Random Matrices, Integrals and Space-time Systems

Babak HassibiCalifornia Institute of TechnologyDIMACS Workshop on Algebraic Coding and Information

Theory, Dec 15-18, 2003

Outline

• Overview of multi-antenna systems• Random matrices• Rotational-invariance• Eigendistributions• Orthogonal polynomials• Some important integrals• Applications• Open problems

IntroductionWe will be interested in multi-antenna systems of the form:

,VSHM

X +=ρ

where NTNMMTNT VHSX ×××× ∈∈∈∈ CCCC ,,, are the receive, transmit, channel, and noise matrices, respectively.Moreover, are the number of transmit/receive antennasrespectively, is the coherence interval and is the SNR.

The entries of are iid and the entries of are also , but they may be correlated.

NM ,T ρ

V )1,0(CN H)1,0(CN

Some Questions

• What is the capacity?• What are the capacity-achieving input distributions?• For specific input distributions, what is the mutual

information and/or cut-off rates?• What are the (pairwise) probability of errors?

We will be interested in two cases. The coherent case, whereis known to the receiver and the non-coherent case, whereis unknown to the receiver.

The following questions are natural to ask.

HH

Random Matrices

A random matrix is simply described by the joint pdf of its entries,

An example is the family of Gaussian random matrices, where the entries are jointly Gaussian.

nm× A

),...,1;,...,1;()( njmiapAp ij ===

Rotational-InvarianceAn important class of random matrices are (left- and right-) rotationally-invariant ones, with the property that their pdf is invariant to (pre- and post-) multiplication by any and

unitary matrices and .mm×

nn× Θ Ψ),()( ApAp =Θ

andmI=ΘΘ=ΘΘ **

),()( ApAp =Ψ nI=ΨΨ=ΨΨ **

If a random matrix is both right- and left- rotationally-invariantwe will simply call it isotropically-random (i.r.).If is a random matrix with iid Gaussian entries, then it is i.r.,as are all of the matrices:G

*11

*1

12

*21

12121

1** ,)(,,,,,, AGGGGGGGGGGGGGGGG p −−−+

Isotropically-Random Unitary Matrices

A random unitary matrix is one for which the pdf is given by

).()()( *mIfp −ΘΘΘ=Θ δ

When the unitary matrix is i.r., then it is not hard to show that

).()1()...()( *2/)1( mmm Imp −ΘΘ

ΓΓ=Θ + δ

πTherefore an i.r. unitary matrix has a uniform distribution overthe Stiefel manifold (space of unitary matrices). It is also calledthe Haar measure.

A Fourier Representation

If we denote the columns of by thenΘ ,,..,1, mkk =θ

))(Im())(Re()( ***kllkkllk

lkmI δθθδδθθδδ −−=−ΘΘ Π

≥

Using the Fourier representation of the delta function

xjedx ωωπ

δ ∫=21)(

It follows that we can write

∫Ω=Ω

−ΘΘΩ+ ΩΓΓ

=−ΘΘ*

* )(2/)13(

*

2)1()...()( mIjtr

mmmm edmIπ

δ

A Few TheoremsI.r. unitary matrices come up in many applications.

Theorem 1 Let be an i.r. random matrix and consider the svd Then the following two equivalent statements hold:1. are independent random matrices and and

are i.r. unitary.2. The pdf of only depends on

Idea of Proof: and have the same distribution forany unitary and

A nm×.*VUA Σ=

VU ,,Σ U V

A :Σ).()( Σ= fAp

*ΨΘA AΘ ....Ψ

Theorem 2 Let A be an i.r. Hermitian matrix and consider theeigendecomposition . Then the following two equivalent statements are true.1. are independent random matrices and is i.r. unitary.2. The pdf of A is independent of U:

Theorem 3 Let A be a left rotationally-invariant random matrixand consider the QR decomposition, A=QR. Then the matricesQ and R are independent and Q is i.r. unitary.

*UUA Λ=

Λ,U U

).()( Λ= fAp

Some JacobiansThe decompositions and can be consideredas coordinate transformations. Their corresponding Jacobianscan be computed to be:

and

for some constant c.

Note that both Jacobians are independent of U and Q.

*UUA Λ= QRA =

)()(!1 *2

mlklk

IUUm

dUddA −−Λ= Π>

δλλ

)( *

1m

kmkk

m

kIQQrdQdRcdA −= −

=Π δ

EigendistributionsThus for an i.r. Hermitian A with pdf we have),(⋅Ap

).()(!1)(),( *2

mlk

lkA IUUm

pUp −−Λ=Λ ∏>

δλλ

Integrating out the eigenvectors yields:

Theorem 4 Let A be an i.r. Hermitian matrix with pdfThen

Note that , a Vandermonde determinant.

