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Optimum Selection Combining for M -QAM on Fading Channels M. Surendra Raju∗ , Ramesh Annavajjala‡ and A. Chockalingam† ∗ Insilica Semiconductors India Pvt. Ltd, Bangalore-560001, India ‡ Department of ECE, University of California, San Diego, La Jolla, CA 92093, U.S.A † Department of ECE, Indian Institute of Science, Bangalore 560012, India Abstract—In this paper, we present the optimum selection com- call this as the ‘bit LLR’ - BLLR) on each diversity branch. bining (SC) scheme for M -QAM which minimizes the average bit For a given bit in a QAM symbol, the diversity branch having error rate on fading channels. We show that the selection com- the largest magnitude of the BLLR is chosen. We show that bining scheme where each bit in a QAM symbol selects the diver- sity branch with the largest magnitude of the log-likelihood ratio the above BLLR based diversity branch selection minimizes (LLR) of that bit is optimum in the sense that it minimizes the the average BER for M -QAM, and hence is optimum. In this average bit error rate (BER). In this optimum SC scheme, dif- optimum SC scheme, it can be noted that different bits in a ferent bits in a given QAM symbol may select different diversity given QAM symbol may select different diversity branches branches (since the largest LLRs for different bits may occur on (since the largest LLRs for different bits may occur on differ- different diversity branches), and hence its complexity is high. However, this scheme provides the best possible BER perfor- ent diversity branches), and hence its complexity is high, i.e., mance for M -QAM with selection combining, and can serve as the scheme needs all the L receive RF chains to be present a benchmark to compare the performance of other SC schemes for the bits to choose their respective best antennas. We how- (e.g., selection based on maximum SNR). We compare the BER ever note that this scheme provides the best possible BER performance of this optimum SC scheme with other SC schemes performance for M -QAM with selection combining, and can where the diversity selection is done based on maximum SNR and maximum symbol LLR. serve as a benchmark to compare the performance of other SC schemes (e.g., selection based on maximum SNR). Keywords – M -QAM, bit log-likelihood ratio, selection combining. We present a BER performance comparison of the BLLR based optimum SC scheme with other SC schemes where the I. I NTRODUCTION diversity branch selection is done based on maximum SNR and maximum symbol LLR. We show that, for 16-QAM with Multilevel quadrature amplitude modulation (M -QAM) is an one transmit antenna and L receive antennas, at a BER of attractive modulation scheme for wireless communications 10−2 , maximum SLLR based SC performance is away from due to the high spectral efﬁciency it provides [1]. Diversity the BLLR based optimum SC performance by 0.9 dB for reception is a well known technique for mitigating the ef- L = 2, by 1.4 dB for L = 3, and by 1.6 dB for L = 4. fects of fading on wireless channels [3],[2]. Typical diversity- Likewise, the maximum SNR based SC performance is away combining schemes include maximal ratio combining (MRC), from the optimum SC performance by 1.4 dB for L = 2, by equal gain combining (EGC), selection combining (SC), and 2.1 dB for L = 3, and by 2.6 dB for L = 4. We also provide generalized selection combining (GSC). Selection