Asymptotic Random Matrix Applications in Communicationweb.mit.edu/sea06/agenda/talks/Bliss.pdf ·...

MIT Lincoln LaboratorymimoTechSem-1

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Asymptotic Random Matrix Applications in Communication:

MIMO and More

Dr. Daniel W. Bliss

MIT-Lincoln Laboratory781-981-3300

[email protected]

This work was sponsored by the Air Force under Air Force Contract FA8721-05-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the United States Government.


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Topics

• Multiple-Input Multiple-Output (MIMO)– Introduction– Asymptotic results

• Survey of Communication Examples– Code-Division Multiple Access (CDMA)– Receiver performance in rich interference networks

• Wish List


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Complicated MultipathEnvironment

MIMO CommunicationMultiple-Input Multiple-Output

• Single transmitted data stream• Single received data stream• Employ array of antennas at both

transmitter and receiver• Employ multiple modes through

environment

Data…01110111001…

Data…01110111001…

TransmitAntenna

ArraySpace-Time

Coding

MultipleInput

ReceiveAntenna

Array

bounce offbuilding

Space-TimeReceiver

MultipleOutput


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The Channel Matrix

• Channel matrix, H, contains complex attenuation between each transmit and receive antenna

• Large channel matrix singular values are useful

Rel

ativ

e Po

wer

Channel Matrix Singular Values

HighCapacity

EnvironmentLowCapacity

Environment

Sorted by Mode Strength

Few Useful Modes

Many Useful Modes

Many Useful Modes

Scatterers


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Rich Scattering Channel

Rich ScatteringEnvironment

• Very complicated scattering environment

• Independent Rayleigh fading for each transmit-receive antenna pair

• Entries in matrix independently sampled from unit-norm complex Gaussian distribution

Scatterer

G

RMS antenna pairattenuation

H = a G

G =

g11 g12

g21 g22

gmn

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟


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MIMO Taxonomy

• Stationarity required to pass channel estimate to transmitter

• Capacity formulation drives choice of transmitter rank

– Low Spec. Eff. ⇒ low rank TX– High Spec. Eff. ⇒ high rank TX

MIMO LinkTransmitter

Receiver MIMO System Selection

Spectral Efficiency

Stat

iona

rity

Stat

icD

ynam

ic

Low High

Low RankInformed Transmitter Informed Transmitter

Uninformed TransmitterUninformed Transmitter

Channel Matrix Estimate?


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Information Theoretic Capacity

Mutual Information

Receive-Signal Entropy Noise-Like Entropy

Signal Model

In Interference Environment

InterferenceCovariance

Matrix

Informed Transmitter Capacity

; Whitened Channel Matrix

Uninformed Transmitter Capacity


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Informed Transmitter MIMOWater Filling

• Model

• Modes in channel matrix live in singular values

• Number of employed modes depends on SNR– Higher SNR ⇒ Many modes– Lowest SNR ⇒ Single mode

ReceiveBeamformers

TransmitBeamformers

Mode

High SNR

Low SNR

Water-Filling Analogy

Single ModeInformed Transmitter

MIMO


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Topics



• Wish List


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Convergence to the ContinuumHow Big is Big?

• Eigenvalue probability density of

• Ensemble of finite NxN eigenvalue spectra

• Converges quickly

Sort

edEi

genv

alue

[dB

]

Fraction of Eigenvalues

Sort

edEi

genv

alue

[dB

]


Sort

edEi

genv

alue

[dB

]


Eigenvalue Spectrum

Eigenvalue SpectrumEigenvalue Spectrum

4x4

16x168x8

HH† = a2GG†


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Continuous Uninformed Transmitter Capacity Approximation

• Replace discrete matrix with continuous approximation

• Shape of random finite matrixeigenvalue distributions converge quickly

• Use number of receivers normalized capacity

CUT = log2 I +Po

nTx

H H†

= log2 I + a2 PoGG†

nTx

= log2 I + a2Po μmm=1

nRx

∑

≈ nRx Φ a2 Po ;r( )

Φ x ; r( ) ≡ dμ fr (μ) log2 (1 + μ0

∞

∫ x)

Total TransmitPower

(noise-normalized)ChannelMatrix

EigenvalueDensity

continuous approximation

D. W. Bliss, K. W. Forsythe, and A. F. Yegulalp, “MIMO communication capacity using infinite dimension random matrix eigenvalue distributions,” Asilomar Conference, Nov., 2001


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Continuous Informed Transmitter Capacity Approximation

