Here, we derive some key aspects of the complex pseudo-Wishart distribution

The Pseudo-Wishart Distribution and its Application to MIMO Systems

Ranjan K. Mallik, Senior Member, IEEE

Abstract—The pseudo-Wishart distribution arises when a Hermitian matrix generated from a complex Gaussian ensemble is not full-rank.

It plays an important role in the analysis of communication systems using diversity in Rayleigh fading. However, it has not been extensively studied like the Wishart distribution. Here, we derive some key aspects of the complex pseudo-Wishart distribution. Pseudo-Wishart and Wishart distributions are treated as special forms of a Wishart-type distribution of a random Hermitian matrix generated from independent zero-mean complex Gaussian vectors with arbitrary covariance matrices. Using a linear algebraic technique, we derive an expression for the probability density function (pdf) of a complex pseudo-Wishart distributed matrix, both for the independent and identically distributed (i.i.d.) and non i.i.d.

Gaussian ensembles. We then analyze the pseudo-Wishart distribution of a rank-one Hermitian matrix. The distribution of eigenvalues of Wishart and pseudo-Wishart matrices is next considered. For a matrix generated from an i.i.d. Gaussian ensemble, we obtain an expression for the char- acteristic function (cf) of eigenvalues in terms of a sum of determinants.

We present applications of these results to the analysis of multiple-input multiple-output (MIMO) systems. The purpose of this work is to make researchers aware of the pseudo-Wishart distribution and its implication in the case of MIMO systems in Rayleigh fading, where the transmitted signals are independent but the received signals are correlated. The results obtained provide a powerful analytical tool for the study of MIMO systems with correlated received signals, like systems using diversity and optimum combining, space—time systems, and multiple-antenna systems.

Index Terms—Characteristic function, diversity and optimum com- bining, multiple-input multiple-output (MIMO) systems, probability density function (pdf), pseudo-Wishart distribution, Wishart distribution.

I. INTRODUCTION

The pseudo-Wishart distribution [1] arises when a Hermitian matrix generated from a complex Gaussian ensemble is not full-rank.

Although it is an incomplete form of the Wishart distribution, it plays an important role in the analysis of communication systems using diversity in Rayleigh fading. It can appear, for example, in the analysis of optimum combining [2], space-time systems [3], and multiple-antenna systems [4], [5]. However, it has not been extensively studied like the Wishart distribution. Here we derive some key aspects of the complex pseudo-Wishart distribution. We derive an expression for the probability density function (pdf) of a complex pseudo-Wishart distributed matrix, both for the independent and identically distributed (i.i.d.) and non-i.i.d. Gaussian ensembles. The distribution of eigen- values of Wishart and pseudo-Wishart matrices is next considered.

For a matrix generated from an i.i.d. Gaussian ensemble, we obtain an expression for the characteristic function (cf) of eigenvalues in terms of a sum of determinants. We present applications of these results to the analysis of multiple-input multiple-output (MIMO) systems. The purpose of this work is to make researchers aware of the pseudo-Wishart distribution and its implication in the case of MIMO systems in Rayleigh fading, where the transmitted signals are independent but the received signals are correlated.

Manuscript received October 15, 2002; revised June 25, 2003.

The author is with the Department of Electrical Engineering, Indian Insti- tute of Technology-Delhi, Hauz Khas, New Delhi 110016, India (e-mail: rk- mallik@ee.iitd.ernet.in).

Communicated by B. Hassibi, Associate Editor for Communications.

Digital Object Identifier 10.1109/TIT.2003.817465

The correspondence is organized as follows. Section II presents the cf of a Wishart-type distribution of a random Hermitian matrix gener- ated from independent zero-mean complex Gaussian vectors with ar- bitrary covariance matrices. The pdf of a complex pseudo-Wishart dis- tributed matrix generated from an i.i.d. Gaussian ensemble is derived in Section III. Section IV gives the complex pseudo-Wishart pdf in case of a non-i.i.d. Gaussian ensemble. The cf of eigenvalues of a matrix generated from an i.i.d. Gaussian ensemble is presented in Section V.

Applications of the results to the analysis of MIMO systems are dis- cussed in Section VI. Section VII give some concluding remarks.

II. CHARACTERISTIC FUNCTION OF A WISHART-TYPE DISTRIBUTION

Let F\, . . . , Fj\r be L x 1 independent zero-mean complex circular Gaussian random vectors such that for each n = 1, . . . . N, Fn has a CJ\f(O, Kn) distribution, Kn being a Hermitian positive-definite ma- trix. This implies

E[Fn] = 0i xi

= o

= Kⁿ (1)

where E[-\ denotes the expectation operator, (-)T the transpose oper- ator, and (-)H the Hermitian (conjugate transpose) operator. Consider the random Hermitian matrix A given by

(2) The rank of the L x L matrix A is niin(A~, L) with probability one.

