EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONSEuro. Trans. Telecomms. (in press)Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/ett.1135

Information Theory

Outage probability in multiple antenna systems†

Eduard A. Jorswieck∗ and Holger Boche

Fraunhofer Institute for Telecommunications, Heinrich-Hertz-Institut, Einsteinufer 37, D-10587 Berlin, Germany

SUMMARY

Multiple transmit antennas improve the ergodic and outage capacity of wireless systems. Spatial proper-ties of the channel including the transmit antenna array affect the optimum transmission strategy as wellas the achievable capacity and average throughput. First, we study the outage probability of a multipleinput single output (MISO) system with perfect channel state information (CSI) at the receiver under dif-ferent types of CSI at transmitter and with transmit antenna correlation. We prove the conjecture givenin Telatar’s seminal paper and complete the analysis of the optimum transmit strategy without CSI at thetransmitter and uncorrelated antennas. Furthermore, we show how the impact of correlation on the out-age probability depends on the transmission rate and SNR. We show that the behaviour of the outageprobability differs from the behaviour of the ergodic capacity. In terms of ergodic capacity there are clearinstructions what is the optimum transmission strategy and what is the impact of correlation. In contrast,the outage probability behaves chameleonic. If the transmitter is aware of the channel correlation matri-ces, the optimum transmit strategy is to transmit along the eigenvectors of the known correlation matrix.The remaining power allocation problem is difficult since it is a non-convex optimisation problem. How-ever, necessary conditions characterise the optimal allocation. Finally, we analyse the outage probabilityfor the general multiple input multiple output (MIMO) system with spatially correlated transmit and re-ceive antennas in asymptotic high and low SNR regime. The theoretical results are illustrated by numericalsimulations. Copyright© 2006 AEIT

1. INTRODUCTION

Multiple-element antenna arrays can improve the perfor-mance and capacity of a wireless communication systemin a fading environment [1–4]. In recent years, it wasdiscovered that multiple input multiple output (MIMO)systems have the ability to reach higher transmission ratesthan one-sided array links [5, 1]. The achievable ergodiccapacity depends on the spatial properties of the wirelesschannel: It was shown that correlation of the transmitantennas reduces the achievable capacity of the channel bynumerical simulations [6], for asymptotic many antennas

* Correspondence to: Eduard A. Jorswieck, Fraunhofer-Institute for Telecommunications, Heinrich-Hertz-Institut, Broadband Mobile CommunicationNetworks, Einsteinufer 37, 10587 Berlin, Germany. E-mail: jorswieck@hhi.de†Part of this work was presented at IEEE Information Theory Workshop, April 2003 and IEEE International Zurich Seminar on Communications,February 2004.

Contract/grant sponsor: Deutsche Forschungsgemeinschaft (DFG).

in MIMO systems [7], and in general for multiple inputsingle output (MISO) systems [8] and MIMO systems [9]for uninformed transmitters. Otherwise, it depends, forexample on the SNR if correlations are beneficial or not[8]. In Reference [10], the authors study effects within thechannel, so called ’key-wholes’.

In literature there are different models which measurethe correlation of the transmit and receive antennas inMIMO systems. We use the well-established model fromReference [7]. This model is well suited for Rayleigh andRicean MIMO channels which naturally arise in a rich scat-tering environment. In Reference [6] a model for correlation

Received 23 February 2005Revised 5 October 2005

Copyright© 2006 AEIT Accepted 10 March 2006

E. A. JORSWIECK AND H. BOCHE

of the multi element antenna was developed and the ergodicand outage capacity was computed under correlation. InReference [7] the impact of correlation was analysed bystudying the asymptotic eigenvalue distribution of thechannel matrix for a large number of transmit and receiveantennas. The theory of majorisation for discrete vectorswas extended to continuous probability density functionsand it was shown that correlation decreases the ergodiccapacity. In Reference [11], an approximative expressionfor the capacity of correlated MIMO channels has beenpresented. In Reference [12] analytical results for themoments of the mutual information under correlation werederived for large antenna arrays, too. Using the tight capac-ity bound which was developed in Reference [13] for highSNR values, the impact of correlation can easily be anal-ysed. All analytical results assume either a large number ofantennas or high SNR values. For the MISO case, a proofwas derived in Reference [14]: The ergodic capacity of aMISO system with no CSI at the transmitter decreases if thetransmit correlation increases. Furthermore, it was shownin Reference [14], that the capacity loss due to correlationis bounded by some small constant. The optimum transmis-sion strategy in MISO channels for different types of CSI atthe transmit array was studied in Reference [8]. For no CSI,the optimum transmission strategy with respect to ergodiccapacity is equal power allocation and the ergodic capacityis a Schur-concave function with respect to the channelcovariance matrix eigenvalues, that is the more correlatedthe channel is, the less ergodic capacity. For covariancefeedback, the optimum transmission strategy can be itera-tively computed and the ergodic capacity is a Schur-convexfunction with respect to the channel correlation.

In contrast to the ergodic capacity which is a measurefor the amount of average information error-free transmit-ted, the outage probability is a more subtle measure for theprobability of successful transmission while the channel isin a channel state. Since the instantaneous capacity dependson the channel state, it is itself a random variable. The firstmoment corresponds to the ergodic capacity. The cumu-lative distribution function (cdf) is the outage probability.The outage probability gives the probability that a giventransmission rate cannot be achieved in one fading block.Recently, the outage probability was studied for multipleantenna channel and space-time codes [15, 16]. The prop-erties of the optimum transmission strategies change, if wereplace the ergodic capacity as objective function with theoutage probability. For example for no CSI at the transmit-ter, the optimum transmission strategy is to use only a frac-tion of the available number of transmit antennas. Telatarhas already conjectured this in Reference [1]. In Reference

[2], a part of this conjecture is verified. Furthermore, analgorithm which finds the optimum number of active an-tennas was proposed. In addition to this, in Reference [2],a necessary and sufficient condition for the optimality ofsingle-antenna processing was derived.

