Error probability and error stream properties in channel with slow Rician fading

Transcription

1 Error probablty ad error stream propertes chael wth slow Rca fadg Paper Krystya M. Noga Abstract I a rado commucato chael wave parameters fluctuate radomly. The sgal evelope udergoes deep fades. Whe bary formato s trasmtted through such a chael, fadg causes radom varato of probabltes of error assocated wth the detecto of dvdual elemetary sgals, whch produces a clusterg of errors. The paper presets a aalytcal descrpto of the probablty of bt error the chael wth very slow Rca fadg ad Gaussa ose for ocoheret ad coheret detecto. Dgtal systems employg error detecto or error correcto codg are geerally based o the trasmsso of blocks of N sequetal bts. Expressos are gve for the probablty of errors occurrg N bts (weghted spectrum of errors) ad the probablty of more tha errors a block of N bts (block error probablty) for ocoheret frequecy shft keyg (NCFSK). Also the calculatos are preseted graphcally. Keywords Rca fadg chaels, multpath propagato, bt error probablty, stream of errors, weghted spectrum of errors, block error probablty.. Itroducto Trasmsso of sgal dgtal rado commucato systems takes place the presece of radom addtve ad radom multplcatve dsturbaces; the multplcatve dsturbaces called ordary as fadg. We assume that the addtve dsturbaces are represeted by whte Gaussa ose wth zero mea value. The fadg s cosdered as ofrequecy selectve. Ths s vald for most cases of moble data commucatos wth moderate bt rates. The fadg process s assumed to be statoary ad slowly varyg compared wth the N bts durato; t s costat durg data block durato. We assume that the fadg s descrbed by the Rca dstrbuto. It s oe of the doubleparameter dstrbuto of the sgal evelope allowg to descrbe propagato codtos exstg rado chael greater detal tha a smpler ad more frequetly appled sgle parameter Raylegh dstrbuto. The Raylegh ad Rca dstrbuto are oly specal case solutos of the radom vector problem. The Nakagam dstrbuto provdes a more geeral soluto. The Rca dstrbuto s a aalytcal model suffcet for a chael where the useful sgal s(t) s a sum of the statoary dffuse Gausssa sgal x(t)=a x cos[ω t +ϕ x (t)] wth zero mea value ad the drect harmoc sgal A cos t,.e.: s(t) =A cosω t + a x cos Λ ω t + ϕ x (t) = = r cos ω t + ϕ s (t)λ : () The Rca model covers the superposto of a radom Raylegh sgal wth the fxed oradom sgal. The Rca dstrbuto ca also be closely approxmated by the Nakagam dstrbuto. The Rca fadg model apples mcrocellular ad satellte rado commucato chaels. I rado commucato chael rado waves parameters fluctuate radomly. Data trasmsso from ad to moble termals suffers from fadg effects caused by multpath propagato. The sgal evelope udergoes deep fades. Whe bary formato s trasmtted through such a chael, fadg causes radom varato of probabltes of error assocated wth the detecto of dvdual elemetary sgals. Deep fades cause bursts of bt errors the trasmtted data,.e. errors dgtal trasmsso over fadg chaels occur bursts. I dgtal commucato a mportat quatty s the bt error probablty. Ths paper presets a resso for the average probablty of bt error for bary trasmsso Rca chael wth ocoheret frequecy shft keyg (NCFSK), dfferetally coheret phase shft keyg (DPSK), coheret frequecy shft keyg (CFSK) ad coheret phase shft keyg (CPSK). Formulas for the average weghted spectrum of errors ad the average block error probablty for NCFSK are also preseted.. Average probablty of bt error The statc bt error probablty for several commo bary modulato schemes wth optmum detecto of ofadg sgals Gaussa ose s gve by the formula 8 ( αρ) >< P s (ρ) = p for NCFSK, DPSK () >: erfc( αρ) for CFSK, CPSK where: ρ s the stataeous sgal-to-ose power rato (SNR), erfc( p x) deotes the error fucto [4], α = :5 for NCFSK, CFSK ad α = for DPSK, CPSK. Sce the Eq. () the argumet of the error fucto appears the lower lmt of the tegral, t s aalytcally 3