combining similar comparisons for 16-QAM with two transmit anten- is the simplest of all, as it processes only one of the diversity nas using Alamouti code [5] and L receive antennas. For 16- branches. In this paper, we are concerned with selection com- QAM with two transmit antennas and L receive antennas, the bining for M -QAM. SLLR based SC performance is away from the optimum SC The diversity branch selection in SC schemes can be done in performance by 1.1 dB for L = 2, by 1.6 dB for L = 3, and several ways. One way is to choose the diversity branch with by 1.9 dB for L = 4 at a BER of 10−2 . We present similar the largest instantaneous SNR. It is known that choosing the performance comparison for 32-QAM as well. Although the diversity branch with the maximum SNR is not the optimum. results are shown only for 16- and 32-QAM in this paper, the An alternate way is to choose the branch with the largest mag- method for BLLR derivation can be extended for any M -ary nitude of the log-likelihood ratio (LLR) of the transmitted QAM. symbol (we call this as the ‘symbol LLR’ - SLLR), as pro- The rest of the paper is organized as follows. In Section 2, posed in [4], where the authors show that choosing the branch we derive BLLR expressions for 16-QAM on a given receive with the largest SLLR minimizes the symbol error rate (SER) antenna in a system with one-Tx/two-Tx antennas. In Section for M -ary signals. 3, we show that the SC scheme that chooses the branch with We, in this paper, obtain the optimum selection combining the largest BLLR minimizes the BER, and hence is optimum. scheme for M -QAM which minimizes the average bit error Simulation results of the BER performance of the optimum rate (BER), rather than minimizing the SER. In our scheme, SC scheme in comparison with the performance of other SC we compute the LLR for each bit in a given QAM symbol (we schemes are presented in Section 4. Conclusions are given in Section 5. This work was supported in part by the Indo-French Centre for Promotion of Advanced Research, New Delhi, under Project 2900-IT. (0) symbols with ri = 1 and Si comprises symbols with ri = 0 in the constellation. Then, from (2), we have α∈Si (1) Pr{a = α|yl , hl } LLRl (ri ) = log . (3) β∈Si (0) Pr{a = β|yl , hl } Assume that all the symbols are equally likely and that fading is independent of the transmitted symbols. Using Bayes’ rule, we then have α∈Si (1) fyl |hl ,a (yl |hl , a = α) LLRl (ri ) = log . (4) β∈Si (0) fyl |hl ,a (yl |hl , a = β) 1 −1 2 Since fyl |hl ,a {yl |hl , a = α} = πσ 2 exp σ2 y l − hl α , (4) can be written as Fig. 1. 16-QAM Constellation −1 2 α∈Si (1) exp σ2 yl − hl α LLRl (ri ) = log −1 2 . (5) β∈Si (0) exp σ2 yl − hl β II. B IT L OG -L IKELIHOOD R ATIOS In this section, we derive expressions for the BLLRs for 16- Using log j exp(−Xj ) ≈ − minj (Xj ), which is a good QAM (i.e., M = 16)1 scheme shown in Fig. 1, where 4 bits approximation [7], we can approximate (5) as (r1 , r2 , r3 , r4 ) are mapped on to a complex symbol a = aI + jaQ . The horizontal/vertical line pieces in Fig. 1 denote that 1 2 2 LLRl (ri ) = min yl − hl β − min yl − hl α . (6) all bits under these lines take the value 1, and the rest take the σ2 (0) β∈Si (1) α∈Si value 0. For example, the symbol with coordinates (−3d, 3d) maps the 4-bit combination r1 = 1, r2 = 0, r3 = r4 = 1. Deﬁne zl as ∆ yl nl A. 1-Tx and L-Rx Antennas zl = = a+ = a + nl , (7) hl hl First, consider