).(⋅Ap

∏>

−

−ΛΓ+Γ

=Λlk

lkA

mm

pm

p 2)1(

)()()1()...1(

)( λλπ

∏>

Λ=−lk

lk V )(det)( 22λλ

Some Examples• Wishart matrices, , where G is

• Ratio of Wishart matrices,

• I.r. unitary matrix. Eigenvalues are on the unit circle and the distribution of the phases are:

GGA *= ., nmnm ≥×

).(det)( 221

1

Λ=Λ−

=

−∏ Vecp kn

k

nmk

λλ

:121−= AAA

).(det)11()( 22

1

Λ+

=Λ ∏=

Vcp n

k

n

k λ

.)(sin),...,( 21 ∏

>

−=lk

lkm cp αααα

The Marginal DistributionNote that all the previous eigendistributions were of the form:

For such pdf’s the marginal can be computed using an eleganttrick due to Wigner.

Define the Hankel matrix

Note that Assume that Then we can perform theCholesky decomposition F=LL*, with L lower triangular.

∏=

Λ=Λm

kk Vfcp

1

2 ).(det)()( λ

[ ].11

)( 1

1

−

−∫

= m

m

fdF λλ

λλ LM

.0≥F .0>F

Note that implies that the polynomials

are orthonormal wrt to the weighting function f(.):

Now the marginal distribution of one eigenvalue is given by

But

mIFLL =−− *1

=

−

−

−1

1

1

0 1

)(

)(

mm

Lg

g

λλ

λMM

∫ = .)()()( kllk ggfd δλλλλ

)(det)()( 2

121 Λ== ∫ ∏

=

Vfddcpm

kkm λλλλλ L

21

1212 ))((det)(

detΛ= −

=− ∫ ∏ VLfddLc m

kkm λλλ L

)()()(

)()(11)(

111

010

111

11 Λ=

=

=Λ

−−−−

−−G

mmm

m

mm

m

Vgg

ggLVL

λλ

λλ

λλ L

MOM

L

L

MOM

L

Now upon expanding out and integrating over the variables the only terms that do not vanish are thosefor which the indices of the orthonormal polynomials coincide.

Thus, after the smoke clears

In fact, we have the following result.

Theorem 5 Let A be an i.r. Hermitian matrix with Then the marginal distribution of the eigenvalues of A is

)(det 2 ΛGV

mλλ L2

∑−

=

=1

0

2 ).()()(m

kkgfcp λλλ

∏= ).()( kA fAp λ

∑−

=

=1

0

2 ).()(1)(m

kkgf

mp λλλ

Orthogonal Polynomials• What was just described was the connection between

random matrices and orthogonal polynomials.• For Wishart matrices, Laguerre polynomials arise. For

ratios of Wishart matrices it is Jacobi polynomials, and for i.r. unitary matrices it is the complex exponential functions (orthogonal on the unit circle).

• Theorem 5 gives a Christoffel-Darboux sum and so

• The above sum gives a uniform way to obtain the asymptotic distribution of the marginal pdf and to obtain results such as Wigner’s semi-circle law.

))()()()(()()( '11

'1 λλλλλλ mmmmm

m ggggaa

mfp −−

− −⋅=

RemarkThe attentive audience will have discerned that my choice ofthe Cholesky factorization of F and the resulting orthogonal polynomials was rather arbitrary.

It is possible to find the marginal distribution without resortingto orthogonal polynomials. The result is given below.

[ ]

=

−

−−

1

11

11)(1)(

m

m Ffm

pλ

λλλ ML

Coherent ChannelsLet us now return to the multi-antenna model

where we will assume that the channel H is known. We will assume that where are the correlationmatrices at the transmitter and receiver and G has iid CN(0,1)entries. Note that can be assumed diagonal wlog.

According to Foschini&Telatar:

1. When

,VSHM

X +=ρ

rtGDDH = rt DD ,

rt DD ,

:, NrMt IDID ==

∑ +=+= ))(1log()det(log*

*

MGGEGG

MIEC N ρλρ

2. When

3. When

4. In the general case:

Cases 1-3 are readily dealt with using the techniques developed so far, since the matrices are rotationally-invariant.

Therefore we will do something more interesting and compute the characteristic function (not just the mean). This requires more machinery, as does Case 4, which we now develop.

:Mt ID =

∑ +=+= ))(1log()det(log*2

*2

MGGD

EGGDM

IEC rrM ρλρ

:Nr ID =

∑ +=+===

))(1log(max)det(logmax*

)(

*

)( MGPDDG

GPDDGM

IEC tt

MPtrttNMPtrρλρ

∑ +==

))(1log(max*

)( MGDPDDGD

EC rttr

MPtrρλ

A Useful Integral FormulaUsing a generalization of the technique used to prove Theorem5, we can show the following result.

Theorem 6 Let functions begiven and define the matrices

Then

where

1,,0),(),(),( −= mkhgf kk Lλλλ

=Λ

=Λ

−−−− )()(

)()()(,

)()(

)()()(

111

010

111

010

mmm

m

H

mmm

m

G

hh

hhV

gg

ggV

λλ

λλ

λλ

λλ

L

MOM

L

L

MOM

L

FmVVfd H

m

kGk det!)(det)(det)(

1

=ΛΛΛ∫ ∏=

λ

[ ].)()()(

)()( 10

1

0

λλλ

λλλ −

−

∫

= m

m

hhg

gfdF LM

Theorem 6 was apparently first shown by Andreief in 1883.

