On the decoding of matrix C in the WiMAX standard

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  • 1. 3560 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 11, NOVEMBER 2011On the Decoding of Matrix C in the WiMAX StandardYoung Gil Kim, Member, IEEE, and Norman C. Beaulieu, Fellow, IEEEAbstract—The decoding of Matrix C in the WiMAX standardis investigated. We propose an exhaustive search and zero-forcing(ES-ZF) decoder and an exhaustive search and nulling canceling(ES-NC) decoder for uncoded systems. The computational com-plexity of the ES-ZF decoder for Matrix C decoding is shownto be the same as the complexity of the ZF decoder for MatrixB decoding with twice the number of receive antennas timesthe complexity of the maximum likelihood (ML) decoder forMatrix B decoding with twice the number of receive antennas.Matrix C can be implemented in a 2×2 multiple-input multiple-output (MIMO) system using the ES-NC decoder with reducedcomplexity compared to ML decoding with no performance loss.For coded systems, double pruned trees using a zero-forcing(DPT-ZF) algorithm or nulling canceling (DPT-NC) algorithmare proposed. The DPT-NC decoder is shown to be implementedin a 2×2 MIMO coded system with reduced complexity comparedto the Max-Log decoding, with no performance loss, by computersimulations.Index Terms—Matrix C, space-time codes, WiMAX standard.I. INTRODUCTIONMULTIPLE-input multiple-output (MIMO) techniquesbased on using multiple antennas at both transmitterand receiver can provide spatial diversity, multiplexing gain,and interference suppression, as well as make various tradeoffsamong them. These techniques have been incorporated in allof the recently developed wireless communications systemspecifications including the IEEE 802.11n standard for localarea networks and the IEEE 802.16e-2005 standard for mo-bile broadband wireless access systems [1]. Mobile WiMAXsystems are based on the scalable OFDMA mode of IEEE802.16e-2005 specifications and use a subset of the differentoptions. Regarding MIMO options, the WiMAX Forum hasspecified two mandatory profiles for use on the downlink. Oneof them is based on the space-time code (STC) proposed byAlamouti for transmit diversity [2]. Denoted in the WiMAXstandard as Matrix A, this code provides perfect second-order diversity when used with a single receive antenna andfourth-order diversity when used with two receive antennas.However, it is only half-rate, because it only transmits twosymbols using two time slots and two transmit antennas. Theother profile, denoted Matrix B in the WiMAX standard, isspatial multiplexing (SM), which uses two transmit antennasto transmit two independent data streams. This scheme is full-rate, but it does not benefit from any diversity gain at theManuscript received January 26, 2010; revised November 9, 2010; acceptedJanuary 11, 2011. The associate editor coordinating the review of this paperand approving it for publication was A. Yener.Y. G. Kim is with the Department of Electrical and Computer Engineering,University of Seoul, 