Capacity of multi-antenna Gaussian Channels, I. E. Telatar By: Imad Jabbour MIT 6.441 May 11, 2006.

Capacity of multi-antenna Gaussian Channels, I. E. Telatar

By: Imad Jabbour

MIT 6.441

May 11, 2006

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


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


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)


MIMO Channel Model (2)

System representation

Telatar: the fading matrix H can beDeterministicRandom and changes over timeRandom, but fixed once chosen

Transmitter Receiver


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



Equivalent parallel channel [nmin=min(r,t)]

Tx. must know H to pre-process it, and Rx. must know H to post-process it



Result of SVDParallel channel with sub-channels

Water-filling maximizes capacityCapacity is

o Optimal power allocation o is chosen to meet total power constraint


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



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



MIMO capacity versus SNR (from [2])


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!


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.


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



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



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



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)



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



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?

Thank you!Any questions?

Capacity of multi-antenna Gaussian Channels, I. E. Telatar By: Imad Jabbour MIT 6.441 May 11, 2006.

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Transcript of Capacity of multi-antenna Gaussian Channels, I. E. Telatar By: Imad Jabbour MIT 6.441 May 11, 2006.