The paper at hand is to be understood as an answer to theconjecture by Telatar in Reference [1]. The contributions inthis paper are described in the following:

1. In MISO systems with uninformed transmitter, perfectCSI at the receiver, and with an uncorrelated channel, theoptimal transmit strategy minimising the outage proba-bility is to use a fraction ofk out of all nT availabletransmit antennas and perform equal power allocation(Theorem 3 in Section 3.1). The SNR range in whichequal power allocation across all available antennas isoptimal is for all SNR greater than or equal to

ρ = 2R − 1

Furthermore, the SNR range in which only one transmitantenna should be used is

0 < ρ ≤ ρ = 2(2R − 1)

−2Lw(−1, −1/2 exp(−1/2)) − 1

Lw is the LambertW function. The Lambert W-function,also called the omega function, is the inverse function off (W) = W exp(W) [17].

2. In MISO systems with perfect CSI at the receiver, andequal power allocation across all available transmit an-tennas, the impact of correlation on the outage proba-bility is characterised using majorisation theory. For allSNR values smaller than or equal to

ρ = 2R − 1

2

the outage probability as a function of the correlation isSchur-concave, that is correlation decreases the outageprobability. For all SNR values greater than or equal to

ρ = 2R − 1

the outage probability as a function of the correlationis Schur-convex, that is correlation increases the outageprobability (Theorem 4 in Section 3.2).

3. In MISO systems with perfect CSI at the receiver, andtransmitter knowing the channel correlation, the neces-sary optimality conditions for the optimal power alloca-tions are characterised in Section 3.3.

Copyright© 2006 AEIT Euro. Trans. Telecomms. (in press)

OUTAGE PROBABILITY IN MULTIPLE ANTENNA SYSTEMS

4. In Section 5, the results from the MISO case are extendedto the MIMO case by analysing the asymptotic outageprobability for small and high SNR values.

1.1. Notation

The space of real numbers is represented by the symbolR.Lw is the LambertW function [17]. Boldface uppercaseA denotes matrices. The matrixI is the identity matrix.Boldface lowercasea corresponds to a column vector. trA

denotes the trace of the matrixA, detA denotes the de-terminant of the matrixA. We writex � y for a vectorxmajorisingy.

Pr[A] denotes the probability of an eventA. The ex-pectation operator is denoted byE. The matrixW denotes arandom matrix with independent and identically distributed(iid) zero-mean complex Gaussian entries with varianceone. The vectorw without index has iid zero-mean complexGaussian distributed entries. Finally, the scalarsw1, . . . , wn

are iid zero-mean complex Gaussian random variables andthe random variabless1, . . . , snT are standard exponentiallydistributed.χ2

d denotes a chi-square distributed random vari-able withd degrees of freedom and unit variance.

2. RECENT RESULTS AND PROBLEMSTATEMENTS

2.1. Channel Model

Consider the standard MIMO block flat-fading channelmodel

y = Hx + n (1)

with complexnT × 1 transmit vectorx, channel matrixHwith nR × nT entries, circularly symmetric complex Gaus-

sian noisen with varianceσ2

N

2 I per dimension. We assumethat the receiver knowsH perfectly. In the MISO case, thechannel is simple a vectorh of dimension 1× nT and thereceived signal as well as the noise is a scalar.

Let us first describe the signal processing at the transmitantenna array. In the following, we will say that a transmitstrategy is a covariance matrix is given byQ = E

(xxH

).

Using the eigenvalue decomposition ofQ = UQ�QUHQ, it

becomes obvious how one can construct a particular trans-mit covariance matrix. The input data streamd(k) is splitinto m parallel data streamsd1(k), . . . , dm(k). Each par-allel data stream is multiplied by a factor

√p1, . . . ,

√pm

and then weighted by a beamforming vectoru1, . . . ,um,

respectively. The number of parallel data streams is lessor equal to the number of transmit antennas (m ≤ nT).The beamforming vectors have size 1× nT with nT asthe number of transmit antennas. ThenT signals of eachweighted data streamxi(k) = di(k) · √

pi · ui are added upx(k) =∑m

i=1 xi(k) and sent. By omitting the time indexkfor convenience we obtain in front of the transmit antennasx =∑m

l=1 dl · √pl · ul. The transmit signal inx has a co-

variance matrixQ with eigenvaluesp1, . . . , pm, 0, . . . , 0and eigenvectorsu1, . . . , um. In order to construct a trans-mit signal with a given covariance matrix, two signal pro-cessing steps are necessary: the power controlp1, . . . , pm

and the beamformersu1, . . . , um. A Gaussian codebookfor d(k) is optimal [18]. We assume a transmit power con-straintP and normalised transmit covariance matricesQ,that isP · trQ = P ·∑m

k=1 pk ≤ P . The SNR is defined asρ = P

σ2n.

Next, we study the stochastic properties of the channel.The correlation of the channel vectors arises in the commondownlink transmission scenario in which the base station isun-obstructed [6]. We follow the model in Reference [19]where the subspaces and directions of the paths betweenthe transmit antennas and the receive cluster change slowlerthan the actual attenuation of each path.

The channel matrixH for the case in which we have cor-related transmit and correlated receive antennas is modeledas

H = R12R · W · R

12T (2)

with transmit correlation matrixRT = UT�TUHT and re-

ceive correlation matrixRR = UR�RUHR . UT andUR are

the matrices with the eigenvectors ofRT andRR respec-tively, and�T, �R are diagonal matrices with the eigen-values of the matrixRT andRR, respectively. The randommatrixW has zero-mean iid complex Gaussian entries, thatis W ∼ CN(0, I). In the MISO case, the transmit antennasare correlated and the channel vector is described as

hH = wH · R12T

The eigenvalues of the transmit covariance matrix andof the channel correlation matrices play an important rolefor analysing the outage probability. Therefore, we orderthe eigenvalues of the transmit covariance matrixQ indescending order, that isp1 ≥ p2 ≥ · · · ≥ pnT , and theeigenvalues of the channel correlation matricesRT andRR, in descending order, that isλT

1 ≥ λT2 ≥ · · · ≥ λT

nTand

λR1 ≥ λR

2 ≥ · · · ≥ λRnR

.

