# Sébastien Aubert - Lattice-Reduction-Aided Sphere-Detector

April 29, 2010

## Transcript

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 29/11/2010 1 Supelec Rennes Signal, Communication et Électronique Embarquée (SCEE)
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
3. ### 1) System introduction  Multiple-Input Multiple-Output (MIMO) Spatial Multiplexing case

 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 +
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)): xML = argmin ||y-Hx||2, for all x in set of possibly transmit symbols vectors + Optimal performance - Exponential complexity (MnT) 29/11/2010 4
5. ### 1) Problem statement (2)  Linear-Equalization  ZF: G=(HHH)-1HH (=H†)

=> Gy=G(Hx+n)=x+Gn  MMSE: G=(HHH+1/SNR I)-1H H + Polynomial complexity - noise amplification for ZF, no diversity in reception  Successive-Interference Canceller (SIC)  QRD-based: H = QR, with QHQ=I and R is upper triangular xSIC = argmin ||QHy-Rx||2 + Polynomial complexity - Error propagation 29/11/2010 5
6. ### 2) Sphere Decoder (1) 29/11/2010 6  General principle of

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
7. ### 2) Sphere Decoder (2) 29/11/2010 7 Layer by layer Partial

Euclidean Distance (PED) minimization ||Re||2 = Σi=nT, …, 1 PEDi PEDi = R’i,i |xi -yZF,i + Σj=i+1, …, nT R’i,j (xest,i -yZF,i )|2 = R’i,i |xi -zi |2 Cumulated Euclidean Distance (CED) CEDi = PEDi + CEDi-1 Try xi at each layer zi is a constant. + Implementation interest for PED computation
8. ### 2) Sphere Decoder (2) 29/11/2010 8 Layer by layer Partial

Euclidean Distance (PED) minimization ||Re||2 = Σi=nT, …, 1 |Ri,: ei |2 = Σi=nT, …, 1 |Ri,i ei + Σj=i+1, …, nT Ri,j ej |2 = Σi=nT, …, 1 |Ri,i |2|ei + Σj=i+1, …, nT Ri,j /Ri,i ej |2 = Σi=nT, …, 1 R’i,i |ei + Σj=i+1, …, nT R’i,j ej |2 = Σi=nT, …, 1 PEDi PEDi = R’i,i |xi -zi |2 Cumulated Euclidean Distance (CED) CEDi = PEDi + CEDi-1 Try xi at each layer zi is a constant. + Implementation interest for PED computation
9. ### 2) Sphere Decoder (3)  Constant radius, Fincke-Pohst enumeration 

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 }
10. ### 2) Sphere Decoder (4)  K-Best Sphere Decoder  The

K candidates with the smallest Euclidean distance are stored + Fixed Complexity, parallel algorithm - K value for high order constellations (16QAM, 64QAM), non-optimal detector 29/11/2010 10 00 01 10 11 01 10 11 00 00 01 10 11 01 10 11 00 01 10 11 00 root 01 10 11 00 00 01 10 11 xnT xnT-1 xnT-2
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 29/11/2010 11
12. ### 2) Sphere Decoder (6)  MIMO-SM 4x4, Rayleigh channel, QPSK/16QAM

 QRD-based K-Best, SQRD-based K-Best 29/11/2010 12
13. ### 2) Sphere Decoder (6)  MIMO-SM 4x4, Rayleigh channel, QPSK/16QAM

 QRD-based K-Best, SQRD-based K-Best 29/11/2010 13
14. ### 2) Sphere Decoder (6)  MIMO-SM 4x4, Rayleigh channel, QPSK/16QAM

 QRD-based K-Best, SQRD-based K-Best 29/11/2010 14
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 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
16. ### 3) Lattice-Reduction (1)  General principle of Lattice-Reduction-Aided algorithms 

