Quadratic Forms on Complex Random Matrices andMulti-Antenna Channel Capacity

Tharmalingam Ratnarajah and R. VaillancourtUniversity of Ottawaphone: 613-271-6151

email: t.ratnarajah@ieee.org

Abstract Quadratic forms on complex random matrices and their joint eigenvaluedensities are derived for applications in information theory. These densities arerepresented by complex hypergeometric functions of matrix arguments, which canbe expressed in terms of complex zonal polynomials. The derived densities areused to evaluate the most important information-theoretic measures, the so-calledergodic channel capacity and capacity versus outage of multiple-input multiple-output(MIMO) Rayleigh-distributed wireless communication channels. Both correlated anduncorrelated channels are considered and the corresponding information-theoreticmeasure formulas are derived. It is shown how channel correlation degrades thecommunication system capacity.

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QUADRATIC FORMS ON COMPLEX RANDOM MATRICES AND MULTI-ANTENNACHANNEL CAPACITY

T. Ratnarajah and R. Vaillancourt

Department of Mathematics and StatisticsUniversity of Ottawa

585 King Edward Ave.Ottawa ON KIN 6N5 Canada

e-mail: t. ratnarajah@ieee. org

ABSTRACT The theory of quadratic forms on complex random ma-

Quadratic forms on complex random matrices and their trices is used to evaluate the capacity of multiple-input,joint eigenvalue densities are derived for applications in in- multiple-output (MIMO) wireless communication systems.

jointLet us denote the number of inputs (or transmitters) and

formation theory. These densities are represented by com- the number of outputs (or receivers) of the MIMO wireless

plex hypergeometric functions of matrix arguments, which communcton sys by r s an tie and as-can e epresed n trmsof cmplx znal olyomils. communication system by nt and nr, respectively, and as-

can be expressed in terms of complex zonal polynomials. sume that the channel coefficients are distributed as com-The derived densities are used to evaluate the most impor- plex Gaussian and correlated at both the transmitter and thetant information-theoretic measures, the so-called ergodic receiver ends. Then the MIMO channel can be representedchannel capacity and capacity versus outage of multiple- by an n, ×nt complex random matrix H - CN(0, Er®Et),input multiple-output (MIMO) Rayleigh distributed wire- bynrxnt co prendmtrexh c n( at)hwhere Zr and Zt represent the channel correlations at theless communication channels. Both (spatially) correlated receiver and transmitter ends, respectively. This means thatand uncorrelated channels are considered and the corre- the covariance matrices of the columns and rows of H aresponding information-theoretic measure formulas are de- denoted by Er and Et, respectively. If Er = In, (or a 2

1nu)

rived. It is shown how channel correlation degrades the and Et = a 2 Int (or Int) then the channel is called uncor-communication system capacity. related Rayleigh distributed channel. The information pro-

cessed by this random channel (or mutual information of1. INTRODUCTION this random channel) is a random quantity which can be

measured in two ways, namely, ergodic capacity (or aver-Let an n x m complex Gaussian random matrix X be dis- age mutual information) and capacity versus outage (or xtributed as X ,-, CN(O, E1 0 E2) with mean &{X} = 0 percent outage).and covariance cov{X} = E 1 0 E 2, where E1 E Cnxn

and E 2 E Cr`m are positive definite Hermitian matrices. Recent studies show that a MIMO uncorrelated

Then the quadratic form on X associated with the positive Rayleigh distributed channel achieves almost n more bits

definite Hermitian matrix A is defined by per hertz for every 3dB increase in signal-to-noise ratio(SNR) compared to a single-input single-output (SISO) sys-

S = XHAX. tem, which achieves only one additional bit per hertz forevery 3dB increase in SNR, where n = min{nt, nr}, see