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).Digital Object Identifier 10.1109/TWC.2011.092011.100110transmitter, and at best, it provides second-order diversity withtwo receive antennas.Experience has shown that the desired quality of servicecannot always be achieved using Matrix A or Matrix B.For future evolutions of the WiMAX standard, it is highlydesirable to include a new code combining the respectiveadvantages of the Alamouti code and the SM, while avoidingtheir drawbacks. Such a code actually already exists in theIEEE 802.16e-2005 specifications, where it is referred to asMatrix C [3]. The Matrix C code is a threaded algebraic space-time (TAST) code [4]- [5], which is known to be one ofthe best STCs of size 2x2. However, this code has a highdecoding complexity which grows with the fourth power ofthe modulation order for maximum likelihood (ML) decoding.This decoding complexity has likely hindered the adoption ofMatrix C by the industry, despite its superior performance overMatrix 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 thecomplexity of the ZF decoder for Matrix B decoding withtwice the number of receive antennas times the complexity ofthe ML decoder for Matrix B decoding with twice the numberof receive antennas. Similarly, the computational complexityof the ES-NC decoder for Matrix C decoding is shown to bethe same as the complexity of the NC decoder for Matrix Bdecoding with twice the number of receive antennas times thecomplexity of the ML decoder for Matrix B decoding withtwice the number of receive antennas. The ES-ZF decoderand the ES-NC decoder for Matrix C decoding in uncodedsystems can be implemented using existing hardware adoptedfor Matrix B decoding. Thus, compared with the ML decoderfor Matrix C decoding, the ES-ZF decoder and the ES-NCdecoder can reduce not only computational complexity butalso chip size for hardware implementation. By computersimulation, the ES-ZF decoder and the ES-NC decoder areshown to give up 4.2 dB and 2.1 dB power gain, respectively,to ML decoding for a single receive antenna and target valueof bit error probability (BEP) = 10−3for uncoded systems.When two receive antennas are used, the ES-ZF decoder andthe ES-NC decoder give up 1.4 dB and 0 dB, respectively,to ML decoding for BEP = 10−3for uncoded systems. Forcoded systems, double pruned trees using the zero-forcing(DPT-ZF) algorithm and double pruned trees using the nullingcanceling (DPT-NC) algorithm are proposed. A pruned tree inthe DPT-ZF decoder and the DPT-NC decoder has only one-half of the number of leaf nodes as the tree for the Max-Logdecoding of Matrix C, since ZF and NC algorithms reducethe number of leaf nodes by hard decision of two symbols1536-1276/11$25.00 c⃝ 2011 IEEE
