9. MIMO III: Diversity-Multiplexing Tradeoff

1. 9. MIMO III: Diversity-Multiplexing Tradeoff

2. Slow Fading MIMO Channel • So far we have emphasized the spatial multiplexing aspect of MIMO channels. • But we also learnt that in slow fading scenario, diversity is an important thing. • How do the two aspects interact? • It turns out that you can get both in a slow fading channel but there is a fundamental tradeoff. • We characterize the optimal diversity-multiplexing tradeoff and find schemes that approach the optimal tradeoff.

3. Diversity and Freedom (Review) Two fundamental resources of a MIMO fading channel: • diversity • degrees of freedom

4. Diversity A channel with more diversity has smaller probability in deep fades.

5. Diversity • Additional independent fading channels increase diversity. • Spatial diversity: • For a nt by nr channel, maximum diversity is nt¢ nr. receive, transmit, or both.

6. Degrees of Freedom • Signals arrive in multiple directions provide multiple degrees of freedom for communication. • Same effect can be obtained via scattering even when antennas are close together. • In a nt by nr channel, there are min{nt,nr} degrees of freedom.

7. Diversity and Freedom In a MIMO channel with rich scattering: maximum diversity = nt¢ nr degrees of freedom = min{nt,nr} The name of the game in space-time coding is to design schemes which exploit as much of both these resources as possible.

8. Space-Time Code Examples: 2 x 1Channel Repetition Scheme: Alamouti Scheme: diversity: 2 data rate: 1/2 sym/s/Hz diversity: 2 data rate: 1 sym/s/Hz

9. Performance Summary: 2 x 1Channel

10. Space-Time Code Examples: 2 x 2 Channel Repetition Scheme: Alamouti Scheme: diversity: 4 data rate: 1/2 sym/s/Hz diversity: 4 data rate: 1 sym/s/Hz But the 2x2 channel has 2 degrees of freedom !

11. V-BLAST with Nulling Send two independent uncoded streams over the two transmit antennas. Demodulate each stream by nulling out the other stream. Data rate: 2 sym/s/Hz Diversity: 1 Winters et al 93: Nulling out k interferers using nr receive antennas yields a diversity gain of nr -k.

12. Performance Summary: 2 x 2 Channel Questions: • Alamouti is clearly better than repetition, but how can it be compared to V-Blast? • How does one quantify the ``optimal'' performance achievable by any scheme? • We need to make the notions of “diversity gain” and “d.o.f. utilized” precise and enrich them.

13. Classical Diversity Gain Motivation: PAM General Definition: A space-time coding scheme achieves (classical) diversity gain dmax, if for a fixed data rate. i.e. error probability deceases by 1/2dmax for every 3 dB increase in SNR, by 1/4dmax for every 6 dB increase, etc.

14. Example: PAM in 1 by 1Channel Every 6 dB increase in SNR doubles the distance between constellation points for a given rate. Both PAM and QAM have the same (classical) diversity gain of 1. (classical) diversity gain does not say anything about the d.o.f. utilized by the scheme.

15. Ask a Dual Question Every 6 dB doubles the constellation size for a given reliability, for PAM. But for QAM, every 6 dB quadruples the constellation size.

16. Degrees of Freedom Utilized Definition: A space-time coding scheme utilizes rmax degrees of freedom/s/Hz if the data rate scales like for a fixed error probability (reliability). In a 1x1 channel, rmax = 1/2 for PAM, rmax = 1 for QAM. Note: A space-time coding scheme is a family of modulations within a certain structure, with varying symbol alphabet as a function of SNR.

17. Diversity-Multiplexing Tradeoff Every 3 dB increase in SNR yields either a 1/2dmax decrease in error probability for a fixed rate; or rmax additional bits/s/Hz for a fixed reliability. But these are two extremes of a rate-reliability tradeoff. More generally, one wants to increase reliability and the data rate at the same time.