the case of one transmit antenna and L receive where nl is complex Gaussian with variance σ 2 / hl 2 . Us- antennas. Assuming that the transmitted symbol a undergoes ing (7) in (6), and normalizing LLRl (ri ) by 4/σ 2 , we get multiplicative and independent fading on each diversity path (the fading is assumed to be slow, frequency non-selective 2 and remain constant over one symbol interval on each diver- hl 2 2 LLRl (ri ) = min zl − β − min zl − α . (8) sity path), the received signal yl at the lth receive antenna 4 β∈Si (0) (1) α∈Si corresponding to the transmitted symbol a can be written as Further simpliﬁcation of (8) gives yl = hl a + nl , l = 0, · · · , L − 1 (1) 2 hl 2 where hl , l = 0, · · · , L − 1, is the complex channel coefﬁ- LLRl (ri ) = min β − 2zlI βI − 2zlQ βQ 4 β∈Si (0) cient on the lth receive antenna with E{ hl 2 } = 1 and the r.v’s hl ’s are assumed to be i.i.d, and nl = nlI + j nlQ is a 2 complex Gaussian noise r.v of zero mean and variance σ 2 /2 − min α − 2zlI αI − 2zlQ αQ , (9) (1) α∈Si per dimension. We deﬁne the log-likelihood ratio of bit ri , i = 1, 2, 3, 4, of where zl = zlI + jzlQ , α = αI + jαQ and β = βI + jβQ . (1) (0) the received symbol on the lth antenna, LLRl (ri ), as [6] Note that the set partitions Si and Si are delimited by hor- izontal or vertical boundaries. As a consequence, two sym- Pr{ri = 1|yl , hl } bols in different sets closest to the received symbol always lie LLRl (ri ) = log . (2) Pr{ri = 0|yl , hl } either on the same row (if the delimiting boundaries are ver- tical) or on the same column (if the delimiting boundaries are Clearly, the optimum decision rule for the lth branch is to horizontal). Using the above fact, the LLRs for bit r1 , r2 , r3 decide ri = 1 if LLRl (ri ) ≥ 0, and 0 otherwise. Deﬁne ˆ and r4 are given by (1) (0) (1) two set partitions, Si and Si , such that Si comprises − hl 2 zlI d |zlI | ≤ 2d 1 BLLR LLRl (r1 ) = 2 hl 2 d(d − zlI ) zlI > 2d expressions for other values of M can be derived likewise. −2 hl 2 d(d + zlI ) zlI < −2d, − hl 2 zlQ d |zlQ | ≤ 2d LLRl (r2 ) = 2 hl 2 d(d − zlQ ) zlQ > 2d − h2l−1 2 + h2l 2 ˆQ zjl d zQ |ˆjl | ≤ 2d −2 hl 2 d(d + zlQ ) zlQ < −2d, a LLRl j (r2 ) = 2 h2l−1 2 + h2l 2 ˆQ d(d − zjl ) ˆQ zjl > 2d LLRl (r3 ) = hl 2 d(|zlI | − 2d), −2 h2l−1 2 + h2l 2 d(d + ˆQ zjl ) ˆQ zjl < −2d, 2 LLRl (r4 ) = hl d(|zlQ | − 2d) (10) a 2 2 LLRl j (r3 ) = h2l−1 + h2l zI d |ˆjl | − 2d , where 2d is the minimum distance between pairs of signal points. a 2 2 LLRl j (r4 ) = h2l−1 + h2l zQ d |ˆjl | − 2d . (14) B. 2-Tx and L-Rx Antennas ˆI ˆQ In the above equations, zjl and zjl are the real and imaginary ˆ ˆ parts of zjl , respectively, where zjl is given by Next, we consider the case of two transmit antennas and L receive antennas. During a given symbol interval, two sym- ˆ ajl bols are transmitted simultaneously on the two antennas using ˆ zjl = 2+ h 2 . (15) h2l−1 2l Alamouti code [5]. Let a1 , −a∗ be the symbols transmitted 2 on the ﬁrst and the second transmit antennas, respectively, It is noted that the LLRs of the various bits in any M -QAM during a symbol interval. During the next symbol interval, constellation of order M and for any arbitrary mapping of a2 , a∗ are transmitted on the ﬁrst and the second transmit an- bits to the M -QAM symbols can be derived following similar 1 tennas, respectively [5]. We denote the fading coefﬁcients as steps given above for 16-QAM. follows: h2l−1 represents the fading