Your SlideShare is downloading. ×
New decoding Algorithms for matrix c in the 802.16e WiMAX standard
New decoding Algorithms for matrix c in the 802.16e WiMAX standard
New decoding Algorithms for matrix c in the 802.16e WiMAX standard
New decoding Algorithms for matrix c in the 802.16e WiMAX standard
New decoding Algorithms for matrix c in the 802.16e WiMAX standard
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×
Saving this for later? Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.
Text the download link to your phone
Standard text messaging rates apply

New decoding Algorithms for matrix c in the 802.16e WiMAX standard

70

Published on

For more projects visit @ www.nanocdac.com

For more projects visit @ www.nanocdac.com

Published in: Technology, Business
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
70
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
Downloads
0
Comments
0
Likes
0
Embeds 0
No embeds

Report content
Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
No notes for slide

Transcript

  • 1. New Decoding Algorithms for Matrix C in the802.16e WiMAX StandardYoung Gil Kim, Member, IEEE and Norman C. Beaulieu, Fellow, IEEEAbstract—We examine the decoding of Matrix C in the 802.16eWiMAX standard. An exhaustive search and zero-forcing (ES-ZF) decoder and an exhaustive search and nulling canceling (ES-NC) decoder are proposed for uncoded systems. The computa-tional complexity of Matrix C decoding using the ES-ZF decoder isshown to be the same as the complexity of the ZF decoder for Ma-trix B decoding with twice the number of receive antennas timesthe complexity of the maximum likelihood (ML) decoder for Ma-trix B decoding with twice the number of receive antennas. MatrixC can be implemented in a 2 × 2 multiple-input multiple-output(MIMO) system using the ES-NC decoder with reduced complex-ity compared to ML decoding with no performance loss. For codedsystems, double pruned trees using a zero-forcing (DPT-ZF) algo-rithm or nulling canceling (DPT-NC) algorithm are proposed. TheDPT-NC decoder can be implemented in a 2×2 MIMO coded sys-tem with reduced complexity compared to the Max-Log decodingwith no performance loss.Index Terms—Matrix C, space-time codes, WiMAX standard.I. INTRODUCTIONMobile WiMAX systems are based on the scalable OFDMAmode of IEEE 802.16e-2005 specifications and use a subset ofthe different options. Multiple-input multiple-output (MIMO)techniques based on using multiple antennas at both transmit-ter and receiver can provide spatial diversity, multiplexing gain,and interference suppression, as well as make various trade-offs among them. These techniques have been incorporatedin all of the recently developed wireless communications sys-tem specifications including the IEEE 802.11n standard for lo-cal area networks and the IEEE 802.16e-2005 standard for mo-bile broadband wireless access systems [1]. Regarding MIMOoptions, the WiMAX Forum has specified two mandatory pro-files for use on the downlink. Denoted in the WiMAX stan-dard as Matrix A, one of them is based on the space-time code(STC) proposed by Alamouti for transmit diversity [2]. Thiscode provides perfect second-order diversity when used witha single receive antenna and fourth-order diversity when usedwith two receive antennas. However, it is only half-rate, be-cause it only transmits two symbols using two time slots andtwo transmit antennas. Denoted Matrix B in the WiMAX stan-dard, the other profile is spatial multiplexing (SM), which usestwo transmit antennas to transmit two independent data streams.This scheme is full-rate, but it does not benefit from any diver-Y. G. Kim is with the Department of Electrical