Here, we study the distribution of S, denoted by [2] and [12]. In this context, we may mention a MIMOCQn,m(A, El, E 2), and its application to information the- strategy that offers a tremendous potential to increase theory. We also derive the joint eigenvalue densities of information capacity of single user wireless communica-S, which are represented by complex zonal polynomials. tion systems, namely, the Bell-Labs Layered Space-TimeComplex zonal polynomials are symmetric polynomials in (BLAST) architecture, see [8] and references therein. Butthe eigenvalues of a complex matrix, see [5], and they en- in many practical situation, channel correlation exists dueable us to represent the derived densities as infinite series, to poor scattering conditions, which degrades capacity (seeIf A = In, E1 = In and E 2 = E, then S = XHX [1], [9], [10] and references therein). In [1], the authorsis said to have a complex Wishart distribution, denoted by studied the correlated channel matrix with large dimensionCWm (n, E), see [3], [6] and references therein. (asymptotic analysis), which is only an approximation to the

practical correlated channel matrix with finite dimension. In and X[,,] (X) is the character of the representation [r,] of the[10], the effect of channel correlations on MIMO capacity is linear group given byquantified by employing an abstract scattering model. Morerecently, in [9] the authors have derived an exact ergodic det Ak3+m-j

capacity expression for an uncorrelated Rayleigh MIMO X[,] (X) =

channels, which is different from the work of Telatar [12], det [(Am-j)]and have also derived an upper bound to ergodic capacityfor correlated Rayleigh MIMO channels. In this paper, we as a symmetric function of the eigenvalues, A1, ... , Am, of

first derive the densities of quadratic forms on complex ran- X. Note that real and complex zonal polynomials are partic-

dom matrices and their joint eigenvalue densities. Then, ular cases of (general a) Jack polynomials, 0(a) (X), where

using these densities we evaluate the capacity degradation a = 1 for complex, and a = 2 for real, zonal polynomials,

for the correlated channel matrices H -, CN(0, ir ® Et) respectively. In this paper we only consider the complex

with nr > nt, nr = nt and nt > nr, This will be done by case; therefore, for notational simplicity we drop the super-deriving closed-form ergodic capacity and outage capacity script, a, of Jack polynomials, as was done in equation (2),

formulas for correlated channels and their numerical evalu- i.e., we write C,. (X) :C(1) (X). Finally, we haveation. Note that this work is an extension of our early work[6], where we have extensively studied the complex Wishart C(In) 22 k 1 rj'<(2k -2k -i+j)

matrices and uncorrelated/correlated (only at the transmitter \(=2,( ) 1 7j'(2ki + r - i)! '

end) MIMO channel capacities.

This paper is organized as follows. Quadratic forms on where

complex random matrices are studied in Section 2. The ca- H . (1n +pacity of a MIMO channel and the computational methods (2 2=) kiare given in Sections 3 and 4, respectively, and the partition r, of k has r nonzero parts. Here (a)k =

a(a+ 1)...(a+ k- 1) and (a)o = 1.

2. QUADRATIC FORMS ON COMPLEX RANDOM The next theorem gives the density of quadratic forms

MATRICES on complex random matrices S = XHAX, see [7].

Theorem 1 Let X be an n x m (n > m) complex GaussianIn this section, the densities of quadratic forms on complex random matrix distributed as X ,-, CN(O, El ® E2), whererandom matrices are given and their joint eigenvalue den- El E C`~ and E2 E Cm`m are positive definite Hermi-sities are derived. The probability distributions of random tian matrices. Then the density function of S = XHAX ismatrices are often derived in terms of hypergeometric func- given bytions of matrix arguments. In the sequel, we need to use thefollowing complex hypergeometric function of two matrix f(S) 1(det

arguments, CFm(n)(det Ei)m (det E2 A) (

xoRo(m) (B, -E21S), (4)

00 C,.(X) C,.(Y)02oF~om)(X,Y) : F__, _

k!C.(IX) ' (1) where A E Cn n is a positive definite Hermitian matrixk=O K and B = A-12E• A-12.

where X E Crmxm, y E Cnxn and n > m. Moreover, The distribution of the matrix S is denoted byr, = (ki,... , km) denotes a partition of the integer k with CQn,m(A, E, 7E2). It should be noted that the density givenk, >_ ... _ km > 0 and k = k, + "". + kin, and E. in Theorem 1 is different from the one given in [4]. Fromdenotes summation over all partitions K of k. The complex this generalized density we can easily derived other well-zonal polynomial, C. (X), of a complex matrix X defined known densities. Special cases of density (4) are:in [3] is (i) If A = In, then the density of S = XHX

