Optmum Seecton Combnng for M-QAM on Fadng Channes M. Surendra Raju, Ramesh Annavajjaa and A. Chockangam Insca Semconductors Inda Pvt. Ltd, Bangaore-56000, Inda Department of ECE, Unversty of Caforna, San Dego, La Joa, CA 92093, U.S.A Department of ECE, Indan Insttute of Scence, Bangaore 56002, Inda Abstract In ths paper, we present the optmum seecton combnng SC) scheme for M-QAM whch mnmzes the average bt error rate on fadng channes. We show that the seecton combnng scheme where each bt n a QAM symbo seects the dversty branch wth the argest magntude of the og-kehood rato LLR) of that bt s optmum n the sense that t mnmzes the average bt error rate BER). In ths optmum SC scheme, dfferent bts n a gven QAM symbo may seect dfferent dversty branches snce the argest LLRs for dfferent bts may occur on dfferent dversty branches), and hence ts compexty s hgh. However, ths scheme provdes the best possbe BER performance for M-QAM wth seecton combnng, and can serve as a benchmark to compare the performance of other SC schemes e.g., seecton based on maxmum SNR). We compare the BER performance of ths optmum SC scheme wth other SC schemes where the dversty seecton s done based on maxmum SNR and maxmum symbo LLR. Keywords M-QAM, bt og-kehood rato, seecton combnng. I. INTRODUCTION Muteve quadrature amptude moduaton M-QAM)san attractve moduaton scheme for wreess communcatons due to the hgh spectra effcency t provdes []. Dversty recepton s a we known technque for mtgatng the effects of fadng on wreess channes [3],[2]. Typca dverstycombnng schemes ncude maxma rato combnng MRC), equa gan combnng EGC), seecton combnng SC), and generazed seecton combnng GSC). Seecton combnng s the smpest of a, as t processes ony one of the dversty branches. In ths paper, we are concerned wth seecton combnng for M-QAM. The dversty branch seecton n SC schemes can be done n severa ways. One way s to choose the dversty branch wth the argest nstantaneous SNR. It s known that choosng the dversty branch wth the maxmum SNR s not the optmum. An aternate way s to choose the branch wth the argest magntude of the og-kehood rato LLR) of the transmtted symbo we ca ths as the symbo LLR - SLLR), as proposed n [4], where the authors show that choosng the branch wth the argest SLLR mnmzes the symbo error rate SER) for M-ary sgnas. We, n ths paper, obtan the optmum seecton combnng scheme for M-QAM whch mnmzes the average bt error rate BER), rather than mnmzng the SER. In our scheme, we compute the LLR for each bt n a gven QAM symbo we Ths work was supported n part by the Indo-French Centre for Promoton of Advanced Research, New Deh, under Project 2900-IT. ca ths as the bt LLR - BLLR) on each dversty branch. For a gven bt n a QAM symbo, the dversty branch havng the argest magntude of the BLLR s chosen. We show that the above BLLR based dversty branch seecton mnmzes the average BER for M-QAM, and hence s optmum. In ths optmum SC scheme, t can be noted that dfferent bts n a gven QAM symbo may seect dfferent dversty branches snce the argest LLRs for dfferent bts may occur on dfferent dversty branches), and hence ts compexty s hgh,.e., the scheme needs a the L receve RF chans to be present for the bts to choose ther respectve best antennas. We however note that ths scheme provdes the best possbe BER performance for M-QAM wth seecton combnng, and can serve as a benchmark to compare the performance of other SC schemes e.g., seecton based on maxmum SNR). We present a BER performance comparson of the BLLR based optmum SC scheme wth other SC schemes where the dversty branch seecton s done based on maxmum SNR and maxmum symbo LLR. We show that, for 6-QAM wth one transmt antenna and L receve antennas, at a BER of 0 2, maxmum SLLR based SC performance s away from the BLLR based optmum SC performance by 0.9 db for L 2,by.4dBforL 3, and by.6 db for L 4. Lkewse, the maxmum SNR based SC performance s away from the optmum SC performance by.4 db for L 2,by 2. db for L 3, and by 2.6 db for L 4. We aso provde smar comparsons for 6-QAM wth two transmt antennas usng Aamout code [5] and L receve antennas. For 6- QAM wth two transmt antennas and L receve antennas, the SLLR based SC performance s away from the optmum SC performance by. db for L 2,by.6dB for L 3, and by.9 db for L 4at a BER of 0 2. We present smar performance comparson for 32-QAM as we. Athough the resuts are shown ony for 6- and 32-QAM n ths paper, the method for BLLR dervaton can be extended for any M-ary QAM. The rest of the paper s organzed as foows. In Secton 2, we derve BLLR expressons for 6-QAM on a gven receve antenna n a system wth one-tx/two-tx antennas. In Secton 3, we show that the SC scheme that chooses the branch wth the argest BLLR mnmzes the BER, and hence s optmum. Smuaton resuts of the BER performance of the optmum SC scheme n comparson wth the performance of other SC schemes are presented n Secton 4. Concusons are gven n Secton 5.
