WHILST QPSK is equivalent to two parallel independent

Transcription

1 38 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., JANUARY 009 Large- Error Probability Analysis of BICM with Unifor Interleaing in Faing Channels Alfonso Martinez, Meber, IEEE an Albert Guillén i Fàbregas, Meber, IEEE Abstract This paper stuies the aerage error probability of bit-interleae coe oulation with unifor interleaing in fully-interleae faing channels. At large signal-to-noise ratio, the oinant pairwise error eents are appe into sybols with Haing weight larger than one, causing a flatteningof the error probability. Close-for expressions for the error probability with general oulations are proie. For interleaers of practical length, the flattening is noticeable only at ery low alues of the error probability. Inex Ters Bit-interleae coe oulation, error probability, higher-orer oulation, salepoint approxiation, binary phase-shift keying BPSK, quaternary phase-shift keying QPSK. I. MOTIVATION AND SUMMARY WHILST QPSK is equialent to two parallel inepenent BPSK channels in the Gaussian channel, the equialence fails in faing channels because of the statistical epenence between the quarature coponents introuce by the faing coefficients. This epenence ensures that the error probability is oinate by the nuber of QPSK sybols with Haing weight two. The asyptotic slope of the error probability is reuce an an error floor results, a phenoenon siilarinnaturetothefloor appearing in turbo coes [], []. More generally, bit-interleae coe oulation BICM [3] is affecte by the sae phenoenon. For large signal-tonoise ratio, the error probability is eterine by error eents with a high iersity, so that the iniu Haing weight of the coe is istribute across the largest possible nuber of oulation sybols. Although worse error eents with saller iersity are present, they are weighte by a low error probability, an thus reain hien for ost practical purposes. We analyze this behaiour by first stuying the union boun to the error probability for QPSK oulation with Gray labeling an fully-interleae faing, an then extening the results to general constellations. The salepoint approxiation allows us to erie close-for expressions for the pairwise error probability, which highlight the aforeentione floor effect. Finally, we estiate of the threshol at which the error probability changes slope. Manuscript receie Noeber 0, 007; reise May 4, 008 an Septeber 5, 008; accepte Noeber 4, 008. The associate eitor coorinating the reiew of this paper an approing it for publication was H. Nguyen. This work was supporte by the International Incoing Short Visits Schee 007/R of the Royal Society. A. Martinez is with Centru Wiskune & Inforatica CWI, Astera, The Netherlans e-ail: alfonso.artinez@ieee.org. A. Guillén i Fàbregas is with the Departent of Engineering, Uniersity of Cabrige, Cabrige, UK e-ail: guillen@ieee.org. Digital Object Ientifier 0.09/T-WC II. CHANNEL MODEL At the transitter, a linear binary coe of rate R c is use to generate coewors c =c,c,...,c l of length l, which are interleae before oulation. Then, consecutie groups of interleae bits c k +,...,c k,fork =,...,n,are appe onto a oulation sybol x k using soe apping rule, such as binary reflecte Gray labeling. We enote the oulation signal set by X, its carinality by X, anthe nuber of bits per sybol = log X. The signal set is noralize to unit aerage energy. Subinices r an i respectiely enote real an iaginary parts. We also assue that n l is an integer. We enote the inerse apping function for labeling position j as c j : X {0, }, naely c j x is the j-th bit of sybol x. ThesetsXc j,...,j,...,c contain the sybols with bit labels in positions j,...,j equal to c,...,c. For each input sybol the corresponing channel output y k is gien by /09$5.00 c 009 IEEE y k = h k xk + z k, k =,...,n where is the aerage receie signal-to-noise ratio, z k are inepenent saples of circularly-syetric Gaussian noise of ariance, an h k are i. i.. faing coefficients. We assue that the faing coefficient is known at the receier, which iplies that the phase of h k is irreleant thanks to the circular syetry of the noise. We also assue that the faing coefficients h k are rawn fro a Nakagai istribution of paraeter f > 0 [4]. This istribution encopasses Rayleigh an AWGN channels, an approxiates Rician faing. III. UNION BOUND AND AVERAGE PAIRWISE ERROR PROBABILITY FOR BICM At the receier, we consier the axiu-etric BICM ecoer which selects the coewor with largest etric n k= qx k,y k [3]. The real-alue etric function qx k,y k is copute by the eoulator for each sybol accoring to the forula qx k,y k = q j cj x k,y k, j= where the bit etric q j cj x =c, y is gien by q j cj x =c, y = py x,h, 3 x X j c where py x, h = π e y h x is the channel transition probability ensity. The wor error rate, enote by P e, is the probability of selecting at the ecoer a coewor ifferent fro the Authorize license use liite to: CAMBRIDGE UNIV. Downloae on February 0, 009 at : fro IEEE Xplore. Restrictions apply.

