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

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

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

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

Wireless Channel Capacity Fundamental Limit on Data Rates 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 Capacity: The set of simultaneously achievable rates {R 1,…,R n } R1R1 R2R2 R3R3 R1R1 R2R2 R3R3

MIMO Channel Model x1x1 x2x2 x3x3 y1y1 y2y2 y3y3 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

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 ( H T QH )=min(n,m) What happens when these assumptions are relaxed?

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?

Partial Channel Knowledge Model channel as H~N( ,  ) Receiver knows channel H perfectly Transmitter has partial information  about H Channel Receiver Transmitter

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

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

Optimality of Beamforming Mean Information

Optimality of Beamforming Covariance Information

No Tx or Rx Knowledge Increasing n T beyond coherence time  T 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

Gaussian Broadcast and Multiple Access Channels 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) Transmit power constraint Perfect Tx and Rx knowledge x h 22 (t)

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 Comparison of MAC and BC P P1P1 P2P2

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

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)

The BC from the MAC Blue = BC Red = MAC P 1 =1, P 2 =1 P 1 =1.5, P 2 =0.5 P 1 =0.5, P 2 =1.5 MAC with sum-power constraint

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

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

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

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

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

Duality Applies to Different Fading 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

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

Dirty Paper Coding (Costa’83) Dirty Paper Coding Clean ChannelDirty Channel Dirty Paper Coding 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

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] … -50 S X …

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

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 * l Computationally very complex MIMO version of the Sato upper bound * Extended by Yu/Cioffi

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

Dirty Paper Coding in Cellular

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

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

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 (R 1,…R N ) has a given input covariance matrix and encoding order We find the corresponding PSD covariance matrix and encoding order to achieve (R 1,…,R N ) with DPC on BC l The rank-preserving transform “flips the effective channel” and reverses the order l Side result: beamforming is optimal for BC with 1 Rx antenna at each mobile DPC BC Sum MAC

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 l “Flip the effective channel” and reverse order DPC BC Sum MAC

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

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

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-case noise correlation Explicit formula for worst-case noise covariance By Lagrangian duality, cooperative BC region equals the sum-rate capacity region of MIMO MAC

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

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?

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

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 + +

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

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

A New Characterization of Channel Capacity Capacity using Lyapunov exponents Similar definitions hold for (Y) and (X;Y) Matrices B Y i and B X i Y i depend on input and channel where the Lyapunov exponent for B X i a random matrix whose entries depend on the input symbol X i

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

Lyapunov Direction Vector The vector p n is the “direction” associated with (X) for any . 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 (X) (X)

Computing Lyapunov Exponents Define  as the stationary distribution of the “direction vector” p n p n 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  pnpn p n+1 p n+2

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

Sensor Networks Energy is a driving constraint. Data flows to centralized location. Low per-node rates but up to 100,000 nodes. Data highly correlated in time and space. Nodes can cooperate in transmission and reception.

Energy-Constrained Network Design Each node can only send a finite number of bits Transmit energy per bit minimized by sending each bit over many dimensions (time/bandwidth product) Delay vs. energy tradeoffs for each bit Short-range networks must consider both transmit, analog HW, and processing energy Sophisticated techniques for modulation, coding, etc., not necessarily energy-efficient Sleep modes save energy but complicate networking New network design paradigm: Bit allocation must be optimized across all protocols Delay vs. throughput vs. node/network lifetime tradeoffs Optimization of node cooperation (coding, MIMO, etc.)

Results to Date Modulation Optimization Adaptive MQAM vs. MFSK for given delay and rate Takes into account RF hardware/processing tradeoffs MIMO vs. MISO vs. SISO for constrained energy SISO has best performance at short distances (<100m) Optimal Adaptation with Delay/Energy Constraints Minimum Energy Routing

Conclusions Shannon capacity gives fundamental data rate limits for wireless channels Many open capacity problems for time-varying multiuser MIMO channels Duality and dirty paper coding are powerful tools to solve new capacity problems and simplify computation Lyapunov exponents a powerful new tool for solving capacity problems Cooperative communications in sensor networks is an interesting new area of research