symbos wth r and S 0) comprses symbos wth r 0 n the consteaton. Then, from 2), we have ) Pra α y,h LLR r ) og. 3) Pra β y,h Fg.. 6-QAM Consteaton II. BIT LOG-LIKELIHOOD RATIOS In ths secton, we derve expressons for the BLLRs for 6- QAM.e., M 6) scheme shown n Fg., where 4 bts r,r 2,r 3,r 4 ) are mapped on to a compex symbo a a I + ja Q. The horzonta/vertca ne peces n Fg. denote that a bts under these nes take the vaue, and the rest take the vaue 0. For exampe, the symbo wth coordnates 3d, 3d) maps the 4-bt combnaton r, r 2 0, r 3 r 4. A. -Tx and L-Rx Antennas Frst, consder the case of one transmt antenna and L receve antennas. Assumng that the transmtted symbo a undergoes mutpcatve and ndependent fadng on each dversty path the fadng s assumed to be sow, frequency non-seectve and reman constant over one symbo nterva on each dversty path), the receved sgna y at the th receve antenna correspondng to the transmtted symbo a can be wrtten as y h a + n, 0,,L ) where h,0,,l, s the compex channe coeffcent on the th receve antenna wth E h 2 and the r.v s h s are assumed to be..d, and n n I + jn Q s a compex Gaussan nose r.v of zero mean and varance σ 2 /2 per dmenson. We defne the og-kehood rato of bt r,, 2, 3, 4, of the receved symbo on the th antenna, LLR r ),as[6] ) Prr y,h LLR r ) og. 2) Prr 0 y,h Ceary, the optmum decson rue for the th branch s to decde ˆr f LLR r ) 0, and 0 otherwse. Defne two set parttons, S ) and S 0), such that S ) comprses BLLR expressons for other vaues of M can be derved kewse. Assume that a the symbos are equay key and that fadng s ndependent of the transmtted symbos. Usng Bayes rue, we then have ) f y h,ay h,a α) LLR r ) og. 4) f y h,ay h,a β) Snce f y h,ay h,a α πσ exp 2 σ y 2 h α ),4) 2 can be wrtten as exp σ y LLR r ) og 2 h α 2) ) exp σ y 2 h β 2). 5) Usng og j exp X j) ) mn j X j ), whch s a good approxmaton [7], we can approxmate 5) as LLR r ) σ 2 [ Defne z as mn z y h y h β 2 mn y h α 2 ]. 6) a + n h a + n, 7) where n s compex Gaussan wth varance σ 2 / h 2.Usng 7) n 6), and normazng LLR r ) by 4/σ 2, we get [ LLR r h 2 ) mn z β 2 mn z α ]. 2 8) 4 Further smpfcaton of 8) gves [ LLR r ) h 2 β 2 2z I β I 2z Q β Q 4 mn ] mn α 2 2z I α I 2z Q α Q, 9) where z z I + jz Q, α α I + jα Q and β β I + jβ Q. Note that the set parttons S ) and S 0) are demted by horzonta or vertca boundares. As a consequence, two symbos n dfferent sets cosest to the receved symbo aways e ether on the same row f the demtng boundares are vertca) or on the same coumn f the demtng boundares are horzonta). Usng the above fact, the LLRs for bt r,r 2,r 3 and r 4 are gven by h 2 z I d z I 2d LLR r ) 2 h 2 dd z I ) z I > 2d 2 h 2 dd + z I ) z I < 2d,