  • 2. KIM and BEAULIEU: ON THE DECODING OF MATRIX C IN THE WIMAX STANDARD 3561among four symbols in Matrix C. For a coded system using a1/2 convolutional code, the DPT-ZF decoder and the DPT-NCdecoder are shown to give up 1.7 dB and 0.8 dB power gain,respectively, to the Max-Log decoding for a single receiveantenna and target value of bit error probability (BEP) = 10−3.When two receive antennas are used, the DPT-ZF decodergives up 0.1 dB to the Max-Log decoder for BEP = 10−3.This paper consists of six sections. In Section II, wedescribe the system model. In Section III, we investigate theES-ZF decoder and the ES-NC decoder for Matrix C decodingin uncoded systems. In Section IV, the DPT-ZF decoder andthe DPT-NC decoder for Matrix C decoding of coded systemsare investigated. In Section V, numerical results are presented.Some conclusions are drawn in Section VI.II. SYSTEM MODELWe consider a MIMO system with two transmit antennasand 𝑀r receive antennas. Matrix C in the WiMAX standardis given as [3]𝐶 =1√1 + 𝑟2(𝑠1 + 𝑗𝑟 ⋅ 𝑠4 𝑟 ⋅ 𝑠2 + 𝑠3𝑠2 − 𝑟 ⋅ 𝑠3 𝑗𝑟 ⋅ 𝑠1 + 𝑠4)(1)where 𝑠1, 𝑠2, 𝑠3, 𝑠4 are 𝑀-ary quadrature amplitude modu-lation (QAM) signals with average symbol energy 𝐸 𝑠, 𝑟 =−1+√52 , and 𝑗 =√−1. Since we compare the complexity ofMatrix C decoding with that of Matrix B decoding, Matrix Bis given as𝐵 =(𝑠1𝑠2). (2)We will denote the average bit energy by 𝐸 𝑏, which is equalto 𝐸 𝑠/ 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 𝑙threceive antenna. The gains ℎ𝑙𝑚 are independent and identi-cally distributed (iid) circularly symmetric complex Gaussianrandom variables with unit variance. The vector n is a 𝑀r ×2additive white Gaussian noise (AWGN) vector with variance𝑁0/2 per dimension. We use the symbol ∣∣ ⋅ ∣∣ for vector two-norm.III. MATRIX C DECODING FOR UNCODED SYSTEMSA. One Receive AntennaIn this subsection, the ES-ZF decoder and the ES-NCdecoder with one receive antenna for uncoded systems aredescribed. First, the ES-ZF decoder is described. The receivedsignal can be written as𝑅1 =1√1 + 𝑟2⋅ (ℎ1 𝑠1 + ℎ1 𝑗𝑟𝑠4 + ℎ2 𝑠2 − ℎ2 𝑟𝑠3) + 𝑛1 (4a)𝑅2 =1√1 + 𝑟2⋅ (ℎ1 𝑟𝑠2 + ℎ1 𝑠3 + ℎ2 𝑗𝑟𝑠1 + ℎ2 𝑠4) + 𝑛2 (4b)where 𝑅1 and 𝑅2 are the received signal at the first andthe second time slot, respectively, and ℎ1 and ℎ2 are thechannel gain from the first and the second transmit antennas,respectively. Then, we fix the values of 𝑠1 and 𝑠4 to anysymbol in the signal constellation, so one has𝑅1 −1√1 + 𝑟2(ℎ1 𝑠1 + ℎ1 𝑗𝑟𝑠4) =1√1 + 𝑟2(ℎ2 𝑠2 − ℎ2 𝑟𝑠3) + 𝑛1𝑅2 −1√1 + 𝑟2(ℎ2 𝑗𝑟𝑠1 + ℎ2 𝑠4) =1√1 + 𝑟2(ℎ1 𝑟𝑠2 + ℎ1 𝑠3) + 𝑛2.Defining 𝛽1 = 𝑅1 − 1√1+𝑟2(ℎ1 𝑠1 + ℎ1 𝑗𝑟𝑠4) and 𝛽2 = 𝑅2 −1√1+𝑟2(ℎ2 𝑗𝑟𝑠1 + ℎ2 𝑠4),(𝛽1𝛽2)=1√1 + 𝑟2(ℎ2 −ℎ2 𝑟ℎ1 𝑟 ℎ1) (𝑠2𝑠3)+(𝑛1𝑛2).(5)Given 𝛽1 and 𝛽2, (5) can be solved using ZF decoding.Multiplying the inverse of the matrix of(ℎ2 −ℎ2 𝑟ℎ1 𝑟 ℎ1)onboth sides of (5) gives1ℎ1ℎ2√1 + 𝑟2(ℎ1 ℎ2 𝑟−ℎ1 𝑟 ℎ2) (𝛽1𝛽2)= (6)(𝑠2𝑠3)+1ℎ1ℎ2√1 + 𝑟2(ℎ1 ℎ2 𝑟−ℎ1 𝑟 ℎ2) (𝑛1𝑛2).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 that minimize∣∣𝑅1 −1√1 + 𝑟2(ℎ1 𝑠1 + ℎ1 𝑗𝑟𝑠4 + ℎ2ˆ𝑠2 − ℎ2 𝑟ˆ𝑠3) ∣∣2+∣∣𝑅2 −1√1 + 𝑟2(ℎ1 𝑟ˆ𝑠2 + ℎ1ˆ𝑠3 + ℎ2 𝑗𝑟𝑠1 + ℎ2 𝑠4) ∣∣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 antennas,𝐾ZF(𝑀r = 2, 𝐵), times the complexity of the ML decoderfor Matrix B decoding with two receive antennas, 𝐾ML(𝑀r =2, 𝐵). That is,𝐾ES−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 following NCdecoding [6]. Then, we repeat NC decoding for every 𝑠1 and𝑠4 in the signal constellation. Symbols 𝑠1 and 𝑠4 are detectedusing exhaustive search. For the ES-NC decoder, a similarcomplexity equality holds 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 complexityof the NC decoder for Matrix B decoding with two receiveantennas, 𝐾NC(𝑀r = 2, 𝐵), times the complexity of the ML