and Computer Engineer-ing, University of Seoul, Seoul, 130-743, Korea. (e-mail: ygkim@ieee.org).N. C. Beaulieu is with the iCORE Wireless Communications Labora-tory, University of Alberta, Edmonton, AB, T6G 2V4, Canada. (e-mail:beaulieu@ece.ualberta.ca).sity gain at the transmitter, and at best, it provides second-orderdiversity with two receive antennas.Reports have indicated that the desired quality of service can-not always be achieved using Matrix A or Matrix B. For futureevolutions of the WiMAX standard, it is highly desirable to in-clude a new code combining the respective advantages of theAlamouti code and the SM, while avoiding their drawbacks.Such a code actually already exists in the IEEE 802.16e-2005specifications, where it is referred to as Matrix C [3]. TheMatrix C code is known to be one of the best STCs of size2x2. However, this code has a high decoding complexity whichgrows with the fourth power of the modulation order for maxi-mum likelihood (ML) decoding. The decoding complexity haslikely hindered the adoption of Matrix C, despite its superiorperformance over Matrix A and Matrix B.In this paper, we propose an exhaustive search and zero-forcing (ES-ZF) decoder and an exhaustive search and nullingcanceling (ES-NC) decoder for Matrix C decoding in uncodedsystems. The computational complexity of the ES-ZF decoderfor Matrix C decoding is shown to be the same as the com-plexity of the ZF decoder for Matrix B decoding with twice thenumber of receive antennas multiplied by the complexity of theML decoder for Matrix B decoding with twice the number ofreceive antennas. Similarly, the computational complexity ofthe ES-NC decoder for Matrix C decoding is shown to be thesame as the complexity of the NC decoder for Matrix B decod-ing with twice the number of receive antennas multiplied bythe complexity of the ML decoder for Matrix B decoding withtwice the number of receive antennas. The ES-ZF decoder andthe ES-NC decoder for Matrix C decoding in uncoded systemscan be implemented using existing hardware adopted for MatrixB decoding. Thus, compared with the ML decoder for MatrixC decoding, the ES-ZF decoder and the ES-NC decoder can re-duce not only computational complexity but also chip size forhardware implementation. Using computer simulation, the ES-ZF decoder and the ES-NC decoder are shown to give up 4.2dB and 2.1 dB power gain, respectively, to ML decoding for asingle receive antenna at a target value of bit error probability(BEP) = 10−3for uncoded systems. When two receive anten-nas are used, the ES-ZF decoder and the ES-NC decoder give up1.4 dB and 0 dB, respectively, to ML decoding for BEP = 10−3for uncoded systems. For coded systems, double pruned treesusing the zero-forcing (DPT-ZF) algorithm and double prunedtrees using the nulling canceling (DPT-NC) algorithm are pro-posed. A pruned tree in the DPT-ZF decoder and the DPT-NCdecoder has only one-half of the number of leaf nodes as thetree for the Max-Log decoding of Matrix C, since ZF and NC978-1-4673-1881-5/12/$31.00 ©2012 IEEE
  • 2. algorithms reduce the number of leaf nodes by hard decisionof two symbols among four symbols in Matrix C. For a codedsystem using a 1/2 convolutional code, the DPT-ZF decoderand the DPT-NC decoder are shown to give up 1.7 dB and 0.8dB power gain, respectively, to the Max-Log decoding for asingle receive antenna and target value of bit error probability(BEP) = 10−3. When two receive antennas are used, the DPT-ZF decoder gives up 0.1 dB to the Max-Log decoder for BEP= 10−3.This paper has the following organization. In Section II, thesystem model is described. In Section III, the ES-ZF decoderand the ES-NC decoder for Matrix