BER Analyss of QAM wth Transmt Dversty n Raylegh Fadng Channels M. Surendra Raju,A. Ramesh and A. Chockalngam Inslca Semconductors Inda Pvt. Ltd.,Bangalore 56000,INDIA Department of ECE,Unversty of Calforna,San Dego,La Jolla,CA 9093,U.S.A Department of ECE,Indan Insttute of Scence,Bangalore 5600,INDIA Abstract In ths paper, we present a log-lkelhood rato LLR based approach to analyze the bt error rate BER performance of quadrature ampltude modulaton QAM on Raylegh fadng channels wthout and wth transmt dversty. We derve LLRs for the ndvdual bts formng a QAM symbol both on flat fadng channels wthout dversty as well as on channels wth transmt dversty usng two transmt antennas Alamout s scheme and multple receve antennas. Usng the LLRs of the ndvdual bts formng the QAM symbol, we derve expressons for the probablty of error for varous bts n the QAM symbol, and hence the average BER. In addton to beng used n the BER analyss, the LLRs derved can be used as soft nputs to decoders for varous coded QAM schemes ncludng turbo coded QAM wth transmt dversty, as n hgh speed downlnk packet access HSDPA n 3G. Keywords QAM, BER analyss, transmt dversty, log-lkelhood rato. I. INTRODUCTION Multlevel quadrature ampltude modulaton M-QAM s an attractve modulaton scheme for wreless communcatons due to the hgh spectral effcency t provdes. Several works have been reported on the performance analyss of M-QAM n fadng channels,where manly the symbol error rate SER performance has been derved. In addton to the SER analyss,bt error rate BER analyss s also of nterest n multlevel modulaton schemes. Recent works reported n -3 provde expressons to compute the BER for M-QAM on channels. In,Vtthaladevun and Aloun provde BER analyss for herarchcal /M-QAM on fadng channels. In the /M- QAM scheme n,hgher order M QAM constellatons are embedded by a lower order QAM constellaton -QAM,and the M-QAM BER s obtaned by usng the results of the underlyng -QAM constellaton. Our focus n ths paper s on the analytcal evaluaton of the BER performance of QAM on Raylegh fadng channels wthout and wth transmt dversty. The key contrbutons n ths paper are two fold frst,we present an alternate method of dervng the BER for QAM on fadng channels usng log-lkelhood ratos LLRs of the ndvdual bts that form the QAM symbol,and second,usng the LLRs,we derve the BER expressons for QAM on Raylegh fadng channels wthout and wth transmt dversty usng two transmt antennas Alamout s scheme 5 and multple receve antennas. We derve the LLRs and BER expressons for Ths work was supported n part by the Swarnajayant Fellowshp from the Department of Scence and Technology,Government of Inda,New Delh, under scheme Ref: No. 6/3/00-S.F 6-QAM scheme n ths paper. The analytcal technque,however,s applcable to any hgh order M >6 QAM constellaton and for any arbtrary mappng of bts to QAM symbols. Another major usefulness of the results n ths paper s that the derved LLRs provde a soft metrc for each bt n the mappng,whch can be used as soft nputs to decoders for varous coded QAM schemes. Examples of such systems nclude turbo coded QAM wth transmt dversty n hgh speed downlnk packet access HSDPA n 3G,and convolutonally coded QAM wth OFDM n dgtal vdeo broadcastng DVB and IEEE 80.a. The rest of the paper s organzed as follows. We present the dervaton of LLRs and BER expresson for 6-QAM on flat Raylegh fadng channels n Secton II. The dervaton of the LLRs and BER expresson for the case of transmt dversty s presented n Secton III. Conclusons are gven n Secton IV. II. LLR AND BER IN FLAT FADING Consder the M-QAM M =6 scheme as shown n Fg.,where log M =bts r,r,r 3,r are mapped on to a complex symbol a = a I + ja Q. The horzontal/vertcal lne peces