( nt ) f ( nt ). (7) t t n Th two man valu ar not th am. Th proof of thir rlationhip i ( nt ) f ( T ) t t n T f ( T ) T f ( T )...( T ) f ( T ) t t t T f ( T ) T f ( T ) T f (3 T )... T f ( T ) t t t t T f ( T ) T f (3 T )...( ) T f ( T ) t t t T f ( nt ) ( nt ) f (( n ) T ) t t n n0 T t 0 (8) Thrfor, th man valu pn on th tarting valu of th icrt ranom variabl of th nity function, a w ai in th prviou ction, an can vary with th uration of icrt intrval T. It i important to hav thi iffrnc in min pcially in th ca whn w ar oing imulation of th icrt lay valu. Namly, in th ca whn th firt nity function valu i fin for n = 0, an th icrt variabl valu (variat) n to b gnrat, thn th variat valu in th firt intrval from 0 to T will b quat with zro, an th valu from T to T will b quat with T an o on, until (-)T i rach. Thrfor, if it i not important to notify an tak into account all th lay ini th firt T intrval thi prntation of th nity function will b u. Howvr, if all lay in th firt T intrval n to b takn into account, thn th firt ampl of th nity truncat function houl b aign to th firt icrt tim intant T. Propoition: Th man an varianc of th icrt truncat ranom variabl, whn th firt icrt valu with th firt nity valu i at n =, ar xpr in thi form ( ) ( ) t. (9) Thi valu i for on gratr than th man valu η t0 of th icrt nity with th firt nity valu at E-IN: Volum 6, 07

6 tvan Brbr n = 0. Thrfor, th man valu η t0 in clo form i ( ) ( ) t0. (0) Thu, in practical application, for th fin man valu /λ of th continuou xponntial itribution, th man valu of icrt xponntial η t can b foun an compar. Proof: Ba on th xprion (3) for th icrt nity function w may hav ( ) n f ( ) ( n) t t n ( ) n ( ) n n Th um can b foun in a clo form from thi xprion n ( ) ( ) ( ) n ( ) Having availabl valu a w may calculat th man ( ) ( ) ( ) ( ) t ( ) ( ) which confirm (9). Uing (8) for T =, w may riv th xprion for th man of th ranom variabl that tart with n = 0 an prov (0). In imilar way, a it wa prnt in Giucananu papr [8] th varianc i E{ } t t ( ) ( ) ( ) ( ),. ( ) () Th two graph, th man an varianc a a function of paramtr /λ, tarting with /λ = an finihing with /λ = 30, for = 40, ar prnt in Fig.. longi with th graph, th graph of th man an varianc for icrt non-truncat nity function, continuou (non-truncat) nity function an continuou truncat nity function ar prnt. Thr i obviou iffrnc in th man an varianc valu btwn non-truncat an truncat function that houl b takn into account in thortical analyi an imulation of icrt tim ytm. To upport thi tatmnt, in th nxt ction w will prnt th procur of gnrating variat of a icrt truncat xponntial nity function Man Dic Trunc Dic Non-Trunc Cont Non-Trunc Cont Trunc /λ Varianc Dic Trunc Dic Non-Trunc Cont Non-Trunc Cont Trunc /λ Figur Th man an varianc of icrt truncat xponntial function..3 Gnrating variat of th icrt truncat xponntial itribution In thi ction, a procur of gnrating ranom variat of th truncat xponntial icrt variabl will b prnt uing th invr tranformation mtho. ccoring to thi mtho, variat of a uniform itribution F will b gnrat, continuou lay valu τ will b calculat an thn a icrt variat valu τ v will b aign. Th valu of th truncat icrt xponntial itribution function, for particular lay τ v = τ, can b calculat for T = an xpr in thi form Ft( ) ( ) F. () Th lay i xpr a a function of th itribution function valu a ln( F / ), (3) whr i a contant, i.., / ( ). Thn th uniform continuou valu variat F ar gnrat an th lay valu ar calculat accoring to (3). Bcau th calculat lay E-IN: Volum 6, 07