Copyright© 2006 AEIT Euro. Trans. Telecomms. (in press)

E. A. JORSWIECK AND H. BOCHE

2.2. Properties of the outage probability

The instantaneous mutual information of the MISO systemwith transmit strategyQ is given byC(h, Q, ρ) = log(1+ρhHQh). All logarithms in the paper are base 2 loga-rithms. The average mutual information of the MISO systemwith transmit strategy is given asC(Q, ρ) = Eh log(1+ρhHQh). The outage probability of the MISO system is

Pout(Q, ρ, R) = Pr[C(h, Q, ρ) ≤ R]

Let us write the outage probability as a function that de-pends on the SNRρ, the transmission rateR, the channelcovariance matrixRT, and the transmit covariance matrixQ, that is

Pout(ρ, R, RT, Q) =

Pr

[log(1 + ρwHR

1/2T QR

1/2T w

)≤ R

](3)

The optimum transmit strategy depends on the objectivefunction and the CSI at the transmitter. For ergodic capac-ity maximisation, if we have perfect CSI at the transmitterwe perform beamforming. If we have the knowledge of thechannel covariance matrix at the transmitter, the optimumtransmit strategy is to send parallel data streams in directionof the eigenvectors of the known covariance matrix [18].The optimum power allocation with respect to the ergodiccapacity of this strategy is discussed in Reference [20]. Forthe case in which the transmitter has no or only covarianceCSI, the outage probability can be further simplified. Us-ing the result from Reference [18], the ergodic capacity aswell as the outage probability does not depend on the direc-tions ofRT andQ, that is the optimal eigenvectors of thetransmit covariance matrix correspond to the eigenvectorsof the correlation matrix and the problem is diagonalised.Therefore, we can write the outage probability as

Pout(ρ, R, λT , p) =

Pr

[log

(1 + ρ

nT∑k=1

λTk pksk

)≤ R

](4)

with power allocationp = [p1, . . . , pnT ], channel matrixeigenvaluesλT = [λT

1 , . . . , λTnT

] and standard exponentialidd random variabless1, . . . , snT , rateR and SNRρ andnumber of transmit antennasnT.

The outage probability in MIMO systems depends onthe SNRρ, the transmission rateR, the transmitRT and

receiveRR correlation, and the transmit strategyQ. Theoutage probability can be written as

Pout(ρ, Q, RT, RR, R) =Pr[det(I + ρR

1/2R WR

1/2T QR

1/2T WHR

1/2R

)< 2R

](5)

The probability is with respect to the random matrixW .This function is studied in Section 5. If we have no CSIat the transmitter and want to maximise the average mutualinformation, we chooseQ = 1/nTI. Observe the symmetryin Equation (5), if this policy is plugged in, that is

Pr[det(I + ρR

1/2R WR

1/2T R

1/2T WHR

1/2R

)< 2R

]=

Pr[det(I + ρR

1/2T WHR

1/2R R

1/2R WR

1/2T

)< 2R

]since det(I + AB) = det(I + BA). It follows that the im-pact of transmit correlation on the outage probability withequal power allocation behaves similar to the impact of thereceive correlation and vice versa.

2.3. Problem statements

First, we solve the following two problems: The first han-dles the optimum transmission strategy if the channel real-isations are iid, that is if the channel covariance matrix isRT = I. For the maximisation of the ergodic capacity, inthis case, the optimal strategy is equal power allocation [1].

2.3.1. Problem 1.

Assume there is one receive antenna, the channel vectorsare iid, the channel is spatially uncorrelated, for fixed rateR and SNRρ. What is the optimal transmission strategywhich minimizes the outage probability?

The problem statement was introduced in Reference [1].For fixed rateR andρ power allocation of the kindQd =1/dId were discussed for variousn. We complete the workin Reference [2] and derive the proof of the optimal trans-mission strategy. The second problem discusses the impactof correlation on the outage probability if equal power isallocated to all available transmit antennas.

2.3.2. Problem 2.

Assume that the transmitter is uninformed and applies equalpower allocation across all available transmit antennas.There is one receive antenna. What is the impact of cor-relation on the outage probability?

Copyright© 2006 AEIT Euro. Trans. Telecomms. (in press)

OUTAGE PROBABILITY IN MULTIPLE ANTENNA SYSTEMS

Next, we discuss briefly the following problem and char-acterise the locally optimal power allocation.

2.3.3. Problem 3.

Assume that the channel is spatially correlated, the trans-mitter knows the transmit correlation, there is one receiveantenna. What is the optimal transmit strategy for givenSNRρ and transmission rateR?

Finally, the extension of the developed techniques to theMIMO case is discussed.

2.3.4. Problem 4.

How can the results be carried over to the MIMO case?Is there a SNR/rate range in which the outage probabilityis Schur-convex or Schur-concave with respect to the fad-ing correlation for uninformed transmitter and equal powerallocation?

2.4. Preliminaries

For two vectorsx, y ∈ Rn one says that the vectorx ma-

jorises the vectory and writesx � y if∑m

k=1 xk ≥∑mk=1 yk

for all m = 1, . . . , n − 1 and∑n

k=1 xk =∑nk=1 yk.

A real-valued function� defined onA ⊂ Rn is said to

beSchur-convex onA if from x � y onA follows �(x) ≥�(y).

Similarly, � is said to beSchur-concave onA if fromx � y onA follows �(x) ≤ �(y).

For further information about majorisation theory seeReference [21]. We need the following result (see Reference[21, Theorem 3.A.4]) which is sometimes called Schur’scondition. It provides an approach for testing whether somevector valued function is Schur-convex or not.