n Fg. denote that all bts under these lnes take the value,and the rest take the value 0. For example,the symbol wth coordnates 3d, 3d maps the -bt combnaton r =, r =0, r 3 = r =. Assumng that the transmtted symbol a undergoes multplcatve fadng the fadng s assumed to be slow,frequency non-selectve and reman constant over one symbol nterval,the receved sgnal y correspondng to the transmtted symbol a can be wrtten as y = ha + n, where h s the complex fadng channel coeffcent wth E h } =and the r.v s h s for dfferent symbols are assumed to be..d Raylegh dstrbuted,and n = n I + jn Q s a complex Gaussan r.v of zero mean and varance σ / per dmenson. A. Log-Lkelhood Ratos We defne the log-lkelhood rato LLR of bt r,=,, 3, of the receved symbol as Prr = y, h} LLRr = log. Prr =0 y, h} Clearly,the optmum decson rule s to decde, ˆr = f LLRr 0,and 0 otherwse. Defne two set parttons, GLOBECOM 003-6 - 0-7803-797-8/03/$7.00 003 IEEE
S and S 0,such that S comprses symbols wth r = and S 0 comprses symbols wth r = 0 n the constellaton. Then,from,we have α S Pra = α y, h} LLRr =log. 3 Pra = β y, h} Assume that all the symbols are equally lkely and that fadng s ndependent of the transmtted symbols. Usng Bayes rule, we then have α S f y h,a y h, a = α} LLRr =log. f y h,a y h, a = β} Snce f y h,a y h, a = α} = σ π exp σ y hα, can be wrtten as α S exp σ y hα LLRr =log exp σ y hβ. 5 Usng the approxmaton log j exp X j mn j X j, we can approxmate 5 as LLRr = mn y hβ mn y hα }. 6 σ α S Defne z as z = y h = a + n h = a + n,where n s a complex Gaussan r.v. wth varance σ / h. Usng the above defnton of z nto 6 and normalzng LLRr by /σ, LLRr = h = h mn mn z β mn α S z α }. β z I β I z Q β Q } } mn α z I α I z Q α Q, 7 α S where z = z I + jz Q, α = α I + jα Q and βk =β I + jβ Q. Note that the set parttons S and S 0 are delmted by horzontal or vertcal boundares. As a consequence,two symbols n dfferent sets closest to the receved symbol always le ether on the same row f the delmtng boundares are vertcal or on the same column f the delmtng boundares are horzontal. Then,for bt r,the two constellaton symbols n S and S 0 havng closest dstances to the receved symbol satsfy the condton α Q = β Q. Hence,for bt r h z Id z I d LLRr = h dd z I z I > d 8 h dd + z I z I < d, where d s the mnmum dstance between pars of sgnal ponts. Followng smlar steps for bts r, r 3,and r,we get h z Qd z Q d LLRr = h dd z Q z Q > d 9 h dd + z Q z Q < d, Ths s qute a standard approxmaton 7,and,as we wll see n Sec. II-B, the analytcal BER evaluated usng ths approxmate LLR s almost the same as the BER evaluated through smulatons wthout ths approxmaton. Fg.. 6-QAM Constellaton LLRr 3= h d z I d} 0 LLRr = h d z Q d}. B. Dervaton of Probablty of Bt Error Usng the LLRr s obtaned above,we derve the analytcal expresson for the probablty of error for the bts r, =,, 3,. The probablty of error for bt r, P b,s gven by P b = Pb r = + P b r =0. Snce r =mples that the real part of the transmtted symbol, a I,can take ether values d or 3d,and r =0mples that a I can take ether values +d or +3d,we can wrte the above equaton as P b = P b ai = d. Pra I = d} + P b ai = 3d. Pra I = 3d} + P b ai =d. Pra I = d} + P b ai = d. Pra I =3d}, 3 where P b ai =m s the probablty of error for bt r gven that the real part of the transmtted symbol takes the value m. Now P b ai = d,h s gven by P b ai = d,h = PrLLRr < 0 a I = d, h} d h = Prˆn I d}, where σi = σ d /. Usng the fact that s the energy per transmtted bt,we have P b ai = d,h = E b h E b,where E b. 5 Uncondtonng on the r.v. h,t can easly be shown that P b ai = d E b h E b /No = 5No 5+E b /No On smlar lnes, P b ai = 3d can be shown to be equal to P b ai = 3d 36E b h 8E b /No 5No = 5+8E b /No. 6. 7 It can be shown that P b ai = d = P b ai =d and P b ai = 3d = P b ai =3d. Hence, P b s gven by P b = E b /N o 8E b /N o. 8 5+E b /N o 5+8E b /N o GLOBECOM 003-6 - 0-7803-797-8/03/$7.00 003 IEEE
