Log-likelihood values and Monte Carlo simulation - some fundamental results Land, Ingmar Rüdiger; Hoeher, Peter; Sorger, Ulrich

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1 Aalborg Univrsitt Log-liklihood valus and Mont Carlo simulation - som fundamntal rsults Land, Ingmar Rüdigr; Hohr, Ptr; Sorgr, Ulrich Publishd in: Ikk angivt Publication dat: 2000 Link to publication from Aalborg Univrsity Citation for pulishd vrsion APA): Land, I., Hohr, P., & Sorgr, U. 2000). Log-liklihood valus and Mont Carlo simulation - som fundamntal rsults. In Ikk angivt: Int. Symp. on Turbo Cods & Rl. Topics, Brst, Franc. pp ). Sptmbr Gnral rights Copyright and moral rights for th publications mad accssibl in th public portal ar rtaind by th authors and/or othr copyright ownrs and it is a condition of accssing publications that usrs rcognis and abid by th lgal rquirmnts associatd with ths rights. Usrs may download and print on copy of any publication from th public portal for th purpos of privat study or rsarch. You may not furthr distribut th matrial or us it for any profit-making activity or commrcial gain You may frly distribut th URL idntifying th publication in th public portal? Tak down policy If you bliv that this documnt brachs copyright plas contact us at vbn@aub.aau.dk providing dtails, and w will rmov accss to th work immdiatly and invstigat your claim. Downloadd from vbn.aau.dk on: Octobr 07, 204

2 Log-Liklihood Valus and Mont Carlo Simulation Som Fundamntal Rsults Ptr Hohr and Ingmar Land Information and Coding Thory Lab Univrsity of Kil, Grmany Abstract: Th purpos of this papr is twofold: W driv som fundamntal proprtis of log-liklihood ratio LLR) valus and propos two novl soft-dcision Mont Carlo simulation tchniqus basd on probabilitis or LLR valus. Spcifically, w prov that th pdf of LLR valus is xponntial symmtric and dmonstrat that soft-dcision simulation outprforms convntional bit rror rat simulations with rspct to accuracy and/or simulation tim. Kywords: Mont Carlo simulation, log-liklihood valus, a postriori probability dcoding.. INTRODUCTION In th contxt of itrativ dcoding, soft-input softoutput dcodrs play an important rol. A suitabl componnt dcodr is th a postriori probability APP) dcodr, which is optimal in trms of minimiing th bit rror rat in th prsnc of a singl cod, and which provids clos to optimal rsults in conjunction with itrativ dcoding in th prsnc of concatnatd cods [], [2]. Th APP dcodr may output probabilitis, s for xampl [3], [], [2], or LLR valus, s for xampl [4], among othr quivalnt altrnativs. Th us of LLR valus offrs practical advantags, such as numrical stability, but also provids thortical insights dcoding ˆ LLR amplification [5]). Th aim of this papr is to xplor furthr, prviously unpublishd, fundamntal proprtis of LLR valus and to us thm in th Mont Carlo simulation of rar rror vnts. W assum binary transmission and th xistnc of an APP dcodr which implis known channl statistics). Th main rsults ar as follows: ) Th avrag bit rror rat BER) can b writtn as lim K K k K,k, whr,k is a singl stimat of th avrag BER and K th numbr of stimats, i.. th numbr of transmittd info bits. In classical Mont Carlo BER simulation s 2.., Mthod ),,k taks on th valus for rror and 0 for no rror, whras in soft-dcision simulation s 2.., Mthod 2),,k +xp{ L k } is usd. As opposd to th classical simulation