h 2 z Q d z Q 2d LLR r 2 ) 2 h 2 dd z Q ) z Q > 2d 2 h 2 dd + z Q ) z Q < 2d, LLR r 3 ) h 2 d z I 2d), LLR r 4 ) h 2 d z Q 2d) 0) where 2d s the mnmum dstance between pars of sgna ponts. B. 2-Tx and L-Rx Antennas Next, we consder the case of two transmt antennas and L receve antennas. Durng a gven symbo nterva, two symbos are transmtted smutaneousy on the two antennas usng Aamout code [5]. Let a, a 2 be the symbos transmtted on the frst and the second transmt antennas, respectvey, durng a symbo nterva. Durng the next symbo nterva, a 2, a are transmtted on the frst and the second transmt antennas, respectvey [5]. We denote the fadng coeffcents as foows: h 2 represents the fadng coeffcent from transmt antenna to receve antenna,,,l, and h 2 represents the fadng coeffcent from transmt antenna 2 to receve antenna,,,l.lety 2 and y 2,,,Lbe the receved sgnas at the th antenna durng two successve symbo ntervas, respectvey. Assumng that the channe reman constant over two consecutve symbo ntervas, the receved sgnas durng the two consecutve symbo ntervas can be wrtten as y 2 a h 2 a 2h 2 + n 2 y 2 a 2 h 2 + a h 2 + n 2, ) where h 2 L and h 2 L are the compex fadng coeffcents and n 2 and n 2 are compex Gaussan random varabes of zero mean and varance σ 2. Assumng perfect knowedge of the fadng coeffcents at the recever, we form â and â 2,forthe th receve branch as â h 2 y 2 + h 2 y2 ) â 2 h 2 y 2 h 2 y2 ). 2) After further smpfcaton, â and â 2 can be rewrtten as â h 2 2 + h 2 2) a + ζ â 2 h 2 2 + h 2 2) a 2 + ζ 2, 3) where ζ and ζ 2 are compex Gaussan random varabes wth of mean and varance h 2 2 + h 2 2 )σ 2. Foowng smar steps as n the case of one transmt and L receve antennas above, the LLR of bts r,, 2, 3, 4 of symbo a j, j, 2 on the th antenna, LLR aj r ),forthe two transmt and L receve antennas, can be derved as r ) h2 2 + h 2 2 ẑj I d ẑi j 2d 2 h 2 2 + h 2 2 dd ẑj I ) ẑi j > 2d 2 h 2 2 + h 2 2 dd +ẑj I ) ẑi j < 2d, h 2 2 + h 2 2) ẑ Q d ẑq j 2d j r 2 ) 2 h 2 2 + h 2 2) dd ẑ Q j ) ẑq j > 2d 2 h 2 2 + h 2 2) dd +ẑ Q j ) ẑq j < 2d, r 3) h 2 2 + h 2 2) d ẑ I j 2d ), r 4) h 2 2 + h 2 2) d ẑ Q j 2d). 4) In the above equatons, ẑj I and ẑq j are the rea and magnary parts of ẑ j, respectvey, where ẑ j s gven by â j ẑ j h 2 2 + h 2 2. 5) It s noted that the LLRs of the varous bts n any M-QAM consteaton of order M and for any arbtrary mappng of bts to the M-QAM symbos can be derved foowng smar steps gven above for 6-QAM. III. BLLR BASED OPTIMUM SC In ths secton, we derve the rue for optma seecton combnng so as to mnmze the BER of each of the bts formng the QAM symbo. We prove that n order to mnmze the BER of bt r, we must seect the dversty branch whch has the argest LLR r ). The proof s as foows. The BER for bt r, P b, s gven as P b Pr ˆr r y, h f y,h dy dh, 6) y,h where y y 0,y 2,,y L ), h h 0,h 2,,h L ), and f y,h s the jont probabty densty functon of y, h. It foows from the above equaton that P b n mnmzed by maxmzng Pr ˆr r y, h for a y, h. Now, Pr ˆr r y, h L 0 Pr ˆr r th branch seected, y, h Pr th branch seected y, h L Pr ˆr r y,h 0 Pr th branch seected y, h max Pr ˆr r y,h. 7) Note that P b s mnmzed by seectng the branch that provdes the maxmum Pr ˆr r y,h, or, equvaenty, seectng the branch that provdes the mnmum Pr ˆr r y,h, whch can be wrtten as Pr ˆr r y,h Pr ˆr,r 0 y,h + Pr ˆr 0,r y,h Pr LLR r ) 0,r 0 y,h + Pr LLR r ) < 0,r y,h. 8)