C decoding in uncoded sys-tems are investigated. In Section IV, the DPT-ZF decoder andthe DPT-NC decoder for Matrix C decoding of coded systemsare examined. Numerical examples are presented in Section V.Conclusions are drawn in Section VI.II. SYSTEM MODELWe consider a MIMO system with two transmit antennas andMr receive antennas. Matrix C in the WiMAX standard is givenby [3]C =1√1 + r2s1 + jr · s4 r · s2 + s3s2 − r · s3 jr · s1 + s4(1)where s1, s2, s3, s4 are M-ary quadrature amplitude modula-tion (QAM) signals with average symbol energy Es, r =−1+√52 , and j =√−1. We will compare the complexity ofMatrix C decoding with that of Matrix B decoding. Matrix B isgiven byB =s1s2. (2)We denote the average bit energy by Eb; it is equal toEs/ log2 M. The received signal is given byR = HC + n (3)where H is the Mr × 2 channel gain matrix whose entries hlmrepresent the gain from the mth transmit antenna to the lth re-ceive antenna. The gains hlm are independent and identicallydistributed (iid) circularly symmetric complex Gaussian ran-dom variables of unit variance. The vector n is a Mr×2 additivewhite Gaussian noise (AWGN) vector with variance N0/2 perdimension. The symbol || · || is used for vector two-norm.III. MATRIX C DECODING IN UNCODED SYSTEMSA. One Receive AntennaIn this subsection, the ES-ZF decoder and the ES-NC de-coder with one receive antenna for uncoded systems are de-scribed. The received signal for the ES-ZF decoder can be writ-ten asR1 =1√1 + r2· (h1s1 + h1jrs4 + h2s2 − h2rs3) + n1 (4a)R2 =1√1 + r2· (h1rs2 + h1s3 + h2jrs1 + h2s4) + n2 (4b)where R1 and R2 are the received signals in the first and thesecond time slot, respectively, and h1 and h2 are the channelgain from the first and the second transmit antennas, respec-tively. We fix the values of s1 and s4 to any symbol in thesignal constellation, so one hasR1 −1√1 + r2(h1s1 + h1jrs4) =1√1 + r2(h2s2 − h2rs3) + n1R2 −1√1 + r2(h2jrs1 + h2s4) =1√1 + r2(h1rs2 + h1s3) + n2.Defining β1 = R1 − 1√1+r2(h1s1 + h1jrs4) and β2 = R2 −1√1+r2(h2jrs1 + h2s4),β1β2=1√1 + r2h2 −h2rh1r h1s2s3+n1n2.(5)Given β1 and β2, eq. (5) can be solved using ZF decoding. Mul-tiplying by the inverse of the matrix ofh2 −h2rh1r h1onboth sides of (5) gives1h1h2√1 + r2h1 h2r−h1r h2β1β2= (6)s2s3+1h1h2√1 + r2h1 h2r−h1r h2n1n2.Symbols s2 and s3 can be detected using (6). We repeat ZFdecoding for every s1 and s4 in the signal constellation. Usingexhaustive search, we find s1 and s4 to minimize||R1 −1√1 + r2(h1s1 + h1jrs4 + h2ˆs2 − h2rˆs3) ||2+||R2 −1√1 + r2(h1rˆs2 + h1ˆs3 + h2jrs1 + h2s4) ||2(7)where ˆs2 and ˆs3 are symbols detected using the ZF decodergiven s1 and s4. Note that (5) can be viewed as a problemof detecting Matrix B with two receive antennas. Eq. (7) canthen be viewed as the distance metric of Matrix B decodingwith two receive antennas. Thus, the complexity of the ES-ZF decoder for Matrix C decoding with one receive antenna,KES−ZF(Mr = 1, C), is the same as the complexity of theZF decoder for Matrix B decoding with two receive anten-nas, KZF(Mr = 2, B), multiplied by the complexity of theML decoder for Matrix B decoding with two receive antennas,KML(Mr = 2, B). That is, one hasKES−ZF(Mr = 1, C) = KZF(Mr = 2, B)·KML(Mr = 2, B).(8)For the ES-NC decoder, we use NC decoding for detectings2 and s3 in (5). If ||h1|| > ||h2||, s3 is detected first. Then,putting the detected ˆs3 into (5), we can detect s2 adopting NCdecoding [5]. Then, we repeat NC decoding for every s1 ands4 in the signal constellation. Symbols s1 and s4 are detected