coefﬁcient from transmit antenna 1 to receive antenna l, l = 1, · · · , L, and h2l repre- III. BLLR BASED O PTIMUM SC sents the fading coefﬁcient from transmit antenna 2 to receive antenna l, i = 1, · · · , L. Let y2l−1 and y2l , l = 1, · · · , L be In this section, we derive the rule for optimal selection com- the received signals at the lth antenna during two successive bining so as to minimize the BER of each of the bits forming symbol intervals, respectively. Assuming that the channel re- the QAM symbol. We prove that in order to minimize the main constant over two consecutive symbol intervals, the re- BER of bit ri , we must select the diversity branch which has ceived signals during the two consecutive symbol intervals the largest |LLRl (ri )|. The proof is as follows. can be written as The BER for bit ri , Pbi , is given as y2l−1 = a1 h2l−1 − a∗ h2l 2 + n2l−1 Pbi = 1 − ˆ Pr ri = ri |y, h fy,h dy dh, (16) y2l = a2 h2l−1 + a∗ h2l 1 + n2l , (11) y,h where {h2l−1 }L and {h2l }L are the complex fading co- where y = (y0 , y2 , · · · , yL−1 ), h = (h0 , h2 , · · · , hL−1 ), and l=1 l=1 efﬁcients and n2l−1 and n2l are complex Gaussian random fy,h is the joint probability density function of y, h. It fol- variables of zero mean and variance σ 2 . Assuming perfect lows from the above equation that Pbi in minimized by max- knowledge of the fading coefﬁcients at the receiver, we form imizing Pr ri = ri |y, h for all y, h. Now, ˆ a1l and a2l , for the lth receive branch as ˆ ˆ L−1 ˆ a1l = h∗ ∗ y2l−1 + h2l y2l ˆ Pr ri = ri |y, h = Pr ri = ri | lth branch selected, y, h ˆ 2l−1 l=0 ˆ a2l = h∗ 2l−1 ∗ y2l − h2l y2l−1 . (12) · Pr{lth branch selected | y, h} ˆ ˆ After further simpliﬁcation, a1l and a2l can be rewritten as L−1 = ˆ Pr ri = ri |yl , hl 2 2 l=0 ˆ a1l = h2l−1 + h2l a1 + ζ1 · Pr{lth branch selected|y, h} 2 2 ˆ a2l = h2l−1 + h2l a2 + ζ2 , (13) ≤ max Pr ri = ri |yl , hl . ˆ (17) l where ζ1 and ζ2 are complex Gaussian random variables with Note that Pbi is minimized by selecting the branch that pro- of mean and variance ( h2l−1 2 + h2l 2 )σ 2 . vides the maximum Pr ri = ri |yl , hl , or, equivalently, se- ˆ Following similar steps as in the case of one transmit and L lecting the branch that provides the minimum Pr ri = ri |yl , hl , ˆ receive antennas above, the LLR of bits ri , i = 1, 2, 3, 4 of which can be written as a symbol aj , j = 1, 2 on the lth antenna, LLRl j (ri ), for the two transmit and L receive antennas, can be derived as ˆ Pr ri = ri |yl , hl ˆ = Pr ri = 1, ri = 0 |yl , hl ˆ + Pr ri = 0, ri = 1 |yl , hl zI −{ h2l−1 2 + h2l 2 }ˆjl d zI |ˆjl | ≤ 2d a LLRl j (r1 ) = ˆI 2{ h2l−1 2 + h2l 2 }d(d − zjl ) ˆI zjl > 2d = Pr LLRl (ri ) ≥ 0, ri = 0 |yl , hl ˆI −2{ h2l−1 2 + h2l 2 }d(d + zjl ) ˆI zjl < −2d, + Pr LLRl (ri ) < 0, ri = 1 |yl , hl . (18) If LLRl (ri ) ≥ 0, then 0 10 ˆ Pr ri = ri |yl , hl = Pr ri = 0 |yl , hl −1 10 1 = LLRl (ri ) . (19) 1+e −2 10 Average Bit Error Rate If LLRl (ri ) < 0, then ˆ Pr ri = ri |yl , hl = Pr ri = 1 |yl , hl −3 10 L=1, No SC L=2, BLLR based Opt. SC 1 L=2, SLLR based SC = L=2, SNR based SC . (20) L=3, BLLR based Opt. SC 1 + e−LLRl (ri ) −4 10 L=3, SLLR based SC L=3, SNR based SC L=4, BLLR based Opt. SC L=4, SLLR based SC Hence, we have L=4, SNR based SC 1 −5 10 4 6 8 10 12 14 16 18 20 Pr ri = ri |yl , hl ˆ = . (21) Average Received SNR per Branch (dB) 1+ e|LLRl (ri )| Fig. 2. Comparison of various selection combining schemes for 16-QAM ˆ Therefore, to