Complex singular wishart matrices and applications

(1)

ELSEVIER

An International Joumal

Available onhne at www.sciencedirect.com

computers &

mathematics

with applications

Computers and Mathematzcs with Applications 50 (2005) 399-411

www elsevier com/locate/camwa

Complex Singular Wishart

Matrices and Applications

T. R A T N A R A J A H A N D R . V A I L L A N C O U R T Department of Mathematics and Statistics

University of Ottawa 585 King Edward Ave Ottawa K1N 6N5, ON, Canada

T. Ratnaraj ah©zeee, org r e m i © u o t t a w a . c a

(Recewed October 2004; rewsed and accepted Apml 2005)

A b s t r a c t - - I n this paper, complex singular Wlshart matrices and their applications to reformation theory are investigated In partacular, a volume element on the space of positive semidefinite m × m

Hermltian matrices of rank n < rn is introduced and some transformation properties are estabhshed. The Jacoblan for the change of variables in the singular value decomposition of general m × n complex matrices as derived. Then, the denmty functions are formulated for all rank n complex singular Wmhart distributions From this, the joint eigenvalue density of low rank complex Wmhart matrices are derived. The derwed densities are used to evaluate the most important informatlon-theoretm measure, the so-called ergod~c channel capaezty of multiple-input multiple-output (MIMO) spatially correlated Raylelgh distributed wireless communication channels. (~) 2005 Elsevier Lid All rights reserved.

K e y w o r d s - - C o m p l e x singular Wishart distribution, Complex singular Wmhart matrices, Singular

value decomposition, Raylelgh distributed MIMO channel, Channel capacity

1. I N T R O D U C T I O N

T h e present c o m m u n i c a t i o n s r e v o l u t i o n is due, in part, to advances in wireless communications, such as wireless i n t e r n e t and m u l t i m e d i a communications. As t h e wireless i n d u s t r y becomes ubiq- uitous and popular, t h e need for a h i g h - d a t a r a t e and large user c a p a c i t y on a wireless p l a t f o r m is t h e key driver in developing robust c o m m u n i c a t i o n s techniques, which offer a substantially increased i n f o r m a t i o n capacity. N o t e t h a t t h e c a p a c i t y of a c o m m u n i c a t i o n channel expresses t h e m a x i m u m r a t e at which i n f o r m a t i o n can be reliably conveyed by t h e channel [1]. T h e recent i n f o r m a t i o n t h e o r y result showed t h a t an enormous s p e c t r a l efficiency can be achieved t h r o u g h t h e use of m u l t i p l e i n p u t multiple o u t p u t (MIMO) a n t e n n a system, see [2-4]. In o t h e r words, this wireless c o m m u n i c a t i o n s y s t e m uses multiple a n t e n n a s at t h e t r a n s m i t t e r and receiver ends. In a wireless c o m m u n i c a t i o n system, d a t a is delivered from a t r a n s m i t t e r to a receiver using This work was partially supported by NSERC of Canada.

The anonymous referee is to be thanked for many corrections and suggestions that considerably improved this

paper

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400 T RATNARAJAH AND l~ VAILLANCOURT

radio waves or other electromagnetic waves. The waves, however, may be reflected off objects in the environment and scattered randomly while propagating from the transmitter to the receiver. Therefore, transmitted signals are attenuated and phase shifted during the transmission. This channel response can be modeled by the complex channel coefficients.

Let an n × m complex Caussian (or normal) random matrix 1 A be distributed as A CN(M, In ® E) with mean g{A} = M and covariance coy{A} = I,~ ® E. Here, we read the symbol "~" as "is distributed as", CN denotes the complex normal distribution, and ® denotes the Kroneeker product. Then, the matrix W = A H A is called a complex noncentral Wishart matrix. If M = 0, then W is called a complex central Wishart matrix. The complex central and noncentral Wishart distributions are denoted by CWm(n, E) and CWm(n, E, ~), respectively, where fl = E-1MHM. The complex Wishart matrices are well studied in the hterature only for n > m, for example, see [5-7].

