Lattice-Reduction-Aided Sphere-Detector as a Solution for Near-Optimal MIMO Detection in Spatial Multiplexing Systems

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1 Lattice-Reduction-Aided Sphere-Detector as a Solution for Near-Optimal MIMO Detection in Spatial Multiplexing Systems Sébastien Aubert ST-ERICSSON Sophia & INSA IETR Rennes sebastien.aubert@stericsson.com Supelec Rennes Signal, Communication et Électronique Embarquée (SCEE) 29/11/2010 1

2 Contents 1) System introduction and problem statement 2) Near-ML techniques: Sphere Decoder (SD) 3) Near-ML techniques: Lattice Reduction (LR) 4) LR-Aided Sphere-Decoder (LRA-SD) 29/11/2010 2

n R xn T complex matrix Receive symbol vector y, complex vector of size n R Additive White Gaussian Noise (AWGN) n components

3 Notations and assumptions: Transmit symbol vector x, each symbol is mapped onto a constellation, complex vector of size n T Memoryless H known at receiver, i.i.d. n R xn T complex matrix Receive symbol vector y, complex vector of size n R Additive White Gaussian Noise (AWGN) n components are i.i.d. 29/11/ y = Hx+n 1) System introduction Multiple-Input Multiple-Output (MIMO) Spatial Multiplexing case System introduction (Narrowband model) n F E D C B A E C A x H + y Receiver x est F E D C B A F D B +

4 1) Problem statement (1) MIMO Detection step is either The dominant source of complexity, or The dominant source of performance loss, or BOTH! Joint detection (Maximum Likelihood (ML)): x ML = argmin y-hx 2, for all x in set of possibly transmit symbols vectors + Optimal performance - Exponential complexity (M n T) 29/11/2010 4

5 1) Problem statement (2) Linear-Equalization ZF: G=(H H H) -1 H H (=H ) => Gy=G(Hx+n)=x+Gn MMSE: G=(H H H+1/SNR I) -1 H H + Polynomial complexity - noise amplification for ZF, no diversity in reception Successive-Interference Canceller (SIC) QRD-based: H = QR, with Q H Q=I and R is upper triangular x SIC = argmin Q H y-rx 2 + Polynomial complexity - Error propagation 29/11/2010 5

6 2) Sphere Decoder (1) General principle of Sphere Decoder [AEVZ02] Neighborhood study, inside a radius d QRD-based: x SD = argmin Q H y-rx 2 < d 2 Unconstrained ZF solution centered: y ZF [WTCM02] y-hx 2 = HH y-hx 2 = H(y ZF x) 2 = Re 2 Layer by layer Partial Euclidean Distance (PED) minimization R 1, R 1, n T... 1 R 1, n T... e R n T 1, n 0 T 1 R R n n T T 1, n, n T T e e n n T T 1 29/11/2010 6

7 2) Sphere Decoder (2) Layer by layer Partial Euclidean Distance (PED) minimization Re 2 = Σ i=n T,, 1 PED i PED i = R i,i x i -y ZF,i + Σ j=i+1,, n T R i,j(x est,i -y ZF,i ) 2 = R i,i x i -z i 2 Cumulated Euclidean Distance (CED) CED i = PED i + CED i-1 Try x i at each layer z i is a constant. + Implementation interest for PED computation 29/11/2010 7

8 2) Sphere Decoder (2) Layer by layer Partial Euclidean Distance (PED) minimization Re 2 = Σ i=n T,, 1 R i,: e i 2 = Σ i=n T,, 1 R i,i e i + Σ j=i+1,, n T R i,je j 2 = Σ i=n T,, 1 R i,i 2 e i + Σ j=i+1,, n T R i,j/r i,i e j 2 = Σ i=n T,, 1 R i,i e i + Σ j=i+1,, n T R i,je j 2 = Σ i=n T,, 1 PED i PED i = R i,i x i -z i 2 Cumulated Euclidean Distance (CED) CED i = PED i + CED i-1 Try x i at each layer z i is a constant. + Implementation interest for PED computation 29/11/2010 8

9 2) Sphere Decoder (3) Constant radius, Fincke-Pohst enumeration Arbitrary constellation exploration [HV05] d min Q{y SIC } y SIC + 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] x ML d max y SIC Q{y SIC } + 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/ x ML