minimize Pr ri = ri |yl , hl , we need to maxi- for 1 Tx antenna and L = 1, 2, 3, 4 Rx antennas – BLLR based optimum SC, SLLR based SC, and SNR based SC. mize the denominator in (21), or, equivalently, maximize the term, |LLRl (ri )|. Hence, by selecting the branch that pro- vides the largest magnitude of LLRl (ri ), we minimize the based SC is preferred over SNR based SC since it achieves BER, Pbi , and hence minimize the average BER. BER performance closer to that of the BLLR based optimum It is noted that different bits in a given symbol may choose SC different antennas, since the largest BLLRs for different bits Figure 3 shows similar comparison for 16-QAM with two may occur on different antennas, and hence will require that transmit antennas using Alamouti code and L receive anten- all the L receive RF chains are present for the bits to choose nas. It is pointed out that the plots corresponding to the SLLR their respective best antennas. This scheme however provides based selection in this ﬁgure has been obtained by deriving the best possible BER performance of M -QAM with selec- the expressions for the symbol LLRs for the two transmit and tion combining, and can serve as a benchmark to compare the L receive antennas case (i.e., by extending derivation in [4] performance of other SC schemes (as illustrated in the next to the 2 Tx antennas case using Alamouti code). From Fig. section). 3, it is observed that, for 2-Tx and L = 4 Rx antennas, the SLLR based SC performance is away from the BLLR based IV. S IMULATION R ESULTS optimum SC performance by 1.1 dB for L = 2, by 1.6 dB for L = 3, and by 1.9 dB for L = 4, at a BER of 10−2 . In this section, we present the simulated BER performance of Similarly, the SNR based SC performance is away from the the BLLR optimum SC scheme derived in the previous sec- optimum SC performance by 1.5 dB for L = 2, by 2.6 for tion in comparison with the performance of other SC schemes L = 3, and by 3.1 for L = 4, at a BER of 10−2 . where the diversity branch selection is done based on maxi- mum SNR (i.e., choose the branch with largest instantaneous We also derived the BLLR expressions for 32-QAM (deriva- SNR) and maximum SLLR (i.e., choose the branch with the tion not given in this paper), and evaluated the BER perfor- largest magnitude of the symbol LLR). The channel gain co- mance of the three SC schemes for 32-QAM. Figure 4 shows efﬁcients hl ’s are taken to be i.i.d complex Gaussian (i.e., fade the BER performance of the three SC schemes for 32-QAM amplitudes are Rayleigh distributed) with zero mean and for 1-Tx and L = 2, 4 Rx antennas. It can be observed that E{||hl ||2 } = 1. Figure 2 shows the simulated average BER for 32-QAM, L = 4, at a BER of 10−2 , the SLLR based SC is performance as function of average SNR per branch for the worse by 1.6 dB and the SNR based SC by 2.5 dB compared following a) BLLR based optimum SC scheme, b) SNR based to the BLLR based optimum SC. SC scheme, and c) SLLR based optimum SC scheme, for 16- QAM with one transmit and L = 1, 2, 3, 4 receive antennas. V. C ONCLUSIONS From Fig. 2, it is observed that, at a BER of 10−2 , the SLLR based SC performance is away from the BLLR based opti- We presented the optimum selection combining (SC) scheme mum SC performance by 0.9 dB for L = 2, by 1.4 dB for for M -QAM which minimize the average bit error rate (BER) L = 3, and by 1.6 dB for L = 4. Likewise, the SNR based on fading