Lemma 1. Let I ⊂ R be an open interval and let f :In → R be continuously differentiable. Necessary and suf-ficient conditions for f to be Schur-convex on In are

(xi − xj)

(∂f

∂xi

− ∂f

∂xj

)≥ 0 for all 1 ≤ i, j ≤ n (6)

Schur’s condition for the Schur-convexity of asymmetric† functionf (x) is given as [21, p. 57]

(x1 − x2)

(∂f

∂x1− ∂f

∂x2

)≥ 0 (7)

† A function is called symmetric if the argument vector can be arbitrarilypermuted without changing the value of the function.

Furthermore,f (x) is a Schur-concave function onIn iff (x) is symmetric and

(x1 − x2)

(∂f

∂x1− ∂f

∂x2

)≤ 0 (8)

The following definition provides a measure for compar-ison of two correlation matrices.

Definition 1. The transmit correlation matrix R1T is more

correlated than R2T if and only if

m∑l=1

λT,1l ≥

m∑l=1

λT,2l for m = 1 . . . nT,

andnT∑l=1

λT,11 =

nT∑l=1

λT,22 . (9)

One says that the vector consisting of the ordered eigenval-ues λT

1 majorises λT2 , and this relationship can be written

as λT1 � λT

2 .

It can be shown that vectors with more than two com-ponents cannot be totally ordered. So there are examplesof correlation vectors that cannot be compared using ourDefinition 1, for exampleη1 = [0.6, 0.25, 0.15] andη2 =[0.55, 0.4, 0.05]. This is a problem of all possible orders forcomparing correlation vectors. Majorisation induces only apartial order.

The case in which the transmit antennas are completelycorrelated corresponds toλT

1 = nT, λT2 = · · · = λT

nT= 0.

The case in which the transmit antennas are uncorrelatedcorresponds toλT

1 = λT2 = · · · = λT

nT= 1.

Note that our definition of correlation in Definition 1 dif-fers from the usual definition in statistics. In statistics adiagonal covariance matrix indicates that the random vari-ables are uncorrelated. This is independent of the auto-covariances on the diagonal. In our definition, we say thatthe antennas are uncorrelated if in addition to statistical in-dependence, the auto-covariances of all entries are equal.This difference to statistics occurs because the direction,that is the unitary matrices of the correlation have no im-pact on our measure of correlation. Imagine the scenario inwhich all transmit antennas are uncorrelated, but have dif-ferent average transmit powers because of their amplifiers.In a statistical sense, one would say the antennas are uncor-related. Our measure of correlation says that the antennasare correlated, because they have different transmit powers.The measure of correlation in Definition 1 is more suitable

Copyright© 2006 AEIT Euro. Trans. Telecomms. (in press)

E. A. JORSWIECK AND H. BOCHE

for the analysis of the performance of multiple antenna sys-tems, because different transmit powers at the antennas ob-viously have a strong impact on the performance. In thiswork, these effects are considered.

2.5. Properties of Gaussian quadratic forms

In the following paper, we will use recent results fromReference [22]. Our approach independently developedin References [2,3] uses very similar techniques and justlike Reference [22] does not provide a general frameworkbut exploit essentially the properties of the exponential dis-tribution. The derivations are very tedious and elementary.Therefore, we present and reuse these recent results when-ever possible. Since [22] studies quadratic forms of realGaussian distributed random variables and we will studyquadratic forms of complex Gaussian distributed randomvariables, we have to extend and generalise the results inReference [22]. At some points, we will provide shortcutsthat simplify the tedious derivations in Reference [22]. At-tempts to solve the problems for Rice or other fading dis-tributions by the same technique shows no success up tonow. Generalised approaches to solve the problem for ar-bitrary or certain classes of fading distributions are muchappreciated.

We present the theorems from Reference [22] for com-pleteness. Consider the quadratic formQn =∑n

i=1 λiw2i

wherewi are zero-mean real normal iid andλi are eigenval-ues of some covariance matrixR.R is positive semidefinite,implyingλi ≥ 0, 1 ≤ i ≤ n. The following constraint is ap-pliedE(Qn) =∑n

k=1 λk = 1. Furthermore, letχ2d a random

variable that isχ2 distributed withd degrees of freedomand unit variance. Note that [22] studies quadratic formsof real Gaussian distributed random variables, whereas weare interested in quadratic forms of complex Gaussian dis-tributed random variables. Therefore, we need to extend theresults provided below at certain points. The main Theorem1 proved in Reference [22] is

Theorem 1 (Theorem 1 in Reference [22]).

inf {Qn≥0|E(Qn)=1} P(Qn ≤ x) ={Pr[

1dχ2

d ≤ x], ∀x ∈ [x(d), x(d − 1)], d = 1, . . . , n − 1

Pr[

1nχ2

n ≤ x], ∀x ∈ [0, x(n − 1)],

where x(0) = ∞ by definition.

The interval boundariesx(d) are characterised by the fol-lowing two propositions.

Proposition 1 (Proposition 1 in Reference [22]). Denoteby gd(x) the difference between the cdf’s of (d + 1)−1χ2

d+1and d−1χ2

d .

(i) For everyd ≥ 1, there exists a unique pointx(d) suchthatgd(x) < 0 for 0 < x < x(d) andgd(x) > 0 forx >

x(d),(ii) x(d) > 1, for all d ≥ 1

(iii) x(1) > x(2) > . . ., that is the sequencex(d) is strictlydecreasing,

(iv) d(x(d) − 1) → 23 for d → ∞.

Proposition 2 (Proposition 1’ in Reference [22]). Forany integers n > m > 0, introduce gn,m(x) = Pr[n−1χ2

n ≤x] − Pr[m−1χ2

m].