2 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., JANUARY transitte one. Siilarly, the bit error rate P b is the aerage nuber of input bits in error out of the possible lr c corresponing to a coewor of length l. Exact expressions for P e or P b are ifficult to obtain an one often resorts to bouning, such as the union boun [4]. In the union boun, the probability of an error eent is boune by the su of the probabilities of all possible pairwise error eents, where a coewor c other than the transitte c has a larger etric. We efine the pairwise score as ξ pw n k= j= log q jc k +j,y k q j c k +j,y k. 4 The pairwise error probability PEPc, c between a reference coewor c an the copetitor coewor c is gien by PEPc, c =Pr { ξ pw > 0 }. By construction, the pairwise score is gien by the su of n sybol scores, ξk s log q jc k +j,y k q j c k +j,y k. 5 j= If the coewors hae Haing istance, atost sybol scores are non zero. Further, each sybol score is in turn gien by the su of bit scores ξk,j b log q jc k +j,y k q j c k +j,y k. 6 Clearly, we only nee to consier the non-zero bit an sybol scores. These scores are rano ariables whose ensity function epens on all the rano eleents in the channel, as well as the transitte bits, their position in the sybol an the bit pattern. In orer to aoi this epenence, an as one by Yeh et al. [5], a unifor interleaer [] is ae between the binary coe an the apper, so that the pairwise error probability is aerage oer all possible ways of placing the bits in the n oulation sybols. We istinguish these alternatie placeents by counting the nuber of sybols with weight, where0 w an w =in,. Denoting the nuber of sybols of weight by n, the sybol pattern ρ n is gien by ρ n =n 0,...,n w. Wealso hae that w =0 n = n an = w = n. For finite-length interleaing, the conitional pairwise error probability PEP, ρ n aries for eery possible pattern. As one by Yeh et al. [5], aeraging oer all possible ways of choosing locations in a coewor we hae the following union bouns P e A P ρ n PEP, ρ n, 7 ρ n P b A P ρ n PEP, ρ n, 8 ρ n where A is the nuber of binary coewors of Haing weight, A = j j R A cl j,, with A j, being the nuber of coewors of Haing weight generate with an input If a ore etaile coe spectru were known, naely A,ρn an A j,,ρn, respectiely enoting the nuber of coewors with Haing weight appe onto the pattern ρ n an the nuber of such coewors with input weight j, siilar expressions to those in Eq. 7 coul be written without the aeraging operation. essage of weight j, anp ρ n is the probability of a particular pattern ρ n. A counting arguent [5] gies the probability of the pattern ρ n, P ρ n Pr ρ n =n 0,...,n w,as P ρ n = n n nw w n n! n 0!n!n!...n w! We reoe the epenence of the pairwise error probability on the specific choice of oulation sybols by aeraging oer all possible such choices. This etho consists of aing to eery transitte coewor c Ca rano binary wor {0, } n known by the receier. This is equialent to scrabling the output of the encoer by a sequence known at the receier. Scrabling guarantees the sybols corresponing to two -bit sequences c,...,c an c,...,c are appe to all possible pairs of sybols iffering in a gien Haing weight, hence aking the channel syetric. In [3], [5], the scrabler role was playe by a rano choice between a apping rule μ an its copleent μ with probability / at eery channel use. Scrabling is the natural extension of this rano choice to weights larger than. The cuulant transfor [6] is an equialent representation of the probability istribution of a rano ariable; the istribution can be recoere by an inerse Fourier transfor. Consier a non-zero sybol score Ξ s of Haing weight, i.e., the