A Low-Complexity Soft QAM De-Mapper Based on First-Order Linear Approximation

Transcription

1 5 IEEE 5th Internatonal Symposum on Personal, Indoor and Moble Rado Communcatons - (PIMRC): Fundamentals and A Low-Complexty Soft QAM De-Mapper Based on Frst-Order Lnear Approxmaton Yanbn Yao, Yongtao Su, Jngln Sh and Jangnan Ln Wreless Communcaton echnology Research Center, Insttute of Computng echnology, Chnese Academy of Scences, Bejng, Chna Bejng Key Laboratory of Moble Computng and Pervasve Devce, Bejng, Chna Unversty of Chnese Academy of Scences, Bejng, Chna Emal: {yaoyanbn, ysu, sjl, lnjangnan}@ct.ac.cn Abstract Hgh-order QAM modulaton schemes are recommended n varous communcaton systems. However, soft demodulatng a hgh-order QAM constellaton s complcated when applyng tradtonal methods. In ths paper, a novel de-mapper s proposed based on the fact that the clpped LLR can also offer suffcent soft-nformaton, meanwhle the theoretcal LLR curves are approxmately lnear n ths clpped range. Hence a frst-order lnear functon s desgned to approxmate the theoretcal value. Smulaton results and complexty analyss demonstrate that the proposed de-mapper outperforms tradtonal approaches n terms of BER performance, and also keeps low complexty. I. INRODUCION Hgh-order modulaton schemes, capable of offerng hgh bandwdth effcency, have been wdely employed n many communcaton systems such as Evolved Hgh-Speed Packet Access (HSPA+), Long erm Evoluton (LE), IEEE 8., etc. By vrtue of ts low symbol error rate (SER) and ease of soft de-mappng [], Quadrature Ampltude Modulaton (QAM) scheme, Gray-labeled n partcular, has attracted wdespread attenton. For nstance, 6QAM has been appled n the above-mentoned wreless communcaton standards. In addton, 56QAM, provdng much hgher data throughput than the former, s ntroduced n the latest Wreless Local Area Network (WLAN) protocol IEEE 8.ac []. When t comes to de-mappng QAM symbols, hgh-order schemes wll nduce consderable computatonal complexty. Prevous channel decoders use a hard-decson strategy, where de-mappers only need to provde hard symbol-to-bt results va selectng the constellaton pont closest to the receved symbol. Wth the emergence of soft-decson decoders, a hghcomplexty soft de-mapper [3] s requred to provde Log- Lkelhood Rato (LLR) nformaton, where numerous logarthmc and onental operatons are nvolved. Conventonally, reled on a max functon, these complex operatons can be elmnated. However, as ths well known MAX-Log-MAP demapper (For convenence, we call t MAX de-mapper) needs to compare m (m stands for the bts mapped onto a symbol) constellaton ponts, the complexty s stll respectable when encounter a hgh-order scheme. In ths paper, we focus on the smplfed Gray-labeled QAM de-mapper. Dfferent de-mappers have been proposed to amelorate MAX de-mapper, among whch the approach n [, 5] reduces comparson operatons by smplfyng the max functon n the MAX de-mapper. A further smplfcaton s depcted n [6]. However, the feasblty of ths smplfcaton s doubtful snce the calculated LLRs can only approxmate to the theoretcal values perfectly on certan nose condton, consequently leadng to a BER performance degradaton when nose condton changes. A smlar approach can be found n [7], where a boundary called Decson-hreshold (DH) lne s appled to partton the symbols, and LLR s calculated based on the dstance between the receved symbol and the DH lnes. Lkewse, [8, 9] use almost the same strategy. Most recently, a unversal low-complexty