(i) For everyn > m positive integers, there exists a uniquepoint x(n, m) > 0 such thatgn,m(x) < 0 for 0 < x <

x(n, m), andg(x) > 0 for x > x(n, m),(ii) 2 > x(m) ≥ x(n, m) ≥ x(n − 1) > 1,

(iii) x(m) = x(m + 1, m) > x(m + 2, m) > . . .,(iv) x(n, 1) > x(n, 2) > · · · > x(n, n − 1) = x(n − 1).

We will need the following corollary that describes the in-terval boundaries if the quadratic form is in complex Gaus-sian random variables.

Corollary 1. Using the notation in Proposition 2 it holds

x(2d + 2, 2d) > x(2d + 4, 2d + 2)

Proof. Starting withx(2d + 4, 2d + 2) it holds

x(2d + 4, 2d + 2) <a x(2d + 3, 2d + 2)

<b x(2d + 2, 2d + 1) <c x(2d + 2, 2d)

where stepa follows from (iii) in Proposition 2, stepbfrom (iii) in Proposition 1 and inequalityc follows from(iv) in Proposition 2. �

Using the same steps as in the proof of Corollary 1, themonotony ofx(2d + 2, 2d) can be extended tox(2d +2n, 2d) > x(2d + 4n, 2d + 2n).

In addition to Theorem 1, the two propositions, and thecorollary, the following corollary is interesting.

Corollary 2 (Corollary 3 in Reference [22]). If x ≥ 2, andλ1 majorises λ2, then

Pr

[n∑

i=1

λi,1w2i ≤ x

]≤ Pr

[n∑

i=1

λi,2w2i ≤ x

]

Copyright© 2006 AEIT Euro. Trans. Telecomms. (in press)

OUTAGE PROBABILITY IN MULTIPLE ANTENNA SYSTEMS

and conversely, if this inequality holds for all positive inte-gers n and for any λ1 that majorises λ2, then x ≥ 2.

A pdf p(x) is called unimodal if there exists one numberx = a such thatp(x) is nondecreasing forx < a andp(x) isnonincreasing forx > a. The pointa is called the mode ofp(x).

Theorem 2 (Theorem 4 in [22]). The pdf of Qn =∑nk=1 λiw

2i is unimodal for arbitrary λi, i ≥ 1.

2.6. Alternative representation of outage probability

The following representation of the outage probability byits Fourier transform shows to be very convenient for furtheranalysis.

Lemma 2. The outage probability as a function of thepower allocation p, rate R, and SNR ρ can be written as

Pout(p, z) = 1

2π

∫ ∞

−∞1

jω

nT∏k=1

1

1 + jpkωexp(jωz)dω

+ 1

2(10)

where z = 2R − 1/ρ.

Proof. Note, that the outage probability can be written as

Pout(ρ, R,1, p) = Pr

nT∑k=1

pksk ≤ 2R − 1

ρ︸ ︷︷ ︸z

= Pout(p, z) (11)

The outage probability in Equation (11) depends only on thepower allocationp and the parameterz which contains theSNRρ and the rateR. Denote the pdf of the random variable∑nT

k=1 pksk asφ(x) and its Fourier transform as�(ω), thatis

�(p, ω) =nT∏

k=1

1

1 + jpkω(12)

Then, the outage probability can be rewritten as

Pout(p, z) =∫ z

0φ(x)dx (13)

= 1

2π

∫ ∞

−∞

nT∏k=1

1

1 + jpkω

∫ z

0exp(jωt)dtdω

= 1

2π

∫ ∞

−∞

nT∏k=1

1

1 + jpkω

exp(jωz) − 1

jωdω

The RHS in Equation (13) is further simplified by the fol-lowing identity

1 = limz→∞ Pout(p, z) = lim

z→∞

∫ z

0φ(x)dx (14)

= limz→∞

1

2π

∫ ∞

−∞

nT∏k=1

1

1 + jpkω

exp(jωz) − 1

jωdω

Choose a constantA > 0 andq(ω) =nT∏

k=1

11+jpkω

. The first

summand in Equation (14) is

1

2π

∞∫−∞

nT∏k=1

1

1 + jpkω

exp(jωz)

jωdω

= 1

2π

∞∫0

1

jω(q(ω) exp(jωz) + q(−ω) exp(−jωz)) dω

= 1

2π

A∫0

1

jωq(ω) exp(jωz) + q(−ω) exp(−jωz)dω

+ 1

2π

∞∫A

1

jωq(ω) exp(jωz) + q(−ω) exp(−jωz)dω.

(15)

The second integral on the RHS in (15) is zero forz → ∞.The first integral is transformed by Taylor’s Theorem to splitthe functionq(ω) into two partsq(0) + ωR(ω). Note thatq(0) = 1. Hence, the first integral on the RHS in Equation(15) can be written as

1

π

∫ A

0

exp(jωz) − exp(−jωz)

2jωdω

+ 1

2πj

∫ A

0

[R(ω) exp(jωz) + R(−ω) exp(−jωz)

]dω.

(16)

R is bounded and continous on [0, A]. Therefore, the secondterm in Equation (16) fanishes forz → ∞. Finally, the first

Copyright© 2006 AEIT Euro. Trans. Telecomms. (in press)

E. A. JORSWIECK AND H. BOCHE

term reads

limz→∞

1

π

∫ A

0

sin(ωz)

ωdω = lim

z→∞1

π

∫ Az

0

sin(ω)

ωdω = 1

2

and combining this with Equations (14) and (16) shows

1

2π

∫ ∞

−∞

nT∏k=1

1

1 + jpkω

1

jωdω = −1

2.

This completes the proof. �

3. OUTAGE PROBABILITY ANALYSIS

In this section, we derive the main results regarding theminimisation of the outage probability and the impact ofcorrelation and solve the two problem statements for theMISO case using in part the results from the Section 2.5.