In this paper, we extend the study of complex central Wlshart distributions to the singular case, w h e r e 0 < n < m a n d n , m E Z. Thus, the rank o f W E C mxm i s n provided the rank of A C C '~xm is n. A volume element on the space of positive semidefinite rn x m Hermitian matrices of rank n < m is introduced (see Theorem 1). The Jacoblan of the change of variables in the singular value decomposition of general m x n complex matrices is derived (see Theorem 2). The density is derived for rank-n complex central Wishart distributions for all integers n, 0 < n < m (see Theorem 3). The 3oint eigenvalue density of low rank complex Wishart matrices are derived (see Theorem 4). It should be noted that singular Wishart and beta distributions are studied in [8] for real random matrices. The singular value distribution of Gaussian random matrices is given in [9].

The theory of complex random matrices is used to evaluate the capacity of MIMO wireless communication systems with nt inputs (or transmitters) and nr outputs (or receivers), under the assumption that the channel coefficients are distributed as complex Gaussian and correlated at both the transmitter and the recelver ends. Then, the MIMO Rayleigh flat fading channel can be represented by an nr x nt complex random matrix H ~ g N ( 0 , Er ® Et), where E~ and Et are positive definite Hermitian matrices which represent the channel correlation at the receiver and transmitter ends, respectively. This means the covariance matrices of the columns and rows of H are denoted by Er and Et, respectively. If Er = (72In~ (or I ~ ) and Et = Int (or z2J[nt) , then the channel is said to be an uneorrelated Rayleigh d~stmbuted channel, and the complex nonsingular Wishart matrix theory can be used to compute the capacities for both cases nr ~ nt and nt > n~ [2]. However, if E~ = In~ and Et = Z then the complex nonsingular and singular Wishart matrix theomes are needed to compute the capacities for the cases n~ > nt and nt > n~, respectively See [10] for the case nr > nt. In this paper, we assume that Z~ = I,~, Et = E, and nt > nr, which leads us to represent the channel capacity in the form of a complex singular Wishart matrix. This is the motivation behind this study.

This paper is organized as follows. Section 2 provides the necessary tools for deriving the complex singular Wishart distribution theory. Complex singular Wishart matrices are studied in Section 3 The capacity of a MIMO channel and the computational method are given in Section 4.

2. N E C E S S A R Y T O O L S

In this section, we derive necessary tools for studying the singular Wishart distribution theory and MIMO channel capacity. If 0 < n < m, then the density does not exist for W ~ CWm(n, E) on the space of Hermitian m × m matrices because W is singular and of rank n almost surely. It can be shown t h a t the density does exist on the (2ran - n2)-dimensional manifold of rank n, CSm,n, of positive semidefimte m × m Hermitian matrices W with n distinct positive eigenvalues. Moreover, the set of all m × n matrices E1 with orthonormal columns is called the Stiefel man,fold,

1Hlstomcally, r a n d o m m a t r i c e s (or variables) have often been denoted by bold t y p e a n d their realizations have been d e n o t e d by nonbold t y p e Hence, n o n r a n d o m matrices are denoted by nonbold t y p e

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Singular W l s h a r t Matrices 401

denoted by C Y n , m . Thus,

CV.,m = {El E C'~X~; EHE1 = I n } . (1)

The elements of E1 can be regarded as the coordinates of a point on a (2ran - n2)-dimensional surface in the 2ran-dimensional Euclidean space.

Theorems 1 and 2 below are derived in terms of the exterior product of differential forms which is a convenient way of computing the determinant of a matrix of partial derivatives (so-called Jacobian). See [11, p. 57] for the definition of the exterior product (dX) for real matrices X, such as symmetric, skew-symmetric, upper-triangular, and arbitrary matrices. On the other hand, if X = X r + iXc is a complex matrix, its exterior product is (dX) = (dXr) A (dXc). Jacobian formulas for some important complex matrix factorizations are given in [12].

The volume element (dW) in the reduced spectral decomposition W = E 1 A E H is given by the following theorem.

THEOREM 1. Let m and n be two positive integers, such that 0 < n < m and consider an m x m positive semidefinite Hermltian matrix W E CSm,n of rank n with decomposition W = E 1 A E H, where the diagonal elements o f A = diag (A1,. • •, Am) are positive eigenvalues in decreasing order, A1 > . . . > A, > O, and E1 E CVn,m. Then, the volume element is

(dW) = (2~r) -~ " k 2 m - 2 n H (Ak - Az) 2 (dA) A ( E H d E 1 ) , (2)

k = l k<l

where

A

A m

(dA) = dAk, ( E l dE1) =

A

de ,

k=l k = l l=k

and matrrx E1 is appended with an m x (m - n) matr/x E2, such that the compound m x m matmx, E = [El : E2] = [ e l , . . . , en : e n + l , . . . , era] is unitary.