Haing weight between the binary labels of the reference an copetitor sybols is. The cuulant transfor of Ξ s is κ s log E [e s Ξs] =log j 9 [ i= E q j i c ji,y s ], c {0,} i= q j i c ji,y s 0 where j =j,...,j is a sequence of bit inices, the bit c is the binary copleent of c, any are the channel outputs with bit -tuple c transitte at positions in j. The cuulant transfor of the pairwise score Ξ pw ρ n is gien by w κ pw s, ρ n log E[e sξpwρn ]= n κ s. = The expectation in 0 is one accoring to p j y c = x X j,...,j py x, h. c,...,c An iportant feature of the cuulant transfor is that the tail probability is to great extent eterine by the cuulant transfor aroun the salepoint ŝ, efine as the alue of s for which κ ŝ κs s =0. Inee, in our notation the Chernoff boun is gien by PEP, ρ n e κpwŝ,ρn. Then the salepoint approxiation to PEP, ρ n is gien by PEP, ρ n e κpwŝ,ρn, ŝ πκ pw ŝ, ρ n where the salepoint ŝ is the root of the equation κ pws, ρ n =0. A particularly iportant case arises when =, i.e., when the binary labels of the reference an copetitor sybols in the sybol pairwise score iffer only by a single bit an all ifferent bits are appe onto ifferent constellation Authorize license use liite to: CAMBRIDGE UNIV. Downloae on February 0, 009 at : fro IEEE Xplore. Restrictions apply.

3 40 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., JANUARY 009 Pb l = B E b N 0 Siulation Union Boun Union Boun n Union Boun n = ax{0, n} l =40 Fig.. Error probability of QPSK an the 5, 7 8 conolutional coe with a unifor interleaer of length l = 40, 00 in a fully-interleae faing channel with f =0.5. Diaons correspon to bit-error rate siulation, the soli line correspons to the union boun, ashe lines correspon to the union boun for n =ax{0, n} the upper one correspons to l =00 an otte lines correspon to the union boun for n = n. sybols. This is the case for interleaers of practical length. As notice in [3], [7], this siplifies significantly the analysis. The cuulant transfor of the sybol score with =, enote by Ξ b,isgienby κ s log E [ e s ] Ξb =log j= c {0,} [ qj c, Y s ] E q j c, Y s. 3 We enote the corresponing pairwise error probability by PEP. As notice in [7], This expression is relate, but not ientical to, the ones appearing in [3], [5]. These authors consier an approxiation to the bit ecoing etric q j c, y whereby only one of the suans in Eq. 3 is kept. Een though this approxiation slightly siplifies the analysis, it leas to large inaccuracies for appings other than Gray apping. In this paper, we consier the full bit etric. As shown in Appenix A, the probability that all bit scores are inepenent approaches as l for large interleaer lengths. In general, the effect of the epenence across the bits belonging to the sae sybol ust be taken into account. IV. INEQUIVALENCE BETWEEN BPSK AND QPSK WITH GRAY LABELING Obtaining close-for expressions for the cuulant transfors can be ifficult, an nuerical ethos are neee. Notable exceptions are BPSK an QPSK with Gray labeling, where the pattern ρ n is uniquely eterine by the nuber of QPSK sybols whose bits are at Haing istance an, respectiely enote by n an n.thenn +n =. It is also clear that ax{0, n} n. We enote the pairwise error probability by PEP, n. Reference [8] gae the union bouns to the aerage error probability for blockfaing channels, of which QPSK can be seen as a particular case. In any case, haing n 0 iplies that we hae n sybols with Haing weight two that fae with the sae faing coefficient, an therefore, QPSK behaes ifferently fro two inepenent BPSK channels. A non-zero sybol score ξ s is the result of haing a Haing istance of either or bits. In the first case, it