2 Krystya M. Noga dffcult to perform averages of ths equato. Aother form for statc bt error probablty s preseted []: P s (ρ) = (aρ)b Γ (b) Z Π= cos ϕ (sϕ) b+ aρ s dϕ ; (3) ϕ where a ad b are the coeffcets whch deped o the partcular form of modulato ad detecto,.e. a = b = = :5 for CFSK, a = :5, b = for NCFSK, a =, b = :5 for CPSK, a = b = for DPSK. Assumg Rca fadg, the evelope r of the useful sgal s(t), descrbed by Eq. (), has the probablty desty fucto: r p(r) = r + A σ x Ar I ; r ; (4) where: A s the evelope of the drect compoet of the useful sgal s(t), s the varace of the dffused compoet x(t) of the useful sgal s(t), I (α) s the zero order modfed Bessel fucto of the frst kd [4]. Probablty desty fucto of the square of the evelope r(t) of the useful sgal s gve by the formula: p(r )= σ x r + A σ x Ar I ; r : (5) The average value of the square of the evelope r(t) ca be wrtte as E(r )=A + σ x ; (6) where E(x) deotes the ected value of the argumet. From Eq. (5) we ca descrbe the dstrbuto of radom varable, whch represets ρ defed as the stataeous rato power of the useful sgal to average power N of the addtve Gaussa ose,.e. the stataeous sgalto-ose power rato: ρ = r N : (7) The probablty desty fucto of ρ Rca chael s deoted by the formula: p(ρ)= N σ x N ρ +A σ x ψ p! A N ρ I ; ρ : (8) Wth suffcetly slow fadg, the average probablty of bt error (A; σ x ; N ),.e. the dyamc probablty of bt error, equals to P s (ρ) averaged over the dstrbuto of SNR ρ: (A; σ x ; N )= Z P s (ρ) p(ρ)dρ = E [P s (ρ)] : (9) Isertg Eqs. () ad (8) to (9) gves the average probablty of bt error Rca chael [5]: (A; σ x ; N A )= Γ(b) A k= σ x k Γ(b + k + ) (k!) B g (k + ; b); () where: Γ(b) s the gamma fucto [4], g = N aσ x +N ad B g (x; y) s the complete beta fucto [4]. Let us troduce addtoal factors, whch descrbe Rca chael ad ca be defed as ρ = A ; ρ = σ x N ; ρ 3 = A N = ρ ρ : () The, the ected value of the SNR ρ we ca ress as E(ρ) =ρ = ρ 3 + ρ : () I lterature the factor ρ (drect sgal to dffused sgal power rato) sometmes s deoted as K ad ρ (average SNR) s deoted as Γ []. I the ed, the formula for the average probablty of bt error Rca chael ca be wrtte as = Γ(b) ( ρ ) (ρ ; ρ )= k= ρ k Γ(b+k+) (k!) B g (k+;b); (3) where g = aρ + or usg the relato () (ρ ; ρ )= ρ Γ(b) ρ k Γ(b + k + ) k= (k!) B g (k + ; b); (4) where g = +ρ aρ ++ρ. The error-rate formulas (), (3), (4) are vald for four prcpal cases of bary modulato Rca chaels. The result over a Raylegh chael ca be obtaed from our results whe A =. However, the border case whe ρ =, we have the formula for bt error probablty chael wthout fadg. Fgure presets the average bt error probablty for bary NCFSK modulato Rca chael as a fucto of ρ for varous values ρ. For detecto the ocoheret systems (whe b = ) the formula for the average probablty of bt error Rca chael ca be ressed as» N (A; σ x ; N )= (α + N A α ) (α + N (5) ) or (ρ ;ρ )= (αρ αρ ρ + ) αρ + (6) 4