CIT = maxtr{P}= Po

log2 I + HP H†

= log2

Po + 1a2 M

tr Λ−1{ }L

a2 M Λ

≈ g M log2

a2 Po + dμ' f1(μ' ) 1μ'μcut

∞

∫g

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

+ M dμ f1(μ) log2 (μ)μcut

∞

∫

g = LM

≈ dμ f1(μ)μ cut

∞

∫ , μ cut =

dμ' f1(μ' )μ cut

∞

∫

a2Po + dμ f1(μ) 1μμ cut

∞

∫

EmployedEigenvalues

SmallestEigenvalueEmployed

Fraction of EigenvaluesEmployed

continuous approximation

• For this example assume that

• Employ top modes of contained in

M = nRx = nTx , r =1

L HH†

Λ

ImplicitlyDefined


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Large Random Matrix MIMO CapacityInformed vs. Uninformed Transmitter

Antenna Number Normalized MIMO Capacity

• Rich scattering environment

• M x M channel

• capacity using asymptotic eigenvalue distribution

• Low-SNR informed capacity higher

Informed

Uninformed

Mean Single Antenna Link SNR

x 4


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MIMO Capacity In InterferenceLarge Dimension Limit (UT)

• Capacity in presence of interference

• Equivalent MIMO Interferer

• Interference type– Cooperative– Uncooperative

• Cooperative interferencecan be demodulated

• Uninformed transmitter

• Square

Antenna Number Normalized MIMO Capacity

Mean Single Antenna Link SNR

No Interference

Uncooperative

H

CooperativeInterference


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Topics



• Wish List


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Multiple Access

• Multiple users trying to talk to the base station at once• Separate users

– Frequency (frequency division multiple access)– Time (time division multiple access)– Code (code division multiple access)

FrequencyFDMA

ω

|F{s(t)}|2

TimeTDMA

�

|s(t)|2

CodeCDMA

1 → 00001111

1 → 11001100

1 → 10101010


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CDMA Performance Bounds

• Verdu’s multiuser detection bounds• Asymptotic in number of users and

spreading• Eigenvalue distribution in code

cross-correlation

Verdu and Shamai, “Spectral Efficiency of CDMA with Random Spreading,” IEEE Trans. On IT, March, 1999

Users

Spreading

Loading

SNR

Spectral Efficiency Bounds

Spec

tral

Effi

cien

cy (b

/chi

p)

MMSE

Optimal


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Multiple-Antenna Performance Bounds

• Random distribution of interferers on a plane• Spatial interference mitigation• Propagation

– Power law attenuation,– Rayleigh fading

• High interference, low noise regime

S. Govindasamy, et al, “The performance of linear multiple-antenna receivers with interferers distributed on a plain,” IEEE Workshop on SWAWC, June, 2005

LinkLength

InterferenceDensity

Number ofReceive

Antennas

Cap

acity

(b/s

/Hz/

user

)

Performance Bound

α = 3α = 4

α= 5

TX RX

InterferenceDensity


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Topics


• Survey of Communication Examples– Code-Division Multiple Access (CDMA)– Ultra-Wideband (UWB)– Receiver performance in rich interference networks

• Wish List


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Wish List

What I would like for the holidays

• Densities for functions of random complex matrices – Finite or infinite

• Ratios, Products, Sums, Differences, Det’s, Traces– Signal in the mean– Colored matrices

• Make it easy for me…


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Backup Slides


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The Channel MatrixA Toy Model

• Toy MIMO channel model– 2x2 – line of sight

• Resolving individual antennas increases eigenvalue

• MIMO systems in real environments employ scatterers to increase effective aperture

• Toy MIMO channel model– 2x2 – line of sight

• Resolving individual antennas increases eigenvalue

• MIMO systems in real environments employ scatterers to increase effective aperture Channel Matrix,

Eigenvalues of Channel Matrix2x2 MIMO System

Vary

Ape

rtur

e

b =

2π

arccosh 1

†h 2

h 1

h 2

H = h 1 h 2( )= 2 a v 1

v 2( )Unit norm

steering vector


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Beamforming

• Phase multiple antenna elements to control direction of transmitted or received power

• Narrowband model - can’t resolve antennas,

wavefronts

k

x 1

x 2

x 3

x n

SteeringVector

c (τn − τ1)


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Multi-Antenna ProcessingSimple Example

How it works:• Adapt antenna array beam pattern• Distinct beam pattern for each user• Done simultaneously for all users• Spatial equivalent to adaptive filtering

in frequency domain

How it works:• Adapt antenna array beam pattern• Distinct beam pattern for each user• Done simultaneously for all users• Spatial equivalent to adaptive filtering

in frequency domain

Adaptive Spatial Beamforming User of

Interest

InterferingUser

Null inBeam Pattern

c2

c1

Σ


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Beamforming Pattern(Receive or Transmit)

• Phase multiple antenna elements to control direction of transmitted or received power

• Narrowband model• n = 10

v † k (θ = 0);

x 1 …

x n( ) v k (θ );

x 1 …

x n( )