To specify the pdf of A, the number of real random variables required L²-(L-Nf,

L².

when N < L when N > L.

Since A is a Hermitian matrix given by (2), its diagonal elements are real and positive with probability one. Let Aij denote the element in the ith row and jth column of A. When N < L, A has rank N with probability one, and we can specify the pdf of A by the joint pdf of part of the diagonal and lower diagonal elements

R.e{A^i:j}, Im{.4;,; j = 1, miu(* - 1, N). (3) On the other hand, when A* > L, A has full-rank L with probability one, and we can specify the pdf of A by the joint pdf of all the diagonal and lower diagonal elements

Re{A;, j } , Im{Ai, j } , j = 1, . . . , i - 1, « = 2, . . . , L. W

Thus, we need the joint pdf of at most L2 real random variables to specify the pdf of A.

We shall say that A has a Wishart-type distribution. In particular, when Ki = • • • = Kx, then A has a Wishart distribution when Ar >

L and a pseudo-Wishart distribution when Ar < L [1].

Let us look at the cf of A when it has a Wishart-type distribution.

To specify the cf of A, we consider a n l x l Hermitian matrix Q of deterministic variables given by

'•=ltoi,j]Z^{i=l (5)} such that

when Ar < L

i = 1. Ar. are real

,,j, ilj. i j = 1, .... m i n ( i — 1. N), i = 2, . . . . L, a r e complex

,i.j, £lj. i, i, j = N + 1, . . . . L, are zero;

{Grow,, }%=i for the set {FIOWk }k = 1 using Gram-Schmidt orthogo- nalization [8].

We can express i^row!; • • •, FIOV,,L in terms of Gr o w i, . . . , Gr o w v

as when N > L

i = 1, . . . , L, are real

min( i. N) FrOw,- = / t. j, Q,j, i, j = 1, . . . . i — 1, i = 2. . . . , L, are complex. ¹⁼¹

(12) (6)

The cf of A can then be expressed as

= £ expjjtr

(7) where j = \/—l. Using the result of [6] for the cf of Hermitian quadratic forms in complex Gaussian random variables, we obtain from (7) the expression

*A(jfi) = — (8)

f[det(I

-jQK

)

where II denotes the L x L identity matrix.

III. DISTRIBUTION WHEN A ' I = • • • = KN = II

Consider the case when F\, .... FN are i.i.d., each with a CA'(O, IL) distribution. Let M + {L) denote the set of all L x L complex positive-definite matrices.

When A* > L, A has the complex Wishart distribution with pdf given by [1]

{det(a)}N~L e x p { - t r ( o ) }

where Bi, 1, . . . . B^, N > 0 with probability one. Note that Bi, 1 =

| | Fr o w i ||, || • || denoting the Euclidean norm. Define matrices Bo and B\ in terms of the random coefficients {B,,;} as

i , i 0 0

0 BN.NI

(13)

BL,i ••• BL,N

Thus, Bo is an Ar x Ar complex lower triangular matrix with real pos- itive diagonal elements while Bi is an (L — N) x Ar complex matrix.

From (12) and (13), we can now express F as Bo

-•row^v - Define the L x A" complex matrix B as

„ A \B0

(14)

(15) Using the fact that {GroWfc } 'k = 1 is an orthonormal set, we can write A as

A = FF^H = BB^H. (16) a e M+(L) . (9) Note that the matrix B is specified by

L(L-l) -

This pdf can also be expressed as the joint pdf of all diagonal and lower diagonal elements of A given by (4).

When Ar < L, A is not a full-rank matrix and has a pdf which is different from (9). In this case, A has a complex pseudo-Wishart distribution. The pdf of A can be specified by the joint pdf of part of the diagonal and lower diagonal elements of A given by (3).

To obtain the pdf of A, we use a linear algebraic technique. We focus on the L x Ar matrix F whose columns are F i , . . . . FN • Thus, F is defined as

F = [Fi, ••-, FN]- (10)

Let Fr o w i. • • •, Fr o w i, denote Ar x 1 column vectors corresponding to the L rows of F. Then we can rewrite F as

Re{Z?i,j}, Iui{Bij}, j = 1. . . . , mind - 1, Ar), (17) i = 2, . . . , L.