CQn,m(In, E1, E2) is given by

C, (X) = X[,] (1)X[,] (X), (2) 1

where X[K] (1) is the dimension of the representation [r,] of = CFm(n)(det Ei)m(det •2)n (det S)nn

the symmetric group given by x 0 F(m)(E-1, -E-'S). (5)

X[•] (1) = k! -mHj(ki-k - i+j) (ii) If A = In, El = In and E2 = E, then S = XHXk -i, (ki + m - i)! is said to have a complex Wishart distribution, denoted by

CWm(n, E), with density XHX - CWm(n, u 2 Im) is given by

1 7rm(m-1)m(O2)-nm m

f(S) = C-m(n)(detZ)n (detS)n-metr(-E-lS), (6) f(A) rm(m-)1(m)n I m(A -A 2C17 () de E nC1m~)CPMm)k=1l k<1

where etr denotes the exponential of the trace, etr(.) = x exp 1 _ Akexp(tr(.)). x2 ExI-L )(1(iii) If A = In, El = In and E2 = 9T2 Im, then S = k=1

XHX - CWm(n, U2 1m) is a complex Wishart matrix with Next, we extend the study of complex central Wishart

density distribution to the singular case, where 0 < n < m andn,m E Z. Thus the rankof S E Cmxm isn. IfO < n < n,

(0"2)-mn S( 1 ) then the density does not exist for S - CWm(n, E) on the) - Cm (det - W2 . space of Hermitian m x m matrices, because S is singu-

lar and of rank n almost surely. However, it can be shownThe next theorem gives the joint eigenvalue density of that the density does exist on the (2mn - n 2)-dimensional

quadratic forms on complex random matrices. This theorem manifold, CSm,n, of rank n of positive semidefinite m x mis one of the key contributions of this paper. Moreover, from Hermitian matrices S with n distinct positive eigenvalues.this theorem we can easily derived other joint eigenvalue Moreover, the set of all m x n matrices, El, with or-densities of complex random matrices, see [7]. thonormal columns is called the Stiefel manifold, denoted

Theorem 2 Consider the n x n positive definite Hermitian byCVnm. Thus,

matrix S = XHAX - CQn,m(A,Ei,E 2), where A C CVn,m = {Ein(m x n);E'HE1 = In}.Cn x n is a positive definite Hermitian matrix. Then the jointdensity of the eigenvalues, A, > A2 > ... > Am > 0, of S The application of the next two theorems is another key con-is tribution of this paper. The first of these theorems gives the

complex singular Wishart density, see [7].

f(A)= Cwm(n)Crm(m)(detm-1)(det m2A)-n I - (Ak At2 Theorem 3 The density of S = xHx CWm(n,E) onk=1 k<) the space CSm,n of rank n (0 < n < m) positive semidefi-

0 C(B)C.(-_21)C. (A) nite in x in Hermitian matrices is given byY- Y-_ MC I)'.(m (8) A )= 7n(n-m) _ S)(eAnmk=O f(S) = CFn(n)(det )n etr (-E-'S) (detA)n m ,

where B = A-1/ 2 E'A-1/ 2 . (12)

Special cases of Theorem 2 are: where S = E 1 AEH, E1 E CVn,m and A =(i) If A = In, then the joint eigenvalue density of S = diag(Ai,... , An).XHX - CQnmIn, , E 17E2) is given by The second of these theorems gives the joint eigenvalue

)-n m m density of a complex singular Wishart matrix, see [7].f(A) = Cm (n)Cfm(m)((det 2 I An m f(Ak _ Ak)2 Theorem 4 Let S = xHx CWm(n, E) with 0 < n <

k00 k<1 m so that S is an m x in positive semidefinite Hermitian

X E E 1C,2( )C,,(_2) A(9) matrix with npositive eigenvalues. Then the joint density of

k=O k.C,(In)C,(Im) the eigenvalues, A1 ,... An, of S is7,n(n-1) (detE)-n n -n H(aA)2

(ii) If A = In, E, = In and E2 = E, then the joint eigen- f(A) = -n(n1) n(de 1 (Ak - Atvalue density of the complex Wishart matrix S = XHX ,ff(n)ff(i) k1 k<1

CWm (n, E) is given by XoFo)(--,)A), (13)

7rm(mf_ )(detE)_ n m n-m ( m -- A) 2 where S = E 1 AE1H, E 1 E CVn,m, and A =f()= Crm(n)CrFm(m) Hk= k< diag(A,. ,An).1~ k<1 l, ,l

xoF(m) (-E-1 A). (10) In the next section, we shall use these densities to evalu-ate the two most important information-theoretic measures,

(iii) If A = In, El = In and E2 = 9 2 Im, then the namely, the ergodic channel capacity and the capacity ver-joint eigenvalue density of the complex Wishart matrix S = sus outage of MIMO Rayleigh distributed channels.

