System introduction (Narrowband model) Notations and assumptions: • Transmit symbol vector x, each symbol is mapped onto a constellation, complex vector of size nT • Memoryless H known at receiver, i.i.d. nR xnT complex matrix • Receive symbol vector y, complex vector of size nR • Additive White Gaussian Noise (AWGN) n components are i.i.d. y = Hx+n 29/11/2010 3 F E D C B A E C A F D B H Receiver F E D C B A x y xest + n +
The dominant source of complexity, or The dominant source of performance loss, or … BOTH! Joint detection (Maximum Likelihood (ML)): xML = argmin ||y-Hx||2, for all x in set of possibly transmit symbols vectors + Optimal performance - Exponential complexity (MnT) 29/11/2010 4
Sphere Decoder [AEVZ02] Neighborhood study, inside a radius d QRD-based: xSD = argmin ||QHy-Rx||2 < d2 Unconstrained ZF solution centered: yZF [WTCM02] ||y-Hx||2 = ||HH†y-Hx||2 = ||H(yZF –x)||2 = ||Re||2 Layer by layer Partial Euclidean Distance (PED) minimization 2 1 1 , , 1 1 , 1 , 1 1 , 1 1 , 1 ... 0 0 0 0 0 ... ... ... ... ... T T T T T T T T T T n n n n n n n n n n e e e R R R R R R
Arbitrary constellation exploration [HV05] + Complexity limitation of ML algorithm, Optimal detector - Problem of radius choice on performance Shrank radius, Schnorr-Euchner enumeration Increasing Euclidean distance at each layer [GN04] + Reduced complexity, independent of radius d, Optimal detector - Problem of variable complexity, complexity depends on SNR and channel conditions, and depth-first search 29/11/2010 9 ySIC xML xML ySIC dmax dmin Q{ySIC } Q{ySIC }
[WMPF03] + Early termination of the tree search - Maximal complexity remains unchanged Layers-reordered K-Best [WTCM02] ZF-ordering, re-order antennas by reducing SNR [WBKK03] MMSE-ordering, re-order antennas by reducing SINR [WBKK03] + Combats errors propagation - Still not the ML diversity for high order constellations, K must be chosen very large for low SNR symbols and would be chosen small for high SNR symbols 29/11/2010 11
K in early stages and smaller K in later stages [LW08], particularly efficient with SQRD + Performance: Avoid missing the ML solution in the first layers (most likely case of global error), Reduced complexity - How to set K, Still too complex for high order constellations Particular case: Fixed-Throughput Sphere-Decoder Full-ML at k top layers, Linear Equalizer (LE) at nT -k bottom layers [BT08] S. Aubert, F. Nouvel, and A. Nafkha, ″Complexity gain of QR Decomposition based Sphere Decoder in LTE receiver,″ Vehicular Technology Conference, IEEE , pp. 1-5, Sept. 2009. 29/11/2010 15 00 01 01 00 root 01 00 10 00 00 11 00 00 xnT xnT-1 xnT-2
Lattice definition: L = HZC nT, ZC =Z+jZ Z is the set of integers H=[h1 , …, hnT ] is a generator basis Interest: a basis is not unique y=Hx+n rewrites y=HTT-1x+n=Hred z+n. Why not realizing equalization or detection through a better conditioned matrix Hred ? What is a better conditioned matrix? Shorter, more orthogonal 29/11/2010 16 H† Hred †
| < 1/2 (μ=<hi ,hj >/<hj ,hj >) Size reduction operation makes vectors shorter and more orthogonal Short norms condition ||hi ||2+ μi,i-1 2||hi-1 ||2 > δ||hi-1 ||2 Swapping operation if condition violated T unimodular (contains Gaussian integers (ZC ) and |det{T}|=1) The reduced constellation z Є ZC nT The nT -parallelotope nT -volume formed by the basis remains unchanged (same channel impact (SNR)) + Worst case polynomial complexity, complexity reduction through the (necessary) SQRD starting point, no channel knowledge at transmitter - Random complexity, iterative algorithm 29/11/2010 18
both Get the LRA technique diversity and reduce the SNR offset through a neighborhood study xLRA-SD = argmin ||y - HTT-1x||2 = argmin ||y - Hred z||2 Problem of neighborhood generation: Zall =T-1Xall , ML complexity 29/11/2010 28
algorithm Qi, Holt algorithm [QH07] Shift-scale-normalization: y’=(y+Hd)/2, d=1/2 Shift-scale-normalization: x’=(x+d)/2, z = T-1x’ xLRA-SD = argmin ||Qred Hy’-Rred z||2 Neighborhood exploration through a predetermined set of displacements around SIC solution: [δ1 , …, δN ], N>K + Improve performances (exploits reduced lattice advantages concerning channel conditions) with low number of candidates - No limitation of number of explored symbols (infinite lattice), zest could give non-existing xest in the original constellation (increase complexity or decrease performance) 29/11/2010 29
algorithm Candidate generation limitation [RGAV09] x’min/max and T are known => zmin/max are known zmax (l) = x’max Σ(T-1)(l,:)>0+x’min Σ(T-1)(l,:)<0, zmin (l) = x’min Σ(T-1)(l,:)>0+x’max Σ(T-1)(l,:)<0 + Complexity reduction without performance loss 29/11/2010 30
