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Andrea Goldsmith Stanford University

Capacity Limits of Wireless Channels with Multiple Antennas: Challenges, Insights, and New Mathematical Methods. Andrea Goldsmith Stanford University. CoAuthors: T. Holliday, S. Jafar, N. Jindal, S. Vishwanath. Princeton-Rutgers Seminar Series Rutgers University April 23, 2003.

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Andrea Goldsmith Stanford University

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  1. Capacity Limits of Wireless Channels with Multiple Antennas:Challenges, Insights, and New Mathematical Methods Andrea Goldsmith Stanford University CoAuthors: T. Holliday, S. Jafar, N. Jindal, S. Vishwanath Princeton-Rutgers Seminar Series Rutgers University April 23, 2003

  2. Future Wireless Systems Ubiquitous Communication Among People and Devices Nth Generation Cellular Nth Generation WLANs Wireless Entertainment Wireless Ad Hoc Networks Sensor Networks Smart Homes/Appliances Automated Cars/Factories Telemedicine/Learning All this and more…

  3. Challenges • The wireless channel is a randomly-varying broadcast medium with limited bandwidth. • Fundamental capacity limits and good protocol designs for wireless networks are open problems. • Hard energy and delay constraints change fundamental design principles • Many applications fail miserably with a “generic” network approach: need for crosslayer design

  4. Outline • Wireless Channel Capacity • Capacity of MIMO Channels • Imperfect channel information • Channel correlations • Multiuser MIMO Channels • Duality and Dirty Paper Coding • Lyapunov Exponents and Capacity

  5. Wireless Channel CapacityFundamental Limit on Data Rates Capacity: The set of simultaneously achievable rates {R1,…,Rn} • Main drivers of channel capacity • Bandwidth and power • Statistics of the channel • Channel knowledge and how it is used • Number of antennas at TX and RX R3 R2 R3 R2 R1 R1

  6. MIMO Channel Model n TX antennas m RX antennas h11 x1 y1 h12 h21 h31 h22 x2 y2 h32 h13 h23 h33 x3 y3 Model applies to any channel described by a matrix (e.g. ISI channels)

  7. What’s so great about MIMO? • Fantastic capacity gains (Foschini/Gans’96, Telatar’99) • Capacity grows linearly with antennas when channel known perfectly at Tx and Rx • Vector codes (or scalar codes with SIC) optimal • Assumptions: • Perfect channel knowledge • Spatially uncorrelated fading: Rank (HTQH)=min(n,m) What happens when these assumptions are relaxed?

  8. Realistic Assumptions • No transmitter knowledge of H • Capacity is much smaller • No receiver knowledge of H • Capacity does not increase as the number of antennas increases (Marzetta/Hochwald’99) • Will the promise of MIMO be realized in practice?

  9. Partial Channel Knowledge Channel • Model channel as H~N(m,S) • Receiver knows channel H perfectly • Transmitter has partial information q about H Transmitter Receiver

  10. Partial Information Models • Channel mean information • Mean is measured, Covariance unknown • Channel covariance information • Mean unknown, measure covariance • We have developed necessary and sufficient conditions for the optimality of beamforming • Obtained for both MISO and MIMO channels • Optimal transmission strategy also known

  11. Beamforming • Scalar codes with transmit precoding Receiver • Transforms the MIMO system into a SISO system. • Greatly simplifies encoding and decoding. • Channel indicates the best direction to beamform • Need “sufficient” knowledge for optimality

  12. Optimality of BeamformingMean Information

  13. Optimality of BeamformingCovariance Information

  14. No Tx or Rx Knowledge • Increasing nT beyond coherence time aT in a block fading channel does not increase capacity (Marzetta/Hochwald’99) • Assumes uncorrelated fading. • We have shown that with correlated fading, adding Tx antennas always increases capacity • Small transmit antenna spacing is good! • Impact of spatial correlations on channel capacity • Perfect Rx and Tx knowledge: hurts (Boche/Jorswieck’03) • Perfect Rx knowledge, no Tx knowledge: hurts (BJ’03) • Perfect Rx knowledge, Tx knows correlation: helps • TX and Rx only know correlation: helps

  15. Broadcast (BC): One Transmitter to Many Receivers. Multiple Access (MAC): Many Transmitters to One Receiver. x x x x h1(t) h21(t) h3(t) h22(t) Gaussian Broadcast and Multiple Access Channels • Transmit power constraint • Perfect Tx and Rx knowledge