3. MULTIPLE-ANTENNA SYSTEMShll

In recent years, multiple-antenna techniques have become X+ 1 ................... h ........... Yia pervasive idea that promises extremely high spectral effi- - 2 2 1ciency for wireless communications. The basic information V2

theory result reported in the pioneering papers by Foschiniand Gans [2] and Telatar [12] showed that enormous spec- X 2 .........tral efficiency can be achieved through the use of multiple- h 2n

antenna systems, provided that the complex-valued propa-gation coefficients between all pairs of transmitter and re- hntl .. .. '... 1in rVnceiver antennas are statistically independent and known to- h +

the receiver antenna array. However, in practical systems f. ' t2 hni

the channel coefficients from two different transmitter an- X, " Yr

tennas to a single receiver antenna can be correlated (at thetransmitter end with covariance matrix Et) and/or from a Fig. 1. A MIMO communication system.single transmitter antenna to two different receiver anten-nas can be correlated (at the receiver end with covariance 3.1. Ergodic channel capacitymatrix Er). Such channel correlation, which degrades ca-pacity (see Chuah et al. [1]), depends on the physical pa- We assume that H is a complex Gaussian random matrix

rameters of the MIMO system and the scatterer charac- whose realization is known to the receiver, or equivalently,

teristics. The physical parameters include the antenna ar- the channel output consists of the pair (y, H). Note that the

rangement and spacing, the angle spread, the angle of ar- transmitter does not know the channel and the input powerrival, etc. One of the objectives of this paper is to evaluate is distributed equally over all transmitting antennas. More-this capacity degradation for the correlated channel matrix over, if we assume a block-fading model and coding overH - CN(O, Er 0 Et) with nr > nt, nr = nt and nt > nr. many independent fading intervals, then the Shannon or er-

The complex signal received at the jth output can be godic capacity of the random MIMO channel [12] is given

written as by

nt C = SH {logdet (Int + (p/nt)HHH)}

yj = hijxi +vj, (14) = SH l{ogdet (In,+ (p/nt)HHH)}, (16)2=1

where the expectation is evaluated using a complex Gaus-

where hij is the complex channel coefficient between input sian density, i.e., H ,-, CN(0, Zr 0 Et). Let S = HHH.i and output j, xi is the complex signal at the ith input and Then the channel capacity can be written asvj is complex Gaussian noise with unit variance, as shown C = Ss {logdet (Int + (p/nt)S)}, (17)in Figure 1. The signal vector received at the output can bewritten as where the expectation is evaluated using the density given in

(5). Note that if the channel is correlated only at the trans-Yi = ... hn[ ] [ vi ] mitter end and nt> nr then S = HHH is a complex sin-

1 i. 1 • , gular Wishart matrix and its density is given in Theorem 3.Similarly, if the channel is correlated only at the receiver

Yn hn "'" hn, n Xnt Vnr end and nr > nt then S = HHH is a complex singular

i.e., in vector notation, Wishart matrix. For the other situations, S is full rank andthe distributions are chosen according to the spatial correla-

y =Hx + v, (15) tion types, e.g., (5) - (7) (see Table 1 in [7]).Let A1 > ... > Ant be the eigenvalues of S and A =

where y,v E Cnr, H c Cnr Xnt, X E Cnt and v - diag(Aq,... ,An). Thenthe capacity canalsobe computed

CN(O, 'nr). The total power of the input is constrained to using the joint eigenvalue density, f (A), i.e.,

£{xE x} • P tr&{xxH} < P. C = { S log 1 + P Ak)} (18)__ or tr(f nt

We shall deal exclusively with the linear model (15) and Again, the expectation in (18) is evaluated using the densityderive the capacity of MIMO channel models in this section. according to the spatial correlation types, e.g., (9) - (11) and

(13).