QPSK/16QAM LRA-KBest in original/reduced constellation + Performances are independent of constellation order - Benefits are limited in QPSK case, less sensitive to ill-conditioned channel and many inexistent symbols vectors [ZG06] 29/11/2010 32
Rayleigh channel, QPSK/16QAM LRA-KBest in original/reduced constellation + Performances are independent of constellation order - Benefits are limited in QPSK case, less sensitive to ill-conditioned channel and many inexistent symbols vectors [ZG06]
Rayleigh channel, QPSK/16QAM LRA-KBest in original/reduced constellation + Performances are independent of constellation order - Benefits are limited in QPSK case, less sensitive to ill-conditioned channel and many inexistent symbols vectors [ZG06]
Rayleigh channel, QPSK/16QAM LRA-KBest in original/reduced constellation + Performances are independent of constellation order - Benefits are limited in QPSK case, less sensitive to ill-conditioned channel and many inexistent symbols vectors [ZG06]
Rayleigh channel, QPSK/16QAM LRA-KBest in original/reduced constellation + Performances are independent of constellation order - Benefits are limited in QPSK case, less sensitive to ill-conditioned channel and many inexistent symbols vectors [ZG06]
Rayleigh channel, QPSK/16QAM LRA-KBest in original/reduced constellation + Performances are independent of constellation order - Benefits are limited in QPSK case, less sensitive to ill-conditioned channel and many inexistent symbols vectors [ZG06]
Full detector computational complexity study is necessary Soft-Decision extension Closed-loop and OFDM case calibration Throughput objectives of LTE-A norm must be shown to be reached 29/11/2010 38
″Factoring Polynomials with Rational Coefficients,″ Mathematica Annalen, vol. 261, pp. 515-534, 1982. [Sey93] M. Seysen, ″Simultaneous Reduction of a Lattice Basis and its Reciprocal Basis,″ Combitanorica, vol. 13, pp. 363-376, 1993 [AEVZ02] E. Agrell, T. Eriksson, A. Vardy, and K. Zeger, ″Closest Point Search in Lattice,″ Information theory, IEEE Transactions on, vol. 48, no. 8, pp. 2201-2214, Nov. 2002. [WTCM02] K.-W. Wong, C.-Y. Tsui, R.S.-K. Cheng, and W.-H. Mow, ″A VLSI Architecture of a K-Best Lattice Decoding Algorithm For MIMO Channels,″ Symposium on Circuits and Systems, IEEE International, vol. 3, pp 273–276, 2002. [WBKK03] D. Wübben, R. Böhnke, V. Kühn, and K.-D. Kammeyer, ″ MMSE Extension of V-BLAST Based on Sorted QR Decomposition,″ Vehicular Technology Conference, IEEE , vol. 1, pp. 508–512, Oct. 2003. [WBKK04] D. Wübben, R. Bohnke, V. Kuhn, and K.-D. Kammeyer, ″Near-Maximum-Likelihood Detection of MIMO Systems using MMSE-based Lattice Reduction,″ International Conference on Communications, IEEE, vol.2, pp. 798-802, June 2004. [GN04] Z. Guo, and P. Nilsson, ″ A VLSI Architecture of the Schnorr-Euchner Decoder for MIMO Systems,″ Circuits and Systems Symposium on Emerging Technologies: Frontiers of Mobile and Wireless Communication, IEEE, vol. 1, pp. 65-68, June 2004. 29/11/2010 40
Sphere-Decoding Algorithm I. Expected Complexity,″, Signal Processing, IEEE Transactions on, 2005. [ZG06] W. Zhao, and G. B. Giannakis, ″Reduced Complexity Closest Point Decoding Algorithms for Random Lattices,″ Wireless Communications, IEEE Transactions on, 5(1):101–111, Jan. 2006. [GLM06] Y.H. Gan, C. Ling, and W.H. Mow, ″ Complex Lattice Reduction Algorithm for Low- Complexity MIMO Detection,″ … [ZM07] W. Zhang, and X. Ma, ″Approaching Optimal Performance By Lattice-Reduction Aided Soft Detectors, ″ Information Sciences and Systems, Conference on, pages 818–822, Mar. 2007. [QH07] X.-F. Qi, and K. Holt, ″A Lattice-Reduction-Aided Soft Demapper for High-Rate Coded MIMO-OFDM Systems, ″ Signal Processing Letters, IEEE, 14(5):305 –308, May 2007. [LW08] Q. Li, and Z. Wang, ″Reduced Complexity K-Best Sphere Decoder Design For MIMO Systems,″ Circuits Systems and Signal Processing, vol. 27, no. 4, pp. 491-505, June 2008. [BT08] L. Barbero, and J. Thompson, ″Fixing the Complexity of the Sphere-Decoder for MIMO Detection,″ Wireless Communications, IEEE Transactions on, vol. 7, no. 6, pp. 2131-2142, June 2008. [Bar08] J.R. Barry, ″MIMO Detection – Theory and Practice,″ IEEE Personal, Indoor and Mobile Radio Conference tutorial, 2008. [RGAV09] S. Roger, A. Gonzalez, V. Almenar, and A.M. Vidal, ″ On Decreasing the Complexity of Lattice- Reduction-Aided K-Best MIMO Detectors,″ European Signal Processing Conference, Aug. 2009. 29/11/2010 41