Capacity Limits of Wireless Channels


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  • Summary of what we’ve done so far
  • Channel on the left is the downlink (BC), channel on the right is the uplink (MAC) – notice dual channels have the same channel gains
  • Caire and Shamai solved the 2 transmit antenna, 2 receiver case with only a single receive antenna – found the achievable region and showed it was optimal at the sum rate point. We consider the general K-user case, where there are an arbitrary number of transmit antennas and arbitrary number of antennas at each receiver – most general scenario. We will develop a duality between the downlink and uplink and show that the achievable region is sum rate optimal.
  • Can transmit to user 2 as if a single-user channel (I.e. as if user 1 does not exist) due to precoding, dirty paper coding. Region found by considering all covariance matrices sigma1 and sigma2, subject to a trace constraint (aka power constraint) on the sum of sigma1 and sigma2. Can also encode in the other order – easily extends to K users.
  • Analogy of relationship between powers for scalar case ( now need covariances because have vector signals) A and B are matrices that represent the interference experiences in the BC and MAC, respectively
  • Analogy of relationship between powers for scalar case ( now need covariances because have vector signals) A and B are matrices that represent the interference experiences in the BC and MAC, respectively
  • If we allow all receivers to cooperate, the system looks like a single-user multiple-antenna link – clearly the capacity of this link is an upper bound to the sum rate capacity of the BC (where the receivers are not allowed to cooperate)
  • In fact, we can make this bound much tighter by introducing noise correlation between the different receivers (that doesn’t affect the BC capacity) – Sato upper bound to the BC
  • In fact, we can make this bound much tighter by introducing noise correlation between the different receivers (that doesn’t affect the BC capacity) – Sato upper bound to the BC
  • Notice that the Sato upper bound is tight (as we proved), and also it is trivial to see that the single-user capacity bounds (i.e. the bounds by considering a single-user channel) are also Tight – not too much room between achievable region and capacity bounds
  • In fact, we can make this bound much tighter by introducing noise correlation between the different receivers (that doesn’t affect the BC capacity) – Sato upper bound to the BC
  • Capacity Limits of Wireless Channels

    1. 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. 2. Future Wireless Systems 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… Ubiquitous Communication Among People and Devices
    3. 3. Challenges <ul><li>The wireless channel is a randomly-varying broadcast medium with limited bandwidth. </li></ul><ul><li>Fundamental capacity limits and good protocol designs for wireless networks are open problems. </li></ul><ul><li>Hard energy and delay constraints change fundamental design principles </li></ul><ul><li>Many applications fail miserably with a “generic” network approach: need for crosslayer design </li></ul>
    4. 4. Outline <ul><li>Wireless Channel Capacity </li></ul><ul><li>Capacity of MIMO Channels </li></ul><ul><ul><li>Imperfect channel information </li></ul></ul><ul><ul><li>Channel correlations </li></ul></ul><ul><li>Multiuser MIMO Channels </li></ul><ul><ul><li>Duality and Dirty Paper Coding </li></ul></ul><ul><li>Lyapunov Exponents and Capacity </li></ul>
    5. 5. Wireless Channel Capacity Fundamental Limit on Data Rates <ul><li>Main drivers of channel capacity </li></ul><ul><ul><li>Bandwidth and power </li></ul></ul><ul><ul><li>Statistics of the channel </li></ul></ul><ul><ul><li>Channel knowledge and how it is used </li></ul></ul><ul><ul><li>Number of antennas at TX and RX </li></ul></ul>Capacity: The set of simultaneously achievable rates {R 1 ,…,R n } R 1 R 2 R 3 R 1 R 2 R 3
    6. 6. MIMO Channel Model x 1 x 2 x 3 y 1 y 2 y 3 h 11 h 21 h 31 h 12 h 22 h 32 h 13 h 23 h 33 Model applies to any channel described by a matrix (e.g. ISI channels) n TX antennas m RX antennas
    7. 7. What’s so great about MIMO? <ul><li>Fantastic capacity gains (Foschini/Gans’96, Telatar’99) </li></ul><ul><ul><li>Capacity grows linearly with antennas when channel known perfectly at Tx and Rx </li></ul></ul><ul><li>Vector codes (or scalar codes with SIC) optimal </li></ul><ul><li>Assumptions: </li></ul><ul><ul><li>Perfect channel knowledge </li></ul></ul><ul><ul><li>Spatially uncorrelated fading: Rank ( H T QH )=min(n,m) </li></ul></ul>What happens when these assumptions are relaxed?
