capacity of multi-antenna gaussian channels, i. e. telatar by: imad jabbour mit 6.441 may 11, 2006
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Capacity of multi-antenna Gaussian Channels, I. E. Telatar
By: Imad Jabbour
MIT 6.441
May 11, 2006
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MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour
Introduction
MIMO systems in wireless comm.Recently subject of extensive researchCan significantly increase data rates and
reduce BER
Telatar’s paperBell Labs (1995)Information-theoretic aspect of single-user
MIMO systemsClassical paper in the field
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MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour
Preliminaries
Wireless fading scalar channelDT Representation:H is the complex channel fading coefficientW is the complex noise,Rayleigh fading: , such that |H| is
Rayleigh distributed
Circularly-symmetric Gaussiani.i.d. real and imaginary partsDistribution invariant to rotations
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MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour
MIMO Channel Model (1)
I/O relationship
Design parameterso t Tx. antennas and r Rx. antennaso Fading matrix o Noise
Power constraint:
AssumptionH known at Rx. (CSIR)
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MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour
MIMO Channel Model (2)
System representation
Telatar: the fading matrix H can beDeterministicRandom and changes over timeRandom, but fixed once chosen
Transmitter Receiver
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MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour
Deterministic Fading Channel (1)
Fading matrix is not randomKnown to both Tx. and Rx.Idea: Convert vector channel to a parallel one
Singular value decomposition of HSVD: , for U and V unitary, and D
diagonalEquivalent system: , where
Entries of D are the singular values of Ho There are singular values
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MIT 6.441 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
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MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour
Deterministic Fading Channel (3)
Result of SVDParallel channel with sub-channels
Water-filling maximizes capacityCapacity is
o Optimal power allocation o is chosen to meet total power constraint
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MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour
Random Varying Channel (1)
Random channel matrix HIndependent of both X and W, and
memorylessMatrix entries
Fast fadingChannel varies much faster than delay
requirementCoherence time (Tc): period of variation of
channel
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MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour
Random Varying Channel (2)
Information-theoretic aspectCodeword length should average out both
additive noise and channel fluctuations
Assume that Rx. tracks channel perfectlyCapacity is Equal power allocation at Tx.Can show thatAt high power, C scales linearly with nmin
Results also apply for any ergodic H
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MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour
Random Varying Channel (3)
MIMO capacity versus SNR (from [2])
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MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour
Random Fixed Channel (1)
Slow fadingChannel varies much slower than delay
requirementH 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 rateIn this sense, capacity is zero!
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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 iso Largest R for which o Optimal power allocation is equal allocation
across only a subset of the Tx. antennas.
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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-BLASTIf H is known at Tx. also (full CSI)
o Power gain over CSIRIf H is unknown at both Tx. and Rx (non-
coherent model)o At high SNR, solution given by Marzetta &
Hochwald, and ZhengReceiver architectures to achieve capacityOther open problems
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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
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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
timeCan show optimal power allocation is P/nmin
Capacity is
What are we gaining?o Power gain of nt/nmin as compared to CSIR case
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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 itBlock fading modelAt high SNR, capacity gain is equal to (Zheng)
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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 channelsCaveat: Complexity of joint decodingSolution: simpler linear decoders
o Zero-forcing receiver (decorrelator)o MMSE receivero MMSE can achieve capacity if SIC is used
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MIT 6.441 Capacity of multi-antenna Gaussian channels (Telatar) Imad Jabbour
Discussion and Analysis (6)
Open research topicsAlternative fading modelsDiversity/multiplexing tradeoff (Zheng & Tse)
ConclusionMIMO can greatly increase capacityFor coherent high SNR, How many antennas are we using?Can we “beat” the AWGN capacity?
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Thank you!Any questions?