whr rrors ar countd i.., only th signs ar valuatd), this formula suggsts to us th APPs or th absolut valus of th LLR valus, rspctivly. Ulrich Sorgr Institut for Communications Tchnology Darmstadt Univrsity of Tchnology, Grmany uli@nsi.tu-darmstadt.d 2) Th pdf of LLR valus L k IR is xponntialsymmtric: pl k ) xp{l k } p L k ). This rsults holds for any APP dcodr and channl. Th xponntial-symmtric proprty is not only intrsting from a thortical point of viw but also from a practical, sinc it can b xploitd for furthr improvmnt of BER simulation s 2.., Mthod 3). 2. BIT ERROR PROBABILITY AND LOG-LIKELIHOOD RATIOS For th simulation of th BER, w us th stup dpictd in Figur. Th info bits U {+, } ar channl ncodd ENC) and transmittd ovr a discrt mmorylss channl DMC). Th rcivd valus Y IR ar fd into a channl dcodr DEC) which computs th a postriori LLRs L IR of th info bits as L LU y) log P U + y) P U y) log P U + y) P U + y) log P U+) + log p Y y + ). ) P U ) p Y y ) a priori information channl information For an fficint implmntation, th so-calld LogAPP algorithm symbol-by-symbol LogMAP algorithm [4], forward-backward algorithm, Log-BCJR algorithm) can b applid. Finally, th snt info bits ar stimatd according to th sign of thir rspctiv LLR. Th avrag bit rror rat of th transmission systm can b stimatd in at last) thr diffrnt ways. 2.. BER Simulation Mthods 2... Mthod Compar th sign of th info bit U and th sign of its LLR L at th dcodr output. If thy agr, st th bit rror indicator E to 0, othrwis st it to : { 0 if sgnu) sgnl), E ls. Th BER is th xpctd valu of th random variabl E: E [E] P E ), 2) Th xpctd valu is dnotd by E [.] throughout this papr.

3 3 )+*,.-/ "!# $%# &' )+*,.-0 )+*,.-2 Figur : Simulation stup. with {0, } dnoting raliations of E. Whn Mthod is applid to a sampl of K transmittd bits, an stimat of th BER is ˆ K K k E k Mthod 2 Tak th absolut valu Λ L of th LLR L and comput Z as Z + L + Λ. This is th probability that th hard dcision of th info bit is wrong, i.. that th signs of U and L diffr. This bcoms obvious, whn ) is xprssd as P U + y) P U y) + L + )Λ, + +L + +)Λ. Du to this proprty, Z can b rgardd as a soft bit rror indicator. Lik in th prvious mthod, th BER is th xpctd valu of Z: E [Z] p Z ) d, 3) with 0 dnoting raliations of th random variabl Z. Whn Mthod 2 is applid to a sampl of K transmittd bits, an stimat of th BER is ˆ K K k Z k Mthod 3 Tak th absolut valu Λ L of th LLR L. Thn, th BER is th stimatd valu of a function of Λ, namly [ ] E λ + λ p Λ λ) dλ, 4) + λ with λ 0 dnoting raliations of th random variabl Λ. This mthod fficintly utilis th xponntialsymmtric proprty of th LLR s 3.3.). Whn Mthod 3 is applid to a sampl of K transmittd bits, an stimat ˆp Λ λ) of p Λ λ) must b computd.g. by mans of a histogram), bfor th intgral in 4) can b solvd numrically to obtain an stimat ˆ of th BER. This corrsponds to block procssing. Comparison: For ach of ths mthods, th task of stimating th BER is quivalntly to th task of stimating th probability distribution of th rspctiv random λ variabl. Sinc p Λ λ) typically is smoothr than P E ) and p Z ), this might b mor fficint for Λ Mthod 3) than for E Mthod ) and Z Mthod 2). Whras in classical simulation Mthod ) th snt info bits hav to b known, this additional information is not ncssary in soft-dcision simulation Mthod 2 and Mthod 3). Sinc th lattr two mthods do not nd th