Thus, there are L2 — (L — Ar)2 real random variables which specify B. It can be shown after some algebra that the Jaeobian of the transformation B —• A is given by

(18)

(19)

FT L *• row L J

(11)

It is clear from (12) that B,j = G"OWl Fr o W i. Now consider Bi,, = G?OWlFIOWl, l = 1, . . . , miu(-i - 1, N),

i = 2, . . . , L,

that is, all nontrivial elements of B except B\, i, . . . . BN,N- From the property of Gram-Schmidt orthogonalization, GrOw, depends only on F r o w i, • • • , FIOV!l. As a result, B%,\ depends only on frowi • • • • , f row, and Frow,- •

Given Fr o Wi , • • •, Frow;_i, Bi,\ = G ^W iFr o w, is a zero-mean complex circular Gaussian random variable with unit variance, that is, Bi, l has a CA'(0, 1) distribution with pdf

Now Fr o w i, • • • • FIOV!L are i.i.d. complex Gaussian random vectors each with a CA'(0, IN) distribution. Since Ar < L, frown • • •' FIOV,-L are linearly dependent. However, any set of Ar

vectors from the set {FIOV!f.}k_1 will be linearly independent with probability one.

We choose FIOVII, ..., FIovlN, which are linearly independent with probability one, and proceed to obtain an orthonormal basis

exp a EC (20)

where C denotes the set of complex numbers. Since Frowi, • • •, Frow;.! does not appear in the pdf of Bt,;, each B,.i, l = 1. . . . . min(i — 1. Ar), is independent of FIOWI... FIOWi_1 . Using the orthogonality of the set {Glawk}k=1, it can be shown that B,, i,l = 1, . . . . min(i — 1, Ar), are independent. Therefore, Bt.;,

l = 1, . . . . min(i — 1, N), are i.i.d. CM (0, 1) random variables and are independent of FIOW1, ..., FIovli_L . Further, using the i.i.d. property of FIOW1, . . . . FIOV,,L, it can be shown that the sets {Bi,i}^l{i~1>N\i = 2, . . . , L, are independent of each other.

Thus, we can conclude that B,j,l = 1. . . . , miii(i — 1, N),i = 2. . . . . L, are i.i.d. CM (0, 1) random variables, and are independent

OI .f row i . • • • ? •* row ^ •

Now, from (12), we have

which implies

i = 1. Ar.

(21)

(22)

1=1

Since ||FrOw, \\2 is a sum of squares of 2N real i.i.d. A' ((), l / \ / 2 ) random variables, it is \/2(2Ar) distributed with mean A*. Similarly, 5Z;=i l-^».'|2 is \2(2(? — 1)) distributed with mean?' — 1. By Fischer's lemma [9, p. 379], Bf , is \2(2(Ar - i + 1)) distributed with mean N-t + 1.

Since the variables FIOV/1, . . . , FIOvri_1 do not appear in the distri- bution of Bf ;, we conclude that Bf t,i = 1. . . . . A", are independent, Bfti being a \ 2 (2 (Ar — i + 1)) random variable with mean Ar — i + 1.

The pdf of B,,; is given by

v > 0. (23) It is also to be noted that {B,;t;, i' = 1. . . . , A*} is independent of

{B,. (, / = 1, . . . , min(« - 1, N), i = 2, ..., L}.

Therefore, the pdf of B, which is the joint pdf of {Bi.i, i = 1, . . . . N\ and

{Bi, (, l = 1. . . . , miii(i - 1, Ar), i = 2, . . . , 1}

can be expressed using (20) and (23) as fB(b) =

2A exp -{ —tr

L²-(L-N)²-N -1- (N-i)l

/I ^ (— 1

Using the Jaeobian given by (18), we get the pdf of A as fB(b)

fA(a) =

(24)

(25)

Let A[jvj denote the first principal A* x Ar submatrix of A, given by and let M+(L, N) denote the set of all L x L complex Hermitian rank-N matrices each with positive-definite N X Ar principal subma- trices. Substituting (18) and (24) in (25), we get

= \ exp{-tr(o)|

Jif(2L-JV-l) -

Thus, (27) is the complex that it is a joint pdf of

a G ..Vi+(I, N). (27) en A* < L. Note L² -(L-N)² = A ' ( 2 1 - N)

real r a n d o m variables.