10 2) Sphere Decoder (4) K-Best Sphere Decoder The K candidates with the smallest Euclidean distance are stored root x n T x n T x n T Fixed Complexity, parallel algorithm - K value for high order constellations (16QAM, 64QAM), non-optimal detector 29/11/

11 2) Sphere Decoder (5) Symbols-reordered K-Best Schnorr-Euchner strategy [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/

12 2) Sphere Decoder (6) MIMO-SM 4x4, Rayleigh channel, QPSK/16QAM QRD-based K-Best, SQRD-based K-Best 29/11/

13 2) Sphere Decoder (6) MIMO-SM 4x4, Rayleigh channel, QPSK/16QAM QRD-based K-Best, SQRD-based K-Best 29/11/

14 2) Sphere Decoder (6) MIMO-SM 4x4, Rayleigh channel, QPSK/16QAM QRD-based K-Best, SQRD-based K-Best 29/11/

15 2) Sphere Decoder (7) Dynamic K-Best Use larger 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 n T -k bottom layers [BT08] root x n T x n T x n T S. Aubert, F. Nouvel, and A. Nafkha, Complexity gain of QR Decomposition based Sphere Decoder in LTE receiver, 29/11/ Vehicular Technology Conference, IEEE, pp. 1-5, Sept

16 3) Lattice-Reduction (1) General principle of Lattice-Reduction-Aided algorithms Lattice definition: L = HZ C n T, Z C =Z+jZ Z is the set of integers H=[h 1,, h nt ] is a generator basis Interest: a basis is not unique y=hx+n rewrites y=htt -1 x+n=h red z+n. Why not realizing equalization or detection through a better conditioned matrix H red? H H red What is a better conditioned matrix? Shorter, more orthogonal 29/11/

17 3) Lattice-Reduction (2) Lattice-Reduction algorithms Korkine-Zolotareff Lenstra-Lenstra-Lovasz (LLL) [LLL82] Complex LLL (CLLL) imply complexity reduction [GLM06] Seysen [Sey93] SQRD-based LLL less complex, Seysen may be parallelized 29/11/

18 3) Lattice-Reduction (3) LLL [LLL82] Orthogonality condition μ i,j < 1/2 (μ=<h i,h j >/<h j,h j >) Size reduction operation makes vectors shorter and more orthogonal Short norms condition h i 2 + μ i,i-12 h i-1 2 > δ h i-1 2 Swapping operation if condition violated T unimodular (contains Gaussian integers (Z C ) and det{t} =1) The reduced constellation z Є Z C n T The n T -parallelotope n T -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/

19 3) Lattice-Reduction (4) General principle of Lattice-Reduction-Aided algorithms v 1 =[7, 6] T v 2 =[10, 8] T 29/11/

20 3) Lattice-Reduction (4) General principle of Lattice-Reduction-Aided algorithms v 1 =[7, 6] T v 2 =[10, 8] T Size reduction v 1 =[7, 6] T v 2 =[3, 2] T 29/11/

21 3) Lattice-Reduction (4) General principle of Lattice-Reduction-Aided algorithms v 1 =[7, 6] T v 2 =[10, 8] T Size reduction v 1 =[7, 6] T v 2 =[3, 2] T Swapping v 1 =[3, 2] T v 2 =[7, 6] T 29/11/

22 3) Lattice-Reduction (4) General principle of Lattice-Reduction-Aided algorithms v 1 =[7, 6] T v 2 =[10, 8] T Swapping v 1 =[3, 2] T v 2 =[7, 6] T Size reduction v 1 =[3, 2] T v 2 =[1, 2] T 29/11/

23 3) Lattice-Reduction (4) General principle of Lattice-Reduction-Aided algorithms v 1 =[7, 6] T v 2 =[10, 8] T Size reduction v 1 =[3, 2] T v 2 =[1, 2] T Swapping v 1 =[1, 2] T v 2 =[3, 2] T 29/11/

24 3) Lattice-Reduction (4) General principle of Lattice-Reduction-Aided algorithms v 1 =[7, 6] T v 2 =[10, 8] T Swapping v 1 =[1, 2] T v 2 =[3, 2] T Size reduction v 1 =[1, 2] T v 2 =[2, 0] T 29/11/

25 3) Lattice-Reduction (4) General principle of Lattice-Reduction-Aided algorithms v 1 =[7, 6] T v 2 =[10, 8] T Size reduction v 1 =[1, 2] T v 2 =[2, 0] T Swapping v 1 =[2, 0] T v 2 =[1, 2] T (v 1, v 2 ) is LLL-reduced 29/11/