3.1. Optimal power allocation for iid fading

The optimisation problem which corresponds with the firstproblem statement is given by

p∗ = arg minpk≥0, 1≤k≤nT∑nT

k=1 pk≤1

Pr

[log

(1 + ρ

nT∑k=1

pksk

)≤ R

]

Remark. In contrast to ergodic capacity maximisation, theoptimisation problem considered here is non-convex withrespect top. As a result, we have local minima and max-ima. Furthermore, there are in general more than one globalminimum and we have to decide what is the best power al-location. Fortunately, it will be shown that the two globalminima have the same power allocation structure. The op-timal power allocation depends on the rateR and the SNRρ. We have the following theorem.

Theorem 3. The transmitter has nT transmit antennas,the transmission rate is R.

(i) Then there are nT − 1 SNR values ρ1 < ρ2 < · · · <

ρnT−1, such that for all ρ ∈ (ρl, ρl+1) only one optimalpower allocation exists. The optimal power allocationis then given by

Ql = 1

l + 1I l+1, (17)

with I l+1 is a nT × nT diagonal matrix with l + 1 onesand nT − l − 1 zeros on the diagonal, that is l + 1 an-tennas out of nT are used with equal power allocation.For ρ = ρl exist two optimal power allocation. Theseare Ql and Ql+1.

(ii) For all SNR values greater than ρ given by

ρ = 2R − 1

equal power allocation across all available antennasis optimal.

(iii) The single antenna range is given by

ρ = 2(2R − 1)

−2Lw(−1, −1/2 exp(−1/2)) − 1(18)

Remark. Lw is the LambertW function. The LambertW-function, also called the omega function, is the inversefunction off (W) = W exp(W) [17]. Its value for the param-eters−1 and−1/2 exp(−1/2) approximately is−1.756.As a result, the single-antenna region can be written asρ = 2R−1

1.258.

Proof. Let us split the proof in three parts according to thethree statements.

The first statement is proved by Theorem 1 for real val-ued Gaussian random variables. There, it is shown that theminimum of the quadratic form is achieved for all regions[x(d), x(d − 1)] by a chi-square distribution withd degreesof freedom. In our case, the rangex(d) is parameterisedby the SNRρ and we have to deal with complex Gaussiandistributed random variables. Therefore, we have to reviewTheorem 1 and extend the steps of its proof.

We start with Lemma 2 and write the outage probabilityas a function of the power allocation andz containing therateR and SNRρ as

Pout(p, z) = 1

2π

∫ ∞

−∞1

jω

nT∏k=1

1

1 + jpkωejωzdω + 1

2

(19)

Fix the power allocationp3, p4, . . . , pn and take the deriva-tive of Pout in Equation (19) with respect top1 keeping thesump1 + p2 = c constant, that isp2 = c − p1,

∂Pout(p, z)

∂p1= (p1 − p2)

1

2π

∫ ∞

−∞jω

·nT∏

k=1

1

1 + jpkω

1

1 + jp1ω

1

1 + jp2ωexp(jωz)dω

(20)

Copyright© 2006 AEIT Euro. Trans. Telecomms. (in press)

OUTAGE PROBABILITY IN MULTIPLE ANTENNA SYSTEMS

Invert the Fourier transform to get

∂Pout(p, z)

∂p1= (p1 − p2)

· ∂2

∂z2Pr[ nT∑

k=1

pksk + p1sn+1 + p2sn+2 ≤ z]

(21)

Therefore, possible minima ofPout are atp1 = 0, p2 = 0,p1 = p2 or if z = z is the mode of the pdf of

∑nTk=1 pksk +

p1sn+1 + p2sn+2. By the unimodality of the pdf fromTheorem 2 follows that the non-zeropk can take at mosttwo different values. Otherwise the expression in Equation(21) cannot be zero. Therefore, we have shown that

inf|p|≤1

Pout(p, z) =

inf0≤p≤1,K+M≤nT

Pr

[p

K

K∑k=1

sk + 1 − p

M

M∑m=1

sK+m ≤ z

]

(22)

and it remains to show that fork ≥ m the infimum in Equa-tion (22) is achieved forp = 0,p = 1 orp = K/(K + M).In addition to these three points, another extremum is at themodez of the pdf of

p

K

K∑k=1

sk + 1 − p

M

M∑m=1

sK+m + p

KsK+M+1 + 1 − p

MsK+M+2

In order to show that at the mode ¯z can only be a localmaximum we perform two steps. First we prove that theextreme point atp = K/(K + M) is a minimum forz < 1and a maximum forz > 3/2. Second, we show that for allz > 1 the first derivative of the outage probability at thepoint p = 0 is greater than zero and at the pointp = 1 issmaller than zero. From this follows that the extremum atthe mode ¯z can only be a local maximum. It holds

∂2Pout(p)

∂p2

∣∣∣∣∣p= K

K+M

= K + M

2MKπ

∫ ∞

−∞jω

(1

1 + j ωK+M

)K+M+2

ejωzdω

= K + M

MK

∂2

∂z2Pr

[K+M+2∑

k=1

sk ≤ z(K + M)

]

= K + M

MK

∂2

∂z2

(1 − (K + M + 2, z(K + M))

(K + M + 2)

)

= (K + M)K+M+1ZK+M exp(−(K + M)z)

MK(K + M + 2)

· (1 + K + M − z(K + M)) (23)

The sign of Equation (23) is determined by 1+ K + M −z(K + M). The sign is nonnegative ifz ≤ 1 ≤ 1/K + M +1 and nonpositive forz ≥ 3/2 > 1 + 1/(K + M). Thisshows that atp = K/(K + M) is a local minimum forz ≤ 1. Further on, we have

∂Pout(p)

∂p

∣∣∣∣∣p=0

= 1

2πM

∫ ∞

−∞jω

(1

1 + j ωM

)M+1

exp(jωz)dω

= − 1

M

∂2

∂z2Pr

[M+1∑k=1

sk ≤ zM

]

= − 1

M

∂2

∂z2

(1 − (M + 1, zM)

(M + 1)

)

= MMzM−1 exp(−zM)

(M)(z − 1) ≥ 0 for all z ≥ 1

and it holds

∂Pout(p)

∂p

∣∣∣∣∣p=1

= 1

2πK

∫ ∞

−∞jω

(1

1 + j ωK

)K+1

exp(jωz)dω

= 1

K

∂2

∂z2Pr

[K+1∑k=1

sk ≤ zK

]

= 1

K

∂2

∂z2

(1 − (K + 1, zK)

(K + 1)

)

= −KKzK−1 exp(−zK)

(K)(z − 1) ≤ 0 for all z ≥ 1

The ordering of the SNR valuesρ1 < ρ2 < · · · < ρnT−1follows from Corollary 1. This completes the first part ofthe proof.