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 †
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 29/11/2010 17
18. ### 3) Lattice-Reduction (3)  LLL [LLL82]  Orthogonality condition |μi,j

| < 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
19. ### 3) Lattice-Reduction (4)  General principle of Lattice-Reduction-Aided algorithms 

v1 =[7, 6]T  v2 =[10, 8]T 29/11/2010 19
20. ### 3) Lattice-Reduction (4)  General principle of Lattice-Reduction-Aided algorithms 

v1 =[7, 6]T  v2 =[10, 8]T  Size reduction  v1 =[7, 6]T  v2 =[3, 2]T 29/11/2010 20
21. ### 3) Lattice-Reduction (4)  General principle of Lattice-Reduction-Aided algorithms 

v1 =[7, 6]T  v2 =[10, 8]T  Size reduction  v1 =[7, 6]T  v2 =[3, 2]T  Swapping  v1 =[3, 2]T  v2 =[7, 6]T 29/11/2010 21
22. ### 3) Lattice-Reduction (4)  General principle of Lattice-Reduction-Aided algorithms 

v1 =[7, 6]T  v2 =[10, 8]T  Swapping  v1 =[3, 2]T  v2 =[7, 6]T  Size reduction  v1 =[3, 2]T  v2 =[1, 2]T 29/11/2010 22
23. ### 3) Lattice-Reduction (4)  General principle of Lattice-Reduction-Aided algorithms 

v1 =[7, 6]T  v2 =[10, 8]T  Size reduction  v1 =[3, 2]T  v2 =[1, 2]T  Swapping  v1 =[1, 2]T  v2 =[3, 2]T 29/11/2010 23
24. ### 3) Lattice-Reduction (4)  General principle of Lattice-Reduction-Aided algorithms 

v1 =[7, 6]T  v2 =[10, 8]T  Swapping  v1 =[1, 2]T  v2 =[3, 2]T  Size reduction  v1 =[1, 2]T  v2 =[2, 0]T 29/11/2010 24
25. ### 3) Lattice-Reduction (4)  General principle of Lattice-Reduction-Aided algorithms 

v1 =[7, 6]T  v2 =[10, 8]T  Size reduction  v1 =[1, 2]T  v2 =[2, 0]T  Swapping  v1 =[2, 0]T  v2 =[1, 2]T  (v1, v2 ) is LLL-reduced 29/11/2010 25
26. ### 3) Lattice-Reduction (5)  Impact on detection step  Quantification

[Bar08]  Non-existing symbols vectors 29/11/2010 26 H H† x + n y xest Hred † T zest
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] + ML diversity, worst case polynomial complexity, independent of SNR - Additional complexity, SNR offset 29/11/2010 27 QPSK 16QAM
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 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
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-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
30. ### 4) Lattice-Reduction-Aided Sphere Decoder (3)  Reduced constellation neighborhood study

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
31. ### 4) Lattice-Reduction-Aided Sphere Decoder (4)  Original constellation neighborhood study

algorithm  LRA-ZF centered SD [ZM07] xest = argmin ||yLRA-ZF - z||2 = argmin ||yLRA-ZF - T-1x||2 = argmin ||QT-1 yLRA-ZF - RT-1 x||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) 29/11/2010 31
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 channel and many inexistent symbols vectors [ZG06] 29/11/2010 32
33. ### 4) Lattice-Reduction-Aided Sphere Decoder (5) 29/11/2010 33  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]
34. ### 4) Lattice-Reduction-Aided Sphere Decoder (5) 29/11/2010 34  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]
35. ### 4) Lattice-Reduction-Aided Sphere Decoder (6) 29/11/2010 35  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]
36. ### 4) Lattice-Reduction-Aided Sphere Decoder (6) 29/11/2010 36  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]
37. ### 4) Lattice-Reduction-Aided Sphere Decoder (6) 29/11/2010 37  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]
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 29/11/2010 38
39. ### Q&A and discussion  Thank you for your attention 

Questions? 29/11/2010 39
40. ### References (1) [LLL82] A. Lenstra, H. Lenstra, and L. Lovasz,

″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
41. ### References (2) [HV05] B. Hassibi, and H. Vikalo, ″On the

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