    8. 8. Realistic Assumptions <ul><li>No transmitter knowledge of H </li></ul><ul><ul><li>Capacity is much smaller </li></ul></ul><ul><li>No receiver knowledge of H </li></ul><ul><ul><li>Capacity does not increase as the number of antennas increases (Marzetta/Hochwald’99) </li></ul></ul><ul><li>Will the promise of MIMO be realized in practice? </li></ul>
    9. 9. Partial Channel Knowledge <ul><li>Model channel as H~N(  ,  ) </li></ul><ul><li>Receiver knows channel H perfectly </li></ul><ul><li>Transmitter has partial information  about H </li></ul>Channel Receiver Transmitter
    10. 10. Partial Information Models <ul><li>Channel mean information </li></ul><ul><ul><li>Mean is measured, Covariance unknown </li></ul></ul><ul><li>Channel covariance information </li></ul><ul><ul><li>Mean unknown, measure covariance </li></ul></ul><ul><li>We have developed necessary and sufficient conditions for the optimality of beamforming </li></ul><ul><ul><li>Obtained for both MISO and MIMO channels </li></ul></ul><ul><ul><li>Optimal transmission strategy also known </li></ul></ul>
    11. 11. Beamforming <ul><li>Scalar codes with transmit precoding </li></ul>Receiver <ul><li>Transforms the MIMO system into a SISO system. </li></ul><ul><li>Greatly simplifies encoding and decoding. </li></ul><ul><li>Channel indicates the best direction to beamform </li></ul><ul><ul><li>Need “sufficient” knowledge for optimality </li></ul></ul>
    12. 12. Optimality of Beamforming Mean Information
    13. 13. Optimality of Beamforming Covariance Information
    14. 14. No Tx or Rx Knowledge <ul><li>Increasing n T beyond coherence time  T in a block fading channel does not increase capacity (Marzetta/Hochwald’99) </li></ul><ul><ul><li>Assumes uncorrelated fading. </li></ul></ul><ul><li>We have shown that with correlated fading, adding Tx antennas always increases capacity </li></ul><ul><ul><li>Small transmit antenna spacing is good! </li></ul></ul><ul><li>Impact of spatial correlations on channel capacity </li></ul><ul><ul><li>Perfect Rx and Tx knowledge: hurts (Boche/Jorswieck’03) </li></ul></ul><ul><ul><li>Perfect Rx knowledge, no Tx knowledge: hurts (BJ’03) </li></ul></ul><ul><ul><li>Perfect Rx knowledge, Tx knows correlation: helps </li></ul></ul><ul><ul><li>TX and Rx only know correlation: helps </li></ul></ul>
    15. 15. Gaussian Broadcast and Multiple Access Channels • Transmit power constraint • Perfect Tx and Rx knowledge Broadcast (BC): One Transmitter to Many Receivers. Multiple Access (MAC): Many Transmitters to One Receiver. x h 1 (t) x h 21 (t) x h 3 (t) x h 22 (t)
    16. 16. <ul><li>Differences: </li></ul><ul><ul><li>Shared vs. individual power constraints </li></ul></ul><ul><ul><li>Near-far effect in MAC </li></ul></ul><ul><li>Similarities: </li></ul><ul><ul><li>Optimal BC “superposition” coding is also optimal for MAC (sum of Gaussian codewords) </li></ul></ul><ul><ul><li>Both decoders exploit successive decoding and interference cancellation </li></ul></ul>Comparison of MAC and BC P P 1 P 2
    17. 17. MAC-BC Capacity Regions <ul><li>MAC capacity region known for many cases </li></ul><ul><ul><li>Convex optimization problem </li></ul></ul><ul><li>BC capacity region typically only known for (parallel) degraded channels </li></ul><ul><ul><li>Formulas often not convex </li></ul></ul><ul><li>Can we find a connection between the BC and MAC capacity regions? </li></ul>Duality
    18. 18. Dual Broadcast and MAC Channels x x + x x + + Gaussian BC and MAC with same channel gains and same noise power at each receiver Broadcast Channel (BC) Multiple-Access Channel (MAC)
    19. 19. The BC from the MAC Blue = BC Red = MAC P 1 =1, P 2 =1 MAC with sum-power constraint P 1 =1.5, P 2 =0.5 P 1 =0.5, P 2 =1.5
    20. 20. Sum-Power MAC <ul><li>MAC with sum power constraint </li></ul><ul><ul><li>Power pooled between MAC transmitters </li></ul></ul><ul><ul><li>No transmitter coordination </li></ul></ul>MAC BC Same capacity region!
