1. Capacity of multi-antenna Gaussian Channels, I. E. Telatar By: Imad Jabbour MIT 6.441 May 11, 2006

2. MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour Introduction MIMO systems in wireless comm. Recently subject of extensive research Can significantly increase data rates and reduce BER Telatar’s paper Bell Labs (1995) Information-theoretic aspect of single-user MIMO systems Classical paper in the field

3. MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour Preliminaries Wireless fading scalar channel DT Representation: H is the complex channel fading coefficient W is the complex noise, Rayleigh fading:, such that | H | is Rayleigh distributed Circularly-symmetric Gaussian i.i.d. real and imaginary parts Distribution invariant to rotations

4. MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour MIMO Channel Model (1) I/O relationship Design parameters ot Tx. antennas and r Rx. antennas oFading matrix oNoise Power constraint: Assumption H known at Rx. (CSIR)

5. MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour MIMO Channel Model (2) System representation Telatar: the fading matrix H can be Deterministic Random and changes over time Random, but fixed once chosen Transmitter Receiver

6. MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour Deterministic Fading Channel (1) Fading matrix is not random Known to both Tx. and Rx. Idea: Convert vector channel to a parallel one Singular value decomposition of H SVD:, for U and V unitary, and D diagonal Equivalent system:, where Entries of D are the singular values of H oThere are singular values

7. MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour Deterministic Fading Channel (2) Equivalent parallel channel [ n min =min(r,t) ] Tx. must know H to pre-process it, and Rx. must know H to post-process it

8. MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour Deterministic Fading Channel (3) Result of SVD Parallel channel with sub-channels Water-filling maximizes capacity Capacity is oOptimal power allocation o is chosen to meet total power constraint

9. MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour Random Varying Channel (1) Random channel matrix H Independent of both X and W, and memoryless Matrix entries Fast fading Channel varies much faster than delay requirement Coherence time (T c ): period of variation of channel

10. MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour Random Varying Channel (2) Information-theoretic aspect Codeword length should average out both additive noise and channel fluctuations Assume that Rx. tracks channel perfectly Capacity is Equal power allocation at Tx. Can show that At high power, C scales linearly with n min Results also apply for any ergodic H

11. MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour Random Varying Channel (3) MIMO capacity versus SNR (from [2])

12. MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour Random Fixed Channel (1) Slow fading Channel varies much slower than delay requirement H still random, but is constant over transmission duration of codeword What is the capacity of this channel? Non-zero probability that realization of H does not support the data rate In this sense, capacity is zero!

13. MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour Random Fixed Channel (2) Telatar’s solution: outage probability p out p out is probability that R is greater that maximum achievable rate Alternative performance measure is oLargest R for which oOptimal power allocation is equal allocation across only a subset of the Tx. antennas.

14. MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour Discussion and Analysis (1) What’s missing in the picture? If H is unknown at Tx., cannot do SVD oSolution: V-BLAST If H is known at Tx. also (full CSI) oPower gain over CSIR If H is unknown at both Tx. and Rx (non- coherent model) oAt high SNR, solution given by Marzetta & Hochwald, and Zheng Receiver architectures to achieve capacity Other open problems

15. MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour Discussion and Analysis (2) If H unknown at Tx. Idea: multiplex in an arbitrary coordinate system B, and do joint ML decoding at Rx. V-BLAST architecture can achieve capacity

16. MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour Discussion and Analysis (3) If varying H known at Tx. (full CSI) Solution is now water-filling over space and time Can show optimal power allocation is P/n min Capacity is What are we gaining? oPower gain of n t /n min as compared to CSIR case

17. MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour Discussion and Analysis (4) If H unknown at both Rx. and Tx. Non-coherent channel: channel changes very quickly so that Rx. can no more track it Block fading model At high SNR, capacity gain is equal to (Zheng)

18. MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour Discussion and Analysis (5) Receiver architectures [2] V-BLAST can achieve capacity for fast Rayleigh-fading channels Caveat: Complexity of joint decoding Solution: simpler linear decoders oZero-forcing receiver (decorrelator) oMMSE receiver oMMSE can achieve capacity if SIC is used

19. MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour Discussion and Analysis (6) Open research topics Alternative fading models Diversity/multiplexing tradeoff (Zheng & Tse) Conclusion MIMO can greatly increase capacity For coherent high SNR, How many antennas are we using? Can we “beat” the AWGN capacity?

20. Thank you! Any questions?