Copyright© 2006 AEIT Euro. Trans. Telecomms. (in press)

E. A. JORSWIECK AND H. BOCHE

In order to prove the optimality of the equal power allo-cation for high SNR, we show that the objective functionis Schur-convex with respect to the power allocation forall SNR valuesρ > ρ = 2R − 1. We prove this by directlyverifying Schur’s condition in Equation (7).

We concentrate in the following on the first term in Equa-tion (10). The first derivative of Equation (10) with respectto p1 is given by

∂Pout(p, z)

∂p1

= − 1

2π

∫ ∞

−∞

nT∏l=1

1

1 + pljω

1

1 + jωp1︸ ︷︷ ︸Gp1(ω)

ejωzdω

= −∫ z

0φ(τ)gp1(z − τ)dτ (24)

with gp1(τ) = (1/p1) e− τ

p1 and its Fourier transformGp1(ω). The last step follows by translating the productof Fourier transforms into a convolution of correspondingpdfs. Note that the partial derivative depends on the choiceof the parameterp1, . . . , pnT only by the functiongp1. Letus extend the parameter of the functiong to an arbitrarynumberξ, that isgξ and define the function

G(p, z, ξ) = −∫ z

0φ(τ)gξ(z − τ)dτ

= − 1

2π

∫ ∞

−∞1

1 + jωξ

nT∏l=1

1

1 + pljωe−jωzdω

which has the property that forξ = p1 andξ = p2 it holds

G(p, z, p1) = ∂ (p, z)

∂p1andG(p, z, p2) = ∂ (p, z)

∂p2

Since p1 ≥ p2 we have the following implications:If G(p, z, ξ) is monotonic increasing withξ, then∂ (p, z)/∂p1 ≥ ∂ (p, z)/∂p2, and if G(p, z, ξ) ismonotonic decreasing withξ, then ∂ (p, z)/∂p1 ≤∂ (p, z)/∂p2. In this way, Schur’s condition is verified forboth cases. Since the low SNR part is already proved, weconsider the high SNR case. The functionG(p, z, ξ) canbe written as

G(p, z, ξ) = −1

ξexp

(−z

ξ

)∫ z

0exp

(τ

ξ

)φ(τ)dτ (25)

By Lemma 3 in Appendix A, for z ∈ (0, 1), thefunction G in Equation (25) is monotonic increas-ing with ξ. As a result, Schur’s condition is veri-fied and the function is Schur-concae with respectto p. Therefore, the minimal outage probability isachieved with equal power allocation across all availableantennas.

Finally, we prove the last part of the theorem, namely thatfor all ρ ≤ (2 − 2R+1)/2Lw(−1, −1/2 exp(−1/2)) + 1,the outage probability is minimised by one active antenna,that isp1 = 1 andp2 = p3 = · · · = pnT = 0. This followsfrom the zero of the difference between the single-antennaand two-antenna power allocation. We evaluate the outageprobability expression for one and two transmit antennas toobtain

g1(x) = Pr[1

2(s1 + s2) ≤ x

]− Pr

[s1 ≤ x

]= e−x − (1 + 2x) e−2x

The zeros of the functiong1(x0) = 0 are (0, −12 −

Lw(−1, −1/2 exp(−1/2))). �

Remark. In contrast to the ergodic capacity, the outageprobability does not have a fixed transmit strategy which isoptimal for all SNR values. The ergodic capacity is max-imised by equal power allocation [8]; the optimal transmitstrategy for outage probability minimization is to select anumber ofd out of nT arbitrary antennas as a function ofthe SNR and perform equal power allocation across them.For small SNR valuesd = 1 and one antenna is optimal.The higher the SNR values, the more antennas are selected.

Note further, that the single-antenna region does not di-rectly follow from Corollary 1. Indeed, there is a SNR rangein which the outage probability is not Schur-concave, butfor which single-antenna transmission is optimum.

The outage probability in Equation (4) can be rewrittenwith the additional parameter 2d (d is the number of ac-tive antennas times two for complex random variables perantenna) and with equal power allocation as

Pout(ρ, R, d) = Pr

(d∑

k=1

sk < d(2R − 1)

ρ

)

=d

(2R−1)ρ∫

0

χ22d(x)dx (26)

Copyright© 2006 AEIT Euro. Trans. Telecomms. (in press)

OUTAGE PROBABILITY IN MULTIPLE ANTENNA SYSTEMS

The pdf of the sum of independent identically dis-tributed standard exponential random variablessk is aχ2-distribution with 2d degrees of freedom, that isχ2

2d(x) =xd−1 exp(−x)/(d). Hence, we obtain for the integral inEquation (26)

Pout(ρ, R, d) =d

(2R−1)ρ∫

0

xd−1 exp(−x)

(d)dx

= P

(d

2R − 1

ρ, d

)(27)

with the incomplete Gamma functionP(a, x) defined inReference [23, 6.5.1]. Either, the rateR and SNRρ fulfilR ≥ log(1+ ρ) or R ≤ log(1+ ρ), then we immediatelyknow how to allocate powerp, or the functionPout(ρ, R, d)in Equation (27) has to be evaluated for all possible1 ≤ d ≤ nT.

3.2. Impact of correlation on the outage probabilitywith uninformed transmitter

In this section, we assume that the transmitter does neitherknow the instantaneous channel realisation nor the channelcorrelation. Therefore, it performs equal power allocationacross all available antennas. We are going to characterisethe impact of the channel correlation on the outage proba-bility.