information phas th sign), thy could b dnotd as incohrnt simulation. Corrspondingly, th first mthod could b dnotd as cohrnt simulation. Not th practical advantag of incohrnt simulation Analysis and Evaluation of th BER Simulation Mthods Whn Mthod or Mthod 2 is applid, th stimatd BER is th man of K random variabls. Thrfor, th varianc of ˆ is givn by σ 2 E /K or σ2 Z /K, rspctivly. Sinc th variancs ar appropriat mans to compar th quality of ths two mthods, thy ar discussd in th following Varianc of Mthod Th probability distribution of th hard bit rror indicator E is givn by { 0 P E ). Thus, th varianc of this binary random variabl computs as σ 2 E E [E 2 ] E [E]) 2 ). 5) Varianc of Mthod 2 Th computation of σz 2 is a bit mor complicatd. Firstly, th pdf of Z is xprssd by mans of th pdf of Λ. Sinc Z + Λ), or quivalntly Λ, th quation ln Z Z p Z ) ) p Λ ln ) can b drivd by a simpl variabl substitution argumnt. Taking ) into considration, th pdf of Z can

4 b formulatd by mans of th conditiond pdf of L: p Z ) ) 2 p L ln ) + Givn this rsult, th varianc of this positiv, ralvalud random variabl computs as 2.3. Applications σ 2 Z E [Z 2 ] E [Z]) 2. 6)., ± σ.4 x BER Mthod class. sim.) BER Mthod 2 soft dc. sim.) ± σ E Mthod ) ± σ Z Mthod 2) Uncodd systm: For uncodd transmission with BPSK ovr an AWGN channl with nois varianc σn, 2 Q /σ n ) and th varianc of E rsults in σe 2 Q /σ n ) [ Q /σ n )] 2. Th varianc of Z can not b rasonably simplifid; but sinc p Z ) is givn analytically, σz 2 can valuatd numrically. In Figur 2, th standard dviations of E and Z ar plottd vrsus th channl SNR. Th curvs show that th standard dviations diffr by about a factor of 2. Sinc th standard dviation of ˆ dcrass with K, an stimation with Mthod nds four tims mor sampls K than an stimation with Mthod 2 to guarant th sam accuracy, i.. th sam standard dviation numbr of valus x 0 6 Figur 3: Comparison of diffrnt BER simulation tchniqus for convolutionally codd binary signaling on th AWGN channl. 3. PROPERTIES OF LOG-LIKELIHOOD RATIOS 3.. Exponntial Symmtry Th conditiond LLR of th binary random variabl U is dfind in ). Whn th LLR is rgardd as a random variabl L with raliation l and whn LU) 0 is assumd, i.., whn th input U is uniformly distributd 3, thn σ E, σ Z std. dv. of E std. dv. of Z p Y y + ) l p Y y ) 7) follows from th dfinition of th LLR. For a transmission systm comprising a symmtric channl and a linar channl cod, p L l + ) p L l ) p L l ) p L l + ). 8) E b /N 0 [db] Figur 2: Standard dviations of E and Z for uncodd transmission ovr an AWGN channl. Codd systm: Th supriority of Mthod 2 is dmonstratd in Fig. 3 for transmission on th AWGN channl using a mmory 3 convolutional cod and APP dcoding. Th dsird BER is 0 4. Th plot illustrats th standard dviation for both Mthod classical simulation) and Mthod 2 th soft-dcision simulation). Furthrmor, th avrag BER for two spcific raliations is shown. Again, th standard dviations diffr by a factor of about 2. Combining 7) and 8), a spcial proprty of th conditiond pdf of th LLR L bcoms obvious: p L l + ) p Y y + ) dy y:l 7) l p Y y ) dy y:l l p L l ), 9) whr th intgration is to b takn ovr all y that lad to th LLR l according to ). This xponntial-symmtric proprty is summarid in th following quation chain: p L l + ) 9) l p L l ) 8) l p L l + ). 0) 2 Q x) / 2π R x xp t2 /2)dt 3 A gnraliation is possibl.

