1. MIMO Communications and Algorithmic Number Theory G. Matz joint work with D. Seethaler Institute of Communications and Radio-Frequency Engineering Vienna University of Technology (VUT)

2. NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006– 2 – RX antennas Setting the Stage MIMO communications: Algorithmic number theory (ANT) is the study of algorithms that perform number theoretic computations (Source: Wikipedia) Examples: primality test, integer factorization, lattice reduction TXRX... channel TX antennas

3. NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006– 3 – Outline MIMO detection and ANT MIMO precoding and ANT Precoding via Vector Perturbation Approximate Vector Perturbation using Lattice Reduction Lattice Reduction using ANT: Brun‘s Algorithm Simulation Results Conclusions

4. NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006– 4 – Multi-Antenna Broadcast (Downlink) System model: precoding user #1... channel M TX antennas user # k user # K users, each with one antenna Users cannot cooperate shift MIMO processing to TX precoding withMIMO I/O relation: K symbols CSI at TX required

5. NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006– 5 – here, and is an integer perturbation vector precoder performs channel inversion and vector perturbation Vector Perturbation (Peel et al.) TX vector: Receive symbols: follows from RX-SNR equals 1/  choose z such that s(z) is “short“ Remaining RX processing: get rid of z via modulo operation quantization w.r.t. symbol alphabet

6. NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006– 6 – Optimum vector perturbation maximizes RX-SNR: Choice of Perturbation Vector Suboptimum precoding: e.g. Tomlinson-Harashima precoding (THP) For channels with large condition number - sphere encoding has high complexity - THP etc. have poor performance Small condition number: all methods work fast and well Implementation: sphere encoder

7. NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006– 7 – Structure of Channel Singular Values smallest singular value and associated singular vector v cause problems M=K= 4

8. NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006– 8 – Vector Perturbation for Poor Channel Condition Example: BPSK, real-valued channel & noise, M = K =2,   = 2 TX vector perturbed versions of TX vector search TX vector that - is integer - has small length - is orthogonal to v approximate integer relation (ANT)

9. NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006– 9 – Relation of MIMO and ANT Precoding at Tx Detection at Rx Approximate Integer Relations Simultaneous Diophantine Approximations duality MIMO ANT poorly conditioned channels

10. NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006– 10 – Try to find “better“=reduced lattice basis Vector Perturbation Using Lattice Reduction (LR) LR-assisted vector perturbation (, ) - cost function: - solve or use THP approximation - use as perturbation vector View as basis of a lattice All lattice basis are related via a unimodular matrix:

11. NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006– 11 – Lattice Reduction Orthogonality defect (quality of lattice basis): left channel singular vectors channel singular values - LLL-LR assisted THP achieves full diversity - but LLL can be computationally intensive Most popular LR method: Lenstra-Lenstra-Lovász (LLL) algorithm LR: find achieving small and thus small

12. NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006– 12 – Integer Relation Based LR For poorly conditioned channels, only one singular value is small To achieve small, vectors must be sufficiently orthogonal to singular vectors with small singular values find integer vectors that are sufficiently orthogonal to This is the approximate integer relation (IR) problem in ANT IR-LR focuses on one singular vector (in contrast to LLL-LR) - some performance loss - significantly smaller complexity Goal: more efficient LR method

13. NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006– 13 – - small can be made arbitrarily small using long vectors Approximate Integer Relations Approximate IR: achieve small with as short as possible Tradeoff: governed by channel singular values Can be realized very efficiently using Brun‘s algorithm large will increase

14. NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006– 14 – Brun’s Algorithm Initialization: Find Calculate Replace is also updated recursively and can be made arbitrarily small (update of ) Very simple: scalar divisions, quantizations, and vector updates repeat until termination condition is satisfied

15. NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006– 15 – Performance of Brun’s Algorithm average no. of iterations average Example using and averaging over 1000 randomly picked

16. NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006– 16 – terminate if update of does not decrease Lattice Reduction via Brun’s Algorithm at each iteration, is a basis for recall: LR aims at minimizing we are just interested in channels with one small singular value in this case, apply Brun’s algorithm to any column of Termination condition Calculation of

17. NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006– 17 – Simulation Results (1) THP w. LLL SNR THP w. Brun iid Gaussian channel 4-QAM Iterations on average: Symbol Error Rate LR using Brun‘s algorithm can exploit large part of available diversity Sphere encoding (optimal) THP - Brun: 2.5 - LLL: 12.9 A Brun iteration is less complex than an LLL iteration

18. NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006– 18 – Simulation Results (2) THP w. LLL SNR THP w. Brun Symbol Error Rate Sphere encoding (optimal) THP iid Gaussian channel 4-QAM Iterations on average: - Brun: 4.8 - LLL: 42 A Brun iteration is less complex than an LLL iteration LR using Brun‘s algorithm can exploit large part of available diversity

19. NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006– 19 – Conclusions Algorithmic number theory provides useful tools for MIMO detection and MIMO precoding Here: proposed vector perturbation using lattice reduction based on integer relations Efficient implementation: Brun‘s algorithm Good performance at very small complexity

20. NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006– 20 – ML detector: MIMO Detection - exact implementation: sphere decoder - suboptimum detectors: ZF, MMSE, V-BLAST, … If is poorly conditioned: Everything is fine if is close to orthogonal - poor performance (ZF, MMSE, V-BLAST,...) - or high complexity (ML) RX vector:

21. NEWCOM Dept. 1 Meeting - Toulouse - May 15, 2006– 21 – Detection for Poor Channel Condition Example: BPSK, real-valued channel & noise, M=K= 2 v ZF-domain Rx vector: search TX vector that is - integer - close to line y +  v simultaneous Diophantine approximation (ANT)