0 0 Smulated BER usng true LLRs no approx. Analytcal BER usng approx. LLRs complex Gaussan r.v s of zero mean and varance σ. Assumng perfect knowledge of the fadng coeffcents at the recever, we form â and â as Average Probablty of Bt Error 0 0 â = h y + h y = h + h }a + n h + n h, â = h y h y = h + h }a + n h n h. 0 3 0 5 0 5 0 5 Eb/No db Fg.. Comparson of the analytcal BER evaluated usng approxmate LLRs vs the smulated BER usng the LLRs wthout approxmaton. 6-QAM on flat Raylegh fadng. For the 6-QAM constellaton consdered, P b = P b and P b3 = P b. The error probabltes, P b3 and P b can be obtaned as P b3 = P b = + E b /No 5+E b /No 50E b /No 5+50E b /No 8E b /No 5+8E b /No. 9 Usng 8 and 9,we obtan the average BER,P b,as P b = P b +P b3. In Fg.,we compare the analytcal BER evaluated usng the approxmate LLRs derved n the above versus the smulated BER usng the LLRs wthout approxmaton,for 6-QAM on flat Raylegh fadng. It s observed that the analytcally computed BER s almost the same as the smulated BER,ndcatng that the approxmaton to the LLRs results n nsgnfcant dfference between the analytcally computed BER and the true BER. III. LLR AND BER IN TRANSMIT DIVERSITY In ths secton,we derve the LLRs and BER for 6-QAM on Raylegh fadng channels wth transmt dversty. We consder a system wth two transmt antennas Alamout s scheme 5. We frst analyze the case of two transmt antennas and one receve antenna. We then extend the analyss to two transmt antennas and L, L>receve antennas. A. Two Transmt Antennas and One Receve antenna Let a, a be the symbols transmtted on the frst and the second transmt antennas,respectvely,durng a symbol nterval. Durng the next symbol nterval, a, a are transmtted on the frst and the second transmt antennas,respectvely 5. Assumng that the channel remans constant over two consecutve symbol ntervals,the receved sgnals durng the two consecutve symbol ntervals are gven as y = a h a h + n y = a h + a h + n, 0 where h and h are the complex fadng coeffcents on the path from the st and the nd transmt antennas,respectvely, to the receve antenna wth h, h beng Raylegh dstrbuted wth E h } = E h } =,and n and n are In,,we replace n h + n h and n h n h by ζ and ζ,respectvely,where ζ and ζ are complex Gaussan r.v s of zero mean and varance h + h }σ. Then â = h + h }a + ζ â = h + h }a + ζ. 3 Log-Lkelhood Ratos: The dervaton of the LLRs for the bts n symbol a and a s qute smlar to that n Secton II- A. We defne the LLR for the bt r,=,, 3, of symbol a j, j =,,as Prr = y,y,h,h } LLR aj r = log = log Prr =0 y,y,h,h } Prr = â j,h,h } Prr =0 â j,h,h }. Assumng all symbols as equally lkely and that the fadng s ndependent of the transmtted symbols,usng Bayes rule, r = log α S fâj h,h,a â j j h,h,a j = α} fâj h,h,a j â j h,h,a j = β}. 5 Usng the condtonal pdf fâj h,h,a j â j h,h,a j = α}, whch s gven by where ˆσ π exp ˆσ âj h + h }α ˆσ = σ h + h }, we obtan LLR aj r as r = ˆσ mn âj h + h }β mn âj h + h }α. 6 α S Defne two complex varables, ẑ j, j =,,as ẑ j = â j h + h. 7 Usng 7 n 6 and normalzng by /σ,we can wrte r = h + h mn ẑ j β mn ẑ j α. 8 α S Followng smlar steps as n Sec. II-A,we obtan the followng LLRs for bts r,r,r 3,r of the symbol a j. h + h }ẑ ji d ẑ ji d LLR aj r = h + h }dd ẑ ji ẑ ji > d 9 h + h }dd +ẑ ji z ji < d, h + h }ẑ jq d ẑ jq d LLR aj r = h + h }dd ẑ jq ẑ jq > d 30 h + h }dd +ẑ jq ẑ jq < d, LLR aj r 3= h + h }d ẑ ji d}, 3 LLR aj r = h + h }d ẑ jq d}. 3 In the above equatons, ẑ ji and ẑ jq are the real and magnary parts of ẑ j,respectvely. GLOBECOM 003-63 - 0-7803-797-8/03/$7.00 003 IEEE
