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About the Authors

Lin Bai received the B.Sc. degree in Electronic and Information Engineering fromHuazhong University of Science and Technology, Wuhan, China, in 2004, the M.Sc.(with distinction) degree in communication systems from the University of Wales,Swansea, UK, in 2007, and the Ph.D. degree in Advanced Telecommunicationsfrom the School of Engineering, Swansea University, UK, in 2010. He is currentlywith the School of Electronic and Information Engineering, Beihang University(Beijing University of Aeronautics and Astronautics, BUAA), Beijing, China, asan Associate Professor.

His research interests include signal processing of wireless communications,particularly multiple-input–multiple-output systems, array/smart antenna, andlattice-based approaches.

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Jinho Choi (Senior Member of IEEE) was born in Seoul, Korea. He received theB.E. (magna cum laude) degree in electronics engineering from Sogang University,Seoul, in 1989 and the M.S.E. and Ph.D. degrees in electrical engineering from theKorea Advanced Institute of Science and Technology, Daejeon, Korea, in 1991 and1994, respectively.

He is currently with the School of Engineering, Swansea University, Swansea,UK, as a Professor/Chair of Wireless. Since 2005, he has been an Editor of theJournal of Communications and Networks. He was an Associate Editor for the ETRIJournal. Since 2009, he has been on the editorial board of the International Journalof Vehicular Technology. He is the author of two books, which were publishedby Cambridge University Press in 2006 and 2010. His research interests includewireless communications and array/statistical signal processing.

Prof. Choi received the 1999 Best Paper Award for Signal Processing from theEuropean Association for Signal Processing and the 2009 Best Paper Award fromthe International Symposium on Wireless Personal Multimedia Communications.From 2005 to 2007, he was an Associate Editor for the IEEE Transactions onVehicular Technology.

Index

SymbolsA-ary PAM, 3A-ary square QAM, 3d -fold diversity, 322-norm, 143, 187

Aa posteriori probability (APP), 7, 91, 101a priori probability (APRP), 17, 92, 101acronyms, xxiiiadditive white Gaussian noise (AWGN), 3antenna subset (AS), 187

BB-Chase ordering, 47bandwidth, 1base station (BS), 8, 142, 169basis, 7, 56basis updating, 200basis vector, 56beamforming, 167, 170binary, 17binary phase shift keying (BPSK), 3bit error rate (BER), 19bit-level, 17BLAST ordering, 47block fading, 3, 10

Ccandidate vectors, 16, 115Cartesian product, 17, 170, 189cellular system, 9channel capacity, 1, 141, 171channel coding, 129

channel gain, 1channel state information (CSI), 4Chase detector, 7Chernoff bound, 19chi-square, 23, 82, 84, 98, 105, 120, 121, 153,

154, 174, 209Cholesky decomposition, 123circular symmetric complex Gaussian (CSCG),

3CLLL algorithm, 77CLLL-LR, 80, 152CLLL-reduced, 180CMOS, 128column reordering criteria (CRC), 132column reordering index set (CRIS), 133column vector, 16complementary error function, 155, 205complex Gaussian random vector, 25complex multiplications (CMs), 122, 125–127complex-valued, 7, 57, 73, 77complex-valued LLL (CLLL), 77computational complexity, 53, 86, 108, 122,

137, 157, 214conditional error probability, 33, 121constellation points, 3correlation, 30, 69, 156cumulative density function (cdf), 83, 98, 106,

174

Ddata rate, 1, 5decision boundary, 52decision feedback equalizer (DFE), 27decision region, 52, 60degrees of freedom, 23, 82, 84, 98, 105, 120,

154, 174, 209

L. Bai and J. Choi, Low Complexity MIMO Detection, DOI 10.1007/978-1-4419-8583-5,© Springer Science+Business Media, LLC 2012

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228 Index

dimension reduction condition (DRC), 95, 102downlink, 143

Eeffective SNR gain, 52eigenvalue, 136, 148, 190ellipsoid, 118energy per bit to the noise power spectral

density ratio, 36, 110, 210entry, 25, 47EP-CRC, 135error patterns, 33error probability, 5, 9, 84, 118, 120, 135, 150,

209exhaustive search, 6, 117, 172expectation, 23, 34, 143

Ffast fading channel, 161flat-fading, 15, 91, 133floating point operation (flops), 108, 138, 157,

159, 214Frobenius norm, 18, 172full multiuser diversity, 10full receive diversity, 6, 113

GGamma function, 121Gaussian LR, 70, 71, 125generalized sphere decoding (GSD), 8, 145Gram-Schmidt (GS), 123greedy user selection, 186

Hhard decision, 20, 135Hermitian transpose, 15, 143Householder reflection, 124Householder transformation, 123hyper-sphere, 83

Iidentity matrix, 70, 107, 143, 187ill-conditioned matrix, 124interfering signal, 6, 100intersymbol interference (ISI), 7

JJensen’s inequality, 1joint detection, 10

LL-R decomposition, 127lattice, 7, 56lattice basis reduced (LBR), 191lattice decoding, 56lattice points, 53, 119, 170lattice reduced basis, 65lattice reduced matrix, 7, 62, 69, 70lattice reduction (LR), 6, 43Lenstra-Lenstra-LovKasz (LLL) algorithm, 7,