  • 3. 3562 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 11, NOVEMBER 2011decoder 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 AntennasIn this subsection, the ES-ZF decoder and the ES-NCdecoder with two receive antennas for uncoded systems aredescribed. First, the ES-ZF decoder is described. The receivedsignal with two receive antennas can be written as(𝑅11 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 𝑛22)where 𝑅𝑙𝑚 and 𝑛𝑙𝑚 are the received signal and the noisesignal, respectively, at the 𝑙th receive antenna and the 𝑚thtime slot. Then, we fix the values of 𝑠1 and 𝑠4 to any symbolin the signal constellation, and one has(𝑅11 − ℎ11(𝑠1+𝑗𝑟𝑠4)√1+𝑟2𝑅12 − ℎ12(𝑗𝑟𝑠1+𝑠4)√1+𝑟2𝑅21 − ℎ21(𝑠1+𝑗𝑟𝑠4)√1+𝑟2𝑅22 − ℎ22(𝑗𝑟𝑠1+𝑠4)√1+𝑟2)=1√1 + 𝑟2(ℎ12(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)can be written as⎛⎜⎜⎝𝛽11𝛽12𝛽21𝛽22⎞⎟⎟⎠ =1√1 + 𝑟2⎛⎜⎜⎝ℎ12 −ℎ12 𝑟ℎ11 𝑟 ℎ11ℎ22 −ℎ22 𝑟ℎ21 𝑟 ℎ21⎞⎟⎟⎠(𝑠2𝑠3)+⎛⎜⎜⎝𝑛11𝑛12𝑛21𝑛22⎞⎟⎟⎠ . (12)Given 𝛽11, 𝛽12, 𝛽21, and 𝛽22, (12) can be solved using ZFdecoding as in Section III. We repeat detecting 𝑠2 and 𝑠3by ZF decoding for all symbols 𝑠1 and 𝑠4 in the signalconstellation. Using exhaustive search, we detect 𝑠1 and 𝑠4.Note that (12) can be viewed as a problem of detecting MatrixB with four receive antennas. Thus, the complexity of the ES-ZF decoder for Matrix C decoding with two receive antennas,𝐾ES−ZF(𝑀r = 2, 𝐶), is the complexity of the ZF decoderfor Matrix B decoding with four receive antennas, 𝐾ZF(𝑀r =4, 𝐵), times the complexity of the ML decoder for Matrix Bdecoding with four receive antennas, 𝐾ML(𝑀r = 4, 𝐵). Thatis,𝐾ES−ZF(𝑀r = 2, 𝐶) = 𝐾ZF(𝑀r = 4, 𝐵)⋅𝐾ML(𝑀r = 4, 𝐵).(13)For the ES-NC decoder, we use NC decoding for detecting𝑠2 and 𝑠3 in (12). For the ES-NC decoder, a similar complexityTABLE ICOMPUTATIONAL COMPLEXITY OF ML DECODING, ES-ZF DECODING,AND ES-NC DECODING FOR UNCODED SYSTEMSML ES-ZF ES-NCNumber of comparisons 𝑀4 5𝑀2 5𝑀2Number of multiplications 18𝑀4 𝑀r 34𝑀2 𝑀r 40𝑀2 𝑀rNumber of additions 11𝑀4 𝑀r 15𝑀2 𝑀r 18𝑀2 𝑀requality holds as holds for the ES-ZF decoder. The complexityof the ES-NC decoder for Matrix C decoding with two receiveantennas, 𝐾ES−NC(𝑀r = 2, 𝐶), is the complexity of the NCdecoder for Matrix B decoding with four receive antennas,𝐾NC(𝑀r = 4, 𝐵), times the complexity of the ML decoderfor 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)In the Appendix, we show that detecting 𝑠2 and 𝑠3 inMatrix C with 𝑀r receive antennas given 𝑠1 and 𝑠4 isequivalent to detecting Matrix B with 2𝑀r receive antennas.Thus, the complexity of the ES-ZF decoder for Matrix Cdecoding with 𝑀r receive antennas is the complexity ofthe ZF decoder for Matrix B decoding with 2𝑀r receiveantennas, 𝐾ES−ZF(2𝑀r, 𝐵), times the complexity of the MLdecoder for Matrix B decoding 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 comes from the exhaustivesearch operation. However, the complexity of the ML