1. Efficient Statistical Pruning for Maximum Likelihood Decoding Radhika Gowaikar Babak Hassibi California Institute of Technology July 3, 2003

2. Outline Integer Least Squares Problem Probabilistic Setup, Complexity as Random Variable Sphere Decoder Modified Algorithm Statistical Pruning, Expected Complexity Results Analysis Conclusions and Future Work

3. Integer-Least Squares Problems Search space is discrete, perhaps infinite Given a “skewed” lattice Given a vector Find “closest” lattice point Known to be NP-hard

4. Applications in ML Decoding ML detection leads to integer least-squares problems Signal constellation is a subset of a lattice (PAM, QAM) Noise is AWG Eg. Multi-antenna systems

5. Approximate Solutions Zero forcing cancellation Nulling and canceling Nulling and canceling with optimal ordering computation BER comparison – ML vs. Approximate But Bit Error Rate suffers

6. Exact Methods Sphere Decoding : search in a hypersphere centered at (Fincke-Pohst ; Viterbo, Boutros; Vikalo, Hassibi) How do we find the points that are in the hypersphere?

7. To find points without exhaustive search When, this is an interval Use this to go from a -dimensional point to a (k+1) – dimensional point. Search over spheres of radius r and dimensions 1,2,…, N. Use to facilitate this Sphere Decoder

8. Sphere Decoder – How it Works Call

9. How it Works contd. depends only on

10. Solve these successively --- get a tree Complexity depends on the size of the tree Search Space and Tree

11. Reducing Complexity Not ML decoding any more

12. Results Complexity exponent and BER for N=20 with QPSK

13. Probability of Error Let e be the probability that the transmitted point s is not in the search space Can be shown that Need to keep small

14. Finding epsilon can be determined exactly in terms of s Choose s to make as small as desired Theorem:

15. Computational Complexity is the search region at dimension is the constellation Need to find

16. Finding s are independent. Hence Also, can be determined exactly Yet have to employ approximations…

17. Upper Bound For, it needs to satisfy conditions. For upper bound, just the -th condition. is the incomplete gamma function. Easy to compute

18. Approximations Can be shown that where and are functions of The complexity can now be determined by Monte Carlo simulations

19. Simulation Results Complexity exponent and BER for N=20 with QAM

20. Simulation Results Complexity Exponent and BER for N=50 with QAM

21. Conclusions and Future Work Significant reduction in Complexity BER can be made close to optimal Quantify trade-off between BER and Complexity Compare with other decoding algorithms Analyze for signaling schemes with coding Other applications for these techniques…?

23. How it Works contd. Solve these successively…