3.2. Capacity versus outage Here 4 ,out = out(Pout) means that, in any instantiation ofH from the ensemble, we shall obtain a mutual informationAnother useful information-theoretic measure is the 'x per- I7 less than 'out with probability Pout~. Using the density of

cent outage' which is defined to be the minimum mutual the m ut info ron anigive anothe ex sion f

information that occurs in all but x percent of the channel the ergodic capacity as follows:

instantiations. In other words, if we measure the mutual

information of the channel many times - in many instanti- C= [ 1q(1)dl. (23)ations of the random channel-- we would find that a mutual 0information greater than 5% outage would occur 95% of thetime (x = 5). In other words, the mean of the mutual information gives

Let us formulate the outage capacity. For a given instan- the ergodic capacity.tiation of H, if the receiver knows the channel, the mutualinformation IT(x; (y, H)) is given by 4. COMPUTATION OF THE CAPACITIES

IT(x; (y, H)) = log det In, + p HHH In this section, we evaluate the capacities for both correlated

H lnt and uncorrelated Rayleigh 2 x 2 fading channels (nr =

nt / nt = 2). First, we consider a channel with correlation at

Y- - log 1 + t Ak) , (19) both transmitter and receiver ends. A typical example ofk=t ) this situation is a communication between two mobile lap-

tops in poor scattering conditions. In addition, due to phys-where A1, of. , At are the eigenvalues of S = HHH. The ical size constraints it is more difficult to space the antennasdensity q(r ) of the mutual informations in equation (19) far apart at the laptops. These constraints lead to channelover the ensemble of instantiations of H can then be written correlation at both transmitter and receiver ends, i.e., H r,

as CN(O,Xr E Xt) and S = HHH - CQnr,nt (In,, E,, Xt).

q()=The ergodic capacity is given byk=1 (20 -cý =(detEt) 2 f[o[fl dA2 dA1 (log [I + pAl]

where 6 is the Dirac delta function, see Simon and Mous- (det ) 2 i0 i 0

takas [11]. The following theorem gives this density in +log [1 + 2A 2]) (A1 - A2 )2

terms of the density of a single unordered eigenvalue of 00(E-1) (-E-') (A)

S = HHH. x E E X[ rt]

k= k2 + 1) F(k1 + 2) F(k 2 + 1)"Theorem 5 Let H be an nr x nt random matrix channeland g(A) be the density of a single unordered eigenvalue of Second, we consider a channel with correlation only at the

S = HHH. Then the density of the mutual information, transmitter end. A typical example of this situation is an

q(IT), is given by uplink communication from a mobile unit to a base station.Here, the antennas at the base station can be spaced suffi-

q () e--/nt-) g(p[e-Z/n t-i ] (21) ciently far apart to achieve uncorrelation at the receiver end=' but, due to physical size constraints, it is more difficult to

space the antennas far apart at the mobile unit (transmitterwhere p is the total transmitter power. end), which leads to correlation at the transmitter end, i.e.,

H , CN(O, In 0 Xt) and S = HHH , CWn, (nlr, xt) isWe can also express the distribution Q (1) of the mutual H- N ,In0EtadS=HH-Cn (,t)iinformation _Talsfollowss: ta complex Wishart matrix. It should be noted that the jointinformation I as follows:

eigenvalue density of a Wishart matrix depends on the pop-f1 ulation covariance matrix Et only through its eigenvalues,

Q(V) =1 q(IT) dT. (22) V1 ,... ,Vn, i.e.,

The unique inverse function of Q(I) is called the out- oF0onl (-Et1,A) = 0F(n Y (T-1',A)age mutual information and is denoted by out (Pout) where

Pout = Q(IEout). In other words, we define out(Pout) such diag(al , a2). Then from [6] we havethat

hTout = out(Pout) ° 2)(-T-'A) = a 1

where (a2 - a,)(Al - A2)

Pout = Q(Wout). x [e-aX-a2A2 - e-alA2-a2A ]. (24)

The ergodic capacity is given by 5. REFERENCES

cc = 1 dA log 1 + A [1] C. N. Chuah, D. Tse, J. M. Kahn and R. A. Valen-(a2 -a,)Jo 21 2zuela, "Capacity scaling in MIMO wireless systems

x [a2re-ra1(a 2 A - 1) - a2e-a2A (aiA - 1)] under correlated fading," IEEE Trans. on InformationTheory, Vol. 48, pp. 637-650, Mar. 2002.