18. Diversity-Multiplexing Tradeoff of A Scheme Definition A space-time coding scheme achieves a diversity-multiplexing tradeoff curve d(r) if for each multiplexing gain r, simultaneously and The largest multiplexing gain is rmax, the d.o.f. utilized by the scheme. The largest diversity gain is dmax = d(0), the classical diversity gain.

19. Diversity-Multiplexing Tradeoff of the Channel Definition The diversity-multiplexing tradeoff d*(r) of a MIMO channel is the best possible diversity-multiplexing tradeoff achievable by any scheme. r*max is the largest multiplexing gain achievable in the channel. d*max= d*(0) is the largest diversity gain achievable. For a nt x nr MIMO channel, it is not difficult to show: What is more interesting is how the entire curve looks like.

20. Example: 1 x 1 Channel

21. Example: 2 x 1 Channel

22. Example: 2 x 2 Channel

23. ML vs Nulling in V-Blast Winters, Salz and Gitlins 93: Nulling out k interferers using nr receive antennas provides a diversity gain of nr-k. Tse,Viswanath and Zheng 03: Jointly detecting all users provides a diversity gain of nr to each. There is free lunch. (?)

24. Optimal D-M Tradeoff for General nt x nr Channel As long as block length For integer r, it is as though r transmit and r receive antennas were dedicated for multiplexing and the rest provide diversity.

25. D-M Tradeoff Optimal Code Design How does one design space-time codes that are tradeoff optimal? Needs a code design criterion.

26. DM-Tradeoff and Outage • So far we have looked at tradeoff for finite block length S-T codes. • For codes with arbitrary long block length, the optimal tradeoff can be computed from the outage probability: • This turns out to be exactly the same as the D-M tradeoff for block length nt. • This suggests that optimal tradeoff codes can be designed from an outage point of view.

27. Univeral Code Design Criterion Simplifies in the high SNR regime. • Parallel channel: product distance of codeword difference (same as Rayleigh!) • MISO channel: smallest singular value of codeword matrix difference. • MIMO channel: product of min(nt,nr) smallest singular values.

28. Outage Revisited • Outage probability: • Operational interpretation: there is a universal code that can reliably communicate whenever H is not in outage. • A pairwise criterion between two codewords can be derived by looking at the worst-case error probability over all H not in outage. • This is in contrast to the classic code design criterion, which is computed by averaging over the channel statistics. • A more robust approach.

29. Achieving Optimal Diversity-Multiplexing Tradeoff • Hao and Wornell 03: MIMO rotation code (2 x 2 channel only) • Tavildar and Viswanath 04: D-Blast plus permutation code. • El Gamal, Caire and Damen 03: Lattice codes.

30. Tavildar and Viswanath 04 • First use D-BLAST to convert the MIMO channel into a parallel channel. • Then design permutation codes to achieve the optimal diversity-multiplexing tradeoff on the parallel channel.

31. D-BLAST Gains g1 and g2 are correlated and have a complicated distribution but universal code design is oblivious to this.

32. Permutation Codes for Parallel Channel It is shown that there are universal permutation codes that can achieve the optimal diversity-multiplexing tradeoff of the parallel channel.

33. Multiple Antennas in Cellular Systems • In a point-to-point link, multiple antennas provide diversity and degrees of freedom gain. • In a cellular systems, they can also perform SDMA and out-of-cell interference suppression. • The D-M tradeoff framework can be extended to incorporate these gains as well.

34. Example: MIMO + SDMA ? Question: what does adding one more antenna at each mobile buy me? No increase in total d.o.f. if the number of users is greater than the number of receive antennas.

35. DM-Tradeoff Analysis • D.o.f. unchanged, but maximum diversity gain increases from n to 2n. • More generally, it improves the diversity gain d(r) for every r.

36. Summary Diversity-multiplexing tradeoff is a unified way to look at performance of MIMO channels. It puts diversity and multiplexing on an equal footing. It provides a framework to compare existing schemes as well as stimulates the design of new schemes.