3 Error probablty ad error stream propertes chael wth slow Rca fadg The weghted spectrum of errors,.e., the probablty of errors occurrg a trasmsso of N bts, for depedet bt errors s gve by the bomal dstrbuto N P(L N = ) = Ps (ρ)[ P s(ρ)] N = ; ;:::;N : (8) The average probablty of errors a block of N bts (average weghted spectrum of errors) s deoted as [5, 6] = (L N = ) = R P(L N = )p(ρ)dρ = N N N ( ) E Ps + (ρ)λ : (9) = Fg.. The average bt error probablty for NCFSK chael wth Rca fadg: ρ = ; ρ = ; 3 ρ = 5; 4 ρ = 7; 5 ρ = ; 6 ρ = ; 7 ρ = 6 (all values db). ad after usg the relato () we ca rewrte t as (ρ ;ρ )= + ρ (αρ + ρ αρ ρ + ) αρ + ρ + (7) where α = :5 for NCFSK ad α = for DPSK. 3. Average weghted spectrum of errors I case of dgtal trasmsso over a fadg chael, tme varato causes the chage of bt error probablty wth the effect of clusterg errors the receved sgal. Forward error correcto s ofte used to break up the clusterg [3, 7, 8]. Dgtal systems employg error detecto or error correcto codg are geerally based o the trasmsso of blocks of N bts. The average probablty of bt error specfed by Eq. () ether descrbes the umber of errors or ther placemet the stream of errors. I a commucato system whch trasmts data blocks o N bts the probablty of errors a block ad the probablty of more tha errors a block are a mportat quattes whch descrbe the stream of errors chael wth fadg. The average Eq. (9) s formed over the stataeous SNR ρ whch has the probablty desty fucto p(ρ) descrbed by Eq. (8). Assumg that the Rca fadg s very slow, oselectve ad depedet, the the stataeous SNR remas the same over a block of N bts. For ocoheret FSK chael wth Rca fadg the average weghted spectrum of errors s gve by N N N (L N = ) = N ( ) = ψ! + A (+) (+) +N (+)+N Λ : () Usg the relatos () ad () we ca rewrte formula () as N N N (L N = ) = ( ) = + + ρ (+)ρ +( + ρ ρ ρ (+) : () ) ρ (+)+( + ρ ) Equato () has bee used to calculate the average probablty of errors N bts. Fgure shows the (L N = ) for N = 3 ad for varous values of ρ ad ρ ;thevalues of ρ ad ρ are as take as (ρ ;ρ )= 3. Addtoal, Fg. 3 shows the average weghted spectrum of errors N bts for NCFSK for dfferet values of N, ρ ad ρ. The probablty (L N = ) takes to accout oly the total umber of errors ad dsregards ther dstrbuto. It s useful oly for the performace evaluato of radom 5

4 Krystya M. Noga error-correctg codes. It caot be used for burst-errorcorrectg codes. If the radom error-correctg code s used, wth the code capable of correctg up to radom errors, the the probablty of correct decodg s (L N = ). = 4. Average block error probablty Whe the system wth burst-error correctg code s used the probablty of correct decodg caot be ressed oly terms of (L N = ). The burst-error-correcto code ca correct all error vectors wth legth less tha or equal to. I ths case the mportat quatty s the average probablty of more tha errors a block of N bts,.e., average block error probablty. It s deoted by Fg.. Average probablty of errors N = 3 bts for NCFSK: ρ =, ρ = 43; ρ =, ρ = 38; 3 ρ = 5, ρ = 3; 4 ρ = 7, ρ = 9; 5 ρ = 6, ρ = 3 (all values db). = = = = Z N N N = (L N > ) = P s (ρ)[ P s(ρ)] N p(ρ)dρ = N ( ) E P + s (ρ)λ : () Thus, from Eqs. (), (), (8), the average block error probablty for NCFSK we ca ressed as (L N > ) = N = N N = N ( ) + A ( +) ( +) +N [ ( +)+N (3) ] or the form (L N > )= = N N = N +ρ ( +)ρ +(+ρ ρ ρ ( +) ) ( ) + ρ ( +)+(+ρ ) : (4) Fg. 3. Average probablty of errors N bts for NCFSK; sold le N = 48, dot le N = 3: ρ =, ρ = 43; ρ =, ρ = 8; 3 ρ =, ρ = ; 4 ρ = 6, ρ = 3 (all values db). Fgure 4 shows the average block error probablty (L 3 > ) for NCFSK,.e., the probablty of at least oe error N = 3 bts. Whe we plotted the resso (4) preseted refereces [] for D =,.e. for o dversty case, we become the same result as s show Fg. 4. 6