is easy to see that it has a Gaussian istribution of ean h k an ariance 4 h k. This score is equal to that of BPSK oulation with effectie signal-to-noise ratio h k. When both bits are ifferent, the score is a real-alue rano ariable with a Gaussian istribution of ean 4 h k an ariance 8 h k.thisscore coincies with that of a BPSK oulation with signal-to-noise ratio h k. Using the expression for κ s fro [7], we conclue that κ pw s, n =log E [ e sξ] n [ E e sξ ] n 4 =log + s s f n f f + 4s 4s f n. 5 f f Direct calculation shows that the salepoint is locate at ŝ = +,anwefinally approxiate the conitional pairwise error probability PEP, n by the salepoint approxiation PEP, n π f f + n f + f + + f n + f n. f f 6 In the liit f, i.e., for the AWGN channel, this forula respectiely becoes PEP, n π e, 7 which is inepenent of n, as expecte. The error perforance approaches that of two coewors at istance transitte oer BPSK with signal-to-noise ratio. Zuo et al. gae a siilar analysis in their stuy of block-faing channels [8] an of BICM [5]. Our use of the salepoint approxiation gies a siple an tight closefor approxiation to the pairwise error probability. Figure epicts the bit-error probability of QPSK with the 5, 7 8 conolutional coe for l =40, 00 an f =0.5. In all cases, the salepoint approxiation to the union boun soli lines is ery accurate for oerate-to-large. In particular, we obsere the change in slope with respect to the stanar union boun, which assues that all bits in which the coewors iffer are appe onto ifferent sybols inepenent binary channels. The approxiate threshol 0 copute with the iniu istance is th =B for l =40an th = 36 B for l = 00. This floor is absent for inepenent binary parallel channels. In the next section, we proe that the floor appears at ery low error rates i. e. at ery high for practical coes. Authorize license use liite to: CAMBRIDGE UNIV. Downloae on February 0, 009 at : fro IEEE Xplore. Restrictions apply.

4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., JANUARY f =0.5,l= 00 f =0.5,l= 000 f =3,l= 00 f =3,l= 000 istance. As expecte fro Figure, the error probability quickly becoes ery sall, at alues typically below the operating point of coon counication systes. For short packets using weak coes transitte oer channels with significant faing, the threshol ay be of iportance. Pb Fig.. Approxiation to the bit error probability of QPSK at the threshol signal-to-noise ratio th, P b = PEP, as a function of the iniu Haing istance for seeral alues of f an l. In soli line, f = 0.5,l =00; in ashe line, f =0.5,l =000; in ash-otte line, f =3,l=00; an in otte line f =3,l=000. A. Asyptotic Analysis In this section, we concentrate on the asyptotic stuy of the pairwise error probability aerage oer all possible interleaers of length n, using Eq. 9. It sees clear that the best i. e. steepest conitional pairwise error probabilty PEP, n is attaine for n = 0, which correspons to all bits in the pairwise score belonging to ifferent sybols. Siilarly, the worst i. e. flattest conitional pairwise error probability PEP, n is attaine for n =, which correspons to the bits haing axial concentration in the iniu possible nuber of QPSK sybols. For these two cases, we show in Appenix B that the respectie probabilities P n ait the following asyptotic in n approxiations P n =0 an l P n = n le een, + 8 o. ele We eterine the position at which the respectie pairwise probabilities cross by soling P n = 0 PEP, 0 = P n = n PEP, n. 9 Uner the assuptions that an l are large, with the ai of Eq. 8, an keeping only the ter at iniu istance in which we enote without the subscript, to reoe clutter fro the equations, Eq. 9 is