IV. D I S T R I B U T I O N W H E N K\ = ••• = = K

Now consider the case when F\. . . . . FAT are i.i.d., each with a CA'(0, &*) distribution. Since the elements of each .F,, are correlated, the Gaussian ensemble is non-i.i.d. By expressing

K = PPH (28)

where P is a nonsingular matrix, and

Fn=PF'n, n = l,...,N (29)

where F\, . . . , FV are i.i.d. CA'(0, JL ) random vectors, we find that

A = PA'PH (30)

with

(31) It is clear that the pdf of A' is given by (9) with A replaced by A' when Ar > L and by (27) with A replaced by A' when A* < L. Let P be an L X L complex lower triangular matrix with real positive diagonal elements.

A. Case When N < L

Consider the case when A* < L. We can express P as

"Po 0

•Pi P-2.

where Po and P2 are A* x Ar and (L — N) x (L — N) lower triangular matrices, respectively, and Pi is an (L — N) x Ar matrix. Let

A' = B'B'H

where

P = ⁽³²⁾

(33a)

B' = B'o

B\ ^(33b)

B'o being an Ar x Ar complex lower triangular matrix with real posi- tive diagonal elements and D' 1 is an (L — Ar) x Ar complex matrix.

Combining (30), (32), and (33), we get

A = DBH (34a)

where

B = B1 = PB' = POB'O

P1B'0+P2B'l (34b) It is clear from (34b) that Bo is an Ar x N complex lower triangular matrix with real positive diagonal elements and B\ is an (L — N) x N complex matrix. Let Pi.j, B[_ 3, and Bt, ^ denote the element in the ith row and jth column of P, B', and B, respectively. The nontrivial parts of the nth column of B' and B have the relation

Bn,n 0 0

L B

,

J

Pn.n

L PL,n

r B'

,,

B'n+1

Pn 1, n + 1

PL.n

L B'

J

n = 1. ^{N. (35)}

Note that B'nn, Bn,n > 0 with probability one for n = 1. . . . , A*. It can be easily shown from (35) that the Jaeobian of the transformation from the nontrivial part of the rath column of B' to that of the nth column of B, which we denote as ./„, is given by

L

Jn = Pn,n J J P'ti, U= 1., . . . , JV. (36)

i = n + 1

Note that ./„ is the Jaeobian of a transformation of 2 ( i — n) + 1 real variables. The Jaeobian of the transformation B' —> B is, therefore,

Lt=A'+l

n

(37)

From (18) and (37), the Jacobian of the transformation A! —> A is given by

Li=A'+l

n ^

2L — 2i

i = 1. N.

2{L-N)

n

Lt=i Since D = PD', we have

Bi,i=Pi,iB'iti, Substituting (39) in (38), we get

Now

Let A*[/v-] denote the first principal submatrix of K, given by It is clear from (41) and (42) that

det (K

)=(f[Pi,i] , det (A) = (f[ Pi, i

Substituting (43) in (40), we get, when N < L, the Jacobian

,T

^

= {det (K)f {det {K

)}

-

.

The pdf of A is given by fA(a) =

''a' — a

(38)

(39)

(40)

(41)

(42)

(43)

(44)

(45)

C. Pseudo-WishartDistribution When N = 1 and L > 2

Consider the case when N = 1 and L > 2. This implies A = Fi-Ff, where Fi has a CA'(0, &*) distribution. We specify the pdf of A by

i , i } , i = 2 , . . . , L . (49)

(50) where (•)* denotes the complex conjugate. From (46), the pdf of A is given by

Note that A\, 1 > 0 with probability one.

Since A is a rank-one Hermitian matrix, we have A i , j = A %- ] A J- \ i.j = 2 . . . . , L

. -(L-l) / A («) =

Let

where

1 det (A') a G M+(L, 1). (51)

(52)

P = [Pi, j]?, J=1 , Pi,j = 0, for j < j , P,, i > 0 (53) is anLxL complex lower triangular matrix with real positive diagonal elements. Using (50), we can express A as

where

where fAi (•) is given by (27) with A replaced by A'. Substituting (44) in (45) and noting that

{det (A'

)}-

-

_ {det ((AV])'

A

)}~

~

~ " {det(K

)}

~

we get

J

A2,l

(54a)

(54b)

If we put

where

B = PB', A = PAP

^ { n < ^A - o , }

a e M+(L, N). (46)

Thus, (46) is the complex pseudo-Wishart pdf of A = £)£=i FnF"

when N < L and F\, . . . , F^r are i.i.d. CA'(0, A) random vectors.