7 tvan Brbr valu ar ral numbr, thy n to b quat to th intgr valu which ar not mallr than th ral numbr in th argumnt of τ, i.., v ln( F / ). (4) whr δ(.) i Dirac lta function an th rror x function complmntary i rfc( ) / x. Proof: Th probability valu ini th intrval aroun zro can b calculat a Th icrt valu, calculat in thi way, rprnt th variat τ v of an xponntial truncat icrt nity function that ha th firt icrt valu at n =. Howvr, if th firt icrt valu i to b at n = 0, th icrt truncat nity an itribution function ar lightly iffrnt, th icrt lay will b gnrat which i not gratr than th gnrat uniform variat, i.., T / P{ T / T / } f ( ) c T / / / T/ T/ T / T / rfc rfc (7) v ln( F / ). (5) f (τ) P{ T / T / } 3 Th Dicrt Truncat Gauian Dnity Function an It Momnt Thi ction contain baic rivativ of th Gauian icrt an truncat icrt nity function an rlat momnt. 3. Th icrt Gauian nity function Following th gnral procur of a continuou f c (τ) an rlat truncat continuou nity function f ct (τ) valuation w will prnt hr th icrtiation of a Gauian nity function with a zro man valu. Th rlat rivation for any man valu can b rlativly aily obtain. Th procur of icrtizing a continuou Gauian nity function i illutrat in Fig. 3. W will calculat th probability valu ini T intrval an aign it to th icrt valu of th ranom variabl. Th probability ini ha ara in Fig. 3 will b aign to th icrt ranom variabl fin for τ = 0. If w u th intrval T on th lft or on th right th icrt nity function will introuc th man valu that i not qual to zro. For thi raon, w ar tarting with th intrval aroun th origin. Propoition: Th icrt nity function of Gauian ranom variabl, having th valu at th uniformly pac intant T of a ranom variabl τ, can b xpr a (n ) T / (n ) T / f( ) rfc rfc ( nt ) nt (6) τ -nt -T / 0 T / T nt Figur 3 Dicrtiation of Gauian nity function. imilarly, th probability that th ranom variabl i ini any icrt intrval n i P{(n ) T / (n ) T / }. (8) (n ) T / (n ) T / rfc rfc If th calculat probabiliti ar aign a th wight to th Dirac lta function that ar fin at icrt intant τ = nt, thn th obtain function rprnt th icrt Gauian nity function xpr a f ( ) P{(n ) T / (n ) / } ( nt ) nt. (9) (n ) T / (n ) T / rfc rfc ( nt ) nt For th unit intrval, T =, thi nity i (n) (n) f ( ) rfc rfc ( n) n 8 8, (30) Erfc( n) ( n) n whr th Erfc(n) i fin a E-IN: Volum 6, 07

8 tvan Brbr (n) (n) Erfc( n) rfc rfc, (3) 8 8 which complt our proof. pfc pf pft pfc, igma=5 pf, D = 40 pft,=5,7,0 3. Th icrt truncat Gauian nity function In orr to fin thi truncat function, th whol omain of poibl icrt valu τ from 0 to infinity, for th alray riv function, n to b ivi into intrval containing icrt valu. ll corrponing valu in th intrval, tarting with t, (+)th, (+)th, tc. trm, n to b a to obtain th truncat nity function valu for n =,,.,. Howvr, in thi ca thi mtho cannot giv u th xprion for nity function in a clo form. For thi raon, a mtho ba on th finition of th continuou truncat nity function will b u. ccoring to thi mtho th truncat nity function i fin a th conitional nity function on th intrval (-, ) an xpr a f ( ) ft ( ) f ( ) P( ) P( ) n (n) (n) rfc rfc ( n) n 8 8, (3) n (n) (n) rfc rfc n 8 8 n P( ) Erfc( n) ( n) n whr Erfc(n) i fin in (3) an P() i a function fin on a truncation intrval n (n) (n) P( ) rfc rfc. (33) n 8 8 Th truncat nity function, for