If LLR r ) 0, then Pr ˆr r y,h Pr r 0 y,h +e LLR r ). 9) If LLR r ) < 0, then Pr ˆr r y,h Pr r y,h Hence, we have +e LLR r ). 20) Pr ˆr r y,h +e LLR r ). 2) Therefore, to mnmze Pr ˆr r y,h, we need to maxmze the denomnator n 2), or, equvaenty, maxmze the term, LLR r ). Hence, by seectng the branch that provdes the argest magntude of LLR r ), we mnmze the BER, P b, and hence mnmze the average BER. It s noted that dfferent bts n a gven symbo may choose dfferent antennas, snce the argest BLLRs for dfferent bts may occur on dfferent antennas, and hence w requre that a the L receve RF chans are present for the bts to choose ther respectve best antennas. Ths scheme however provdes the best possbe BER performance of M-QAM wth seecton combnng, and can serve as a benchmark to compare the performance of other SC schemes as ustrated n the next secton). IV. SIMULATION RESULTS In ths secton, we present the smuated BER performance of the BLLR optmum SC scheme derved n the prevous secton n comparson wth the performance of other SC schemes where the dversty branch seecton s done based on maxmum SNR.e., choose the branch wth argest nstantaneous SNR) and maxmum SLLR.e., choose the branch wth the argest magntude of the symbo LLR). The channe gan coeffcents h s are taken to be..d compex Gaussan.e., fade amptudes are Rayegh dstrbuted) wth zero mean and E h 2. Fgure 2 shows the smuated average BER performance as functon of average SNR per branch for the foowng a) BLLR based optmum SC scheme, b) SNR based SC scheme, and c) SLLR based optmum SC scheme, for 6- QAM wth one transmt and L, 2, 3, 4 receve antennas. From Fg. 2, t s observed that, at a BER of 0 2, the SLLR based SC performance s away from the BLLR based optmum SC performance by 0.9 db for L 2,by.4dBfor L 3, and by.6 db for L 4. Lkewse, the SNR based SC performance s away from the optmum SC performance by.4 db for L 2,by2.dBforL 3, and by 2.6 db for L 4. Snce both the SNR based SC as we as the SLLR based SC have the same compexty.e., ony one of the dversty branches needs to be processed n both cases), SLLR Average Bt Error Rate 0 0 0 0 2 0 3 L, No SC L2, BLLR based Opt. SC L2, SLLR based SC L2, SNR based SC L3, BLLR based Opt. SC L3, SLLR based SC L3, SNR based SC L4, BLLR based Opt. SC L4, SLLR based SC L4, SNR based SC 0 5 4 6 8 0 2 4 6 8 20 Average Receved SNR per Branch db) Fg. 2. Comparson of varous seecton combnng schemes for 6-QAM for Tx antenna and L, 2, 3, 4 Rx antennas BLLR based optmum SC, SLLR based SC, and SNR based SC. based SC s preferred over SNR based SC snce t acheves BER performance coser to that of the BLLR based optmum SC Fgure 3 shows smar comparson for 6-QAM wth two transmt antennas usng Aamout code and L receve antennas. It s ponted out that the pots correspondng to the SLLR based seecton n ths fgure has been obtaned by dervng the expressons for the symbo LLRs for the two transmt and L receve antennas case.e., by extendng dervaton n [4] to the 2 Tx antennas case usng Aamout code). From Fg. 3, t s observed that, for 2-Tx and L 4Rx antennas, the SLLR based SC performance s away from the BLLR based optmum SC performance by. db for L 2,by.6dB for L 3, and by.9 db for L 4, at a BER of 0 2. Smary, the SNR based SC performance s away from the optmum SC performance by.5 db for L 2,by2.6for L 3, and