    21. 21. BC to MAC: Channel Scaling <ul><li>Scale channel gain by  , power by 1/  </li></ul><ul><li>MAC capacity region unaffected by scaling </li></ul><ul><li>Scaled MAC capacity region is a subset of the scaled BC capacity region for any  </li></ul><ul><li>MAC region inside scaled BC region for any  scaling </li></ul>+ + + MAC BC
    22. 22. The BC from the MAC Blue = Scaled BC Red = MAC
    23. 23. <ul><li>BC in terms of MAC </li></ul><ul><li>MAC in terms of BC </li></ul>Duality: Constant AWGN Channels What is the relationship between the optimal transmission strategies?
    24. 24. <ul><li>Equate rates, solve for powers </li></ul><ul><li>Opposite decoding order </li></ul><ul><ul><li>Stronger user (User 1) decoded last in BC </li></ul></ul><ul><ul><li>Weaker user (User 2) decoded last in MAC </li></ul></ul>Transmission Strategy Transformations
    25. 25. Duality Applies to Different Fading Channel Capacities <ul><li>Ergodic (Shannon) capacity: maximum rate averaged over all fading states. </li></ul><ul><li>Zero-outage capacity: maximum rate that can be maintained in all fading states. </li></ul><ul><li>Outage capacity: maximum rate that can be maintained in all nonoutage fading states. </li></ul><ul><li>Minimum rate capacity: Minimum rate maintained in all states, maximize average rate in excess of minimum </li></ul><ul><ul><li>Explicit transformations between transmission strategies </li></ul></ul>
    26. 26. Duality: Minimum Rate Capacity <ul><ul><li>BC region known </li></ul></ul><ul><ul><li>MAC region can only be obtained by duality </li></ul></ul>Blue = Scaled BC Red = MAC MAC in terms of BC What other unknown capacity regions can be obtained by duality?
    27. 27. Dirty Paper Coding (Costa’83) <ul><li>Basic premise </li></ul><ul><ul><li>If the interference is known, channel capacity same as if there is no interference </li></ul></ul><ul><ul><li>Accomplished by cleverly distributing the writing (codewords) and coloring their ink </li></ul></ul><ul><ul><li>Decoder must know how to read these codewords </li></ul></ul>Dirty Paper Coding Clean Channel Dirty Channel Dirty Paper Coding
    28. 28. Modulo Encoding/Decoding <ul><li>Received signal Y=X+S, -1  X  1 </li></ul><ul><ul><li>S known to transmitter, not receiver </li></ul></ul><ul><li>Modulo operation removes the interference effects </li></ul><ul><ul><li>Set X so that  Y  [-1,1] =desired message (e.g. 0.5) </li></ul></ul><ul><ul><li>Receiver demodulates modulo [-1,1] </li></ul></ul>-1 +3 +5 +1 -3 … -5 0 S +7 -7 … -1 +1 0 -1 +1 0 X
    29. 29. Broadcast MIMO Channel t  1 TX antennas r 1  1 , r 2  1 RX antennas Non-degraded broadcast channel Perfect CSI at TX and RX
    30. 30. Capacity Results <ul><li>Non-degraded broadcast channel </li></ul><ul><ul><li>Receivers not necessarily “better” or “worse” due to multiple transmit/receive antennas </li></ul></ul><ul><ul><li>Capacity region for general case unknown </li></ul></ul><ul><li>Pioneering work by Caire/Shamai (Allerton’00): </li></ul><ul><ul><li>Two TX antennas/two RXs (1 antenna each) </li></ul></ul><ul><ul><li>Dirty paper coding/lattice precoding * </li></ul></ul><ul><ul><ul><li>Computationally very complex </li></ul></ul></ul><ul><ul><li>MIMO version of the Sato upper bound </li></ul></ul>* Extended by Yu/Cioffi