(55b)

then the pdf of A! is given by (27) with A replaced by A! and N = 1, resulting in

B. A" > L j \ exp{—tr(a )

Wheni\r > L, A is full-rank with probability one, and, therefore, B' and B are L x L complex lower triangular matrices with real positive diagonal elements. As a result, we get

a e M+(L. 1). (56) This can be rewritten as

•2L

JA'-A =[Y[Pi,i) = { { J ^ (A')}L. (47) fA,(a') =

i l l exp <j " ( « i,

- L - l

Replacing A by A' in (9) and substituting (47) in (45), we obtain the complex Wishart pdf [1]

{det(o)}j v"L exp {-tr (K^a)}

/ A ( « ) = - ^ - ^ L i ^ J )J-, a G M+(L).

«i,i > 0, a'2,1, •••. «i, 1 e C. (57) It can be easily shown from (57) that

f |a';.i|2

exp i j - — I " 1,1

(48) ci-,1 G C , «,;_! > 0. i = 2, . . . , L. (58)

Therefore, given A\_ 1, A'2, i, . . . . A'Li j are conditionally i.i.d., each with a CA"(0, 4 'u) distribution. Moreover,

0 (59)

Therefore, we can conclude that given Ai,i, Ai /z«5 a CM ({Ai, i/Kiti)Ki. A\,iM2r) distribution, and that Ai.i has an exponential distribution with mean ii 1.1.

Then the pdf of A given by (51) can be rewritten as implying that A[t i has an exponential distribution with unit mean.

Define ( L - l ) x l vectors A[ ,A1,P1,K1, and Mi as

U'

.J

A-2,1

K2,l

/ A (a) =

a i , i > 0, fli G CL (67) where CL 1 denotes the set of (L — 1) x 1 vectors with complex el- ements. Note from (62) that det (K) = A'i, 1 det ( M ^1) . Alterna- tively, we can write

(60a) IA(a) =^exp

the (L — 1) x (L — 1) lower triangular matrix P² as

•P2.2 0 0

exp • 1=2 3=2

Pi,2 Pj.i

0 PL.LI

xM,,, (a,,, - g+Ki.i

(60b) a i , i > 0 , a2,i, . . . , « £ , 1 S C . ( 6 8 )

f when Fi

and the (L — 1) x (I — 1) Hermitian matrices K2 and M2 as

We can then rewrite P as P- and K and M as

A* =

P i

(60c)

(61)

Thus, (67) or (68) is the pseudo-Wishart pdf of A = has a CA'(0, A') distribution.

To obtain the cf of A, note from (6) that .i fif

fi =

where

(69a)

(69b)

AT

J '

Mf"

M, I '

A — T>2 A'

(63) We know that given A'ltl, A[ has a CA'(0, A i ^ / i - i ) distribu- tion. Therefore, from (63), we can say that given A i , i , Ai has a CA' ((Ai, 1/P1,1 ) P i , Ai, 1P2P2) distribution, and that

det(Af)det (A""1 - jil) 1

ATi.idet (M^1) det (M - jil)

1 (70)

exp

{

_{" ^ 7}

implying that A\, 1 has an exponential distribution with mean P{ 1. Now K = PPH, along with (61) and (62), implies

P1.1 = V A ^ , P i = ^ = ^=A'i

V'U.i

• " • 1 , 1

Since Af is the inverse of AT, it can be shown from (62) that

— 1

(65)

(66)

(71)

V. CHARACTERISTIC FUNCTION OF EIGENVALUES WHEN K1 = . . = KN = II

Consider the matrix A = 5Hn=i ^n^n when P i , . . . , F,v are i.i.d. CA'(0, J L ) random vectors.

When Ar < L, A is not full-rank; it has rank N with probability one. As a result, the N x A* matrix 5Zn=i Fn Fn is full-rank with probability one. Let Ai, . . . . A,v denote the eigenvalues of XX=i Fn Fn which are real and positive with probability one. These are also eigenvalues of A. As a result, A* of the eigenvalues of A, which are Ai, . . . , A.v, are real and positive with probability one, and

the remaining L — N of the eigenvalues, denoted as A.v+i, . . . . \L, are zero. The joint pdf of A i , . . . . A A- is then given by [7]

exp

1 < » < J < Y

N\ ]\{N - i)\(L - i)\

! = 1

Ai, . . . , AY > 0 . (72) On the other hand, when N > L, the matrix A, which is a sum of N independent LxL rank-one matrices, is full-rank with probability one.

The L eigenvalues of A, denoted as Ai. . . . , A L , are real and positive with probability one, and have a joint pdf which has the same form as the pdf (72) for N < L with A" and L exchanged.