decoderfor Matrix C decoding is proportional to 𝑀4. A discussionon the complexity of the ZF decoder and the NC decoder isavailable in [7, p. 8]. Another advantage of the ES-ZF decoderand the ES-NC decoder for Matrix C decoding is that theycan reuse the hardware already implemented for Matrix Bdecoding. For example, if NC decoder hardware for Matrix Bdecoding is implemented on a chip, then the additional hard-ware required for the ES-NC decoder for Matrix C decodingis only the ML decoder for searching 𝑀2candidates since theNC decoder hardware for Matrix B decoding can be reused.In Table I, computational complexities of the ML decoder,the ES-ZF decoder, and the ES-NC decoder are shown. TheMatrix C decoding for uncoded systems can be thought as atree-searching problem. For the ML decoder, we have a treewith 𝑀4leaf nodes as in Fig. 1 (a). For the ES-ZF and ES-NCdecoders, we have a pruned tree with only 𝑀2leaf nodes as inFig. 1(b). This fact explains why the computational complexitycan be reduced by adopting the ES-ZF decoder or the ES-NCdecoder.IV. MATRIX C DECODING FOR CODED SYSTEMSIn this section, the DPT-ZF decoder and the DPT-NCdecoder for coded systems are described. For coded systems,
  • 4. KIM and BEAULIEU: ON THE DECODING OF MATRIX C IN THE WIMAX STANDARD 3563(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].an error correction code is used as an outer code and MatrixC can be considered as an inner code. Soft-input for the outererror correction decoder can approximated as [9]𝐿(𝑥 𝑗,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 𝑥 𝑗,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 MatrixC’s that have zero at the 𝑏thbit in the 𝑗thsymbol and viceversa. The signal-to-noise ratio (SNR) loss of the max-logapproximation in (17) is around 0.25 dB over a large range ofSNRs [9]. We will call this max-log approximation, Max-Logdecoding. The Max-Log decoding for coded systems requirescomputation of (17) for 𝑗 = 1, 2, 3, 4, 𝑏 = 1, 2, ..., log2 𝑀.For each 𝑗 and 𝑏, the number of required comparisons is2(𝑀4/2 − 1) since the number of possible Matrix C’s thathave zero at the 𝑏thbit in the 𝑗thsymbol is 𝑀4/2. Notethat Matrix C can be represented as a tree that has 𝑀4leafnodes since Matrix C consists of four 𝑀-ary symbols. Usingthe ZF or NC algorithm, the tree of Matrix C can be prunedto a tree that has only 𝑀2leaf nodes since 𝑠2 and 𝑠3 canbe fixed to certain values as in Section 3. For this prunedtree as in Fig. 1(b), 𝐿(𝑥 𝑗,𝑏) in (17) for 𝑗 = 1, 4 exists for𝑏 = 1, 2, ..., log2 𝑀. However, 𝐿(𝑥 𝑗,𝑏) in (17) for 𝑗 = 2, 3could not exist for some 𝑏 = 1, 2, ..., log2 𝑀. Thus, we makeanother pruned tree so that 𝐿(𝑥 𝑗,𝑏) in (17) for 𝑗 = 2, 3exists for all 𝑏 = 1, 2, ..., log2 𝑀. Using these double prunedtrees, the 𝐿(𝑥 𝑗,𝑏)s in (17) are calculated. For each 𝑗 and 𝑏,the number of required comparisons is 2(𝑀2/2 − 1), whichis much less than 2(𝑀4/2 − 1) of the Max-Log decoding.In Table II, the computational complexities of the Max-Logdecoder, the DPT-ZF decoder, and the DPT-NC decoder areshown. We can see that the number of comparisons of theDPT-ZF decoder and the DPT-NC decoder is reduced fromthe order of 𝑀4to the order of 𝑀2by pruning the tree inFig. 