    31. 31. Dirty-Paper Coding (DPC) for MIMO BC <ul><li>Coding scheme: </li></ul><ul><ul><li>Choose a codeword for user 1 </li></ul></ul><ul><ul><li>Treat this codeword as interference to user 2 </li></ul></ul><ul><ul><li>Pick signal for User 2 using “pre-coding” </li></ul></ul><ul><li>Receiver 2 experiences no interference: </li></ul><ul><li>Signal for Receiver 2 interferes with Receiver 1: </li></ul><ul><li>Encoding order can be switched </li></ul>
    32. 32. Dirty Paper Coding in Cellular
    33. 33. Does DPC achieve capacity? <ul><li>DPC yields MIMO BC achievable region. </li></ul><ul><ul><li>We call this the dirty-paper region </li></ul></ul><ul><li>Is this region the capacity region? </li></ul><ul><li>We use duality, dirty paper coding, and Sato’s upper bound to address this question </li></ul>
    34. 34. MIMO MAC with sum power <ul><li>MAC with sum power: </li></ul><ul><ul><li>Transmitters code independently </li></ul></ul><ul><ul><li>Share power </li></ul></ul><ul><li>Theorem: Dirty-paper BC region equals the dual sum-power MAC region </li></ul>P
    35. 35. Transformations: MAC to BC <ul><li>Show any rate achievable in sum-power MAC also achievable with DPC for BC: </li></ul><ul><ul><li>A sum-power MAC strategy for point (R 1 ,…R N ) has a given input covariance matrix and encoding order </li></ul></ul><ul><ul><li>We find the corresponding PSD covariance matrix and encoding order to achieve (R 1 ,…,R N ) with DPC on BC </li></ul></ul><ul><ul><ul><li>The rank-preserving transform “flips the effective channel” and reverses the order </li></ul></ul></ul><ul><ul><ul><li>Side result: beamforming is optimal for BC with 1 Rx antenna at each mobile </li></ul></ul></ul>DPC BC Sum MAC
    36. 36. Transformations: BC to MAC <ul><li>Show any rate achievable with DPC in BC also achievable in sum-power MAC: </li></ul><ul><ul><li>We find transformation between optimal DPC strategy and optimal sum-power MAC strategy </li></ul></ul><ul><ul><ul><li>“ Flip the effective channel” and reverse order </li></ul></ul></ul>DPC BC Sum MAC
    37. 37. Computing the Capacity Region <ul><li>Hard to compute DPC region (Caire/Shamai’00) </li></ul><ul><li>“ Easy” to compute the MIMO MAC capacity region </li></ul><ul><ul><li>Obtain DPC region by solving for sum-power MAC and applying the theorem </li></ul></ul><ul><ul><li>Fast iterative algorithms have been developed </li></ul></ul><ul><ul><li>Greatly simplifies calculation of the DPC region and the associated transmit strategy </li></ul></ul>
    38. 38.  Based on receiver cooperation  BC sum rate capacity  Cooperative capacity Sato Upper Bound on the BC Capacity Region + + Joint receiver
    39. 39. The Sato Bound for MIMO BC <ul><li>Introduce noise correlation between receivers </li></ul><ul><li>BC capacity region unaffected </li></ul><ul><ul><li>Only depends on noise marginals </li></ul></ul><ul><li>Tight Bound (Caire/Shamai’00) </li></ul><ul><ul><li>Cooperative capacity with worst-case noise correlation </li></ul></ul><ul><li>Explicit formula for worst-case noise covariance </li></ul><ul><li>By Lagrangian duality, cooperative BC region equals the sum-rate capacity region of MIMO MAC </li></ul>
    40. 40. Sum-Rate Proof DPC Achievable Lagrangian Duality Obvious Duality Sato Bound Compute from MAC *Same result by Vishwanath/Tse for 1 Rx antenna
    41. 41. MIMO BC Capacity Bounds Sato Upper Bound Single User Capacity Bounds Dirty Paper Achievable Region BC Sum Rate Point Does the DPC region equal the capacity region?