5 Error probablty ad error stream propertes chael wth slow Rca fadg Also Fg. 5 shows the average block error probablty (L 3 > ) a block of N = 3 bts for NCFSK ad for varous values of ρ ad. 5. Cocluso Fg. 4. Average block error probablty (L 3 > ) for NCFSK ad N = 3: ρ = ; ρ = 5; 3 ρ = 7; 4 ρ = ; 5 ρ = ; 6 ρ = 4 (all values db). The preseted equatos are vald for oselectve, very slow ad depedet Rca fadg. Sce the preseted ressos clude results for the cases of Raylegh fadg ad o fadg, they ca be wdely used to evaluate the performace of error cotrol techques for moble rado. Preseted results ca be useful for error detecto or error correcto codg. I a commucato system that trasmts data blocks of N bts a mportat quatty s the probablty of more tha errors block. If a smple automatc repeat request scheme s used, the throughput ca be determed from (L N > ),.e. from the probablty of at least oe error blocks. However, f the use of forward error correcto s to be vestgated, the kowledge of (L N > ) s requred. I case of error detecto a block s receved correctly oly f all N bts are receved wthout error. I Fgs. 5 umercal results are preseted, the fluece of fadg for error detecto s preseted. The obtaed resso ca easly be programmed usg stadard mathematcal software package such as Mathcad. Refereces Fg. 5. Average block error probablty (L 3 > ) for NCFSK ad N = 3: ρ =, = ; ρ =, = ; 3 ρ =, = 4; 4 ρ = 5, = ; 5 ρ = 5, = ; 6 ρ = 5, = 4; 7 ρ =, = ; 8 ρ =, = ; 9 ρ =, = 4 (all values db). [] F. Adach ad K. Oho, Block error probablty for ocoheret FSK wth dversty recepto moble rado, Electro. Lett., vol. 4, o. 4, pp , 988. [] M. S. Alou ad M. K. Smo, Geerc form for average error probablty of bary sgals over fadg chaels, Electro. Lett., vol. 34, o., pp , 998. [3] R. R. Eaves ad A. H. Levesque, Probablty of block error for very slow Raylegh fadg Gaussa ose, IEEE Tras. Commu., vol. COM-5, o. 3, pp , 977. [4] S. Gradste ad I. M. Ryzhk, Tables of Itegrals, Sums, Seres ad Products. Lodo, 964. [5] K. Noga, Errors stream propertes for bary trasmsso chael wth fadg, Proc. Nat. Cof. Rado Dff. Rado Commu., Pozań, Polad, 998, pp. 7. [6] K. Noga, The performace of bary trasmsso slow Nakagam fadg chaels wth MRC dversty, IEEE Tras. Commu., vol. 46, o. 7, pp , 998. [7] A. Seyoum ad N. C. Beauleu, Semaalytcal smulato for evaluato of block error rates of fadg chaels, IEEE Tras. Commu., vol. 46, o. 7, pp. 96 9, 998. [8] E. Sudberg, Block error probablty for ocoheret FSK wth dversty for very slow Raylegh fadg Gaussa ose, IEEE Tras. Commu., vol. COM-9, o., pp. 57 6, 98. 7

6 Krystya M. Noga Krystya Mara Noga was bor Gdańsk, Polad. She receved the M.S. degree electrcal egeerg from the Uversty of Techology Gdańsk, 977 ad Ph.D. degree electrocs egeerg from the Uversty of Techology Gdańsk, 987. Sce 978 she has bee employg the Isttute of Shp Automato Gdya Martme Uversty, Polad. Her Ph.D. thess was ettled Probablstc characterstcs of bary trasmsso rado commucatos chael wth slowly fadg for sgle ad dversty systems. Her research terest ad techcal erece clude the aalyss of wreless dgtal commucatos over fadg chaels, moble chael modelg, multpath chaels, dversty recepto, sgal detecto ad estmato, chael estmato, statstcal sgal processg. She s the author of umerous orgal research papers. e-mal: agat@vega.wsm.gdya.pl Isttute of Shp Automato Gdya Martme Uversty Morska st 8/ Gdya, Polad 8