1

Capacity of Correlated MIMO Channels: Channel Power and

Multipath Sparsity

Akbar Sayeed(joint work with Vasanthan Raghavan,

UIUC)

Random Matrix Theory and Wireless Communications WorkshopBoulder, CO, July 14 2008

Wireless Communications Research LaboratoryDepartment of Electrical and Computer Engineering

University of Wisconsin-Madison

akbar@engr.wisc.edu , vasanth@uiuc.edu

http://dune.ece.wisc.edu

2

Sergio Verdu – Hard Act to Follow!

Shannon (belly) Dance! (ISIT 2006, Seattle)

3

Sergio Verdu – Model Incognito?

4

Multipath Wireless Channels

• Multipath signal propagation over spatially distributed paths due to signal scattering from multiple objects – Necessitates statistical channel modeling– Accurate and analytically tractable Understanding the physics!

• Fading – fluctuations in received signal strength

• Diversity – statistically independent modes of communication

5

Antenna Arrays: Multiplexing and Energy Capture

Multiplexing – Parallel spatial channels

sec/bitsW

1logW),W(C 2

sec/bits1logNN

N1logN~),N(C 22

Array aperture: Energy capture

Wideband (W):

Multi-antenna (N):

Dramatic linear increase in capacity with number of antennas

6

Key Elements of this Work• Sparse multipath

– i.i.d. model – rich multipath– Seldom true in practice– Physical channels exhibit sparse multipath

• Modeling of sparse MIMO channels – Virtual channel representation (beamspace)– Physically meaningful channel power

normalization– Sparse degrees of freedom– Spatial correlation/coherence

• The Ideal MIMO Channel– Fastest (sub-linear) capacity scaling with N– Capacity-maximization with SNR for fixed N– Multiplexing gain versus received SNR tradeoff– Simple capacity formula for all SNRs (RMT)

• Creating the Ideal MIMO Channel in Practice– Reconfigurable antenna arrays– Three canonical configurations: near-optimum

performance over entire SNR range – Source-channel matching– New capacity formulation

Capaci

ty

MUX IDEAL BF

qC p, p log 1

p

C(N

) sparse C N O Di.i.d.

C N O N

Correlated C N O N

N

7

Virtual Channel Modeling

Spatial sampling commensurate with signal space resolution

Channel statistics induced by the physical scattering environment

Abstract statistical models

Physical models

Virtual Model

Tractable Accurate

Accurate & tractable

Interaction between the signal space and the physical channel

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Narrowband MIMO Channel

Hsx

TR N

2

1

TRRR

T

T

N

2

1

s

s

s

)N,N(H)2,N(H,)1,N(H

)N,2(H)2,2(H,)1,2(H

)N,1(H)2,1(H,)1,1(H

x

x

x

Received signal Transmitted signal

TN

RN

Transmit antennas

Receive antennas

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Uniform Linear Arrays

RR

R

)1N(2j

2j

RR

e

e

1

)(

aReceive response veector

/)sin(d RRR

= Rx antenna spacingRd

TT

T

)1N(2j

2j

TT

e

e

1

)(

aTransmit steeringvector

/)sin(d TTT

= Tx antenna spacingTd

2/2/

Spatial sinusoids: angles frequencies

)sin(

d

d

10

Physical Model

)()( n,THTn,RR

N

1nn

path

aaH

pathN

}{ n,R }{ n,T

}{ n: number of paths : complex path gains

: Angles of Arrival (AoA’s) : Angles of Departure (AoD’s)

Non-linear dependence of H on AoA’s and AoD’s

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Virtual Modeling

T

N

1i

N

1k

HT

RRVn,T

N

1n

HTn,RRn N

k

N

ik,iH

R Tpath

aaaaH

RR

TT N

1,

N

1Spatial array resolutions:

Physical Model Virtual Model

(AS ’02)

Virtual model is linear -- virtual beam angles are fixed

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Antenna Domain and Beamspace

HTVR AHAH T

HRV HAAH

RRR NN: A

TTT NN: A

Two-dimensional unitary (Fourier) transform

Generalization to non-ULA’s (Kotecha & AS ’04 ; Weichselberger et. al. ’04; Tulino, Lozano, Verdu ’05;)

HRRR

HR

RR

E UΛUHHΣ

UA

HTTT

HT

TT

E UΛUHHΣ

UA

Unitary (DFT) matrices

13

Virtual Imaging of Scattering Geometry

2 point scatterers 2 scattering clusters

Diagonal scattering Rich scattering

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Virtual Path Partitioning

pathk,Ti,Rk,i

k,Tk

i,Ri

N,,2,1]SS[SS

T

R

RRn,Ri,R N

)2/1i(,

N

)2/1i(:nS

RN

1

TN

1

TTn,Tk,T N

)2/1k(,

N

)2/1k(:nS

k,Ti,R SSnnV )k,i(H

Distinct virtual coefficients disjoint sets of paths

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Virtual Coefficients are Approximately Independent