Probablty of Bt Error: In ths subsecton,we derve the probablty of error for the bt r when transmt dversty s employed. The bt error probablty for bt r, P b,as n Sec. II-B,can be wrtten as 0 0 0 No Dversty Transmt Dversty Tx, Rx P b = P b aji = d. Pra ji = d} + P b aji = 3d. Pra ji = 3d} +P b aji =d. Pra ji = d} + P b aji = d. Pra ji =3d}, 33 where a ji,j=, represents the real part of a j.now P b aji= d,h,h s gven by P b aji = d,h,h = PrLLR aj r < 0 a Ij = d, h,h } } ζ ji = Pr h + h d d h + h, 3 where σi = σ /. Scalng the sgnal power n proporton to the number of transmt antennas,we have d E = b where E b s the energy per bt per transmt antenna. We then have E b h + h P b aji = d,h,h. 35 Uncondtonng the above on h, h,t can be shown that 6 µ P b aji = d = +µ, 36 where µ s gven by µ = Eb/N o 5+E b/n o. Smlarly,the condtonal error probablty P b aji= 3d,h,h s gven by P b aji = 3d,h,h = PrLLR aj r < 0 a I = 3d, h,h } } ζ ji = Pr h + h 3d 8E b h + h. 37 Uncondtonng the above on h and h,we get µ P b aji = 3d = +µ, 38 where µ s gven by µ = 9Eb/N o 5+9E b/n o. It can further be shown that P b ai = d = P b ai =d and P b ai = 3d = P b ai =3d. Hence,the probablty of error for bt r s gven by P b = µ µ + µ + + µ. 39 For the 6-QAM constellaton used,t can be shown that P b = P b. Usng a smlar approach,we can obtan the error probabltes for bts r 3 and r, P b3 and P b,as P b3 = P b = µ + µ + µ + µ µ3 + µ 3, 0 where µ 3 s gven by µ 3 = 5Eb/N o 5+5E b/n o. Usng 39 and 0, we can wrte the average BER, P b,as P b = P b + P b3. Average Probablty of Bt Error, P b 0 0 3 0 0 5 0 6 0 5 0 5 0 5 E /N db b o Fg. 3. BER performance of uncoded 6-QAM wth transmt dversty. transmt antennas and receve antenna. We computed the average BER from the above expresson and plotted the numercal results n Fg. 3. Fg. 3 shows P b as a functon of E b /N o for 6-QAM wthout and wth transmt dversty transmt, receve antenna. It can be seen that when transmt dversty s employed,the BER performance mproves as expected. B. Two Transmt Antennas and L Receve Antennas We now consder a recever wth L, L > receve antennas. The transmtter remans the same as dscussed n Secton III-A. We denote the channel fadng coeffcents as follows: h represents the fadng coeffcent from transmt antenna to receve antenna, = L,and h represent the fadng coeffcent from transmt antenna to receve antenna, = L. Let y and y,= L be the receved sgnal at the th antenna durng two consecutve symbol ntervals,respectvely. Assumng perfect knowledge of the fadng coeffcents at the recever,we have as n Sec. III-A L â = h y + h y â = = L = h y h y. After further smplfcaton, â and â can be rewrtten as â = â = L = L = h a + ζ 3 h a + ζ, where ζ and ζ are complex Gaussan random varables wth zero mean and varance L = h }σ. Log-Lkelhood Ratos : Followng a smlar approach as n Sec. III-A.,t can be shown that the log-lkelhood ratos for bts r, r, r 3 and r are gven by L = h ẑ ji d ẑ ji d L LLR aj r = = h dd ẑ ji ẑ ji k > d 5 L = h dd +ẑ ji ẑ ji k < d GLOBECOM 003-6 - 0-7803-797-8/03/$7.00 003 IEEE
0 0 0 0 0 0 Average Probablty of Bt Error, P b 0 0 3 0 0 5 Tx, Rx Tx, Rx Tx, 3Rx Tx, Rx Tx, 0Rx Average Probablty of Bt Error, P b 0 0 3 0 0 5 0 6 Fadng No Dversty Tx, Rx Tx, Rx 0 6 0 6 8 0 E b /N o db 0 7.5 3 3.5.5 5 5.5 E b /N o db Fg.. BER performance of uncoded 6-QAM wth transmt dversty. transmt antennas and L receve antennas. L =,, 3,, 0. L = h ẑ jq d ẑ jq d L LLR aj r = = h dd ẑ jq ẑ jq > d 6 L = h dd +ẑ jq ẑ jq < d L LLR aj r 3= h d ẑ ji d}, 7 = L LLR aj r = h d ẑ jq d}. 