2

SteeringVector

θ (degrees)

Rel

ativ

e Po

wer


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Minimize Mean Square Error Example

• 10 antenna array• 4 interfering users (10,-20,30,-40 degrees)• Signal of interest at 0 degrees

f †

v k (θ);

x 1…

x n( )

2

θ (degrees)

Rel

ativ

e Po

wer

InterferingUsers

SpatialFilter


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Minimize Mean Square Error ExampleMoving Interferer

• 10 antenna array• 1 interfering users • Signal of interest at 0 degrees

f †

v k (θ);

x 1…

x n( )

2

Rel

ativ

e Po

wer

θ (degrees)

InterferingUser

QuickTime™ and a Animation decompressor are needed to see this picture.


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Channel Complexity

Scatterers• Scatterers act as virtual elements in antenna array

• Channel matrix complexity increases with effective virtual aperture

• Virtual scattering array resolves antenna of other antenna array

VirtualArray


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The Rich Scattering ChannelLarge Dimension Limit

Eigenvalue Density,H H†

• Eigenvalue probability density of

• Entries in are drawn from unit norm complex Gaussian

• Eigenvalues probability density of

• PDF dependent upon,

fr=1(x) dx =x (1− x)

Rel

ativ

e Ei

genv

alue

(dB

)

Eigenvalue Spectrum


π xdx

HH† = a2GG†

(1/ nTx) GG†

G

r = n

PeakNormalized

Rx / nTx

Prob

abili

ty D

ensi

ty

Eigenvalue [dB]


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MIMO Capacity Bound

• MIMO bound follows different theoretical limit• Divide total energy amongst transmitters

avoiding compressive regime of SISO Shannon limit

Spectral Efficiency(b/s/Hz)

E b/N

0(d

B)

Shannon Limit

SISO

4x4 MIMO

8x8 MIMO TransmitPowerMatrixChannel

Matrix

Determinant

ChannelLoss

BandwidthNormalized

Capacity Transmit Power(noise normalized)

Assuming FlatMIMO Channel

Assuming FlatMIMO Channel


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Uncooperative External Interference Uninformed Transmitter Redux

˜ C UT ,Un = log2 I + Po

nTx

˜ H ˜ H †

= log2 R +Po

nTx

H H† − log2 R

= log2 I +Pint

nint

JJ† +Po

nTx

HH† − log2 I +Pint

nint

JJ†

a2Po

nTx

=aint

2 Pint

nint

˜ C UT ,Un,Eq = log2 I +a2Po

nTx

AA† − log2 I +a2 Po

nTx

Γ Γ† ; A = G Γ( )

Assume same equal average receive power per antenna

Interference plus noise can written in the form

R = I +

Pint

nint

JJ† = I +aint

2 Pint

nint

Γ Γ†

Re-Express bound for uncooperative interference


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Uncooperative External Interference Continuous Uninformed Transmitter

˜ C UT ,Un,Eq = log2 I +a2Po

nTx

AA† − log2 I +a2 Po

nTx

Γ Γ†

≈ nRx Φ a2PonTx + nint

nTx

;nRx

nTx + nint

⎛ ⎝ ⎜

⎞ ⎠ ⎟ − Φ a2Po

nint

nTx

;nRx

nint

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

Continuous approximation

• Continuous form can be expressed as the difference of two UT bounds

• Assumptions– Interference has same channel statistics– Interference has same power per transmit antenna


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ReceiveArray

Cooperative InterferenceOther MIMO Systems

• Cooperative interference affects performance less• Receiver jointly demodulates signals from both interfering MIMO

system and system of interest– Discards information from interfering MIMO system

• Cooperative interference affects performance less• Receiver jointly demodulates signals from both interfering MIMO

system and system of interest– Discards information from interfering MIMO system

TransmitArray

OtherReceive

Array

InterferingTransmit

Array

Interfering MIMO System


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Cooperative External Interference Continuous Uninformed Transmitter

˜ C UT ,Co = γ log2 I +Po

nTx

H H† +Pint

nint

J J†

˜ C UT ,Co ,Eq =nTx

nTx + nint

log2 I +a2 Po

nTx

G Γ( ) G Γ( )† ;a2Po

nTx

=aint

2 Pint

nint

≈ nRx nTx

nTx + nint

Φ a2 PonTx + nint

nTx

; nRx

nTx + nint

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Continuous approximation

• Interfering and intended MIMO systems allowed to coordinate

• Assumptions– Interference has same channel statistics– Interference has same power per transmit antenna

InterferingChannel Matrix

Fraction of data associatedwith system of interest

Asymptotic Random Matrix Applications in Communicationweb.mit.edu/sea06/agenda/talks/Bliss.pdf ·...

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