iffrnt valu of th truncation intrval, i prnt in Fig. 4 alongi with th continuou an icrt nity function. Whn th truncation intrval incra th truncat varianc incra an i alway mallr that th varianc of continuou nity. Th D not th omain of icrt function bfor truncation an fin th truncation intrval, i.., th omain of truncat variabl n Figur 4 Continuou, icrt an icrt truncat Gauian niti. D i th intrval of continuou an icrt ranom variabl valu. σ = 5 i th tanar viation of continuou variabl an i truncation intrval. Propoition: Th man of th truncat icrt nity function i zro. Proof: By finition n f ( ) P( ) Erfc( n) ( n) t t n n n P( ) Erfc( n) ( n) P( ) n Erfc( n) n n (34) Th trm of th um for n = 0 i zro. Th corrponing trm for ngativ an poitiv n ar canclling ach othr. Thn, w may hav P( ) n Erfc( n) t n n n n P( ) n Erfc( n) P( ) n Erfc( n) 0 n n which complt our proof. Propoition: Th varianc of thi nity i n t P n, (35) ( ) n Erfc( n). (36) Proof: By finition n t t n f ( ) P( ) Erfc( n) ( n). (37) P n ( ) n Erfc( n) n E-IN: Volum 6, 07

9 tvan Brbr Th trm for n = 0 i zro. Th corrponing trm for ngativ an poitiv n ar a to ach othr. In aition, th w know that P(), thu, having in min th varianc for th icrt Gauian ranom variabl, w may hav f (τ) P{ T / T / } n n t P( ) n Erfc( n) P( ) n Erfc( n) n n, (38) which complt our proof. 4 Th Dicrt Truncat Uniform Dnity Function an It Momnt 4. Th icrt uniform nity function Propoition: Th icrt uniform nity function i fin by thi xprion n f( ) ( nt), (39) n an, for a unit intrval T =, it i n f ( ) ( n). (40) n Proof: uppo th uniform continuou nity function i xpr a fc( ) / T, fin ini c th intrval Tc T. If it i icrti in c rpct to τ, a hown in Fig. 5, with th intrval of icrtiation of T, th probability valu in th firt intrval aroun zro, n = 0, can b xpr a P{ T / T / } T. (4) T imilarly, th probability valu in any intrval n i P{(n ) T / (n ) T / } T. (4) T Th probabiliti can b unrtoo a th wight of Dirac lta function that fin th icrt nity function, which can b xpr a T f( ) ( nt). (43) T n c c c -T c -T -T / 0 T / T T T c Figur 5 Dicrtiation of th uniform nity function. In th ca th numbr of poitiv an ngativ intrval i, th whol intrval i Tc T T, an th rlation btwn th valu T c, T an, which will b u in thi ction, can b foun in th form Tc T, Tc T Tc. (44) T T Now, ba on (4) an (44) th probability that th ranom variabl i in th n-th intrval can b xpr a P{(n ) T / (n ) T / } T T () c τ (45) Thrfor, th icrt nity function (43) can b xpr a tat by (39) for any T incluing T =. Th calculat probability in T intrval (for xampl th ha intrval in Fig. 5) i aign a th wight of a lta function fin at th origin. Th probability valu can b calculat in ach T intrval an aign to th right or lft of th intrval a w icu for th xponntial function, with th imilar conqunc rlat to th ymmtry an momnt of th truncat icrt function. Propoition: Th man, man quar an varianc ar xpr a an 0, ( ) E{ } T, 3 ( ) E{ } E{ } T, (46) 3 n, for th unit intrval T =, th varianc i E-IN: Volum 6, 07