1(a).V. NUMERICAL RESULTSIn this section, we present numerical results for the BEPperformance. The results are obtained by computer simulation.TABLE IICOMPUTATIONAL COMPLEXITY OF MAX-LOG DECODING, DPT-ZFDECODING, AND DPT-NC DECODING FOR CODED SYSTEMSMax-Log DPT-ZF DPT-NCNumber of comparisons 𝑀4 9𝑀2 9𝑀2Number of multiplications 18𝑀4 𝑀r 68𝑀2 𝑀r 80𝑀2 𝑀rNumber of additions 11𝑀4 𝑀r 30𝑀2 𝑀r 36𝑀2 𝑀r0.0010.010.110 2 4 6 8 10 12 14 16 18 20 22 24 26BEPEb/N0ES-ZFES-NCMLM r =2M =1rFig. 2. The BEP as a function of 𝐸 𝑏/𝑁0 for uncoded systems with 𝑀r =1, 2 and QPSK signaling.The FFT size is chosen as 1024 which corresponds to a systembandwidth of 10 MHz in the mobile WiMAX OFDMA PHYspecifications [3], [8]. We have used downlink Partial Usageof SubChannel (PUSC) permutation for a 10 MHz bandwidthas in [8]. Jake’s channel model is used in a Pedestrian Benvironment with a speed of 3 km/h [8] in the simulations.In Fig. 2, we show the BEP vs. 𝐸 𝑏/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 to the ML decoder in returnfor its reduced complexity compared to the ML decoder forBEP = 10−3. However, the ES-NC decoder gives up only2.1 dB in power gain to the ML decoder for BEP = 10−3when one receive antenna is used, while offering worthwhilereduced complexity. When two receive antennas are used, theES-ZF decoder gives up only 1.4 dB to the ML decoderfor BEP = 10−3. Significantly, the ES-NC decoder providesvirtually the same performance as that of the ML decoder, atreduced complexity. This is an important result as it impliesthat Matrix C can be implemented in 2 × 2 MIMO systems,greatly improving performance over Matrix A and Matrix B,with reduced decoding complexity using the ES-NC decoderwith no performance loss relative to optimal ML decoding.In Fig. 3, we show the BEP vs. 𝐸 𝑏/𝑁0 for 𝑀r = 1, 2 andQPSK signaling. A 1/2 convolutional code with a constraintlength 7 that has the generator polynomials 171 and 133 inoctal format is used for coded system simulations with soft-input Viterbi decoding. For one receive antenna, the DPT-ZFdecoder gives up 1.7 dB in power gain to the Max-Log decoderin return for its reduced complexity compared to the Max-Log decoder for BEP = 10−3. However, the DPT-NC decodergives up only 0.8 dB in power gain to the Max-Log decoder
  • 5. 3564 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 11, NOVEMBER 20110.00010.0010.010.110 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 𝐸 𝑏/𝑁0 for coded systems with 1/2convolutional code, 𝑀r = 1, 2, and QPSK signaling.for BEP = 10−3when one receive antenna is used, whileoffering worthwhile reduced complexity. When two receiveantennas are used, the DPT-ZF decoder gives up only 0.1 dBto the ML decoder for BEP = 10−3. Significantly, the DPT-NC decoder provides virtually the same performance as thatof the Max-Log decoder, at reduced complexity.VI. CONCLUSIONIn this paper, we have proposed the ES-ZF decoder andthe ES-NC decoder for decoding Matrix C in the WiMAXstandard for uncoded