  • 3. by exhaustive search. For the ES-NC decoder, a similar com-plexity equality is valid as holds for the ES-ZF decoder. Thecomplexity of the ES-NC decoder for Matrix C decoding withone receive antenna, KNCD(Mr = 1, C), is the complexity ofthe NC decoder for Matrix B decoding with two receive an-tennas, KNC(Mr = 2, B), multiplied by the complexity of theML decoder for Matrix B decoding with two receive antennas,KML(Mr = 2, B). That is,KES−NC(Mr = 1, C) = KNC(Mr = 2, B)·KML(Mr = 2, B).(9)B. Two Receive AntennasNow, the ES-ZF decoder and the ES-NC decoder with tworeceive antennas for uncoded systems are described. First, theES-ZF decoder is described. The received signal with two re-ceive antennas is written asR11 R12R21 R22=1√1 + r2×h11(s1 + jrs4) + h12(s2 − rs3)h21(s1 + jrs4) + h22(s2 − rs3)(10)h11(rs2 + s3) + h12(jrs1 + s4)h21(rs2 + s3) + h22(jrs1 + s4)+n11 n12n21 n22where Rlm and nlm are the received signal and the noise signal,respectively, over the lth receive antenna and the mth time slot.Then, fix the values of s1 and s4 to any symbol in the signalconstellation, and one hasR11 − h11(s1+jrs4)√1+r2R12 − h12(jrs1+s4)√1+r2R21 − h21(s1+jrs4)√1+r2R22 − h22(jrs1+s4)√1+r2=1√1 + r2h12(s2 − rs3) h11(rs2 + s3)h22(s2 − rs3) h21(rs2 + s3)+n11 n12n21 n22. (11)Defining β11 = R11 − h11(s1+jrs4)√1+r2, β12 = R12 − h12(jrs1+s4)√1+r2,β21 = R21 − h21(s1+jrs4)√1+r2, and β22 = R22 − h22(jrs1+s4)√1+r2, (11)are written asβ11β12β21β22 =1√1 + r2h12 −h12rh11r h11h22 −h22rh21r h21s2s3+n11n12n21n22 . (12)Eq. (12) can be solved given β11, β12, β21, and β22 using ZFdecoding. We repeat detecting s2 and s3 by ZF decoding for allsymbols s1 and s4 in the signal constellation. Using exhaustivesearch, we detect s1 and s4. Note that (12) can be viewed as de-scribing the problem of detecting Matrix B with four receive an-tennas. Thus, the complexity of the ES-ZF decoder for MatrixC decoding with two receive antennas, KES−ZF(Mr = 2, C),is the complexity of the ZF decoder for Matrix B decodingwith four receive antennas, KZF(Mr = 4, B), multiplied bythe complexity of the ML decoder for Matrix B decoding withfour receive antennas, KML(Mr = 4, B). That is,KES−ZF(Mr = 2, C) = KZF(Mr = 4, B)·KML(Mr = 4, B).(13)For the ES-NC decoder, we adopt NC decoding for detect-ing s2 and s3 in (12). For the ES-NC decoder, a similar com-plexity equality is valid as holds for the ES-ZF decoder. Thecomplexity of the ES-NC decoder for Matrix C decoding withtwo receive antenna, KES−NC(Mr = 2, C), is the complexityof the NC decoder for Matrix B decoding with four receive an-tennas, KNC(Mr = 4, B), multiplied by the complexity of theML decoder for Matrix B decoding with four receive antennas,KML(Mr = 4, B), vizKES−NC(Mr = 2, C) = KNC(Mr = 4, B)·KML(Mr = 4, B).(14)We show that detecting s2 and s3 in Matrix C with Mr re-ceive antennas given s1 and s4 is equivalent to detecting MatrixB with 2Mr receive antennas in the Appendix. Thus, the com-plexity of the ES-ZF decoder for Matrix C decoding with Mrreceive antennas is the complexity of the ZF decoder for Ma-trix B decoding with 2Mr receive antennas, KES−ZF(2Mr, B),multiplied by the complexity of the ML decoder for Matrix Bdecoding with 2Mr receive antennas, KML(2Mr, B). That is,KES−ZF(Mr, C) = KZF(2Mr, B) · KML(2Mr, B). (15)Similarly,KES−NC(Mr, C) = KNC(2Mr, B) · KML(2Mr, B). (16)Since the computational complexity of the ZF decoder andthe NC decoder does not depend on the modulation order M,the complexity of the ES-ZF decoder and the ES-NC decoderis proportional to M2, which originates from the exhaustivesearch operation. However, the complexity of the ML decoderfor Matrix C decoding is proportional to M4. A discussion onthe complexity of the ZF decoder and the NC decoder is foundin [6, p. 8]. Another advantage of the ES-ZF decoder and theES-NC decoder for Matrix C decoding is that they can