Third, we consider an uncorrelated channel at the transmit-ter and receiver ends, i.e., H ,-, CN(O, In, 0 a 2 In) and [2] J. G. Foschini and M. J. Gans, "On limits of wire-

S = HHH -'s CWn, (n7, a 2In). The ergodic capacity is less communications in a fading environment when

given by using multiple antennas," Wireless Personal Commu-nications, Vol. 6, pp. 311-335, Mar. 1998.

C.= 2 /0 log [ + A A/A 2 A4+ dA. [3] A. T. James, "Distributions of matrix variate and la-La a aj 4tent roots derived from normal samples," Ann. Math.

Statist., Vol. 35, pp. 475-501, 1964.

Figure 2 shows the capacity in nats vs signal-to-noise ratio [4] C. G. Khatri, "On certain distribution problems basedfor correlated/uncorrelated Rayleigh 2 x 2 fading channel on positive definite quadratic functions in normal vec-matrices. The following parameters are used: a 2

= 1 and tors," Ann. Math. Statist., Vol. 37, pp. 468-479, 1966.

E [ 1 0.5 + 0.5i [5] R. J. Muirhead, Aspects of Multivariate Statistical0.5 - 0.5i 1 J Theory, Wiley, New York, 1982.

From this figure we note the following: (i) the capacity is [6] T. Ratnarajah, R. Vaillancourt and M. Alvo, "Com-decreasing with channel correlation. Moreover, correlation plex random matrices and Rayleigh channel ca-at the transmitter and receiver ends causes more capacity pacity," Communications in Information & Sys-degradation compare to correlation at the transmitter end tems, Vol. 3, No. 2, pp. 119-138, Oct. 2003only. (ii) the capacity is increasing with SNR. (http://www.ims.cuhk.edu.hk/-cis/).

10 r.[7] T. Ratnarajah and R. Vaillancourt, "Quadratic forms9 •on complex random matrices and multiple-antenna

systems," Submitted to IEEE Trans. on Information8 Theory, Jan. 2004.

7 C. [8] M. Sellathurai and G. Foschini, "A stratified diagonal

6• layered space-time architecture: Information theoreticand signal processing aspects," IEEE Trans. on Signal

5 /jProcessing, Vol. 51, pp. 2943-2954, Nov. 2003.

[9] H. Shin and J. H. Lee, "Capacity of multiple-antennaS3 fading channels: Spatial fading correlation, double

2 scattering, and keyhole," IEEE Trans. on InformationTheory, Vol. 49, pp. 2636-2647, Oct. 2003.

012 [10] D. S. Shin, G. F. Foschini, M. G. Gans and J. M. Kahn,0 5 10 15 20 25 "Fading correlation and its effect on the capacity of

Signal-to-Noise Ratio (in dB) multielement antenna systems," IEEE Trans. on Com-

munications, Vol. 48, pp. 502-513, Mar. 2000.

Fig. 2. Capacity vs SNR for nt = 2 andnr = 2, i.e., [11] S. H. Simon and A. L. Moustakas, "OptimizingRayleigh 2 x 2 channel matrices. Note that C., Cc and [11] antenna A. L. M o vakas, feed-

Cc, denote the capacity of the three environments: uncorre- Ma ntnn sytm wt el co vaince feed-lated, correlated at the transmitter end and correlated at both back," IEEE J. Select. Areas Commun., Vol. 31, pp.transmitter and receiver ends, respectively. 406-417, Apr. 2003.

[12] I. E. Telatar, "Capacity of multi-antenna Gaussian

'In equation (18), if we use loge then the capacity is measured in nats. channels," Eur. Trans. Telecom, Vol. 10, pp. 585-595,If we use log 2 then the capacity is measured in bits. Thus, one nat is equal 1999.to e bits/sec/Hz (e = 2.718... ).