PROOF. First, we note that

d W = d E I A E H + E1 d h E g + E1A dE H and Therefore, we have EH E1 = In, EH E1 = 0 E g d W E = [ EH dE1A + d A + A d E H E 1 , A d E H E 2 , ]

L

E g dE1 A, 0, ] "

The exterior product on the left side of equation (3) is equal to ( E H d W E ) = (det E) 2"~ (dW) = ( d W ) .

T h e / t h r o w of E H dE1A is

[e H d e l . . . e H den] A, n + 1 < l < m,

(3)

and the exterior product of its elements is equal to

(detA)

dek =

e f &k.

(4)

4 0 2 T . I~ATNARAJAH AND R VAILLANCOURT

Therefore, the exterior product of all the elements of E H dE1A is

(4)

Second, we consider E H dE1A + dA + AdEHE1. Since the columns of E1 are orthonormal, we

h a v e

E r E 1 = In ~ E H dE1 = - d E H E 1 = - ( E H dE1)".

This implies that the real and the imaginary parts of E H dE1 are skew-symmetric and symmetric, respectively. Moreover, we have

E H dE1A + dA + AdE1 HE1 = E1H dE1A - hE1H dE1 + dA.

(5)

The exterior product of the diagonal elements of the right side of equation (5) is

(dA) = f dAk. (6)

k = l

Note that the diagonal elements of E H dEIA - A E H dE1 are zeros.

Now, we consider the upper diagonal elements of E H dE1A - A E H dE1. Re(E H dE1) is skew-symmetric, it can be written as

Since the matrix

R e ( E H d E 1 ) =

0 - R e (e H del) . . . Re (e H del)"

Re (e H de1) 0 . . . Re (eft de2)

R e ( e H de1) R e ( e H de2) 0 . . . . R e ( e 5 dea)

Re (e~ ~el)

Re (e~ de2) . . .

0

For k < l, the (k,/)th element of Re (E H dE1)A-ARe (E H dE1) is Re (e H dek)(Ak--Al). Therefore, the exterior product of the upper diagonal elements of

R e ( E H d E 1 ) A - A R e ( E H dE1)

is given by

k < l k < l

Similarly, since the matrix Im (E H dE1) is symmetric, it can be written as

Im (E H dE1) = I m ( e Hdel) - I m ( e Hdel) . . . - I m (e H de1) Im (e H de2) . . . - I m (e Hdel) - I m (e Hde2) . . . ; ; • . - I m (e $ eel) - I m (e $ de2) . . . -Im (e H de1)

]

-Im (e H de2)

I

-Im (ef eel) /

/

~m ( eg ,~en) .J

and the exterior product of the upper diagonal elements of

I m ( E H dE1) A - A I m ( E H dE1)

(7)

is given by

Im

(e / dek) h

(Ak

- A,).

k < l k < l

(5)

S i n g u l a r W m h a r t M a t r m e s 403

Therefore, the exterior p r o d u c t of the elements of the right side of equation (3) is obtained by multiplying equations (4), (6)-(8), i.e.,

ffI )n

(dW) = (21r)-n ;~m-2n H ( A k _ A z ) 2 ( d A ) A ( E H d E I ) .

k = l k<l

(9)

Note t h a t we must divide the volume element by (27c) n to normalize the arbitrary phases of the n

elements in the first row of E l . |

T h e volume element (dW) (or Jacoblan) m the singular value decomposition is given by the following theorem.

THEOREM 2. Let Z be an m x n complex matrix and Z = E, T H the nonsmgular p a r t of the singular value decomposition, where E1 E CV~,,~, H C U(n), and the diagonal dements of T = diag ( v , , . . . , v , ) are positive singular values with vl > ' - - > v , > 0. Then, we have

(dZ) = (27r) - ~ v~ m-2n+l H ( v ~ - v~)2(dT) A (E H dE,) A (HHdH), (10)

k = l k < l

where

m

(dT)

dye, : h, dh ,

(El"dE,)

: A effd ,

k=Z k=l Z=k k=, l=k

and m a t r i x E1 is appended with an m x (m - n) m a t r i x E2 such t h a t the compound rn x m matrix, E = [E, : E2] = [ e l , . . , e , : en+,, , em] is unitary

PROOF. First, we note t h a t

dZ = dE, T H H + E1 d T H H + E I T d H g

and

W E 1 = ±n, E , = o.