channels. We showed that the SC scheme which SC performance is away from the optimum SC performance chooses the diversity branch with the largest magnitude of by 1.4 dB for L = 2, by 2.1 dB for L = 3, and by 2.6 dB for the log-likelihood ratios (LLRs) of the individual bits in the L = 4. Since both the SNR based SC as well as the SLLR QAM symbol minimizes the BER, and hence is optimum. based SC have the same complexity (i.e., only one of the di- It was pointed out that the complexity of this optimum SC versity branches needs to be processed in both cases), SLLR scheme is higher since different bits in a given QAM symbol 0 10 [3] M. K Simon and M. S. Alouini, Digital Communications Over Fading Channels: A Uniﬁed Approach to Performance Analysis, Wiley Series, July 2000. [4] Y. G. Kim and S. W. Kim, “Optimum selection combining for M -ary signals in fading channels”, Proc. IEEE GLOBECOM’2002, November −1 10 2002. [5] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE Jl. Sel. Areas in Commun., vol. 16, no. 8, pp. 1451–1458, October 1998. Average Bit Error Rate −2 [6] R. Pyndiah, A. Picard and A. Glavieux, “Performance of block turbo 10 coded 16-QAM and 64-QAM modulations,” Proc. IEEE GLOBE- 2Tx, 1 Rx, No SC COM’95, pp. 1039–1043, November 1995. 2Tx, 2Rx, BLLR based Opt. SC 2Tx, 2Rx, SLLR based SC [7] A. J. Viterbi, “An intuitive justiﬁcation and a simpliﬁed implementation 2Tx, 2Rx, SNR based SC of the MAP decoder for convolutional codes,” IEEE Jl. Sel. Areas in 2Tx, 3Rx, BLLR based Opt. SC −3 10 2Tx, 3Rx, SLLR based SC Commun., vol. 16, no. 2, pp. 260–264, 1998. 2Tx, 3Rx, SNR based SC 2Tx, 4Rx, BLLR based Opt. SC 2Tx, 4Rx, SLLR based SC 2Tx, 4Rx, SNR based SC −4 10 4 5 6 7 8 9 10 11 12 13 14 Average Received SNR per Branch (dB) Fig. 3. Comparison of various selection combining schemes for 16-QAM for 2 Tx antennas using Alamouti code and L = 1, 2, 3, 4 Rx antennas – BLLR based optimum SC, SLLR based SC, and SNR based SC. 0 10 32−QAM −1 10 Average Bit Error Rate −2 10 L=2, BLLR based Opt. SC −3 L=2, SLLR based SC 10 L=2, SNR based SC L=4, BLLR based Opt. SC L=4, SLLR based SC L=4, SNR based SC −4 10 0 2 4 6 8 10 12 14 16 18 20 Average Received SNR per Branch (dB) Fig. 4. Comparison of various selection combining schemes for 32-QAM for 1 Tx antenna and L = 2, 4 Rx antennas – BLLR based optimum SC, SLLR based SC, and SNR based SC. may select different diversity branches and since the largest LLRs for different bits may occur on different diversity bran- ches. However, this scheme provides the best possible BER performance for M -QAM with selection combining, and can serve as a benchmark to compare the performance of other SC schemes. We presented a BER performance comparison of this optimum SC scheme with other SC schemes where the diversity selection is done based on maximum SNR and maximum symbol LLR. R EFERENCES [1] W. T. Webb and L. Hanzo, Modern Quadrature Amplitude Modulation: Principles and Applications for Fixed and Wireless Channels, IEEE Press, New York, 1994. [2] W. C. Jakes, Microwave Mobile Communications, New York: IEEE Press, 1974.

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selection combining, Rayleigh Fading, IEEE Transactions on Communications, OFDM Systems, channel estimation, Norman C. Beaulieu, Performance Analysis, Rician fading, IEEE Transactions on Wireless Communications, IEEE International Conference

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