26 3) Lattice-Reduction (5) Impact on detection step n y x H + H x est H red Quantification [Bar08] Non-existing symbols vectors z est T 29/11/

27 3) Lattice-Reduction (6) MIMO-SM 4x4, Rayleigh channel, QPSK/16QAM LRA-ZF, LRA-MMSE, LRA-MMSE Extended [WBKK04] LRA-SIC, LRA-OSIC [WBKK04] QPSK 16QAM + ML diversity, worst case polynomial complexity, independent of SNR 29/11/ Additional complexity, SNR offset 27

28 4) Lattice-Reduction-Aided Sphere Decoder (1) Principle of combination of both Get the LRA technique diversity and reduce the SNR offset through a neighborhood study x LRA-SD = argmin y - HTT -1 x 2 = argmin y - H red z 2 Problem of neighborhood generation: Z all =T -1 X all, ML complexity 29/11/

29 4) Lattice-Reduction-Aided Sphere Decoder (2) Reduced constellation neighborhood study algorithm Qi, Holt algorithm [QH07] Shift-scale-normalization: y =(y+hd)/2, d=1/2 Shift-scale-normalization: x =(x+d)/2, z = T -1 x x LRA-SD = argmin Q redh y -R red z 2 Neighborhood exploration through a predetermined set of displacements around SIC solution: [δ 1,, δ N ], 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), z est could give non-existing x est in the original constellation (increase complexity or decrease performance) 29/11/

30 4) Lattice-Reduction-Aided Sphere Decoder (3) Reduced constellation neighborhood study algorithm Candidate generation limitation [RGAV09] x min/max and T are known => z min/max are known z max (l) = x max Σ(T -1 )(l,:)>0+x min Σ(T -1 )(l,:)<0, z min (l) = x min Σ(T -1 )(l,:)>0+x max Σ(T -1 )(l,:)<0 + Complexity reduction without performance loss 29/11/

31 4) Lattice-Reduction-Aided Sphere Decoder (4) Original constellation neighborhood study algorithm LRA-ZF centered SD [ZM07] x est = argmin y LRA-ZF - z 2 = argmin y LRA-ZF - T -1 x 2 = argmin Q T -1y LRA-ZF - R T -1x 2 + Reduced complexity (Although the needed QRD of T -1 needed), Avoid non-existing symbols vectors - Performance (does not exploit reduced lattice advantages concerning channel conditions) 29/11/

32 4) Lattice-Reduction-Aided Sphere Decoder (5) MIMO-SM 4x4, 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 29/11/2010 channel and many inexistent symbols vectors [ZG06] 32

33 4) Lattice-Reduction-Aided Sphere Decoder (5) MIMO-SM 4x4, 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] 29/11/

34 4) Lattice-Reduction-Aided Sphere Decoder (5) MIMO-SM 4x4, 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] 29/11/

35 4) Lattice-Reduction-Aided Sphere Decoder (6) MIMO-SM 4x4, 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] 29/11/

36 4) Lattice-Reduction-Aided Sphere Decoder (6) MIMO-SM 4x4, 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] 29/11/

37 4) Lattice-Reduction-Aided Sphere Decoder (6) MIMO-SM 4x4, 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] 29/11/

38 Conclusion and further studies Hard-Decision performance is near-ml 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/

39 Q&A and discussion Thank you for your attention Questions? 29/11/

40 References (1) [LLL82] A. Lenstra, H. Lenstra, and L. Lovasz, Factoring Polynomials with Rational Coefficients, Mathematica Annalen, vol. 261, pp , [Sey93] M. Seysen, Simultaneous Reduction of a Lattice Basis and its Reciprocal Basis, Combitanorica, vol. 13, pp , 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 , Nov [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 , [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 , Oct [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 , June [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 , June /11/

41 References (2) [HV05] B. Hassibi, and H. Vikalo, On the Sphere-Decoding Algorithm I. Expected Complexity,, Signal Processing, IEEE Transactions on, [ZG06] W. Zhao, and G. B. Giannakis, Reduced Complexity Closest Point Decoding Algorithms for Random Lattices, Wireless Communications, IEEE Transactions on, 5(1): , Jan [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 , Mar [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): , May [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 , June [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 , June [Bar08] J.R. Barry, MIMO Detection Theory and Practice, IEEE Personal, Indoor and Mobile Radio Conference tutorial, [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 /11/

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