8 = In the above equatons, ẑ j,j=,,are gven by â j ẑ j = L = h, 9 and ẑ ji and ẑ jq are the real and magnary parts of ẑ j. Probablty of Bt Error: The probablty of error can be derved followng smlar lnes n Sec. III-A.. The error probabltes for bts r, r r 3 and r can be derved to be: P b = P b = P + P 50 P b3 = P b = P + P P 3, 5 where P,=,, 3,are gven by P = µ L L k=0 L +k k E where µ = b /N o 9E 5L+E b /N o, µ = b /N o 5L+9E b /N o,and µ 3 = 5E b /N o 5L+5E b /N o. +µ k, 5 Fg. provdes the numercal results of the average BER, P b,computed usng the BER expresson derved above,for the case of two transmt and multple receve antennas. The varous values of L consdered are,, 3,,and 0. It s seen that the performance mproves as L ncreases due to the ncreased dversty order. We pont out that the performance of -Tx,L-Rx scheme s same as that of -Tx,L- Rx scheme. Thus our analyss provdes a means to analytcally evaluate the BER of QAM wth receve-only dversty usng MRC when the number of receve antennas s even. Fg. 5. BER performance of rate-/3 turbo coded 6-QAM scheme wth transmt dversty n Raylegh fadng. LLRs of bts n QAM symbols used as soft nputs to the turbo decoder. C. LLRs as Soft Inputs to Decoders We note that,n addton to beng used n the BER analyss above,the derved LLRs for the ndvdual bts n the QAM symbols can be used as soft nputs to the decoders n varous coded QAM schemes. As an example,we employed the LLRs as soft nputs to the turbo decoder n a rate-/3 turbo coded 6-QAM scheme n Raylegh fadng wthout and wth transmt dversty usng Alamout scheme. Fg. 5 shows the smulated BER performance of turbo coded 6-QAM system usng the derved LLRs as soft nputs to the decoder. The turbo code used n the smulatons s the one specfed n the 3GPP standard. Lkewse, the LLRs can be used as soft nputs to decoders n DVB and IEEE 80.a,where convolutonally coded QAM wth OFDM s used. IV. CONCLUSIONS We analyzed the BER performance of QAM schemes n Raylegh fadng channels wthout and wth transmt dversty. The key contrbutons n ths paper are two fold frst,we presented an alternate method of dervng the BER for QAM on fadng channels usng loglkelhood ratos LLRs of the ndvdual bts that form the QAM symbol,and second,usng the LLRs,we derved the BER for QAM wth transmt dversty n a system that uses two transmt antennas and multple receve antennas. Although we derved the LLRs and BER for a 6-QAM scheme n ths paper,the analytcal technque apples to any hgher order M > 6 QAM constellaton and for any arbtrary mappng of bts to QAM symbols. We also ponted out another major applcaton of the LLRs derved; that s,the LLRs provde a soft metrc for each bt n the mappng,whch can be used as soft nputs to decoders for varous coded QAM schemes,ncludng turbo coded QAM wth transmt dversty as specfed n hgh speed downlnk packet access HSDPA n 3G. REFERENCES P. K. Vtthaladevun and M.-S. Aloun, BER computaton of /M- QAM herarchcal constellatons, IEEE Trans. Broadcastng, vol. 7, no. 3,pp. 8-0,September 00. K. Cho and D. Yoon, On the general BER expresson of one and two dmensonal ampltude modulatons, IEEE Trans. Commun., vol. 50, no. 7,pp. 07 080,July 00. 3 L.-L. Yang and L. Hanzo, A recursve algorthm for the error probablty evaluaton of M-QAM, IEEE Comm. Letters, vol.,no. 0,pp. 30 306,October 000. R. Pyndah,A. Pcard and A. Glaveux, Performance of block Turbo coded 6-QAM and 6-QAM modulatons, Proc. IEEE GLOBE- COM 95, pp. 039 03,Sngapore,November 995. 5 S. M. Alamout, A smple transmt dversty technque for wreless communcatons, IEEE Jl. Sel. Areas n Commun., vol. 6,no. 8,pp. 5 58,October 998. 6 J.G.Proaks,Dgtal Communcatons, McGraw-Hll,995. 7 A. J. Vterb, An ntutve justfcaton and a smplfed mplementaton of the MAP decoder for convolutonal codes, IEEE Jl. Sel. Areas n Commun., vol. 6,no.,pp. 60 6,998. GLOBECOM 003-65 - 0-7803-797-8/03/$7.00 003 IEEE