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
• Telatars paper
• Bell Labs (1995)
• Information-theoretic aspect of single-user MIMO
  systems
• Classical paper in the field

3
Preliminaries

• Wireless fading scalar channel
• DT Representation
• H is the complex channel fading coefficient
• 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

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)

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
6
Deterministic Fading Channel (1)

• Fading matrix is not random
• Known to both Tx. 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

7
Deterministic Fading Channel (2)

• Equivalent parallel channel nminmin(r,t)
• Tx. must know H to pre-process it, and Rx. must
  know H to post-process it

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

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

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

11
Random Varying Channel (3)

• MIMO capacity versus SNR (from 2)

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!

13
Random Fixed Channel (2)

• Telatars 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.

14
Discussion and Analysis (1)

• Whats 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

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

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

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)

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

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?

20
Thank you!

• Any questions?