systems. It is shown that Matrix C canbe implemented in a 2 × 2 MIMO system with its superiorperformance without the cost of ML decoding. The ES-NCdecoder used with Matrix C performs virtually as well asML decoding with the advantage of reduced complexity andpower consumption. Also, the DPT-ZF decoder and the DPT-NC decoder have been proposed for Matrix C decoding incoded systems. The computational complexity is reduced bypruning the tree by the ZF or NC algorithm as in an uncodedsystem. Combining the sphere decoding algorithm with theDPT-ZF or the DPT-NC algorithms could be an interestingtopic for future research.APPENDIXIn 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 𝑀rreceive antennas, the received signal for Matrix C transmissionin (3) can be written as(R1 R2)=1√1 + 𝑟2(h1h2)(𝑠1 + 𝑗𝑟 ⋅ 𝑠4 𝑟 ⋅ 𝑠2 + 𝑠3𝑠2 − 𝑟 ⋅ 𝑠3 𝑗𝑟 ⋅ 𝑠1 + 𝑠4)+(n1 n2)(18)where R1 and R2 are the received signal at 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 at the first and the second time slot,respectively. The vectors R1, R2, h1, h2, n1, and n2 havesize 𝑀r × 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 + 𝑟2{(h1(𝑠1 + 𝑗𝑟 ⋅ 𝑠4) h2(𝑗𝑟 ⋅ 𝑠1 + 𝑠4))+(h2(𝑠2 − 𝑟 ⋅ 𝑠3) h1(𝑟 ⋅ 𝑠2 + 𝑠3))}+(n1 n2).Then,(R1 R2)−1√1 + 𝑟2⋅(h1(𝑠1 + 𝑗𝑟 ⋅ 𝑠4) h2(𝑗𝑟 ⋅ 𝑠1 + 𝑠4))=1√1 + 𝑟2(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),(𝜷1 𝜷2)=1√1 + 𝑟2(h2(𝑠2 − 𝑟 ⋅ 𝑠3) h1(𝑟 ⋅ 𝑠2 + 𝑠3))+(n1 n2).Then,(𝜷1𝜷2)=1√1 + 𝑟2(h2(𝑠2 − 𝑟 ⋅ 𝑠3)h1(𝑟 ⋅ 𝑠2 + 𝑠3))+(n1n2)=1√1 + 𝑟2(h2 −𝑟h2𝑟h1 h1) (𝑠2𝑠3)+(n1n2).(19)Eq. (19) can be viewed as as a problem of detecting MatrixB with 2𝑀r receive antennas.REFERENCES[1] S. Sezginer and H. Sari, “Full-rate full-diversity 2×2 space-time codesfor 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 wirelesscommunications,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1451-1458,Oct. 1998.[3] IEEE 802.16e-2005: IEEE Standard for Local and Metropolitan AreaNetworks - Part 16: Air Interface for Fixed and Mobile BroadbandWireless Access Systems - Amendment 2: Physical Layer and MediumAccess Control Layers for Combined Fixed and Mobile Operation inLicensed Bands, Feb. 2006.[4] M. O. Damen, H. E. Gamal, and N. C. Beaulieu, “Linear threadedalgebraic space-time constellations,” IEEE Trans. Inf. Theory, vol. 49,no. 10, pp. 2372-2388, Oct. 2003.[5] H. E. Gamal and M. O. Damen, “Universal space-time coding,” IEEETrans. Inf. Theory, vol. 49, no. 5, pp. 1097-1119, May. 2003.[6] G. B. Giannakis, Z. Liu, X. Ma, and S. Zhou, Space-Time Coding forBroadband Wireless Communications. Wiley, 2007.[7] T. Kailath, H. Vikalo, and B. Hassibi, “MIMO Receive Algorithms,”in Space-Time Wireless Systems: From Array Processing to MIMOCommunications, editors: H. Bolcskei, D. Gesbert, C. Papadias, andA. J. van der Veen. Cambridge University Press, 2005.[8] R. Kobeissi, S. Sezginer, and F. Buda, “Downlink performance analysisof full-rate STCs in 2×2 MIMO WiMAX systems,” in Proc. IEEE Veh.Tech. Conf., pp. 1-5, Apr. 2009.[9] C. Studer, A. Burg, and H. Bolcskei, “Soft-output sphere decoding:algorithms and VLSI implementation,” IEEE J. Sel. Areas Commun.,vol. 26, pp. 290-300, Feb. 2008.