A useful generalization has been noted in Chiani, Win and Zanella (2003).

Theorem 7 Let functionsbe given. Then

where for the tensor we have defined

and the sums are over all possible permutations of the integers 1 to m.

1,,0),(),(),( −= mkhgf kkk Lλλλ

))()()(()(det)(det)(1

λλλλ kjiH

m

kGkk hgfTensorVVfd ∫∫ ∏ =ΛΛΛ

=

ijkA

∑ ∑ ∏=

=µ α

αµαµm

kkkk

AATensor1

)sgn()sgn()(

αµ,

An Exponential IntegralTheorem 8 (Itzyskon and Zuber, 1990) Let A and B be m-dimensional diagonal matrices. Then

where

Idea of Proof: Use induction. Start by partitioning

∫ Γ+Γ=−ΘΘΘ ΘΘ

)(det)(det),(det)1()1()( **

BVAVBAEmIed m

BAtr Lδ

=mmm

m

baba

baba

ee

eeBAE

L

MOM

L

1

111

),(

[ ]

=Θ=Θ

maA

A 11 ,ϑ

Then rewrite so that thedesired integral becomes

trBaBIaAtrBAtr mmm +Θ−Θ=ΘΘ − ))(()( *1111

*

∫ ∫ −ΘΘΘΘ −ΘΘΘ=−ΘΘΘ )()( 11

*11

* *1

'1

*

mBAtrtrBa

mBAtr IedeIed m δδ

∫ −−ΘΘΩ+ΘΘΩΘ= )(1

11*1

*1

'1 mm IjtrBAtrtrBa eddce

∫∏=

Ω−

Ω+Ω= m

kk

jtrtrBa

jAb

edce m

1

' )det(

∫∏=

−

−

+= m

kmk

WjtrAtrBa

jWIb

edWce m

11 )det(

'

)()(det)(

1*2

1

1

1

*'

−

=

−

=

Λ−

−Λ+

Λ= ∫∏∏

mm

klk

m

l

UUjtrAtrBa IUUV

jb

edUdce m δλ

The last integral is over an (m-1)-dimensional i.r. matrix.And so if use the integral formula (at the lower dimension)to do the integral over U, we get

An application of Theorem 6 now gives the result.

)(det),(det)(

1)(det

'

1

1

1' ΛΛ

+Λ= ∫

∏∏

=

−

=

VAEjb

dAV

ec m

klk

m

l

trBam

λ

Characteristic FunctionConsider

The characteristic function is (assuming M=N)

Successive use of Theorems 6 and 8 give the result.

)det(log)det(log ** DGGM

IEGDGM

IEC NMρρ

+=+=

ωρ

ω ρ jN

DGGM

IjDGG

MIEEe N )det( *)det(log *

+=+

trWjN eDW

MIdWc −+= ∫ ωρ )det(

1

)det()(det

−−+= ∫ trWDjNm eW

MIdW

Dc ωρ

)()(det)1()(det

*2

1

1*

MDUtrU

M

kkm IUUVe

MdUd

Dc

−Λ+Λ=−Λ−

=∏∫ δλρ

Non-coherent Channels

Let us now consider the non-coherent channel.

where H is unknown and has iid CN(0,1) entries.

Theorem 9 (Hochwald and Marzetta, 1998) The capacity-achieving distribution is given by S = UD, where U is T-by-Mi.r. unitary and D is an independent diagonal.

Idea of Proof: Write S=UDV*. V* can be absorbed in H and so Is not needed. Optimal S is left rotationally-invariant.

,VSHM

X +=ρ

Mutual InformationDetermining the optimal distribution on D is an open problem.However, given D, one can compute all quantities of interest.The starting point is

The expectation over U is now readily do-able to give p(X|D). (A little tricky since U is not square, but doable using FourierRepresentation of delta functions and Theorems 6 and 8.)

)(det),|(

*2

)( 1*2*

UUDM

I

eDUXpT

NTN

XUUDM

ItrX T

ρπ

ρ

+=

−+−

)(det 2

)( *12**

DM

I

e

MNTN

XUDMIUtrXXtrX M

ρπ

ρ

+=

−−++−

Other Problems• Mutual information for almost any input distribution on D

can be computed.• Cut-off rates for coherent and non-coherent channels for

many input distributions (Gaussian, i.r. unitary, etc.) can be computed.

• Characteristic function for coherent channel capacity in general case can be computed.

• Sum rate capacity of MIMO broadcast channel in some special cases can be computed.

• Diversity of distributed space-time coding in wireless networks can be determined.

Other Work and Open Problems• I did not touch at all upon asymptotic analysis using the

Stieltjes transform.• Open problem include determining the optimal input

distribution for the non-coherent channel and finding the optimal power allocation for coherent channels when there is correlation among the transmit antennas.