    42. 42. Full Capacity Region <ul><li>DPC gives us an achievable region </li></ul><ul><li>Sato bound only touches at sum-rate point </li></ul><ul><li>We need a tighter bound to prove DPC is optimal </li></ul>
    43. 43. A Tighter Upper Bound <ul><li>Give data of one user to other users </li></ul><ul><ul><li>Channel becomes a degraded BC </li></ul></ul><ul><ul><li>Capacity region for degraded BC known </li></ul></ul><ul><ul><li>Tight upper bound on original channel capacity </li></ul></ul><ul><li>This bound and duality prove that DPC achieves capacity under a Gaussian input restriction </li></ul><ul><ul><li>Remains to be shown that Gaussian inputs are optimal </li></ul></ul>+ +
    44. 44. Full Capacity Region Proof Tight Upper Bound Worst Case Noise Diagonalizes Duality Final Result Duality Compute from MAC
    45. 45. Time-varying Channels with Memory <ul><li>Time-varying channels with finite memory induce infinite memory in the channel output. </li></ul><ul><li>Capacity for time-varying infinite memory channels is only known in terms of a limit </li></ul><ul><li>Closed-form capacity solutions only known in a few cases </li></ul><ul><ul><li>Gilbert/Elliot and Finite State Markov Channels </li></ul></ul>
    46. 46. A New Characterization of Channel Capacity <ul><li>Capacity using Lyapunov exponents </li></ul><ul><li>Similar definitions hold for  (Y) and  (X;Y) </li></ul><ul><ul><li>Matrices B Y i and B X i Y i depend on input and channel </li></ul></ul>where the Lyapunov exponent for B X i a random matrix whose entries depend on the input symbol X i
    47. 47. Lyapunov Exponents and Entropy <ul><li>Lyapunov exponent equals entropy under certain conditions </li></ul><ul><ul><li>Entropy as a product of random matrices </li></ul></ul><ul><ul><li>Connection between IT and dynamic systems theory </li></ul></ul><ul><li>Still have a limiting expression for entropy </li></ul><ul><ul><li>Sample entropy has poor convergence properties </li></ul></ul>
    48. 48. Lyapunov Direction Vector <ul><li>The vector p n is the “direction” associated with  (X) for any  . </li></ul><ul><ul><li>Also defines the conditional channel state probability </li></ul></ul><ul><li>Vector has a number of interesting properties </li></ul><ul><ul><li>It is the standard prediction filter in hidden Markov models </li></ul></ul><ul><ul><li>Under certain conditions we can use its stationary distribution to directly compute  (X)  (X) </li></ul></ul>
    49. 49. Computing Lyapunov Exponents <ul><li>Define  as the stationary distribution of the “direction vector” p n p n </li></ul><ul><li>We prove that we can compute these Lyapunov exponents in closed form as </li></ul><ul><li>This result is a significant advance in the theory of Lyapunov exponent computation </li></ul> p n p n+1 p n+2
    50. 50. Computing Capacity <ul><li>Closed-form formula for mutual information </li></ul><ul><li>We prove continuity of the Lyapunov exponents with respect to input distribution and channel </li></ul><ul><ul><li>Can thus maximize mutual information relative to channel input distribution to get capacity </li></ul></ul><ul><ul><li>Numerical results for time-varying SISO and MIMO channel capacity have been obtained </li></ul></ul><ul><li>We also develop a new CLT and confidence interval methodology for sample entropy </li></ul>
    51. 51. Sensor Networks <ul><ul><li>Energy is a driving constraint. </li></ul></ul><ul><ul><li>Data flows to centralized location. </li></ul></ul><ul><ul><li>Low per-node rates but up to 100,000 nodes. </li></ul></ul><ul><ul><li>Data highly correlated in time and space. </li></ul></ul><ul><ul><li>Nodes can cooperate in transmission and reception. </li></ul></ul>
    52. 52. Energy-Constrained Network Design <ul><li>Each node can only send a finite number of bits </li></ul><ul><ul><li>Transmit energy per bit minimized by sending each bit over many dimensions (time/bandwidth product) </li></ul></ul><ul><ul><li>Delay vs. energy tradeoffs for each bit </li></ul></ul><ul><li>Short-range networks must consider both transmit, analog HW, and processing energy </li></ul><ul><ul><li>Sophisticated techniques for modulation, coding, etc., not necessarily energy-efficient </li></ul></ul><ul><ul><li>Sleep modes save energy but complicate networking </li></ul></ul><ul><li>New network design paradigm: </li></ul><ul><ul><li>Bit allocation must be optimized across all protocols </li></ul></ul><ul><ul><li>Delay vs. throughput vs. node/network lifetime tradeoffs </li></ul></ul><ul><ul><li>Optimization of node cooperation (coding, MIMO, etc.) </li></ul></ul>
    53. 53. Results to Date <ul><li>Modulation Optimization </li></ul><ul><ul><li>Adaptive MQAM vs. MFSK for given delay and rate </li></ul></ul><ul><ul><li>Takes into account RF hardware/processing tradeoffs </li></ul></ul><ul><li>MIMO vs. MISO vs. SISO for constrained energy </li></ul><ul><ul><li>SISO has best performance at short distances (<100m) </li></ul></ul><ul><li>Optimal Adaptation with Delay/Energy Constraints </li></ul><ul><li>Minimum Energy Routing </li></ul>
    54. 54. Conclusions <ul><li>Shannon capacity gives fundamental data rate limits for wireless channels </li></ul><ul><li>Many open capacity problems for time-varying multiuser MIMO channels </li></ul><ul><li>Duality and dirty paper coding are powerful tools to solve new capacity problems and simplify computation </li></ul><ul><li>Lyapunov exponents a powerful new tool for solving capacity problems </li></ul><ul><li>Cooperative communications in sensor networks is an interesting new area of research </li></ul>