easily sole to obtain 0. For large, the threshol is approxiately gien by th f 4 le f in both cases. For the special case of Rayleigh faing, f =, th grows linearly with the interleaer length l an is inersely proportional to the Haing istance for ery large. Figure epicts the approxiate alue of P b, coarsely approxiate as PEP, at the threshol for seeral alues of f an l as a function of the iniu Haing V. HIGH-ORDER MODULATIONS: ASYMPTOTIC ANALYSIS In this section, we exten the analysis presente in the preious section for QPSK to general constellations an appings, an estiate the signal-to-noise ratio at which the slope of the error probability changes. Since we are intereste in large alues of, we shall be working with an asyptotic approxiation to the error probability. In particular, our analysis is base on an extension of the results in [3], [7] for the asyptotic behaiour of the error probability in the Rayleighfaing channel. We will proie such asyptotic expressions for the sybols scores of arying weight an use the approach presente for QPSK in the preious section to eterine the alue of at which the arious approxiations to the error probability cross. In [3], Caire et al. consiere the Rayleigh faing at large, an erie the following approxiation to the cuulant transfor of the bit score, κ s log h 4, where h is a haronic istance gien by h = x x, c=0 j= x Xc j where x is the closest sybol in the constellation X j c to x. Eq. ay be use in the Chernoff boun to gie an approxiation to the error probability at large. Itwas foun in [3] that this gies a goo approxiation. Using that ŝ =,thatli κ ŝ =8 f [7] an the salepoint approxiation in Eq., we obtain the large heuristic approxiation PEP π 4 h. 3 In Appenix C we exten the analysis in [3] to the cuulant transfor of the BICM sybol score of weight for Nakagai- f faing, an obtain the following liit κ ŝ f log h 4 f, 4 where h is a generalization of the haronic istance gien by i= h = x x i f f. i= x x i c j x X c j 5 For a gien x, x i is the i-th sybol in the sequence of sybols x,...,x which hae binary label c ji at position j i an for which the ratio i= x x i is iniu aong all i= x x i Authorize license use liite to: CAMBRIDGE UNIV. Downloae on February 0, 009 at : fro IEEE Xplore. Restrictions apply.

5 4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., JANUARY 009 le 4 f 8 th = 4 f le f f e f f + f + f een o. 0 possible such sequences. For f =an =we recoer the haronic istance h aboe. As it happene with the bit score an PEP, Eq. 4 ay be use in the Chernoff boun or in the salepoint approxiation in Eq., to obtain a heuristic approxiation to the pairwise error probability for large, naely n f 4 f PEP H, ρ n π f n = h. 6 For the sake of siplicity, we isregar the effect of the coefficient in our analysis of the threshol. ŝ πκ pwŝ For sufficiently large signal-to-noise ratio, the error probability is eterine by the worst possible istribution pattern ρ n of the bits onto the n sybols, that with the largest tail probability for the pairwise score. Then, since = w = n by construction, we can iew the pattern ρ n = n 0,n,...,n w as a non-unique representation of the integer as a weighte su of the integers {0,,,...,w }. By construction, the su n is the nuber of nonzero Haing weight sybols in the caniate coewor. Clearly, the lowest n gies the worst flattest pairwise error probability in the presence of faing. We obtain an equation for th siilar to Eq. 9 for QPSK for the fully-interleae faing channel, f 4f P ρ 0 h = th f = n th n f 4f P ρ n ρ n:in n = h. 7 The left-han sie correspons to the steepest pairwise error probability, naely PEP, weighte by the probability that all bit