de-mapper was proposed n []. By fndng out two symbols x and x from the constellaton symbol set to replace the symbols assocated wth the current bt beng and n the sets X () and X () respectvely, the complexty can be reduced consderably to O(m). Meanwhle, the BER performance attans almost the same as that of the MAX de-mapper. Generally, the aforementoned optmzaton strateges are based on the MAX de-mapper, whch result n the BER performance smlar to the MAX de-mapper. Furthermore, most of them gnore the ssue that channel decoders don t requre accurate LLR nformaton. It has been demonstrated n many lteratures such as [, ] that decoders can stll acheve almost the same BER performance wth clpped LLR rangng from - to. More mportantly, the theoretcal LLR curves n ths range are approxmately lnear. Inspred by ths result, we propose a novel de-mapper by desgnng a frstorder lnear LLR functon to approxmate the theoretcal value n ths clpped range. he result valdates that the proposed de-mapper obtans nearly the same BER performance as the theoretcal Log-MAP de-mapper, whch outperforms the MAX de-mapper and other modfcatons. What s more, the complexty stll remans on the order of O(m). A. System Model II. BACKGROUND Consderng a MIMO-OFDM system y = Hs + n () where y and s both are N vectors. N specfes the number of recevng and transmttng antennas. H s a N N channel frequency response matrx. And n s a N nose vector, the /5/$3. 5 IEEE

2 5 IEEE 5th Internatonal Symposum on Personal, Indoor and Moble Rado Communcatons - (PIMRC): Fundamentals and entres of whch fulfl Gauss dstrbuton. Furthermore, each entry of s s taken from a fnte alphabet set S defned by a m -ary modulaton scheme, whch contans m symbols attaned through groupng m coded bts (b, b,, b m ) and mappng them onto the constellaton ponts. As for lnear equalzaton, there s a lnear flter matrx W W y = (W H)s + W n = Gs + W n, where G W H. Snce the purpose of equalzaton s to elmnate the nterference between the antennas, the matrx G s smlar to a dagonal matrx. Defne ỹ W y and ñ W n. It s clearly that each entry of ñ fulfls Gauss dstrbuton. he n th entry of ỹ can be llustrated as ỹ(n) = g(n, )s() + + g(n, n)s(n) + + g(n, N )s(n ) + ñ(n) = g(n, n)s(n) + ñ(n) + g(n, )s() + + g(n, N )s(n ) }{{} the resdual nterference = g(n, n)s(n) + n(n), () where n(n) ñ(n)+g(n, )s()+ +g(n, N )s(n ). It s readly recognzed that n(n) s stll Gauss, thus r(n) = s(n) + w(n) (3) where r(n) ỹ(n) g(n,n) and w(n) n(n) g(n,n), w(n) represents a Gauss varance wth the mean of zero and the varance of σ ( σ per dmenson). It should be noted that the proposed de-mapper s sut for lnear equalzaton. For those nonlnear equalzers, the soft nformaton s attaned durng the equalzaton process. See [3, ] for example. B. LLR Defnton and the Log-MAP De-Mapper In the followng, as every symbol r(n) or s(n) undergoes the same process, ther ndexes are omtted for smplcty. LLR s defned as the logarthmc functon of the rato between the probabltes of current bt beng and,.e. LLR(b ) log p(b = r) p(b = r). () Wth the ad of Bayes Formula [6], Equ. () can be further ressed as LLR(b ) = log p(b = r) p(b = r) = log p(s = s k r) p(s = s k r) p(r s = s = log k ) p(r s = s k ) = log [ r s k σ ] [ r s k σ ] (5) where the set S and the set S contan the symbols wth the th bt b beng and, respectvely. Note that S S = S. LLR b LLR b LLR b LLR b LLR b LLR b wth SNR=5dB 5 5 wth SNR=dB 5 5 wth SNR=5dB 5 5 LLR b LLR b LLR b Fg.. he theoretcal LLR functons for 6QAM