'kk'ii*VV k,i'k,'iHk,iHE

k,Ti,R SSnnV )k,i(H

k,Ti,R SSn

2

n

2

V Ek,iHEk,i

Channel power matrix: joint angular power profile

T

R

TN

1

RN

1

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Joint and Marginal Statistics

k,Ti,R SSn

2

n

2

V Ek,iHEk,i

Joint distribution of channel power as a function of transmit and receive virtual angles

Joint statistics:

Marginal statistics:

HvvR E HHΛ vHvT E HHΛ

RN

T Ti 1

k k,k i, k

Λ TN

R Rk 1

i i, i i, k

Λ

Transmit Receive

(diagonal)

V Vvec( )h H

H

V V VE[ ]

diag i, k

R h h

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Kronecker Product Model

RTVRT ΛΛRΣΣR

Independent transmit and receive statistics

Separable angular scattering function (angular power profile)

2V R T

| H |i, k E i, k (i) (k) Separable

arbitrary kronecker

i, k R T{ (i) (k)}

2/1Tiid

2/1RV

2/1Tiid

2/1R ΛHΛHΣHΣH

parameters

1NN RT

parameters

NN RT

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Communication in Eigen (Beam) Space

nHsx

vvvv nsHx xAx

sAsHRv

HTv

Multipath PropagationEnvironmen

t

TA

VsHRA

x Vx

is an image of the far-field of the RX

VxImage of is created in the far-field of TX

Vs

s

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Capacity Maximizing Input

)det(logEmax)det(logEmaxC HVVV)(tr

H)(tr

VHQHIHQHI QQ

nHsx

HToptTopt

opt,V

UΛUQQ

Optimal input covariance matrix is diagonal in the virtual domain:

- Beamforming optimal at low SNR (rank-1 input)- Uniform power input optimal at high SNR (full-rank input)- Uniform power input optimal for regular channels (all SNRs)

2H

E

][E

s

Inn

Veeravalli, Liang, Sayeed (2003); Tulino, Lozano, Verdu (2003); Kotecha and Sayeed (2003)

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Degrees of Freedom

R ,i T ,k

2 2V n

n S S

H i,ki, k E E

Dominant (large power) virtual coefficients

Statistically independent Degrees of Freedom (DoF)

DoF’s are ultimately limited by the number of resolvable paths

D DoF (i, k) : (i, k) 0

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Channel Power and Degrees of Freedom

D(N) = number of dominant non-vanishing virtual coefficients = Degrees of Freedom (DoF) in the channel

L(N)H H 2

c V V ni,k n 1

(N) (i, k) trace(E[ ]) trace(E[ ]) E[| | ]

H H H H

c (N) ~ O(D(N))

The D non-vanishing virtual coefficients are O(1)

Simplifying assumption:

Assume equal number of transmit and receive antennas – RT NNN

Channel power:

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Prevalent Channel Power Normalization

2c N D(N) ~ O N

The channel power/DoF grow quadratically with N

2c T RN N N

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Quadratic Channel Power Scaling?

2c N D(N) ~ O(N ) is physically impossible indefinitely

(received power < transmit power)

N

Total TX power

Total RX powerQuadratic growth in channel power

Linear growth in total received power

Linear capacity scaling

Increasing power coupling between the TX and RX due to increasing array apertures

2TX EP s2

RX TX TX

D(N)P P P NE

N Hs

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Sparse (Resolvable) Multipath

2rich max R TD D O N N O N

2R TN N N

2sparse R TD o N N o N N , 0,2

Deg

rees

of

Freed

om

(D

)

Rich (linear)

Sparse (sub-linear)

Channel Dimension

c (N) ~ O(D(N))

Sub-quadratic power scaling dictates sparsity of DoF

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Capacity Scaling: Sparse MIMO Channels

c ~ D ~ O(N ) , 0 2

For a given channel power/DoF scaling law

what is the fastest achievable capacity scaling?

