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Capacity Limits of Wireless Channels
 

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 Capacity Limits of Wireless Channels Presentation Transcript

  • 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 } R 1 R 2 R 3 R 1 R 2 R 3
  • 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
  • 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 • 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)
    • 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 P 1 P 2
  • 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 MAC with sum-power constraint P 1 =1.5, P 2 =0.5 P 1 =0.5, P 2 =1.5
  • 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)
    • 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 Clean Channel Dirty Channel Dirty Paper Coding
  • 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]
    -1 +3 +5 +1 -3 … -5 0 S +7 -7 … -1 +1 0 -1 +1 0 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 *
        • 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
        • 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
  • 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
  • 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
     p n 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