reusehardware already implemented for Matrix B decoding. For ex-ample, if NC decoder hardware for Matrix B decoding is imple-mented on a chip, then the additional hardware needed for theES-NC decoder for Matrix C decoding is only the ML decoderfor searching M2candidates since the NC decoder hardwarefor Matrix B decoding can be reused. In Table I, the computa-tional complexities of the ML decoder, the ES-ZF decoder, andthe ES-NC decoder are shown. The Matrix C decoding for un-coded systems can be thought as a tree-searching problem. Forthe ML decoder, we have a tree with M4leaf nodes as in Fig. 1(a). For the ES-ZF and ES-NC decoders, we have a pruned treewith only M2leaf nodes as in Fig. 1(b). This fact clarifies whythe computational complexity can be reduced by adopting theES-ZF decoder or the ES-NC decoder.IV. MATRIX C DECODING IN CODED SYSTEMSIn this section, we describe the DPT-ZF decoder and theDPT-NC decoder for coded systems. For coded systems, an
  • 4. error correction code is used as an outer code and Matrix C canbe considered as an inner code. Soft-input for the outer errorcorrection decoder can approximated as [8]L(xj,b) = min{C0j,b|C0j,b∈χ0j,b}||R − HC0j,b||2− min{C1j,b|C1j,b∈χ1j,b}||R − HC1j,b||2(17)where xj,b is the bthbit in the jthsymbol in Matrix C, j =1, 2, 3, 4, b = 1, 2, ..., log2 M, χ0j,b is a set of Matrix C’s thathave zero at the bthbit in the jthsymbol and vice versa. Thesignal-to-noise ratio (SNR) loss of the max-log approximationin (17) is around 0.25 dB over a large range of SNRs [8].We will call this max-log approximation, Max-Log decoding.The Max-Log decoding for coded systems requires computa-tion of (17) for j = 1, 2, 3, 4, b = 1, 2, ..., log2 M. For eachj and b, the number of required comparisons is 2(M4/2 − 1)since the number of possible Matrix C’s that have zero at the bthbit in the jthsymbol is M4/2. Note that Matrix C can be repre-sented as a tree that has M4leaf nodes since Matrix C consistsof four M-ary symbols. Using the ZF or NC algorithm, the treeof Matrix C can be pruned to a tree that has only M2leaf nodessince s2 and s3 can be fixed to certain values as in Section 3.For this pruned tree as in Fig. 1(b), L(xj,b) in (17) for j = 1, 4exists for b = 1, 2, ..., log2 M. However, L(xj,b) in (17) forj = 2, 3 could not exist for some b = 1, 2, ..., log2 M. Thus,we make another pruned tree so that L(xj,b) in (17) for j = 2, 3exists for all b = 1, 2, ..., log2 M. Using these double prunedtrees, L(xj,b)s in (17) are calculated. For each j and b, the num-ber of required comparisons is 2(M2/2−1), which is far fewerthan 2(M4/2 − 1) of the Max-Log decoding. In Table II, thecomputational complexities of the Max-Log decoder, the DPT-ZF decoder, and the DPT-NC decoder are shown. We can seethat the number of comparisons of the DPT-ZF decoder and theDPT-NC decoder is reduced from the order of M4to the orderof M2by pruning the tree in Fig. 1(a).V. NUMERICAL RESULTSIn this section, we present numerical results for the BEP per-formance. The results are obtained by computer simulation.The FFT size is chosen as 1024 which corresponds to a systembandwidth of 10 MHz in the mobile WiMAX OFDMA PHYspecifications [3], [7]. We have used downlink Partial Usage ofSubChannel (PUSC) permutation for a 10 MHz bandwidth asin [7]. Jake’s channel model is used in a Pedestrian B environ-ment with a speed of 3 km/h in [7] in the simulations.In Fig. 2, we show the BEP vs. Eb/N0 for Mr = 1, 2 andquadrature phase shift keying (QPSK) signaling. Observe thatwhen there is one receive antenna, Mr = 1, the ES-ZF decodergives up 4.2 dB in power gain at BEP = 10−3to the ML de-coder in return for its reduced complexity compared