  16. Comparison of MAC and BC P • Differences: • Shared vs. individual power constraints • Near-far effect in MAC • Similarities: • Optimal BC “superposition” coding is also optimal for MAC (sum of Gaussian codewords) • Both decoders exploit successive decoding and interference cancellation P1 P2

  17. MAC-BC Capacity Regions • MAC capacity region known for many cases • Convex optimization problem • BC capacity region typically only known for (parallel) degraded channels • Formulas often not convex • Can we find a connection between the BC and MAC capacity regions? Duality

  18. Dual Broadcast and MAC Channels Gaussian BC and MAC with same channel gains and same noise power at each receiver x + x + x x + Multiple-Access Channel (MAC) Broadcast Channel (BC)

  19. P1=0.5, P2=1.5 P1=1.5, P2=0.5 The BC from the MAC P1=1, P2=1 Blue = BC Red = MAC MAC with sum-power constraint

  20. Sum-Power MAC • MAC with sum power constraint • Power pooled between MAC transmitters • No transmitter coordination Same capacity region! BC MAC

  21. BC to MAC: Channel Scaling • Scale channel gain by a, power by 1/a • MAC capacity region unaffected by scaling • Scaled MAC capacity region is a subset of the scaled BC capacity region for any a • MAC region inside scaled BC region for anyscaling MAC + + + BC

  22. The BC from the MAC Blue = Scaled BC Red = MAC

  23. Duality: Constant AWGN Channels • BC in terms of MAC • MAC in terms of BC What is the relationship between the optimal transmission strategies?

  24. Transmission Strategy Transformations • Equate rates, solve for powers • Opposite decoding order • Stronger user (User 1) decoded last in BC • Weaker user (User 2) decoded last in MAC

  25. Duality Applies to DifferentFading Channel Capacities • Ergodic (Shannon) capacity: maximum rate averaged over all fading states. • Zero-outage capacity: maximum rate that can be maintained in all fading states. • Outage capacity: maximum rate that can be maintained in all nonoutage fading states. • Minimum rate capacity: Minimum rate maintained in all states, maximize average rate in excess of minimum Explicit transformations between transmission strategies

  26. Duality: Minimum Rate Capacity MAC in terms of BC Blue = Scaled BC Red = MAC • BC region known • MAC region can only be obtained by duality What other unknown capacity regions can be obtained by duality?

  27. Dirty Paper Coding (Costa’83) • Basic premise • If the interference is known, channel capacity same as if there is no interference • Accomplished by cleverly distributing the writing (codewords) and coloring their ink • Decoder must know how to read these codewords Dirty Paper Coding Dirty Paper Coding Clean Channel Dirty Channel

  28. -1 0 +1 X -1 0 +1 Modulo Encoding/Decoding • Received signal Y=X+S, -1X1 • S known to transmitter, not receiver • Modulo operation removes the interference effects • Set X so that Y[-1,1]=desired message (e.g. 0.5) • Receiver demodulates modulo [-1,1] … … -5 -3 -1 0 +1 +3 +5 +7 -7 S

  29. Broadcast MIMO Channel t1 TX antennas r11, r21 RX antennas Perfect CSI at TX and RX Non-degraded broadcast channel

  30. Capacity Results • Non-degraded broadcast channel • Receivers not necessarily “better” or “worse” due to multiple transmit/receive antennas • Capacity region for general case unknown • Pioneering work by Caire/Shamai (Allerton’00): • Two TX antennas/two RXs (1 antenna each) • Dirty paper coding/lattice precoding* • Computationally very complex • MIMO version of the Sato upper bound *Extended by Yu/Cioffi

  31. Dirty-Paper Coding (DPC)for MIMO BC • Coding scheme: • Choose a codeword for user 1 • Treat this codeword as interference to user 2 • Pick signal for User 2 using “pre-coding” • Receiver 2 experiences no interference: • Signal for Receiver 2 interferes with Receiver 1: • Encoding order can be switched

  32. Dirty Paper Coding in Cellular

  33. Does DPC achieve capacity? • DPC yields MIMO BC achievable region. • We call this the dirty-paper region • Is this region the capacity region? • We use duality, dirty paper coding, and Sato’s upper bound to address this question