scores are inepenent, enote by P ρ 0. The righthan sie correspons to the largest pairwise error probability with sallest nuber of non-zero sybol scores, that is aong all possible patterns ρ n with iniu n.note that the exponent of is f n, an thus has the lowest possible iersity, as it shoul. Fro Eq. 7, we can extract the alue of th as P ρ n h th 4 f ρ P ρ n:in 0 n h f n n. 8 As expecte, for the specific case of QPSK with Gray apping, coputation of this alue of th h =, h = 4 gies a result which is consistent with the result erie in Section IV-A, naely Eq. 0, with the inor ifference that we use now the Chernoff boun whereas the salepoint approxiation was use in the QPSK case. Pb l = l =90 l = 300 l = B E b N 0 Siulation l =90 Siulation l = 3000 Salepoint UB inf. int. Heuristic Approx. = Heuristic Approx. = Heuristic Approx. = 3 Fig. 3. Bit error probability union bouns an bit-error rate siulations of 8-PSK with the 8-state rate-/3 conolutional coe in a fully-interleae Rayleigh faing channel. Interleaer length l =90circles an l =3000 iaons. In soli lines, the salepoint approxiation union bouns for l =90, l = 300, l = 3000 an for infinite interleaing, with PEP. In ashe, ashe-otte, an otte lines, the heuristic approxiations with weight =,, 3 respectiely. As we obsere in Figure 3, the slope change present for QPSK is also present for 8-PSK with Rayleigh faing an Gray labeling; the coe is the optiu 8-state rate- /3 conolutional coe, with in =4. Again, this effect is ue to the probability of haing sybol scores of Haing weight larger than. The figure epicts siulation results for interleaer sizes l = 90, 3000 together with the salepoint approxiations for finite l for l =90, 300, 3000, infinite interleaing with PEP, an with the heuristic approxiation PEP H ρ n only for l =90an =4. For 8-PSK with Gray apping, ealuation of Eq. 5 gies h = , h =.775, an h 3 =.478. Table I gies the alues of P ρ n for the arious patterns ρ n. Table I also gies the threshol th gien in Eq. 8 for all possible alues of n, not only for the worst case. We obsere that the ain flattening of the error probability takes place at high. This effect essentially isappears for interleaers of practical length: for l = 300 resp. l = 3000 the error probability at the first threshol is about 0 8 resp. 0. The salepoint approxiation is rearkably precise; the heuristic approxiation PEP H in,ρ n also gies ery goo results. VI. CONCLUSIONS We hae stuie the large- behaior of the error probability of BICM oer fully-interleae faing channels. Our Authorize license use liite to: CAMBRIDGE UNIV. Downloae on February 0, 009 at : fro IEEE Xplore. Restrictions apply.

6 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., JANUARY TABLE I ASYMPTOTIC ANALYSIS FOR 8-PSK WITH VARYING INTERLEAVER LENGTH l =3n AND MINIMUM DISTANCE =4. Pattern ρ n l Pρ n Threshol E b N 0 n 4, 4, 0, 0 l = N/A l = N/A l = N/A n 3,,, 0 l = l = l = n, 0,, 0, l = n,, 0, l = l = B analysis reeals that the pairwise error probability is asyptotically oinate by the nuber pairwise error sybols with Haing weight larger than one, yieling an error floor. We hae erie close-for approxiations to this error probability. For practical coe lengths, the error floor appears at ery low error rates. APPENDIX A PROBABILITY OF ALL-ONE SEQUENCE We use Stirling s approxiation to the factorial, n! n n e n πn, to Eq. 9 to obtain P in P n =, n =0,...,n w =0 l n = l n + l l + l l, 9 with the obious siplifications an cobinations. Extracting a factor l in l an l, an cancelling coon powers of l in nuerator an enoinator, we get P in n+ l n l We now take logariths, an use Taylor s expansion of the logarith, log + t t t, in the right-han sie of Eq. 30. Discaring all powers of l higher than l,an cobining coon ters, we obtain log P in l + n + l l =. l 3 Finally, recoering the exponential, P in e l. APPENDIX B ASYMPOTICS OF P n Assue that is een, so n = an n =0.Then!n!n! P n = n!n!. 3! Using Stirling s approxiation to the factorial, an after soe siplifications, we hae that n n P n. 33 n n In the liit of large n, usingthatli n + a n n = e a,an n, wehae P n e =. 