New scaling result: coherent capacity cannot scale faster than

2/c NO)N(O)N(DO~)N(C

and this scaling rate is achievable (Ideal channel)

(AS, Raghavan, Kotecha ITW 2004)

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MIMO Capacity Scaling

N

C(N

)

Correlated channels (kronecker model)Chua et. al. ’02

i.i.d. modelTelatar ’95Foschini ’96 physical channels

(virtual representation)Liu et. al. ’03

RT NNN 2,NOD

2/NODONC AS et. al. ’04

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Sparse Virtual Channels

• Sub-quadratic power scaling dictates sparse virtual channels:

• Capacity scaling depends on the spatial distribution of the D(N) channel DoF in the possible channel dimensions

2D(N) N

2N

28

Simple Model for Sparse MIMO Channels

D N

NNN RT iidv )D( HMH

0/1 mask matrix with D non-zero

entries)D(MΨ

Sparsity in virtual (beam) domain correlation/coherence in the antenna (spatial) domain

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Three Canonical (Regular) Configurations

)1(ND

Beamforming

p = number of parallel channels (multiplexing gain)

q = D/p = DoF’s per parallel channel

Ideal

NDqp Nqq

1pp

max

min

Multiplexing

1qq

Npp

min

max

qpD Consider

p transmit dimensions; r = max(q , p) receive dimensions

Multiplexing gain = p increases

Received SNR = q/p increases

Received SNR = q/p

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Simple Capacity Formula:Multiplexing Gain vs Received-SNR

rx 2

q DC ~ p log 1 p log 1 p log 1

p p

2

crx

(N) q(N)E(N)p(N) p(N)p(N)

Hs Received SNR per parallel channel

bf

rx

C (N) ~ log(1 N)

(N) N

id

rx

C (N) ~ N log(1 )

(N)

mux

rx

C (N) ~ N log(1 / N)

(N) / N 0

Beamforming (BF) Ideal Multiplexing (MUX)

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Morphing Between the Configurations

qpND:]2,0(

:)1,0(

:)2,1[

0min

1min

beamforming)2/,[ min

ideal2/

multiplexing],2/( max)0,1max(min )1,min(max

]1,0[,Nq,Np

max

1max

:2 1min 1max

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Fastest Capacity Scaling: The Ideal Configuration

2rx p 2

p

C(N) p log 1 N log 1 N

pp

D 1D N , p N , q N

p

rx

q

p

)N(;NlogNO~)N(C)2/,[ rx2

bfmin

2prx

2/idid /)N(;NO~)N(C2/

0)N(;NO~)N(C],2/( rxmuxmax

BF regime:

Ideal regime:

MUX regime:

bf mux

id id

C (N) C (N)0 , 0

C (N) C (N) / 2

id cC (N) ~ O N ~ O (N)

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Impact of Transmit SNR on Capacity Scaling

BF:

MUX:

Ideal:

N)N(q)N(p

1)N(q,N)N(p

N)N(q,1)N(p

0min

2/1id

1;N)N(D

1max C

(N)

N

34

Accuracy of Asymptotic Expressions

C(N

)

N

BF and MUX tight atall SNRs

Ideal tight in the low-

or high-SNR regimes

35

Capacity Formula Proofs: RMT

• If H is r x p, coherent ergodic is given by

• If, in addition, H is regular

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Capacity Formula Proofs: RMT

If under broad assumptions on entries of H, the empirical spectral distribution function (Fp) of (normalized by p) converges to a deterministic limit (F)

Limit capacity computation Approach 1: Sometimes this limit can be characterized explicitly

Approach 2: Often, the limit can only be characterized implicitly via the Stieltjes transform. The limit capacity formula is the solution to a set of recursive equations

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Beamforming Configuration

• Two cases:

• In either case,

• Thus, with

38

Ideal Configuration

• Two cases:

Case i) reduces to a q x p i.i.d. channel Case ii) reduces to a q-connected p-dimensional channel [LRS

2003]

Case i) Case ii)

39

Ideal Configuration

• In either case, empirical density of (normalized by q) converges to Case i) Case ii)

• Thus,

Case i) Case ii)

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Multiplexing Configuration

• Previous result due to [Grenander & Silverstein 1977] not applicable • Two cases: • In either case, empirical density is unknown • The implicit characterization of [Tulino, Lozano & Verdu 2005] based

on results due to [Girko] can be easily extended here • Exploiting the regular nature of H

41

The Ideal MIMO Channel: Fixed N

iidv )D( HMH

2ND

NNN RT

0/1 mask matrix with D non-zero

entries)D(MΨ

2NSpatial distribution of the D channel DoF in the possible dimensions (“resolution bins”) that yields the highest capacity

),D(Cmaxarg),D()D(

ideal MMM

42

Optimum Input Rank versus SNR

5.235.34diag

5.05.05.01

5.05.011

5.0111

1111

T

Correlated channels:- beamforming (rank-1 input) optimal at low SNR- uniform power (full-rank, i.i.d.) input optimal at high SNR

1

2

3

4

SNR

ran

k

Vs

s

TAi.i.d. channels: equal power (i.i.d.) input optimal at all SNRs

Loss of precious channel power!