  34. MIMO MAC with sum power • MAC with sum power: • Transmitters code independently • Share power • Theorem: Dirty-paper BC region equals the dual sum-power MAC region P

  35. Transformations: MAC to BC • Show any rate achievable in sum-power MAC also achievable with DPC for BC: • A sum-power MAC strategy for point (R1,…RN) has a given input covariance matrix and encoding order • We find the corresponding PSD covariance matrix and encoding order to achieve (R1,…,RN) with DPC on BC • The rank-preserving transform “flips the effective channel” and reverses the order • Side result: beamforming is optimal for BC with 1 Rx antenna at each mobile DPC BC Sum MAC

  36. Transformations: BC to MAC • Show any rate achievable with DPC in BC also achievable in sum-power MAC: • We find transformation between optimal DPC strategy and optimal sum-power MAC strategy • “Flip the effective channel” and reverse order DPC BC Sum MAC

  37. Computing the Capacity Region • Hard to compute DPC region (Caire/Shamai’00) • “Easy” to compute the MIMO MAC capacity region • Obtain DPC region by solving for sum-power MAC and applying the theorem • Fast iterative algorithms have been developed • Greatly simplifies calculation of the DPC region and the associated transmit strategy

  38. Sato Upper Bound on the BC Capacity Region  Based on receiver cooperation  BC sum rate capacity  Cooperative capacity + Joint receiver +

  39. The Sato Bound for MIMO BC • Introduce noise correlation between receivers • BC capacity region unaffected • Only depends on noise marginals • Tight Bound (Caire/Shamai’00) • Cooperative capacity with worst-casenoise correlation • Explicit formula for worst-case noise covariance • By Lagrangian duality, cooperative BC region equals the sum-rate capacity region of MIMO MAC

  40. Sum-Rate Proof DPC Achievable Duality Obvious Sato Bound *Same result by Vishwanath/Tse for 1 Rx antenna Lagrangian Duality Compute from MAC

  41. MIMO BC Capacity Bounds Single User Capacity Bounds Dirty Paper Achievable Region BC Sum Rate Point Sato Upper Bound Does the DPC region equal the capacity region?

  42. Full Capacity Region • DPC gives us an achievable region • Sato bound only touches at sum-rate point • We need a tighter bound to prove DPC is optimal

  43. A Tighter Upper Bound • Give data of one user to other users • Channel becomes a degraded BC • Capacity region for degraded BC known • Tight upper bound on original channel capacity • This bound and duality prove that DPC achieves capacity under a Gaussian input restriction • Remains to be shown that Gaussian inputs are optimal + +

  44. Full Capacity Region Proof Tight Upper Bound Final Result Duality Duality Compute from MAC Worst Case Noise Diagonalizes

  45. Time-varying Channels with Memory • Time-varying channels with finite memory induce infinite memory in the channel output. • Capacity for time-varying infinite memory channels is only known in terms of a limit • Closed-form capacity solutions only known in a few cases • Gilbert/Elliot and Finite State Markov Channels

  46. A New Characterization of Channel Capacity • Capacity using Lyapunov exponents • Similar definitions hold for l(Y) and l(X;Y) • Matrices BYi and BXiYidepend on input and channel where the Lyapunov exponent for BXi a random matrix whose entries depend on the input symbol Xi

  47. Lyapunov Exponents and Entropy • Lyapunov exponent equals entropy under certain conditions • Entropy as a product of random matrices • Connection between IT and dynamic systems theory • Still have a limiting expression for entropy • Sample entropy has poor convergence properties

  48. Lyapunov Direction Vector • The vector pn is the “direction” associated with l(X) for any m. • Also defines the conditional channel state probability • Vector has a number of interesting properties • It is the standard prediction filter in hidden Markov models • Under certain conditions we can use its stationary distribution to directly compute l(X) l(X)

  49. Computing Lyapunov Exponents • Define p as the stationary distribution of the “direction vector” pnpn • We prove that we can compute these Lyapunov exponents in closed form as • This result is a significant advance in the theory of Lyapunov exponent computation p pn+2 pn pn+1

  50. Computing Capacity • Closed-form formula for mutual information • We prove continuity of the Lyapunov exponents with respect to input distribution and channel • Can thus maximize mutual information relative to channel input distribution to get capacity • Numerical results for time-varying SISO and MIMO channel capacity have been obtained • We also develop a new CLT and confidence interval methodology for sample entropy

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