Consider the case when N < L. Let A be the diagonal matrix of eigenvalues, given by

N

! = 1 N

= det '(H). (77)

A = diag(Ai (73)

Thus, {A(Ai, . . . . A Y ) } is the determinant of an Ar x A" Hankel matrix H whose element in the A:th row and Ith column is given by

N

(H)k t = ^x\+l~2. (78)

1 = 1

For each ii 6 Pi (N), that is, for each ii = 1, . . . , Ar, we can replace the first row of the matrix H with 1, A^, A^. . . . . Af ^1 to form a newmatrixIF'(ii),replacedet (fi^)bydet (H'(ii)) in(76),and sum the result over all ii £ Pi(Ar). The integral is, thus,

To specify the cf of A, consider a diagonal matrix u> of real determin- istic variables, which is expressed as

u) = diag(u/'i, (74)

The cf of A is

* A ( j w ) = £ [ e x p { j t r ( A w ) } ]

= E

co coc 0 JO

L-N

x {A(Ai, . . . , AAO}2 (75)

where A(Ai, . . . . AA7) denotes the Vandermonde determinant of

A i , . . . , AAr.

We consider the evaluation of the integral X(w) given by

Li=i

"jf

(76) L e t P , (N) denote the set of all N\ / (N — i)\ length- i permutations of(1.2, ...,N).

The square of the Vandermonde determinant can be expressed as

{ A ( A i , . . . , XL)}2 = (Afc - A,) 1 XN

> A ' — 1 i A ' - l

E 1 •••/

n

x det (79a)

where

Atl

f Af

Z ^

j = 1

(79b)

The variable Au can now be eliminated from the other rows of the matrix H'(ii) by subtracting a suitable multiple of the first row, re- sulting in another matrix H"(ii) given by

Y Y

E

H"(h)T =

Y

E

E E

Y

E

(80) such that

det {H"(h)) = det {H'(h)) .

For each (ii, i-z) G P2(A"), we can now replace the second row of the matrix H" (ii) with Al 2, A f . . . . , Af to form a new ma- trix H'"(ii.i'z), replace det (H"(ii)), which is the same as det (H' (ii)) . by det (H'"(ii, iz)) in (79a), and sum the result over all (ii, iz) E'P2(N). The integral then becomes

= E

3 = 1

n

x det

'*" exp(-(l -

(81a)

where

H in, «a) =

E .

_ E .

E Af-

The process can be repeated to yield

*(«)= E f - /

^1 i*j )G P.wi1^ )

< A! 2

N

E

E

n

where

mi

j e x p ( - ( l - ju>j)\j

X det (H_(ii, .... iN))d\

1 An

AI 2 Af3

A^2N-2

(82a)

(82b)

Note that the element in the kth row and /th column of the Ar x Ar

matrix H_(ii, ..., IN) is A*+'~2. Since the various rows of SXii > • • • i *AO depend on distinct variables, we can multiply the kth row of H_(i\, . . . , IN) by

Affc~A e x p ( - ( l - ju>lk)\tk)

for A: = 1, . . . . Ar, and express X(w) in terms of the summation of determinants of matrix functions whose elements are integrals. Thus,

d e t

where the element in the fcth row and the /th column of S(ujil, . . . , u>iN) is given by

/•cc

= / Af^"' exy(-(l-juik

Jo

. L-N+k+l-2

X exp( —(1 — j'jj%k )x)dx

(L-N + k + l-2)\

When A* > L, the cf of the eigenvalues Ai, . . . , AL of A has the same form as (84) with Ar and L exchanged.

For example, when Ar = 2, L > 2, the cf of the eigenvalues Ai, A2

is given by

A 1

2 { ( j r

X

- 1 f 21

A2

1

1 (.1-3

+ (

1

S" ^2)!

-Iv'

u,²)^L ( (L-2)!

(L-1)!

l ) L"1 ( 1

2 ( L -

( £

( 1~ 1-3

( 1

L

1)

-IV.

L!

( L - L -3-^1

" ^ 1)!

)L + 1

+ 1

1

^+1

(85) Similarly, when Ar = 3,£ > 3, thecfofthe eigenvalues Ai, A2, A3 is given by

(86)

where

(83b) From (76) and (83a), the cf of the eigenvalues Ai, . . . . AA7 of A when Ar < L is expressed as

1

(1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1)

VI. APPLICATIONS

The results obtained can be applied to the analysis of MIMO com- mmunication systems using diversity. It is to be noted that such appli- cations give rise to analytical expressions; they are not substitutes for Monte Carlo simulation based approaches. We present here two appli- cations.