64, 74likelihood function, 17, 91linear combination, 56linear detector, 6linear filter, 20, 59linear relationship, 24, 69linear transformation, 175linear-list, 43list, 6list decoding, 43, 99list length, 46, 109, 111, 118list sphere detector, 7list-based Chase algorithm, 43, 107list-based Chase detector, 7, 144list-based detector, 6LLL-LR, 80, 152LLL-reduced, 74, 152, 208log-likelihood ratio (LLR), 122, 130logarithms of a posteriori probability ratios

(LAPPR), 101–103lower bound, 98LR domain, 115, 134LR-based detection, 56, 113LR-based greedy (LRG), 193LR-based linear detector, 59, 81, 136, 149, 177LR-based list detector, 7, 113LR-based MMSE, 59, 117, 191LR-based MMSE-SIC, 62, 117, 149, 178, 192LR-based SIC, 84, 136, 177LR-based ZF, 59, 191LR-based ZF-SIC, 62, 178, 191

MMahalanobis distance, 118MAP detection, 17MATLAB, 4, 10, 36, 108, 138, 159, 214matrix, 2matrix inversion, 126max-log approximation, 103max-min diagonal (MD), 136, 149, 179, 192max-min distance (MDist), 173, 190max-min eigenvalue (ME), 136, 148, 173, 190

Index 229

maximum a posteriori probability (MAP), 7,91

maximum likelihood (ML), 6maximum mutual information (MMI), 171mean-square error (MSE), 21MIMO-OFDM, 128min-max mean square error (MMMSE), 177minimum mean square error (MMSE), 6MMSE filtering, 128MMSE-SIC, 28modulation, 16Monte Carlo, 36multiple-input multiple-output (MIMO), 1multiuser, 8, 167multiuser diversity, 9, 167, 169multiuser MIMO, 9, 169, 185mutual information, 171

Nnoise, 1notations, xxvNP-hard, 173nulling and cancellation, 92, 108

OOD-CRC, 135optimal decision region (ODR), 177ordering, 29, 46orthogonal, 56orthogonal basis, 7, 60, 118, 152orthogonal frequency division multiplexing

(OFDM), 128orthogonal vectors, 24orthogonality, 79, 135, 152orthogonality deficiency, 79, 135, 152orthonormal, 178overdetermined, 144

Ppairwise error probability (PEP), 17, 156, 172,

204partial APP, 101partial MAP, 92, 101partial MAP based list detector, 99, 107permutation matrix, 47, 144polynomial complexity, 64, 113, 173postvoting, 142postvoting vector selection (PVS), 142prevoting, 142prevoting vector cancellation (PVC), 142

probability density function (pdf), 31, 83, 156,174, 206

probability of dimension reduction (PDR), 97,102, 105

projection, 23proximity factor, 53, 86pseudo-inverse, 81, 143pulse amplitude modulation (PAM), 3PVC-MIMO, 142

QQ-function, 18QR factorization, 23, 62, 92, 100, 114, 134,

149, 178, 191quadrature amplitude modulation (QAM), 3

Rrandom variable, 15, 23, 31, 84, 98, 105, 120,

174rank-deficient, 7Rayleigh fading, 32real-valued, 7, 56, 70, 75receive diversity order, 6received signal, 1receiver, 1residual vectors, 44rounding operation, 60, 72

SS-Chase, 46sequential detection, 26set minus, 31, 143, 187shortest vector problem (SVP), 173SIC detection, 43SIC-list, 45, 108signal alphabet, 16, 133, 143, 170signal to interference plus noise ratio (SINR),

30signal-to-noise ratio (SNR), 1single-input single-output (SISO), 1, 141size-reduction, 198slow fading channel, 138, 157soft-bit, 122, 129sorted-QR decomposition, 108space division multiple access (SDMA), 167spatial diversity gain, 5spatial multiplexing gain, 5spectral efficiency, 1, 141, 167spectrum, 1sphere decoding, 109, 145statistical independent, 93

230 Index

statistical properties, 25statistically independent, 156, 205sub-CRIS, 134, 135subdetection, 7, 92, 100, 113, 136subdetector, 7, 44, 46, 49, 136submatrix, 45, 94suboptimal, 8subspace, 22subvector, 45, 94successive interference cancellation (SIC), 6,

92, 115sum of squared error (SSE), 45supremum, 86symbol energy, 21symbol error rate (SER), 36symbol level, 20symbol vector, 44

Tthroughput, 10throughput efficiency, 171trace operation, 18transmission rate, 1transmitted symbol, 1transmitter, 1transpose, 2, 143tree search decoder-column reordering

(TSD-CR), 142, 146

UUBLR-based greedy (UBLRG), 197underdetermined, 7, 141

underdetermined integer least squares (UILS),145

unimodular, 62, 116unitary, 25, 114, 134unitary transformation, 124updated basis LR (UBLR), 197uplink, 8, 169, 186upper block triangle, 96upper bound, 18, 82, 153, 154, 172upper trapezoidal, 145upper triangular, 25, 114, 134user selection, 10, 167, 169, 185

Vvariance, 15vector, 2vertical Bell laboratories layered space-time

(V-BLAST), 6, 92very large scale integration (VLSI), 124, 128Voronoi region, 119

WWishart matrix, 120words of memory, 125, 126

Zzero-forcing (ZF), 6ZF-SIC, 24