by 3. for L 4,ataBERof0 2. We aso derved the BLLR expressons for 32-QAM dervaton not gven n ths paper), and evauated the BER performance of the three SC schemes for 32-QAM. Fgure 4 shows the BER performance of the three SC schemes for 32-QAM for -Tx and L 2, 4 Rx antennas. It can be observed that for 32-QAM, L 4, at a BER of 0 2, the SLLR based SC s worse by.6 db and the SNR based SC by 2.5 db compared to the BLLR based optmum SC. V. CONCLUSIONS We presented the optmum seecton combnng SC) scheme for M-QAM whch mnmze the average bt error rate BER) on fadng channes. We showed that the SC scheme whch chooses the dversty branch wth the argest magntude of the og-kehood ratos LLRs) of the ndvdua bts n the QAM symbo mnmzes the BER, and hence s optmum. It was ponted out that the compexty of ths optmum SC scheme s hgher snce dfferent bts n a gven QAM symbo
Average Bt Error Rate 0 0 0 0 2 0 3 2Tx, Rx, No SC 2Tx, 2Rx, BLLR based Opt. SC 2Tx, 2Rx, SLLR based SC 2Tx, 2Rx, SNR based SC 2Tx, 3Rx, BLLR based Opt. SC 2Tx, 3Rx, SLLR based SC 2Tx, 3Rx, SNR based SC 2Tx, 4Rx, BLLR based Opt. SC 2Tx, 4Rx, SLLR based SC 2Tx, 4Rx, SNR based SC [3] M. K Smon and M. S. Aoun, Dgta Communcatons Over Fadng Channes: A Unfed Approach to Performance Anayss, Wey Seres, Juy 2000. [4] Y. G. Km and S. W. Km, Optmum seecton combnng for M-ary sgnas n fadng channes, Proc. IEEE GLOBECOM 2002, November 2002. [5] S. M. Aamout, A smpe transmt dversty technque for wreess communcatons, IEEE J. Se. Areas n Commun., vo. 6, no. 8, pp. 45 458, October 998. [6] R. Pyndah, A. Pcard and A. Gaveux, Performance of bock turbo coded 6-QAM and 64-QAM moduatons, Proc. IEEE GLOBE- COM 95, pp. 039 043, November 995. [7] A. J. Vterb, An ntutve justfcaton and a smpfed mpementaton of the MAP decoder for convoutona codes, IEEE J. Se. Areas n Commun., vo. 6, no. 2, pp. 260 264, 998. 4 5 6 7 8 9 0 2 3 4 Average Receved SNR per Branch db) Fg. 3. Comparson of varous seecton combnng schemes for 6-QAM for 2 Tx antennas usng Aamout code and L, 2, 3, 4 Rx antennas BLLR based optmum SC, SLLR based SC, and SNR based SC. 0 0 32 QAM 0 Average Bt Error Rate 0 2 0 3 L2, BLLR based Opt. SC L2, SLLR based SC L2, SNR based SC L4, BLLR based Opt. SC L4, SLLR based SC L4, SNR based SC 0 2 4 6 8 0 2 4 6 8 20 Average Receved SNR per Branch db) Fg. 4. Comparson of varous seecton combnng schemes for 32-QAM for Tx antenna and L 2, 4 Rx antennas BLLR based optmum SC, SLLR based SC, and SNR based SC. may seect dfferent dversty branches and snce the argest LLRs for dfferent bts may occur on dfferent dversty branches. However, ths scheme provdes the best possbe BER performance for M-QAM wth seecton combnng, and can serve as a benchmark to compare the performance of other SC schemes. We presented a BER performance comparson of ths optmum SC scheme wth other SC schemes where the dversty seecton s done based on maxmum SNR and maxmum symbo LLR. REFERENCES [] W. T. Webb and L. Hanzo, Modern Quadrature Amptude Moduaton: Prncpes and Appcatons for Fxed and Wreess Channes, IEEE Press, New York, 994. [2] W. C. Jakes, Mcrowave Mobe Communcatons, New York: IEEE Press, 974.