A. Application 1: Analysis of Optimum Combining in Rayleigh Fading Consider the case of optimum combining in Rayleigh fading with space diversity using L receive antennas, Ar equipower interferers, and additive white Gaussian noise. The maximum instantaneous output signal-to-interference-plus-noise ratio (SINR) is given by

7 = Ps^B^c. (87) where Ps is the desired signal power, c is the propagation vector of the desired signal, and

N \Y[(N-i)\(L -i)\

x V det (S(^tl, ..., uiN)) (84)

T2IL (88)

where S(u>i1, ..., UJI v ) is given by (83b).

is the interference-plus-noise correlation matrix, Pi being the inter- ferer power, Ft being propagation vector of the ith interferer, and a2

the noise power. Owing to Rayleigh faing, the propagation vectors are zero-mean complex circular Gaussian random vectors. We assume that

A* < L, and F\, . . . . FN are i.i.d., each having a CM (0, K) distri- bution. In addition, c is assumed to have a CA'( 0, A*c) distribution. As a result, the interference correlation matrix A, given by

= [ f i , • • • , FN][F1, . . - , F . , ]f f (89) has a pseudo-Wishart distribution. To analyze the performance of such a system for various digital modulation schemes, we often need the pdf or cf of 7, which, in turn, depends on the distribution of the interference correlation matrix or on the distribution of its eigenvalues.

A convenient method of finding the symbol error probability (SEP) for coherent reception of Af-ary modulated signals like M-PSK, M-QAM, etc., is by evaluation of integrals 1 ( 0 . g) having the form

/ • G

J ^ \ (90)

where ^y(f^j) denotes the cf of 7, and g and 0 are parameters de- pending on the type of modulation. To obtain $T(ju;) we can use one of the following approaches.

1) Find the conditional cf of 7, conditioned on the nonzero eigenvalues of A, and then average this over the joint pdf of the nonzero eigenvalues of A

2) Find the conditional cf of 7, conditioned on the elements of A, and then average this over the joint pdf of the elements of A Approach 1) works well when A' = SIL, * > 0 (*' is an arbitrary positive scale factor), since, in that case, the matrix A, given by

A = [ FU •••, FN]H[ FU ••-, FN] (91) has a Wishart distribution.

However, when A' / SIL , implying spatially correlated interfering signals received from any interferer, A does not follow a Wishart dis- tribution. Although the N nonzero eigenvalues of A and the N eigen- values of A are the same, there exists no convenient analytical expres- sion for the joint pdf of these eigenvalues, and hence approach 1) is not convenient. In such a situation, approach 2) is useful.

To illustrate this, consider the case when Ar = 2. L = 3. The pdf of the Hermitian matrix A is specified by the elements

j , i } , R.e{A,,²},

It can be easily shown that the element .4.3,3 can be expressed in terms of A i , i , A2,-2,A2,i, A3,i, A3, 2 as

A3. 3 =

Ai.i A i , i U i , i A2 | 2- Let

-1 =M = M^2tl M²,² Ml²

M3,i M3:2 Mi,3

(92)

(93)

From (46), the joint pdf of the elements of A is then given by

{_{ai,ia2,2 -} exp{-tr(M«)}

a £ 3, 2) (94a) where

tr (Ma) = Mi. iai.i + M2. 2a2.2

I l « 3 . i| 2

i. 3

" 1 , 1

+ 2 Re {Ml ^{l t t 2}, 1 + Ml jccj, 1 + Ml ²«3,2 } • (94b) The condition a G M.+C&, 2) in (94a) implies

« i , i , a 2:2 > 0 , a2, i , aa, i , a . 3 , 2 G C , a i , i a2,2 > | a2, i |2. ( 9 5 )

From results on quadratic forms in complex Gaussian random variables [6], the conditional cf of 7 in (87), conditioned on A, is given by

*T l ~ det {Is - J^Ps [PiA + a^h]-1 Kc) det (PiA + a2ls)

~ det (PiA + a'2I-j - juPsKc)' Combining (96) and (94a), we get

det (Pia + a2 Lj)

(96)

2) {det ( P / a + o-²/3 - ju,'P^sK^c) {(ii.i(i2,2 — "2.1 "} exp {—tr (Ma)}

dRe(a-j.

2. i)dlin(a2.1

(97) Substituting (97) in (90), we finally obtain

I(Q.g)= /

{«i,ia.2,2 - |«2,i|2} exp { - t r ( M a ) }

{det(M)}-²

«2, 2d R.e(a2,

,i)dlm(a3,i)dRe(a-j,2)dlm(as,2)d0. (98) Although the integral in (98) is a nine-fold integral, it can be evaluated numerically. It is not possible to obtain a simpler or even similar ex- pression for 1 ( 0 , g) using approach 1).