10 tvan Brbr ( ). (47) 3 Proof: Th proof for th man i trivial. Th man quar valu i f ( ) f ( a a) t a ( nt ) f ( ). (5) na P( a a) P( ) n n T n n E{ } ( nt ) n T T ( )( ) ( ) 6 3 n n T n Th varianc i (48) Th valu P() can b calculat a a P ( ) na. (5) a ( a a) ( ) E{ } E{ } T. (49) 3 4. Th icrt truncat uniform nity function In practical application th icrt lay ar taking valu in a limit intrval fin a th truncat intrval (- + a, - a), whr a i a poitiv whol numbr nam th truncation factor. Thrfor, th function that crib th lay itribution i truncat an ha th valu in th truncat intrval, a hown in Fig. 6. f t (τ) (-+a)t 0 (-a)t Figur 6 Dicrt truncat uniform nity function prnt uing Dirac lta function. Propoition: Th nity function i givn in clo form by th xprion an na ft( ) ( nt ), ( a) na na ft( ) ( n). (50) T ( a) na Proof: Ba on th finition of a truncat nity function a a conitional nity function, th truncat icrt uniform nity function can b xpr a τ By inrting thi xprion into (5), th nity function can b xpr a in (50). In th am way, for a unit ampling intrval T =, th ranom variabl τ i a whol numbr from th intrval (- +a, -a), for a 0, an th nity function (5) i xpr a in (50) Th varianc of continuou, icrt an truncat icrt uniform nity function ar prnt in Tabl. Du to icrtiation, th valu of all truncat varianc ar mallr than th varianc of continuou nity. Thi fact n to b takn into account in thortical analyi an imulation of icrt tim ytm. Tabl Varianc xprion Uniform Varianc itribution Continuou c Tc /3 Continuou a truncat ct c Dicrt c ( ) Dicrt 4 a( a ) truncat t c ( ) 4.3 Th icrt aymmtric uniform nity function Th abov-mntion icrt nity function prrv th ymmtry in rpct to th man valu. Howvr, in icrt tim ignal analyi, u to th natur of analogy-to-igital convrion, w ar aling with aymmtric niti. Th nity function can b obtain uing th am procur a for th ymmtric niti crib abov. Th only iffrnc i in th aignmnt of icrt probability valu that will tart at th point -T c = -T that corrpon to th icrt valu a hown in Fig. 7. In impl wor w ar aigning E-IN: Volum 6, 07

11 tvan Brbr th probabiliti to th right i of th intrval fin by T c an making icrt valu on th ngativ τ axi an (-) icrt valu at th poitiv τ axi incluing alo a componnt for τ = 0. P{ T T T } c c f (τ) -T c =-T -T 0 T (-)T T c Figur 7 Dicrtiation of th uniform nity function to obtain an aymmtric icrt nity Following th procur in ubction 4. it can b provn that th icrt probability nity function can b xpr in trm of Dirac lta function a n f( ) ( nt), (53) n Whr th whol intrval of continuou ranom variabl valu can b xpr a Tc T. Thrfor, th corrponing icrt variabl hr can hav ngativ valu an (-) poitiv valu, i.. it will hav th nity function that i aymmtric in rpct to th zro valu. Th conqunc of thi i that th man valu of th icrt nity will b iffrnt from zro u to thi aymmtry. Following th procur prnt in ubction 4. an 4. it i impl to riv th momnt of thi icrt nity function an th rlat truncat nity function. 5 pplication of Driv Exprion FIRT CE: Th application of th prnt thory will b montrat on th olution for th firt problm mntion in th Introuction of thi papr. Bcau th fin tochatic proc X(m) i a icrt tim proc th numbr of pha i to b finit an n to b rprnt by an aymmtric nity function xpr by (53). uppo that th numbr of ampl ini on prio of th inuoial raliation of tochatic proc x(m) hav N valu. Thrfor th numbr of ranom pha θ n will b N all of thm having th poibl ranom valu θ n = πn/n, for n = -N/,, N/-. Thrfor, th nity function of th pha i fin by xprion (53) with th following τ rfin paramtr: T c = π, = N/, T c = T N/. Thn, th icrt nity function (53) ha thi form nn / nn / f ( ) ( ) ( n / N) N n nn / N nn / an th man valu of th tochatic proc can b calculat a ( m) E{ X ( m, )} x( m, ) f ( ) nn/ x( m, ) ( n / N) N nn/ nn/ nn/ in( m ) ( n / N) N nn/ nn/ N