5 Equations 8) and 0) dscrib two fundamntal proprtis of th conditiond pdf of th LLR of a binary random variabl that is transmittd ovr a linar, symmtric channl codd or uncodd). Applying this proprty, th rlation btwn th conditiond pdf p L l + ) of th LLR L and th pdf p Λ λ) of th LLR s absolut valu Λ can b drivd. Firstly, th pdf p L l) can b xprssd by th conditiond pdf of L as: p L l) P U +)p L l + ) + P U )p L l ) and 0) P U +)p L l + ) + P U ) l p L l + ) [ P U +) + l P U ) ] p L l + ) p L l) P U +)p L l + ) + P U )p L l ) 8),0) P U +) l p L l + ) + P U )p L l + ) [ l P U +) + P U ) ] p L l + ). Thn, th pdf p Λ l) of th absolut valu Λ can b computd as p Λ l) p L l) + p L l) [P U +) + P U )][ + l ]p L l + ), which is valid for all l 0. For l < 0, p Λ l) is obviously qual to ro. Thus, th conditiond pdf p L l + ) of L and th pdf p Λ l) of its absolut valu Λ ar linkd by p L l + ) p Λl) + l l 0. ) 3.2. Exampl for Exponntial-Symmtric Distribution As an xampl, lt s look at th LLR of an uncodd BPSK transmission ovr an AWGN channl with th on-sidd nois powr dnsity N 0, i.., with nois varianc σ 2 n N 0 /2. For th fixd input U +u, u > 0, th output Y is Gaussian distributd with man valu µ Y +u and varianc σ 2 Y σ2 n N 0 /2, i.. p Y y + ) 2π σy xp y µ Y ) 2 ) 2 σy 2. Sinc L /σy 2 y, th pdf of L can b computd by th variabl substitution y σy 2 / ) l. This rsults in p L l + ) σ2 Y p Y σ 2 Y l ) 2π σ Y 2 2µ Y l ) 2 ) σy xp 2 2 ) 2, σ Y which is again a Gaussian distribution with man valu µ L 2µ 2 Y /σ2 Y and varianc σ2 L 4µ2 Y /σ2 Y. Sinc this is a conditiond pdf of an LLR, this distribution is xponntial-symmtric. It is asy to prov that actually vry Gaussian distribution with σ 2 2µ shows this proprty Proof of Mthod 3 Givn Equations 8), 0) and particularly ), Mthod 3 can b drivd as follows: P U +) p L l + ) dl +P U ) p L l ) dl 8) p L l + ) } {{} p L l + ) [P U +) + P U )] p L l + ) dl 0) ) l p L l + ) dl p L l + ) dl l + l p Λl) dl + l p Λl) dl. 2) Not that in th drivation of th xponntial-symmtric proprty a uniform input distribution P U u) was assumd, but a gnraliation is possibl. Howvr, Mthod 3 rlis on two additional assumptions: th xistnc of a linar channl cod and a symmtric channl. REFERENCES [] C. Brrou, A. Glaviux, and P. Thitimajshima, Nar Shannon Limit Error-Corrcting Coding and Dcoding: Turbo Cods, Proc. IEEE ICC, pp , May 993. [2] J. Lodg, R. Young, P. Hohr, and J. Hagnaur, Sparabl MAP Filtrs for th Dcoding of Product and Concatnatd Cods, Proc. IEEE ICC, pp , May 993. [3] L.R. Bahl, J. Cock, F. Jlink, and J. Raviv, Optimal Dcoding of Linar Cods for Minimiing Symbol Error Rat, IEEE Trans. Inform. Thory., pp , Mar [4] P. Robrtson, P. Hohr, and E. Villbrun, Optimal and Sub-Optimal Maximum a Postriori Algorithms Suitabl for Turbo Dcoding, ETT, pp. 9 25, Mar./Apr [5] I. Land, P. Hohr, and U. Sorgr, On th Intrprtation of th APP Algorithm as an LLR Filtr, Proc. IEEE ISIT, Jun 2000.