to the MLdecoder. However, the ES-NC decoder gives up only 2.1 dBin power gain at BEP = 10−3to the ML decoder when onereceive antenna is used, while offering needed reduced com-plexity. When two receive antennas are used, the ES-ZF de-coder gives up only 1.4 dB to the ML decoder for BEP = 10−3.Significantly, the ES-NC decoder provides virtually the sameperformance as that of the ML decoder, at reduced complexity.This is an important result as it implies that Matrix C can beimplemented in 2 × 2 MIMO systems, greatly improving per-formance over Matrix A and Matrix B, with reduced decodingcomplexity using the ES-NC decoder with essentially no per-formance loss relative to optimal ML decoding.Fig. 3 shows the BEP vs. Eb/N0 for Mr = 1, 2 and QPSKsignaling. A 1/2 convolutional code with constraint length 7that has the generator polynomials 171 and 133 in octal for-mat is used for coded system simulations employing soft-inputViterbi decoding. For one receive antenna, the DPT-ZF decodergives up 1.7 dB in power gain at BEP = 10−3to the Max-Log decoder in return for its reduced complexity compared tothe Max-Log decoder. However, the DPT-NC decoder gives uponly 0.8 dB in power gain at BEP = 10−3to the Max-Log de-coder when one receive antenna is used, while offering neededreduced complexity. When two receive antennas are used, theDPT-ZF decoder gives up only 0.1 dB to the ML decoder forBEP = 10−3. Significantly, the DPT-NC decoder provides vir-tually the same performance as that of the Max-Log decoder, atreduced complexity.VI. CONCLUSIONThe 802.16e 2005 WiMAX Standard specifies a Matrix Cspace-time code that offers the potential of significantly im-proved QoS, however, Matrix C has not yet been adopted owingto the large power requirements for its decoding. In this paper,we have proposed the ES-ZF decoder and the ES-NC decoderfor decoding Matrix C in the WiMAX standard for uncoded sys-tems. It is shown that Matrix C can be implemented in a 2 × 2MIMO system without the cost of ML decoding. The ES-NCdecoder used with Matrix C performs virtually as well as MLdecoding with the advantage of reduced complexity and powerconsumption. Also, the DPT-ZF decoder and the DPT-NC de-coder have been proposed for Matrix C decoding in coded sys-tems. The computational complexity is reduced by pruning thetree by the ZF or NC algorithm as in an uncoded system.REFERENCES[1] S. Sezginer and H. Sari, “Full-rate full-diversity 2×2 space-timecodes for reduced decoding complexity,” IEEE Commun. Lett.,vol. 11, No. 12, pp. 1-3, Dec. 2007.[2] S. M. Alamouti, “A simple transmit diversity technique for wire-less communications,” IEEE J. Sel. Areas Commun., vol. 16, pp.1451-1458, Oct. 1998.[3] IEEE 802.16e-2005: IEEE Standard for Local and MetropolitanArea Networks - Part 16: Air Interface for Fixed and MobileBroadband Wireless Access Systems, Feb. 2006.[4] J.-C. Belfiore, G. Rekaya, and E. Viterbo, “The golden code: a2×2 full rate space-time code with nonvanishing determinants,”IEEE Trans. Inf. Theory, vol. 51, pp. 1432-1436, April 2005.[5] G. B. Giannakis, Z. Liu, X. Ma, and S. Zhou, Space-Time Codingfor Broadband Wireless Communications. Wiley, 2007.[6] T. Kailath, H. Vikalo, and B. Hassibi, “MIMO Receive Algo-rithms,” in Space-Time Wireless Systems: From Array Processingto MIMO Communications, Cambridge University Press, 2005.[7] R. Kobeissi, S. Sezginer, and F. Buda, “Downlink performanceanalysis of full-rate STCs in 2×2 MIMO WiMAX systems,” inProc. IEEE Veh. Tech. Conf., pp. 1-5, April 2009.[8] C. Studer, A. Burg, and H. Bolcskei, “Soft-output sphere decod-ing: algorithms and VLSI implementation,” IEEE J. Sel. AreasCommun., vol. 26, pp. 290-300, Feb. 2008.