34 n ne If is o, then n = an n =.Then!n!n! P n = n! n +!!. 35 Using again Stirling s approxiation, an after soe siplifications, we hae n n n n P n e n n + + n + n. 36 Again for large n, usingthatli n + a n n = e a,an n, wehae P n e e e + + n + = ene. 37 APPENDIX C ASYMPTOTIC ANALYSIS WITH FULLY-INTERLEAVED NAKAGAMI FADING κ ŝ e We wish to copute the liit l s li, f gien by l s = li f [ ] i= E qj i cj i,ys. j,c i= qj i cj i,ys 38 We can rewrite the enoinator as H q ji c ji,y= X x +Z i= i= e x X j i c ji = e H X x i +Z, 39 x i= where x is one of all possible sequences of oulation sybols, with sybol at inex j i rawn fro the set X c ji ji.a siilar forula hols for the enoinator, now with sybols rawn fro the set Xc ji ji. Expaning the exponent in Eq. 39, we obtain q ji c ji,y i= = x e H i= X x i + Re i= HX x i Z + Z. 40 As one in [3], [7], we keep only the oinant suan in the bit scores q ji,y appearing in nuerator an enoinator. For a gien x, this suan correspons to Authorize license use liite to: CAMBRIDGE UNIV. Downloae on February 0, 009 at : fro IEEE Xplore. Restrictions apply.

7 44 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO., JANUARY 009 the sequence x haing the sallest possible alue of the ratio i= x x i i= x x. In particular, in the enoinator all the i sybols in the sequence coincie with x. We now carry out the expectation oer Z. Copleting squares, an using that the forula for the ensity of Gaussian noise, we hae that π e z e i= s H X X i + ReHX X i z z = e H s i= X X i s i= X X i. 4 In turn, the expectation oer h of this quantity yiels [9] + s X X i s X X i i= i= f. f 4 We next turn back to the liit of large. Wehaethat f f l s = s x x i s j,c x Xc j i= i= x x i f. 43 For each suan, the optiizing s is reaily copute to i= x x i be ŝ =, which gies i= x x i l ŝ = j,c x X j c 4 f i= x f x i i= x x i. 44 REFERENCES [] C. Berrou, A. Glaieux, an P. Thitiajshia, Near Shannon liit errorcorrecting coing an ecoing: turbo-coes. in Proc. IEEE Int. Conf. Coun., Genea, ol., 993. [] S. Beneetto an G. Montorsi, Uneiling turbo coes: soe results on parallel concatenate coing schees, IEEE Trans. Inf. Theory, ol. 4, no., pp , 996. [3] G. Caire, G. Taricco, an E. Biglieri, Bit-interleae coe oulation, IEEE Trans. Infor. Theory, ol. 44, no. 3, pp , 998. [4] J. G. Proakis, Digital Counications. McGraw-Hill, 4th eition, 00. [5] P. Yeh, S. Zuo, an W. Stark, Error probability of bit-interleae coe oulation in wireless enironents, IEEE Trans. Veh. Technol., ol. 55, no., pp. 7 78, 006. [6] R. W. Butler, Salepoint Approxiations with Applications. Cabrige Uniersity Press, 007. [7] A. Martinez, A. Guillén i Fàbregas, an G. Caire, Error probability analysis of bit-interleae coe oulation, IEEE Trans. Inf. Theory, ol. 5, no., pp. 6 7, Jan [8] S. Zuo, P. Yeh, an W. Stark, A union boun on the error probability of binary coes oer block faing channels, IEEE Trans. Veh. Technol., ol. 54, no. 6, pp , 005. [9] A. Martinez, A. Guillén i Fàbregas, an G. Caire, A close-for approxiation for the error probability of BPSK faing channels, IEEE Trans. Wireless Coun., ol. 6, no. 6, pp , June 007. Authorize license use liite to: CAMBRIDGE UNIV. Downloae on February 0, 009 at : fro IEEE Xplore. Restrictions apply.