under dfferent SNR. Note that the range of nterest s a) [ 8, 8] for the bt b, b) [, 8] for b because of an axs of symmetry =, c) [, ] for b because of three axes of symmetry =, = and =. C. he MAX-Log-MAP De-Mapper he calculaton of onental and logarthmc functons s extremely dffcult n Equ. (5), especally when ( beng put ) nto practce. After applyng the well-known log j exj max x j equaton [5], the MAX-Log-MAP de-mapper s acqured LLR(b ) = [ ] σ max ( r s k ) max ( r s k ) (6) As has been mentoned before, the MAX-Log-MAP demapper gets rd of the logarthmc and onental operatons. he approxmaton s relatvely accurate when t works n a hgh sgnal-to-nose rato (SNR). However, when SNR condton gets worse, the approxmaton turns nto naccuracy. III. A FIRS-ORDER LINEAR LLR APPROXIMAION DE-MAPPER Besdes the approxmaton accuracy problem, the complexty of the MAX de-mapper s another ssue whch prevents t from mplementng nto a hgh-order modulaton system. Fortunately, t has been demonstrated n many lteratures (such as [, ]) that the usage of clpped LLR leads to lttle BER performance degradaton. On the other hand, the theoretcal LLR functons of dfferent bts n ths clpped LLR range [, ] are approxmately lnear (See Fg. ), whch motvates us to substtute frst-order lnear functons for them. A. he Proposed De-mapper For the purpose of clarty and smplcty, we use the 6QAM scheme appled n LE [6] as an example to llustrate our proposed de-mapper n the further analyss. It should be noted that the factor for normalzng constellaton ponts s omtted for smplcty. Moreover, as the In-Phase and the Quadrature

3 5 IEEE 5th Internatonal Symposum on Personal, Indoor and Moble Rado Communcatons - (PIMRC): Fundamentals and ABLE I HE PROPOSED DE-MAPPER FOR GRAY-LABELED m -QAM INPU: the receved symbols r(n), n =,,, N the power of the nose σ OUPU: the LLR for each bt for, m/ do α LU (σ ) λ α /σ α : the correcton factor for n, N do β I (n), β Q r Q (n) for, m/ do LLR(b +,n ) λ β I, LLR(b +,n ) λ β Q β I β I m/, β Q β Q m/ part of the symbols n a Gray-labeled 6QAM constellaton are modulated ndependently, we only pay attenton to the In- Phase part, whle the analyss of the Quadrature part s smlar. As for 6QAM scheme, every symbol can be represented by (b, b,, b 5 ), where (b, b, b ) and (b, b 3, b 5 ) defne the In-Phase and the Quadrature part respectvely. ) he bt b As shown n the left three sub-fgures of Fg., the theoretcal LLR functon for the bt b s approxmately lnear n the range of nterest, whch can be substtuted by a frst-order lnear functon.e. LLR(b ) = α σ (7) where α s the correcton factor for calculatng the LLR of current bt b, and α σ defnes the slope of the desgned frst-order lnear functon. ) he bt b As for the bt b, the theoretcal LLR curve s not lnear. However, snce the curve s symmetrcal wth the axs =, when applyng the transformaton β = the frst-order lnear functon can also be mplemented. Hence LLR(b ) = α σ β (8) where α s the correcton factor for b, analogously. 