Therefore, we have

E H d Z H = [ EH dE1TE H+dTdEIT + T d H H H ] . (11)

T h e exterior product of the left side of equation (11) is equal to

( E " d Z H ) = (dZ).

As in the proof of T h e o r e m 1 (see equation (4)), the exterior p r o d u c t of all the elements of E ~ dE1T is

(EH dE1T) = ( ~ I vk 2m-2n~ / m

k = , k = l l = n + l

Second, we consider E H dE, T + dT + T d H H H . Since H is unitary, we have

H H H = I , ~ H H d H = - d H H H = - ( H H d H ) H •

This implies t h a t the real and imaginary parts of H H d H are skew-symmetric and symmetrm, respectively. Similarly, for E1H dE1. Thus, the n x n matrix of differential forms is

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4 0 4 T RATNARAJAH AND R VAILLANCOUI%T

with entries

{ dvk + v k ( I m ( e H d e k ) - I m ( h H d h k ) ) , k = l , Tkl = e H delvl -- vkh H dhl, k > l,

e[ { dekv~ - vkh H dhk, k < l.

The exterior product of the elements of T is

where

k = l k<l

Tk~ A T~k = (v~ - .~) Re ( e l ~e~ A h f dh~)

A (v~ - v~) Im (e H dek A h H dhk) (14)

= (.~ - ~,~)2 ~p d ~ ~ h ; ~h~. Therefore, the exterior product of equation (11) is

(dZ) = (2~) - ~ ( E ~ dE~T) A (T) = (2~) - ~

- ~ - ~

~/~ ~k

.~

(.~-.~)~

d.~ \ k = l / k----1 l:n-bl k----1 k<l k = l k = l l=k k = l Z=k

(15)

/

; (27r) - n v m - ~ + l \ k = l / x ~ I (v~ - v~) 2 (dT) A (E H dE1) A (H g d H ) . k<l

Note that we must divide the volume element by (27r) n to normalize the arbitrary phases of the n

elements in the first row of El, |

The volume of the Stiefel manifold CV.,m is given by

~C 2nTrmn

Vol (CV~,m) = (E H dE1) - CF~ (m)' (16)

Vn,m

where the complex multivariate gamma function is

n

ern (a) = ~ ( ~ - 1 ) / 2 H r (a - k + 1), k = l

Re(a) > n - l ,

and the gamma function is F(n) = f o t n - l e - t dt. The differential form, c r n ( m )

1 (EHdE1) - (E H dE1) (17)

(dE1) - Vol [CV,~,.~] 2'~7r mn

has the property that

~c (dE1) : 1,

Vn,m

and it represents the Haar mvariant probability measure on CV~,.~.

If m = n, then we get a special case of Stiefel manifold, the so-called unitary manifold, defined by

(7)

Singular Wmhart Matrmes 405 that is, the set of unitary n x n matrices. The volume of U ( n ) is given by

2nvra 2

r

Vol [U (n)] = J u ( E H d E ) - (18)

(n) CI~n (f~)'

The probabihty distributions of random matrices are often derived in terms of hypergeometric functions of matrix arguments. The following complex hypergeometric function of two Hermitian matrix arguments is required in the sequel, i.e.,

oF~2)(X,y)=zzC

C.(V)

k=o ~ • (Ira)

where X E C mx'~, Y E C ~xn and 0 < n < m. Moreover, t~ = ( k l , . . . ,kn) denotes a partition of the integer k with kl _> ... >__ k~ >_ 0 and k = kl + ... + k~ and }-~ denotes summation over all partitions ~ of k. The complex zonal polynomial of a Hermitian matrix Y is defined in [6] as

c,~ (y) = x[~] (1) x[,q ( v ) , (19)

where X[~](1) is the dimension of the representation [~] of the symmetric group, n

II (k, - k, - , + 3)

X[,q (1) = k! '<J

IeI (k~ + n - ~ ) !

(20)

and X[~] (Y) is the character of the representation [n] of the linear group given as a symmetric function of the eigenvalues, A1,... , )~n, of Y by

act [ ( A : ' + n - ' ) ]

X[,q (Y) = (21)

deC [ ( A : - J ) ]

Note that both the real and complex zonal polynomials are particular cases of the (general a) Jack polynomials C ( ~ ) ( Y ) , where a = 1 for complex and a = 2 for real zonal polynomials, respectively. See [13,14] for details. In this paper, we only consider the complex case, therefore, for notational simplicity, we drop the superscript of Jack polynomials, as was done in equation (19), i.e., C ~ ( Y ) := C O ) ( Y ) . Finally, we have

G (In) = k!