QUADRATIC FORMS ON COMPLEX RANDOMMATRICES AND MULTI-ANTENNA CHANNEL

CAPACITY

T. -Ratnarajah and R. Vaillancourt

Department of Mathematics and Statistics

University of Ottawa

Ottawa ON KIN 6N5 Canada

e-mail: t .ratnarajah~ieee. org

Mar. 16, 2004

T. Ratnarajah University of Ottawa

lVEVIEW

In this paper, we

"* study the densities of quadratic forms on complex random matrices

"* study the eigenvalue densities of quadratic forms on complex random matrices

"* show that these densities are represented by complex hypergeometric functions

of matrix arguments

"* show that the above can be expressed in terms of complex zonal polynomials

"* derive a close form ergodic capacity and capacity versus outage formula for

spatially correlated MIMO channels

"* show how the channel correlation degrades the capacity of the communication

system

ASAP 2004 1 Mar. 16, 2004

T. Ratnarajah University of Ottawa

MIMO CHANNEL CAPACITY/

1V

hYl

X , ~ ~ ~~~ ... .... .... ..... ..........--:. +.-Y

".. .. ...... 2 n. .... -" "'..... .. ...... --.. .....2 -. ::. ---- h

hh

x •, • ,%'."{. :......................... :.............. . -. , + •yXnt --- - - , Ynr

Figure 1: A MIMO communication system.

The signal vector received at the output can be written as

y =Hx+v, where y,v E Cn, x Cel0, H Cnrxn. (1)

The total power of the input is constrained to p,

{xHx}• < p or trS{xxH} < p.

ASAP 2004 2 Mar. 16, 2004

T. Ratnarajah University of Ottawa

ERGODIC CHANNEL CAPACITY ]

The ergodic capacity of MIMO channel is given byC = maxSH {I(x; y)}

f(x)

- SH {log det (I,, + (p/ntt)HHH) }- Es {log det (1,, + (p/rnt)S)}

- SA {lo0 l [1 + (P/rit)Aki)}

- S• Ak {log(1 + (p/rnt)Ak)}, (2)k=1

where the expectation is evaluated using the density according to the spatial

correlation types, see the table in the next slide.

If S = HHH H' CQn,.,n (In,, 7E,) Et) then the channel is spatially correlated

Rayleigh distributed, where E, and Et represent the channel correlation at the

receiver and transmitter ends, respectively.

ASAP 2004 3 Mar. 16, 2004

T. Ratnarajah University of Ottawa

SPATIAL CORRELATIONS AND CHANNEL DISTRIBUTIONS

Spatial correlation Channel distributions

types Tnr > nt nr = nt nt > 1r

Uncorrelated H - CN(gIo, Ar0 a 22In) H N, Cg(o, a 2 _[n 0 hIn)(i.i.d) S = HHH S=HHH

S CTVn t (T1,r, ' n21 n)- S C CTVnr (TIt, (7 2nr)

Correlated at H , CN(O7 Inr 0 Et) H , CN(O, Inr0 Et)

the transmitter S = HHH S = HHH

S -CV~nt (rir, Et) S -CV~nt (rir,LEt)

Correlated at H , CN(O,L0 E Int) H , CN(O,Lr ®r In)

the receiver S = HHH S=HHH

s CTVnr (TIt, Kr) S f CTVVnr (TIt, Kr)

Correlated at both H , CN(0, Er 0 Et) H U CN(0, Er 0 Et)

transmitter and receiver S = HHH S=HHH

S - CQnr,nt(Inr,YEr,YEt) S - CQnr,nt(Int,,Er ,Et)

ASAP 2004 4 Mar. 16, 2004

T. Ratnarajah University of Ottawa

COMPUTATION OF THE CAPACITY (CONT.)

10

9

8

7 C.

S6

5 CC

. 4

S3

2

00 5 10 15 20 25

Signal-to-Noise Ratio (in dB)

Figure 2: Ergodic capacity vs SNR for nt = 2 and r1 = 2, i.e., Rayleigh 2 x 2 channel

matrices. Note that C•, C, and C,, denote the capacity of the three environments:

uncorrelated, correlated at the transmitter end and correlated at both transmitter

and receiver ends, respectively.

ASAP 2004 5 Mar. 16, 2004