3) he bt b Fnally, the same as the calculaton of b, the curve s symmetrcal wth the axs = agan after applyng transformaton n step ). herefore the monotonc LLR curve can be obtaned by β = β whch can be approxmated by the thrd lnear functon LLR(b ) = α σ β (9) Of partcular note s that the theoretcal curves of LLRs vary on dfferent nose values. hus, LookUp-ables (LUs) should be establshed for storng the correcton factors under Fttng Mean Square Error Fttng Mean Square Error (a) Contnuous α LLR of b LLR of b LLR of b Sgnal to Nose Rato (db) (b) Dscrete α wth an SNR ntervel of db LLR of b LLR of b LLR of b Sgnal to Nose Rato (db) Fg.. Fttng MSE between the theoretcal LLR and the approxmated LLR for dfferent bts n terms of the 6QAM constellaton. (a) under contnuous α; (b) under dscrete α wth an SNR nterval of db dfferent nose condton. he ntegral verson of the proposed de-mapper s concluded n able I. B. How to Acqure the Correcton Factors he crux of the proposed de-mapper les n how to obtan and establsh the LUs for dfferent correcton factors. Frst, we analyze the acquston of correcton factor under certan power nose σ. Defne two vectors a [a()] and b [b()] ( =,, P ), where the elements a() and b() are attaned by samplng the theoretcal LLR functon LLR(r) = log [ r s k σ ] [ r s k σ ] () and the frst-order lnear LLR functon LLR(r) = r () σ at equal nterval, respectvely. And P s the length of two vectors, whch s determned by samplng nterval. he smaller the nterval s, the larger the value of P wll be, and the more accurate value of α we wll get. Note that the range of LLR(r) should be n the clpped range namely, [, ]. After ths modelng, the orgnal problem turns nto a curve fttng problem,.e. ˆα = arg mn α a αb. () After some mathematcal operatons, ˆα turns out to be ˆα = a, b / b (3) For certan power of nose σ (or SNR wth a converson formula SNR = log P s σ, where P s stands for the averaged sgnal power), repeat the above steps, fnally we can establsh a LU wth an argument of σ. However, snce the contnuous value of σ wll result n a large scale of LU, t s mpractcal to desgn a correcton factor for every contnuous value of σ. 3

4 Bt Error Rate Bt Error Rate 5 IEEE 5th Internatonal Symposum on Personal, Indoor and Moble Rado Communcatons - (PIMRC): Fundamentals and - - MIMO, heory MIMO, MAX MIMO, In [6] MIMO, In [] MIMO, Proposed AWGN, heory AWGN, MAX AWGN, In [6] AWGN, In [] AWGN, Proposed - - MIMO, heory MIMO, MAX MIMO, In [6] MIMO, In [] MIMO, Proposed AWGN, heory AWGN, MAX AWGN, In [6] AWGN, In [] AWGN, Proposed Sgnal to Nose Rato (db) Sgnal to Nose Rato (db) Fg. 3. BER performance comparson wth CC Fg.. BER performance comparson wth C Actually, the change rate of α s mnor, whch nspres us to use the same α n a small range of σ. Fg. shows the fttng Mean Square Error (MSE) between the theoretcal LLR and the approxmated LLR for dfferent bts. he top and bottom subfgures llustrate the curves about the fttng MSE versus SNR, whle the former and the latter uses the contnuous α and the dscrete α wth an SNR nterval of db, respectvely. Comparng two subfgures, t s practcal to apply one correcton factor α to replace the contnuous correcton factor needed n an SNR range of db. IV. SIMULAION RESUL In ths secton, the BER performance of the proposed demapper s evaluated by dfferent smulaton scenaros. he smulaton parameters are concluded as follows. Constellaton: Glay-Labeled 56QAM. De-mapper: the Log-map de-mapper (heory), the MAX- Log-MAP de-mapper (MAX), the de-mappers proposed n [6, ] and the proposed de-mapper (Proposed). Channel Codng: a (3,, 7) convolutonal code (CC) wth the generators beng (33), (7) and (65) respectvely, and a urbo code (C) specfed n [7] where extrnsc scalng MAX-Log-MAP urbo decodng algorthm s used to decode soft bts and the teraton tme s set to 8 to ensure convergence. Block length: 6 for CC and 5 for C. Channel: AWGN and MIMO multpath fadng channel. When applyng MIMO channel, LMMSE decoder s carred out to equalze the MIMO channel. From