[ f i

(k,-k,-~+3/] ~

*<3

l~I r(k~ + n - i + 1) fI r ( n - ~ + 1)

~=i z=l 3. C O M P L E X S I N G U L A R W I S H A R T M A T R I C E S

In thls section, we derive the complex singular Wishart density and the joint eigenvalue density of a complex singular Wishart matrix.

THEOREM 3 L e t m and n be two positive integers, such t h a t 0 < n < m T h e d e n s i t y o f W ",~ C W m ( n , ~) on the space CSrn,n O[ m X m positive semidefinite H e r m l t l a n matrices o f r a n k n is given by

7rn(n--m)

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406 T RATNARA3AI-I AND R VAILLANCOURT

where W = EIAE H, E1 e CVn,m,

A = diag (A1,... ,

An), and

etr

denotes

the

exponential

of

the

trace, etr (.) = exp (tr(.)).

PROOF. Let W = A H A , where the n x m m a t r i x A is distributed as A ~ CN(0,

In ® E)

and is an m × m positive definite H e r m i t i a n matrix. T h e density of A is

f ( d ) -- ~r - n m (det E) - ~ etr

( - A E - 1 A H) (dA)

(23) = ~r - n m (det E) - n etr

( - E - 1 A H A ) (dA),

where the volume element

(dA) =_ A~=I A ml=l dakl

is included to facilitate the calculation of Jacobians when we transform A. Transforming

A H -- E I T H as

in T h e o r e m 2 results in the desired p a r a m e t e r i z a t i o n

W = E1AE H,

where A = T 2, and det A = det T 2. Thus, we have

(dA) =

dAk = 2 n

vk A dvk = 2 n

vk

( d T ) . (24)

k = l k = l k = l

From T h e o r e m s 1 and 2 and equation (24) we can write

(dA)

as

(dA) = 2-'~ ( f l A~-m) (dW) (HH dH)

Since YI~=I Ak = det A, the joint density of W and H is

:r -nr~ (det E) - n etr ( - E - 1 W ) 2 - n (det A) n - ' ~

(dW) (HHdH).

Integrating with respect to H over the Stiefel manifold

CVn,m

and using (16), we obtain the

marginal density function of W , given by (22). |

T h e following t h e o r e m gives the joint density of the eigenvalues of a complex singular Wishart matrix.

THEOREM 4.

Let m and n be two positive integers such that 0 < n < m and consider the

m x m positive semidefinite Hermitian matrix W

~ C W m ( n , ~ ) .

The joint density of the positive

eigenvalues,

A 1 , . . . , An,

of W is

~r n(~-l) (det p~)-n A~ -'~ (3,k - Al) 2 f (A) = c r . (m) k < l x / etr ( - E - ' E 1 A E H) ~ a

(dEl) ,

J E

where W =

EIAE H, E1 E CV~,,~, and

A = diag ( A 1 , . . . , An).

Moreover,

(dE )

(26)

err ( - E - I E 1 A E H )

PROOF. Substituting (2) into the Wishart density (22) and integrating with respect to E1 over the Stiefel manifold

CVn,m,

we have

f (A) = Cl-'~ (n) (det E) n A~ -'~ (Ak - Al) 2

k = l _k<l ₍₂₇₎

X ~f~lecv~metr(-E-1EIAEH)(Z g dE1).

Now, using (17), we o b t m n the desired results (25). |

Note that, if E =

a~Im,

then the joint density of the eigenvalues A1,... , A,~ has a simple form which does not require a complex hypergeometric function representation.

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S i n g u l a r W i s h a r t M a t r i c e s 407

COROLLARY 1.