in( m n / N) (54) Du to th proprti of inuoial function th um in (54) i zro an th man valu of th tochatic proc i zro for ach m. Th autocorrlation function n to b calculat a R ( m, l) E{ X ( m, ) X ( l, )} X ( m, ) X ( l, ) f ( ) X nn/ X ( m, ) X ( l, ) ( n / N) N nn/ nn/ nn/ [co( ( m l) co ( m l ) ( n / N) N nn/ co( ( m l) ( n / N) N N nn/ nn/ nn/ co( ( m l n / N) Th um in th firt an i qual to N an th um in th con trm i zro which rult in thi xprion for th autocorrlation function RX ( m, l) co( ( m l). (55) Of cour, th olution prnt in () an () ar formally th am a in (54) an (55). Howvr, th right calculation ar montrat in (54) an (55) bcau thy ar ba on rigorou mathmatical prntation of th nity function an trict mathmatical procur in th rlat olution of intgral. Thrfor, th mtho of prnting icrt ranom variabl I am propoing in thi papr can b u wily. Morovr, uing th niti in th prnt form th intgral E-IN: Volum 6, 07

12 tvan Brbr involv can b olv rlativly ay. In aition to thi, w n to tak car of variou rivativ of icrt niti for th am continuou ranom variabl. For xampl, in th analyi of icrt tim tochatic proc, lik abov, w n to u an aymmtric nity. Howvr, in th ca whn th variabl rprnt ranom pha in communication ytm, which can tak ngativ an corrponing poitiv ranom valu with th am probability, w n to u a ymmtric nity function a can b mntion in th thir ca. ECOND CE: W prnt hr how to fin th man of th ranom function (3) by uing an olving intgral (4). uppo that th ranom lay ini th chip intrval τ i itribut accoring to th truncat icrt xponntial nity function. Inrting xprion (3) for th conitional probability of rror an th xprion for th truncat icrt xponntial nity function (3) into th xprion for th man valu of th probability of rror (4), w may gt th xprion for th probability of rror in a clo form a ( ) n ( ) n ( ) t( ) ( ) n n P P f P n / ( ) n / E b rfc n / n / 4 n / N 0 (56) Thrfor, uing th nity function xprion a uggt in thi papr it i rlativly ay to olv th intgral an avoi th u of numrical intgration. THIRD CE: W can alo fin th probability of rror in a D ytm a th avrag valu of th conitional probability of rror prnt in (4). For thi ytm, it i impoibl to achiv prfct ynchroniation of th praing qunc. uppo th ranom lay τ in th ytm with imprfct ynchroniation i charactri by th uniform truncat nity function xpr a (50). By inrting (5) an (50) into (4) th avrag valu for probability of rror can b foun na P P ( ) f t( ) P ( ) ( n) ( a) P ( n) ( a) 4( a) n na / 4( / 4) 4 E b rfc X ( n) ( X ( n)) n b N 0 (57) Bcau th poibl ranom lay can hav any poitiv an ngativ valu aroun zro w hav bn uing icrt ymmtric nity function. Th xprion (5) prnt th probability of rror in D ytm, which i quivalnt to a ingl-ur co iviion multipl acc (CDM) ytm. If th CDM ytm oprat with N ur, th xprion for th probability of rror i / 4N E b P ( ) rfc X ( ) X ( ) b N 0 (58) whra 4( / 4 N ) /. By inrting (58) an (50) into (4) th avrag valu of th probability of rror can b calculat following th procur prnt in (57), rulting in thi xprion P P ( ) f t ( ) 4( a) / 4N E b rfc X ( n) ( X ( n)) n b N 0. (59) Th final not w may mak: Firtly, th xprion for Gauian an uniform nity function ar riv for th aum zro man valu of ranom variabl. Howvr, it i ay to riv th corrponing xprion for any man valu of th icrt ranom variabl, a wa on for an xampl of aymmtric uniform itribution in ubction 4.3. conly, in thi papr w prnt nity function of icrt