Fg. 3 and, t s readly recognzed that the curves of the Log-MAP de-mapper and the proposed de-mapper are almost overlapped over both AWGN and MIMO channels, whch confrms that our approach has a good approxmaton accuracy. Besdes, t shows that the proposed approach acheves a better BER performance than the MAX de-mapper and other modfcatons. In partcular, at the BER of n Fg., the performance gap between the proposed de-mapper and the approach n [6] s nearly db over MIMO channel. V. COMPLEXIY ANALYSIS A. Basc Operaton Comparson In ths secton, the complexty of the proposed de-mapper s compared wth that of the prevous approaches n terms of basc operatons such as addton (ADD), multplcaton (MUL), acquston of absolute value (ABS) and comparson (CMP). he data s counted by de-mappng a symbol for a Gray-labeled m -QAM scheme (m s even). What s more, as the proposed de-mapper requres an extra memory for storng the correcton factors, the memory occupaton (MEM) s also added for comparson. he result s shown n able II. From the able II, t can be learnt that the complexty of the proposed de-mapper s on the order of O(m) whch means that, the proposed de-mapper can reduce the total operatons by 8.5% for 6QAM compared to the MAX de-mapper. In addton, the proposed de-mapper needs an extra memory wth the sze of mr Bytes, where R stands for the SNR range sze we want to cover, and represents the SNR nterval. As there are m LUs need to be stored, the total number of the correcton factors s R m. Wth each value occupyng Bytes, the fnal result s mr. Partcularly, f the desred SNR range s [, ] and SNR nterval s.5db, then R = 3 and =.5. For 6QAM and 56QAM, the occuped memores are and 36 Bytes respectvely. Compared to those O(m)- complexty de-mappers, t won t become a burden after puttng the proposed de-mapper nto practce. B. DSP Cycles Comparson Moreover, snce the actual mplementaton results vary on dfferent DSP platforms, we also mplement dfferent demappers on two LE-target DSPs. One s the MAOU ASP (Applcaton Specfed Processor) proposed n [8, 9], whch s equpped wth mxed SIMD (Sngle Instructon Multple

5 5 IEEE 5th Internatonal Symposum on Personal, Indoor and Moble Rado Communcatons - (PIMRC): Fundamentals and ABLE II BASIC OPERAION COMPARISON OF DIFFEREN DE-MAPPERS MAX In [5] In [6] In [] Proposed ADD m + + m m m 3m m MUL m + + m m m m m ABS m m m CMP m m + m MEM (Byte) Data) and VLIW (Very Long Instructon Word) archtecture, amng at offerng a hgh volume parallel data processng n LE related applcatons. And the other s I s hgh performance CI668 DSP [] that s able to support a MIMO soluton of 3 Mbps downlnk and 5 Mbps uplnk. Besdes, all of the de-mappers are mplemented n assembly language on MAOU ASP, whle beng realzed n C language on CI668. he results are counted by demodulatng symbols. As lsted n able III, the clock cycles executed by the proposed de-mapper on MAOU ASP s 67, whch s only.6% of the MAX s. Smlar to the MAOU ASP platform, when beng mplemented on the CI668 DSP, the number of clock cycles of the proposed de-mapper s less than that of the MAX de-mapper. hough the executve outcome ndcates that the proposed de-mapper attans almost the same results wth the O(m)-complexty de-mappers, the BER performance of the proposed de-mapper outperforms those de-mappers as has been llustrated n last secton. VI. CONCLUSION In ths paper, a frst-order lnear LLR approxmaton demapper s proposed for demodulatng Gray-labeled QAM, hgh-order n partcular, where a