MIT
6.441
Capacity of multi-antenna
Gaussian Channels, I. E. Telatar
May 11, 2006
By: Imad Jabbour
Introduction
 MIMO systems in wireless comm.
 Recently subject of extensive research
 Can significantly increase data rates and
reduce BER
 Telatar’s paper
 Bell Labs (1995)
 Information-theoretic aspect of single-user
MIMO systems
 Classical paper in the field
MIT 6.441
Capacity of multi-antenna Gaussian channels (Telatar)
Imad Jabbour
Preliminaries
 Wireless fading scalar channel
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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
MIT 6.441
Capacity of multi-antenna Gaussian channels (Telatar)
Imad Jabbour
MIMO Channel Model (1)
 I/O relationship
 Design parameters
o t Tx. antennas and r Rx. antennas
o Fading matrix
o Noise
 Power constraint:
 Assumption
 H known at Rx. (CSIR)
MIT 6.441
Capacity of multi-antenna Gaussian channels (Telatar)
Imad Jabbour
MIMO Channel Model (2)
 System representation
Transmitter
Receiver
 Telatar: the fading matrix H can be
 Deterministic
 Random and changes over time
 Random, but fixed once chosen
MIT 6.441
Capacity of multi-antenna Gaussian channels (Telatar)
Imad Jabbour
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
o There are
MIT 6.441
of H
singular values
Capacity of multi-antenna Gaussian channels (Telatar)
Imad Jabbour
Deterministic Fading Channel (2)
 Equivalent parallel channel [nmin=min(r,t)]
 Tx. must know H to pre-process it, and Rx.
must know H to post-process it
MIT 6.441
Capacity of multi-antenna Gaussian channels (Telatar)
Imad Jabbour
Deterministic Fading Channel (3)
 Result of SVD
 Parallel channel with
sub-channels
 Water-filling maximizes capacity
 Capacity is
o Optimal power allocation
o is chosen to meet total power constraint
MIT 6.441
Capacity of multi-antenna Gaussian channels (Telatar)
Imad Jabbour
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
MIT 6.441
Capacity of multi-antenna Gaussian channels (Telatar)
Imad Jabbour
Random Varying Channel (2)
 Information-theoretic aspect
 Codeword length should average out both
additive noise and channel fluctuations
 Assume that Rx. tracks channel perfectly
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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
MIT 6.441
Capacity of multi-antenna Gaussian channels (Telatar)
Imad Jabbour
Random Varying Channel (3)
 MIMO capacity versus SNR (from [2])
MIT 6.441
Capacity of multi-antenna Gaussian channels (Telatar)
Imad Jabbour
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!
MIT 6.441
Capacity of multi-antenna Gaussian channels (Telatar)
Imad Jabbour
Random Fixed Channel (2)
 Telatar’s solution: outage probability pout
 pout is probability that R is greater that
maximum achievable rate
 Alternative performance measure is
o Largest R for which
o Optimal power allocation is equal allocation
across only a subset of the Tx. antennas.
MIT 6.441
Capacity of multi-antenna Gaussian channels (Telatar)
Imad Jabbour
Discussion and Analysis (1)
 What’s missing in the picture?
 If H is unknown at Tx., cannot do SVD
o Solution: V-BLAST
 If H is known at Tx. also (full CSI)
o Power gain over CSIR
 If H is unknown at both Tx. and Rx (noncoherent model)
o At high SNR, solution given by Marzetta &
Hochwald, and Zheng
 Receiver architectures to achieve capacity
 Other open problems
MIT 6.441
Capacity of multi-antenna Gaussian channels (Telatar)
Imad Jabbour
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
MIT 6.441
Capacity of multi-antenna Gaussian channels (Telatar)
Imad Jabbour
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?
o Power gain of nt/nmin as compared to CSIR case
MIT 6.441
Capacity of multi-antenna Gaussian channels (Telatar)
Imad Jabbour
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)
MIT 6.441
Capacity of multi-antenna Gaussian channels (Telatar)
Imad Jabbour
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
o Zero-forcing receiver (decorrelator)
o MMSE receiver
o MMSE can achieve capacity if SIC is used
MIT 6.441
Capacity of multi-antenna Gaussian channels (Telatar)
Imad Jabbour
Discussion and Analysis (6)
 Open research topics
 Alternative fading models
 Diversity/multiplexing tradeoff (Zheng & Tse)
 Conclusion
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MIMO can greatly increase capacity
For coherent high SNR,
How many antennas are we using?
Can we “beat” the AWGN capacity?
MIT 6.441
Capacity of multi-antenna Gaussian channels (Telatar)
Imad Jabbour
Thank you!
Any questions?