Capacity of multi-antenna Gaussian Channels, I. E. Telatar

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

2. 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 MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

3. Preliminaries • Wireless fading scalar channel • DT Representation: • H is the complex channel fadingcoefficient • 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 MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

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

5. 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 MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

6. Deterministic Fading Channel (1) • Fading matrix is not random • Known to bothTx. 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 • There are singular values MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

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

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

9. 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 (Tc): period of variation of channel MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

10. 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 nmin • Results also apply for any ergodic H MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

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

12. 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! MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

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

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

15. 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 MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

16. 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/nmin • Capacity is • What are we gaining? • Power gain of nt/nmin as compared to CSIR case MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

17. 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) MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

18. 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 • Zero-forcing receiver (decorrelator) • MMSE receiver • MMSE can achieve capacity if SIC is used MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

19. 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? MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour

20. Thank you! Any questions?