lnear functon s desgned to approxmate the complcated theoretcal LLR functons. he smulaton result and complexty analyss llustrate that the complexty of the proposed de-mapper s only on the order of O(m) for a m -QAM scheme, whle also keeps nearly the same BER performance wth the theoretcal Log-MAP demapper. ACKNOWLEDGMEN hs work was funded by Natonal Hgh echnology Research and Development Program of Chna (863 Program: AAA75), Natonal Natural Scence Foundaton for Dstngushed Young Scholars of Chna (63) and Bejng Natural Scence Foundaton Major Project (63). REFERENCES [] J. G. Proaks, Dgtal Communcatons (th Edton). McGraw- Hll Hgher Educaton,, ch. 5. [] L. Verma, M. Fakharzadeh, and S. Cho, Wf on sterods: 8.AC and 8.AD, IEEE Wreless Commun., vol., no. 6, pp. 3 35, 3. [3] Y. Zhou and J. Wang, Downlnk transmsson of broadband OFCDM systems-part IV: soft decson, Selected Areas n mr ABLE III CLOCK CYCLES COMPARISON ON WO DSP PLAFORMS MAX In [5] In [6] In [] Proposed MAOU ASP I s CI Commun., IEEE Journal on, vol., no. 6, pp. 8, 6. [] M. Surendra Raju, A. Ramesh, and A. Chockalngam, BER analyss of QAM wth transmt dversty n Raylegh fadng channels, n proc. of IEEE GLOBECOM, vol.. IEEE, 3, pp [5] M. S. Raju, R. Annavajjala, and A. Chockalngam, BER analyss of QAM on fadng channels wth transmt dversty, IEEE rans. Wreless Commun., vol. 5, no. 3, pp. 8 86, 6. [6] F. osato and P. Bsagla, Smplfed soft-output demapper for bnary nterleaved COFDM wth applcaton to HIPERLAN/, n proc. of IEEE ICC, vol.. IEEE,, pp [7] S. Ryoo, S. Km, and S. P. Lee, Effcent soft demappng method for hgh order modulaton schemes, n proc. of CIC, 3. [8] K. Hyun and D. Yoon, Bt metrc generaton for Gray coded QAM sgnals, n IEE Proc. Commun., vol. 5, no. 6. IE, 5, pp [9] C.-W. Chang, P.-N. Chen, and Y. S. Han, A systematc bt-wse decomposton of M-ary symbol metrc, IEEE rans. Wreless Commun., vol. 5, no., pp. 7 75, 6. [] Q. Wang, Q. Xe, Z. Wang, S. Chen, and L. Hanzo, A unversal low-complexty symbol-to-bt soft-demapper, IEEE rans. Vehcular echnology, vol. 63, no., pp. 9 3,. [] C. Studer, C. Benkeser, S. Belfant, and Q. Huang, Desgn and mplementaton of a parallel turbo-decoder ASIC for 3GPP- LE, IEEE Journal of Sold-State Crcuts, vol. 6, no., pp. 8 7,. [] P. J. Black and. H.-Y. Meng, A -Gb/s, four-state, sldng block Vterb decoder, IEEE Journal of Sold-State Crcuts, vol. 3, no. 6, pp , 997. [3] B. Hochwald and S. ten Brnk, Achevng near-capacty on a multple-antenna channel, Commun., IEEE rans. on, vol. 5, no. 3, pp , 3. [] K.-H. M. N. S. M. Hguch, K., Lkelhood functon for QRM-MLD sutable for soft-decson turbo decodng and ts performance for OFCDM MIMO multplexng n multpath fadng channel, n proc. of IEEE PIMRC,, pp. 8 Vol.. [5] V.-E. H. P. Robertson, P., A comparson of optmal and suboptmal map decodng algorthms operatng n the log doman, n proc. of IEEE ICC, 995, pp [6] 3GPP echncal Specfcaton 36. v8.9. Release 8, Evolved Unversal errestral Rado Access (E-URA); Physcal Channels and Modulaton, 3GPP,. [7] Physcal Layer Standard for cdma Spread Spectrum Systems, 3GPP,. [8] Z. Zyuan,. Shan, S. Yongtao, H. Juan, S. Gang, and S. Jngln, A GOPS ASP based baseband processor for wreless communcaton, n proc. of DAE, March 3, pp.. [9]. Shan, Z. Zyuan, and S. Yongtao, System-level desgn methodology enablng fast development of baseband MP-SoC for G small cell base staton, n proc. of DAE, March, pp. 6. [] Z. Ln and G. Wood, MS3CI668 - I s hgh-performance LE physcal layer soluton, exas Instruments,. 5