Let W

.o tWin(n, G2Xm)

with 0 < n < m. Then,

the

joint density of the

eigenvalues, )~1,... , An, of W

is

7rn(n_ 1) (G2)_nm n n 1

E A k

,

g(A) = A; - A £ exp / (2s)

where W =

E1AE H, El c CV,~,m,

and A = diag (/~1, - - - , "~n)

PROOF. Putting E =

a2Im

in Theorem 4 and noting that

~iccv~, etr ( - - ~ E I A E H ) (dE1) = etr ( - ~ A )

fEl~CVn,m (dE1)

= exp - , ,

(29)

complete the proof. II

4. T H E M I M O C H A N N E L C A P A C I T Y

Recently, in response to the demand for higher bit rates in wireless communications, indus- trial researchers have exploited the use of multiple input multiple output systems, as shown in Figure 1. These studies show that MIMO systems increase capacity significantly over single input single output (SISO) systems. For example, if n = min{nt, nr}, a MIMO uncorrelated Rayleigh distributed channel achieves almost n more bits per hertz for every three-dB increase in signal-to-noise ratio (SNR) compared to a SISO system, which achieves only one additional bit per hertz for every three-dB increase in SNR [2]. But the channel coefficients from different transmitter antennas to a single receiver antenna can be correlated. This channel correlation, which degrades the channel capacity [15], depends on the physical parameters of a MIMO system and the scatterer characteristics. The physical parameters include the antenna arrangement and spacing, the angle spread, the angle of arrival, etc. One of the objectives of this paper is to evaluate this capacity degradation for the channel matrix H ~ CN(0, In~ ® E) with

nt > n~.

This will be done by deriving closed form ergodic capacity formulas for correlated channels and their numerical evaluation

Xl ~ Yl

X2

~ Y2

X

_{n t}

hl 1 \ ~ ~ vl

';...

. . . ....'."~ //

"'"'" ... h21 h12 ... """

I

• ...

... .,:,:.:....'"

...'"

' ~ ~

v2

"°*.# • • • . • , o. o o o • . j [ °

:: 6 : ::. .... : . : : i : : : : : : : ::

"'".... h r 2 ::'::" ...'" " % . * * * " ' ° . 4 . °° ] . . " " ' .

o .°* "..

]

hlnt ..""

..'"':<'"'..

"" • h r l

I v

. , , ° • . . . "" "" • . . , o

.'°,.°.

~n/,

_ _ <

"'::"'"'" h2n t

hnrn t

"'".~:':::. \

/

=

F i g u r e 1 A M I M O c o m m u m c a t l o n s y s t e m

(10)

408 W RATNARAJAH AND R VAILLANCOURT

The complex signal received at the ~th output can be written as

n t

= + (30)

z = l

where h3, is the complex channel coefficient between input ~ and output 3, x~ is the complex signal at the Z th input, and vj is complex Gaussian noise with unit variance. The signal vector received at the output can be written as

i.e., in vector notation,

I !l hll I hi!]Ix11 [v!]

= • _ + ,

y ~ [ h ~ l hn nt X , V r

y = H x + v, (31)

where y , v e C '~, H e C n~×'~', x e C m, and v ~ CN(0, I ~ ) (see Figure 1). It should be noted

t h a t the noise v is independent of the input signal x and channel matrix H. The total power of the input is constrained to p,

E{xHx} < p or

tre{xx ~} ___

p.

In this section, we shall deal exclusively with the linear model (31) and compute the capacity of MIMO channel models for nt > n~.

We assume that H is a complex Gaussian random matrix whose realization is known to the receiver, or equivalently, the channel output consists of the pair (y, H). Note t h a t the transmitter does not know the channel and its statistics (i.e., ~,) and the input power is distributed equally over all transmitting antennas, which is a natural thing to do in this case. Moreover, if we assume a block-fading model and coding over many independent fading intervals, then the Shannon or ergodic capacity of the random MIMO channel [2] is given by

where the expectation is evaluated using a complex Gaussian density. If H ~ CN(O, In~ ®E), then the channel is Rayleigh distributed and correlated at the transmitter end. This is typical of fixed or mobile communication environments. Here, the covariance matrix of the rows of H is denoted by ~, which is an nt × nt positive definite Hermitian matrix. Let W = H H H ~ C W ~ , ( n ~ , E ) and nt > n~. Then the channel capacity can be written as

--PW ,

C = C w { l o g d e t ( I m + n t ) } (33)

where the expectation is evaluated using a complex singular Wishart density given in Theorem 3. Let A1 > " " > A,,~ be the eigenvalues of W and A = diag(A1,... ,A,~). Then, the capacity can also be computed using a joint eigenvalue density, f(A), or the single unordered eigenvalue density, f(A), i.e.,

n~

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Smgutar Wishart Matrices 409 T h e joint eigenvalue density of a complex singular W i s h a r t m a t r i x is given in T h e o r e m 4. From this joint eigenvalue density, we can obtain a single unordered eigenvalue density, f(A), by dividing f ( A ) b y n~! and integrating with respect to A2,... , A ~ .