  • 5. TABLE ICOMPUTATIONAL COMPLEXITY OF ML DECODING, ES-ZF DECODING,AND ES-NC DECODING FOR UNCODED SYSTEMSML ES-ZF ES-NCNumber of comparisons M45M25M2Number of multiplications 18M4Mr 34M2Mr 40M2MrNumber of additions 11M4Mr 15M2Mr 18M2MrAPPENDIXIn this Appendix, we show that detecting s2 and s3 in MatrixC with Mr receive antennas given s1 and s4 is equivalent todetecting Matrix B with 2Mr receive antennas.For a MIMO system with two transmit antennas and Mr re-ceive antennas, the received signal for Matrix C transmissionin (3) can be written asR1 R2 =1√1 + r2h1h2s1 + jr · s4 r · s2 + s3s2 − r · s3 jr · s1 + s4+ n1 n2 (18)where R1 and R2 are the received signals in the first and thesecond time slot, respectively, h1 and h2 are the channel gainfor the first and the second transmit antennas, respectively, andn1 and n2 are AWGN in the first and the second time slot, re-spectively. The vectors R1, R2, h1, h2, n1, and n2 have sizeMr × 1. From (18),R1 R2 =1√1 + r2(h1(s1 + jr · s4) + h2(s2 − r · s3)h1(r · s2 + s3) + h2(jr · s1 + s4)) + n1 n2=1√1 + r2h1(s1 + jr · s4) h2(jr · s1 + s4)+ h2(s2 − r · s3) h1(r · s2 + s3)+ n1 n2 .Defining β1 = R1 − 1√1+r2h1(s1 + jrs4) and β2 = R2 −1√1+r2h2(jrs1 + s4),β1β2=1√1 + r2h2(s2 − r · s3)h1(r · s2 + s3)+n1n2=1√1 + r2h2 −rh2rh1 h1s2s3+n1n2.(19)Eq. (19) can be viewed as describing the problem of detectingMatrix B with 2Mr receive antennas.(a)SSSS1234(b)SSSS1234Fig. 1. (a) A tree for the ML decoding in uncoded systems and the Max-Logdecoding in coded systems (b) A pruned tree for the ES-ZF and the ES-NCdecoders in uncoded systems and the DPT-ZF and the DPT-NC decoders incoded systems. Note that we assume that the modulation order, M, is two inthis figure for simplicity. However, in WiMAX systems, M should be one of4, 16, 64 [3].0.00010.0010.010.110 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32BEPEb/N0ES-ZFES-NCMLM =1M =2rrFig. 2. The BEP as a function of Eb/N0 for uncoded systems with Mr = 1, 2and QPSK signaling.0.00010.0010.010.110 2 4 6 8 10 12 14 16 18 20 22 24BEPEb/N0DPT-ZFDPT-NCMax-LogM =1M =2rrFig. 3. The BEP as a function of Eb/N0 for coded systems with 1/2 convo-lutional code, Mr = 1, 2, and QPSK signaling.TABLE IICOMPUTATIONAL COMPLEXITY OF MAX-LOG DECODING, DPT-ZFDECODING, AND DPT-NC DECODING FOR CODED SYSTEMSMax-Log DPT-ZF DPT-NCNumber of comparisons M49M29M2Number of multiplications 18M4Mr 68M2Mr 80M2MrNumber of additions 11M4Mr 30M2Mr 36M2Mr

×