ranom variabl in trm of Dirac lta function. It i important to not that it i poibl to vlop an u thir xprion uing Kronckr lta function. Thirly, in practic, th icrt ranom variabl tak th valu in limit intrval. Thrfor, th u of truncat nity function i ncary. Thi ncity upport our motivation for writing thi papr. Namly, th man valu an varianc ar gnrally changing pning on th iz of truncating intrval, which can hav ignificant influnc on our thortical analyi, imulation an practical ign of igital vic. 6 Concluion In thi papr th xprion for xponntial, Gauian an uniform icrt an truncat icrt nity function, an thir firt an con momnt, ar riv. Th xprion for th E-IN: Volum 6, 07

13 tvan Brbr momnt ar compar with rlat momnt of th continuou an truncat continuou nity function. It wa confirm that th nity function coul b xpr in trm of Dirac lta function in orr to b appli for th calculation of th man valu of a function of ranom variabl. Th application of riv niti of icrt ranom variabl i montrat on thr xampl. On xampl prnt a rigorou calculation of th man an autocorrlation function of a icrt tim harmonic proc, an two xampl montrat prci calculation of th probability of rror in D communication ytm whr all ignal ar rprnt in icrt tim omain. Rfrnc: [] H. M. Hay, tatitical Digital ignal Procing an Molling, John Wily & on, 996. [] D.. Poularki Dicrt Ranom ignal Procing an Filtring Primr with MTLB, CRC Pr, Boca Raton, 009 [3] Brbr, M..., Probability of Error Drivativ for Binary an Chao-Ba CDM ytm in Wi-Ban Channl. IEEE Tran. on Wirl Communication 3/0, 04, pp [4] Lo, C. M., W. Lam, H., Error probability of binary pha hift kying in Nakagami-m faing channl with pha noi, Elctronic Lttr 36, 000, pp [5] Eng. T., Miltin L. B., Partially Cohrnt D- Prformanc in Frquncy lctiv Multipath Faing, IEEE Tran. on Comm. 45, 997, pp [6] Richar, M.., Cohrnt Intgration Lo Du to Whit Gauian Pha Noi, IEEE ignal Procing Lttr 0, 003, pp [7] Polprart, C., Ritcy, J.., Nakagami Faing Pha Diffrnc Ditribution an it Impact on BER Prformanc, IEEE Tran. on Wirl.Comm. 7, 008, pp [8] ong, X. F., Chng, Y. J., l-dhahir, N., Xu, Z., ubcarrir Pha-hift Kying ytm With Pha Error in Lognormal Turbulnc Channl. Journal of Lightwav Tchnology 33, 05, pp [9] Chanra,., Patra,., Bo, C., Prformanc analyi of PK ytm with pha rror in faing channl: urvy, Phyical Communication 4, 0, pp [0] Roy, D., Th Dicrt Normal Ditribution. Communication in tatitic Thory an Mtho 3:0, 003, pp [] Xkalaki, E., Hazar Function an Lif Ditribution in Dicrt Tim. Communication in tatitic Thory an Mtho :, 983, pp [] Roy, D., Dgupta, T., Dicrtizing pproach for Evaluating Rliability of Complx ytm Unr tr-trngth Mol, IEEE Tran. on Rliability, 50:, 00, pp [3] Roy, D., Dicrt Rayligh Ditribution. IEEE Tran. on Rliab., 53:, 004, pp [4] Ho, K., Chng, K. H., two-imnional Fibonacci buy ytm for ynamic rourc managmnt in a partitionabl mh. ropac an Elctronic Confrnc, NECON, Procing of th IEEE,, 997, pp [5] nanullah, M, Charactritic Proprty of th Exponntial Ditribution. Th nnal of tatitic 5:3, 977, pp [6] o, J. I., Kang,. B., Not on th xponntial half logitic itribution, ppli Mathmatical Molling 39, 05, pp [7] Rachk, M., Moling of magnitu itribution by th gnraliz truncat xponntial itribution, Journal of imology 9, 05, pp [8] Giurcananu, C. D., bywickrama, R. V., Brbr,., Prformanc nalyi for a Chao- Ba CDM ytm in Wi-Ban Channl, Th Journal of Enginring, 05, pp. 9. E-IN: Volum 6, 07