B. Application 2: Capacity of a MIMO System

Consider a MIMO system using transmit-receive diversity in Rayleigh fading and additive white Gaussian noise with A* transmit antennas and L receive antennas, such that Ar < L. Let F denote the L x Ar channel matrix, given by

F=[F^U • • • , F^N] (99)

where Fi, . . . , Fj\- are i.i.d., each having a CA'(0, K) distribution, and K / SIL, *' > 0. This implies that the transmissions are indepen- dent, but the receptions are correlated. We wish to obtain the ergodic capacity of this MIMO channel in bits per second per hertz, which is given by [4], [10]

C = E [log det (l^L + ^FF")] (100)

where p is the average signal-to-noise ratio (SNR) at each receive an- tenna. Here FFH has a pseudo-Wishart distribution, but since K / SIL, FH F does not have a Wishart distribution. In addition, the joint pdf of the N nonzero eigenvalues of FF^H or the joint pdf of the A^r eigenvalues of FHF is not available in a convenient analytical form.

Therefore, although the capacity in (100) can also be written as

C = E [log det (l^N + ^F^HF)] (101)

it is easier to perform the averaging in (100) using the pseudo-Wishart distribution of A = FF given by (46) as compared to the averaging in (101).

In particular, when Ar = 2. L = 3, we get

C= I [log det ( l³ + £aYl

{ai.i«2,2 — |a2.i|² } exp {—tr (Ma)}

2. i)dlin(a²,1

i)dlm(a.3, (102)

where M is given by (93), tv(Ma) by (94b), and the condition a £ M + {3, 2) by (95).

In the case of N = 1, L > 2, we get from (68)

C = [log det [IL + pa)]

exp

L L

exp •

- ^ T E E I " ' .

- ^

^

xMi ,

(ai,, i)dlm(aL.

(103a) where

M, M, •

L (103b)

log det (IL + pa) = log ( en, Note that (103) can alternatively be written as

The cf of Fi Fi is given by

(104)

(105) from which the pdf of F1^Fi needs to be obtained to get the capacity.

C. Discussion

The point we want to make through these examples is as follows.

When F\, . . . , Fj\- are i.i.d. Gaussian random vectors, each having a CA'(O, K) distribution, such that A* is not a scaled version of II and Ar < L, the Hermitian matrix A, given by

FN]H

has a pseudo-Wishart distribution. However, the matrix A, given by A=[F1, - . . , FN]H[FU • • • , FN]

does not have a Wishart distribution. Moreover, although the Ar

nonzero eigenvalues of A and the Ar eigenvalues of A are the same, there exists no convenient analytical expression for the joint pdf of these eigenvalues.

It is only when A' is a scaled version of the identity matrix that for Ar < L, A is pseudo-Wishart and A is Wishart.

Thus, for arbitrary K, which implies, for example, spatially corre- lated interfering signals received from any interferer in a MIMO system with Ar interferers and L receive antennas and subject to Rayleigh fading, pseudo-Wishart and Wishart distributions cannot be exchanged.

In such a situation, we can directly use the joint statistics of the el- ements of the pseudo-Wishart matrix to obtain analytical expressions for performance measures like SEP or capacity.

Numerical evaluation of the integrals given in (98), (102), and (103a) is easier by Monte Carlo simulation. However, these analytical expres- sions can reveal to us some aspects of the problem which cannot be seen through simulation.

VII. CONCLUSION

We have treated pseudo-Wishart and Wishart distributions as spe- cial forms of a Wishart-type distribution of a random Hermitian matrix generated from independent zero-mean complex Gaussian vectors with arbitrary covariance matrices. An expression for the pdf of a complex pseudo-Wishart distributed matrix using a linear algebraic technique has been derived. We have then analyzed the pseudo-Wishart distribu- tion of a rank-one Hermitian matrix. For a matrix generated from an i.i.d. Gaussian ensemble, we have obtained an expression for the cf of eigenvalues in terms of a sum of determinants. The results obtained provide a powerful analytical tool for the study of MIMO systems with correlated received signals, like systems using diversity and optimum combining, space-time systems, and multiple-antenna systems. This work is intended to serve as a beginning to finding answers to some questions in these areas. It is based on the philosophy that analytical results help researchers to look beyond what can be seen through sim- ulation; if not now, then in the future.

ACKNOWLEDGMENT

The author wishes to thank the anonymous reviewers for their valu- able comments.

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