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Ideal Channel: Optimum MG vs SNR tradeoff

Np

)p(1logpp

D1logp

p

q1logp,pC rx2

Capaci

ty

Beamforming:

Ideal:

Multiplexing:

Nq,1p

Nqp

1q,Np

NpqD

high,max

highlow

lowmin

ideal

p

,,2

D

,p

)(p

low

BF

Ideal

high

MUX

Np

2/1

44

Impact of Antenna Spacing on Beamstructure

)sin(d

maxmux dd

Nbeams#

N

1beamwidth

N

dd muxideal

Nbeams#

N

1beamwidth

1beams#

N

dd muxbf

1beamwidth

45

Adaptive-resolution Spatial SignalingIdeal

Medium resolution TX and RX

Multiplexing gain and spatial coherence: Fewer independent streams with wider beamwidths at lower SNRs.

High resolution TX and RX

Multiplexing Beamforming

Low resolutionTX and High-Res. RX

46

Wideband/Low-SNR Capacity Gain

MUXmin,o

b

IDEALmin,o

b

BFmin,o

b

N

E

N

1

N

E

N

1

N

E

N-fold increase in capacity (or reduction in ) via BF configuration at low SNR

minob )N/E(

47

Source-Channel MatchingAdapting the multiplexing gain p via array configuration:

matching the rank of the inputrank of the input to the rank of the effective rank of the effective channel channel

Multiplexing

Full-rank channel

Full-rank input

RX

TX

48

Source-Channel Matching

Ideal

“Square root” rank channel

“Square root” rank input

RX

TX

Adapting the multiplexing gain p via array configuration: matching the rank of the inputrank of the input to the rank of the effective rank of the effective

channel channel

49

Source-Channel Matching

Beamforming

Rank-1 channel

Rank-1 input

TX

RX

Adapting the multiplexing gain p via array configuration: matching the rank of the inputrank of the input to the rank of the effective rank of the effective

channel channel

50

New Capacity Formulation for Reconfigurable MIMO Channels )det(logEmaxmaxC H

VVVpqD:)(tr VV

HQHIHQ

To achieve O(N) MIMO capacity gain at all SNRs

Optimal channel configuration realizable with reconfigurable antenna arrays

Optimum number of antennas: N ~ DC

apaci

ty

p

q1logp,pC

MUX IDEAL BF

51

Summary• Sparse multipath

– i.i.d. model – rich multipath– Seldom true in practice– Physical channels exhibit sparse multipath

• Modeling of sparse MIMO channels – Virtual channel representation (beamspace)– Physically meaningful channel power

normalization– Sparse degrees of freedom– Spatial correlation/coherence

• The Ideal MIMO Channel– Fastest (sub-linear) capacity scaling with N– Capacity-maximization with SNR for fixed N– Multiplexing gain versus received SNR tradeoff– Simple capacity formula for all SNRs (RMT)

• Creating the Ideal MIMO Channel in Practice– Reconfigurable antenna arrays– Three canonical configurations: near-optimum

performance over entire SNR range – Source-Channel Matching– New capacity formulation

Capaci

ty

MUX IDEAL BF

qC p, p log 1

p

C(N

) sparse C N O Di.i.d.

C N O N

Correlated C N O N

N

52

Extensions: Implications of Sparsity

• Relaxing the 0-1 sparsity model

• Non-uniform sparsity

• Wideband MIMO channels/doubly-selective MIMO channels

• Space-time coding

• Reliability (error exponents)

• Impact of TX CSI (full or partial)

• Channel estimation (compressed sensing) and feedback

• Network implications (learning the network CSI)

53

REFERENCES

Beamforming Channel: Bai & Yin: “Convergence to the semicircle law,” Annals Prob., vol. 16, pp. 863-875,

1988

Ideal Channel: Marcenko & Pastur: “Distribution of eigenvalues for some sets of random matrices,”

Math-USSR-Sb., vol. 1, pp. 457-483, 1967 Bai: “Methodologies in spectral analysis of large dimensional random matrices: A

review,” Statistica Sinica, vol. 9, pp. 611-677, 1999 Silverstein & Bai: “On the empirical distribution of eigenvalues of a class of large

dimensional random matrices,” Journal of Multivariate Analysis, vol. 54, no. 2, pp. 175-192, 1995

Grenander & Silverstein: “Spectral analysis of networks with random topologies,” SIAM Journal on Appl. Math., vol. 32, pp. 499-519, 1977

Liu, Raghavan & Sayeed, “Capacity and spectral efficiency of wideband correlated MIMO channels,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2504-2526, Oct. 2003

Multiplexing Channel: Girko: Theory of random determinants, Springer Publishers, 1st edn, 1990 Tulino, Lozano & Verdu: “Impact of antenna correlation on the capacity of multiantenna

channels,” IEEE Trans. Inform. Theory, vol. 51, no. 7, pp. 2491-2509, July 2005

• Multi-antenna Capacity of Sparse Multipath Channels, V. Raghavan and A. Sayeed. http://dune.ece.wisc.edu