As a numerical example, we c o m p u t e the channel capacity of a correlated 2 × 4 channel m a t r i x (n~ = 2 and nt = 4), i.e., H ~ CN(O, I2 @ E), where we assume the eigenvalues of the positive definite H e r m i t i a n m a t r i x E are

1 8090, 1.3090, 0.6910, 0.1910.

In this case, W is 4 x 4 complex singular W i s h a r t m a t r i x with two nonzero eigenvalues A1 and A2. T h e joint eigenvalue distribution is given by

1

f (At, A2) - 12 (det

E) 2 (AIA2)2 ()~1 - )~2) 2 oFO

(2) ( - E - l , A),

(35) where A = diag (A1,

"~2).

T h e capacity of the correlated channel, H ~ CN(0, _72 ® E), is given by

For comparison purpose we

also

c o m p u t e the capacity of the uncorrelated 2 x 4 channel matrix, i.e., H ~ C N ( 0 , / 2 ® a214). In this case, the joint eigenvalue density of the complex singular W i s h a r t m a t r i x is given by

g(A1, A2) = 1-~16 (A1A2)2(A1 - A2)2e -()~1+)~2)/a2 . (37) From (37), we can evaluate the single unordered eigenvalue density, g(A), as

g ( x ) - 12a---v- ~ ~2 + 12 . (38)

T h e capacity of H ~ ON(0, I2 ® a 2 h ) is given by

C,, -- 2 ~o°~ log (I +PA)g(A) dA.

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12 10 , - , 8 ~ 6 0 0 Figure 2 C u i I 1 r 5 10 15 20 Signal-to-noise raUo (SNR) Capacity vs 25

SNR for nt = 4 and nr = 2, le., 2 x 4 channel matrix.

Cu and Cc denote the capacity of uncorrelated and correlated Rayteigh channels, respectively

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4 1 0 T R A T N A R A J A H A N D PL V A I L L A N C O U R T

Figure 2 shows the capacity in nats 2 vs. signal-to-noise ratlo for a 2 x 4 correlated/uncorrelated Rayleigh fading channel matrix. F r o m this figure, w e note the following,

(i) the capacity decreases due to channel correlation, and (ii) the capacity is increasing with increasing S N R .

5. C O N C L U S I O N

In this paper, we studied the complex singular Wishart distribution and its applications. In particular, we derived the complex singular Wishart density and joint eigenvalue density of a complex singular Wishart matrix. Using these distributions, both correlated and uncorrelated MIMO Rayleigh channel capacity formulas were obtained. The capacities of 2 x 4 MIMO Rayleigh channel matrices were computed for both correlated and uncorrelated channels. It was also shown how channel correlation degrades the capacity of communication systems.

A P P E N D I X

T h e r e a l s i n g u l a r W i s h a r t d e n s i t y is d e r i v e d i n [8, T h e o r e m 6]. I n t h i s a p p e n d i x , we give t h e j o i n t e l g e n v a l u e d e n s i t y of a r e a l s i n g u l a r W i s h a r t m a t r i x . N o t e t h a t we follow t h e n o t a t i o n

of [8,11]

THEOREM 5. Let m and n b e two posltive integers, such that 0 < n < m and consider an m × m positive semidefinite s y m m e t r i c Wishart m a t r i x W ,.~ ~)m(n, ~,). Then, the joint density of the positive eigenvalues, )~1, . . . , An, of W as

/ (A) = Fn (n/2) Pn (m/2) A~m (Ak - A~)

k<l

X /E16Vn,metr ( - ~ - I E 1 A E H ) (dE1),

where W =

E1AE1 H, Ez E ]2n,m,

and A = diag (A1,..., An).

Moreover,

/ & e V n , m e t r ( - l ~ - l E , A E H ) ( d E 1 ) = o F ( n ) ( - 2 ~ - Z , A ) •

R E F E R E N C E S

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10

2in equation (34), If we use Iog e then the capacity is measured in nets. If we use log 2 then the capacity IS measured in bits Thus, one nat is equal to 1/iOge(2 ) blts/sec/Hz (e = 2 718 )

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