In order to verify Schur’s condition in Equations (7) and(8), respectively, we compute the first derivative of the out-age probability with respect toλ1 andλ2 for fixed p1 =p2 = · · · = pnT = 1

nT. But this directly corresponds to the

proof of Theorem 3 in which onlyp is swapped forλ. As aresult, we have that the outage probability is Schur-convexfor C(ρ, R) = x < 1 and Schur-concave forC(ρ, R) = x >

2. This is formulated in the following theorem.

Theorem 4. For a MISO system which applies equalpower allocation and for fixed transmission rate R, the out-age probability as a function of the correlation propertiesof the transmit antennas is characterised by the followingstatements:�

for SNR ρ < ρ = 2R−12 , the outage probability is a

Schur-concave function of the channel covariance ma-trix eigenvalues λ1, . . . , λnT , that is correlation de-creases the outage probability and

λ1 � λ2 =⇒Pout(ρ < ρ, R, λ1, 1) ≤ Pout(ρ < ρ, R, λ2, 1)

�

for SNR ρ > ρ = 2R − 1, the outage probability is aSchur-convex function of the channel covariance matrixeigenvalues λ1, . . . , λnT , that is correlation increases theoutage probability and

λ1 � λ2 =⇒Pout(ρ < ρ, R, λ1, 1) ≥ Pout(ρ < ρ, R, λ2, 1)

Remark.

1. For SNR values betweenρ andρ the outage probability isneither Schur-convex nor Schur-concave. The behaviourof the ergodic capacity in MISO systems differs fromthe behaviour of the outage probability since the ergodiccapacity is for no CSI at the transmitter Schur-concavefor all SNR values [14] and the outage probability iseither Schur-convex or Schur-concave for high and smallSNR values.

2. Theorem 3 could lead to the conjecture that for all in-tervals (ρl, ρl+1) the picture of the impact of correlationis clear, that is that in the intervall, the outage proba-bility is Schur-convex with respect to power allocationvectorsp which lie in R

l+. This conjecture is mislead-ing, because the outage probability has its minima at theedge of the interval but the maximum in between, that isthere is no monotony with respect to the partial order ofmajorisation.

3. Another difference between the results from Theorems 3and 4 is that the single-antenna region in Theorem 3 givenby ρ lies above the Schur-concave region in Theorem 4given byρ. This is due to the fact that Schur-convexityof a functionf (x) implies that the maximum is attainedfor x = [1, 0, . . . , 0], but the converse is not true.

3.3. Power allocation with covariance knowledge

For the case, in which the channel covariance matrix is notthe identity matrix, and the transmitter knows the channelcovariance matrix, the power control strategy is an openresearch problem. The results from Theorem do not applybecause the transmitter knows the channel covariancematrix and can adapt its transmit strategy to the channelstatistics. This scenario is difficult, because the objectivefunction is not convex with respect to the power allocation.As a result, the Karush–Kuhn–Tucker (KKT) optimalityconditions are only necessary and not sufficient, that is onepower allocation which fulfils the KKT optimality condi-tions is local optimum but not global. However, an efficientalgorithm which finds a local optimum can be developed.

Copyright© 2006 AEIT Euro. Trans. Telecomms. (in press)

E. A. JORSWIECK AND H. BOCHE

We consider the following programming problem

p∗ = arg minpk≥0, 1≤k≤nT∑nT

k=1 pk=1

Pout(ρ, R, λT , p) (28)

The Lagrangian for optimisation problem Equation (28)is given with the representation in Equation (10) by

L(p, φ, γ) = 1

2π

∫ ∞

−∞1

jω

nT∏k=1

1

1 + jλTk pkω

exp(jωz)

+ 1

2+ γ

(1 −

nT∑k=1

pk

)+

nT∑k=1

φkpk (29)

The first derivative of Equation (29) with respect topl isgiven by

∂L(p, φ, γ)

∂pk

=

−λTk

2π

∫ ∞

−∞

nT∏l=1

1

1 + λTl pljω

ejωz

1 + λTk pkjω

dω

−γ + φk (30)

The necessary KKT optimality conditions are given by∂L(p, φ, γ)/∂pk = 0, pk ≥ 0, φk ≥ 0 for all 1 ≤ k ≤ nTandγ ≥ 0, 1−∑nT

k=1 pk ≥ 0. The Lagrangian multiplierφk for all activek with pk > 0 is zero. Therefore, we havefrom Equation (30) withαk (see below) for all activek, lαk = γ = αl and for allpm = 0, αm = γ − φm < γ = αk.

The (locally) optimal power allocationp∗ for all ac-tive antennas withp∗

k > 0 andp∗l > 0, fulfil the following

equality

p∗k > 0 and p∗

l > 0 =⇒ αk = αl (31)

with

αk = 1

2π

nT∏

j=1j =k

λTk p∗

k

λTk p∗

k − λTj p∗

j

e

− z

λTk

p∗k z

λTk p∗2

k

+ 1

2πλT

k ×nT∑i=1i =k

nT∏

j=1j =k

λTi p∗

i

λTi p∗

i − λTj p∗

j

e

− z

λTk

p∗k − e

− z

λTi

p∗i

λTk p∗

k − λTi p∗

i

Furthermore, all inactive antennasp∗r = 0 have smallerαr,

that is

p∗k > 0 and p∗

l = 0 =⇒ αk > αl (32)

This characterisation can be used to derive an efficientalgorithm which computes the locally optimum power al-location. In nT steps, the algorithms starts with beam-forming, that is all power is allocated to the strongestl = arg maxk=1,...,nT λk and pl = 1, pk = 0 for all k = l.Then the KKT conditions are verified. If they are satisfiedby the power allocation, a locally optimum power alloca-tion has been found. If not, the second (or next) directionis supported and the power allocation that fulfils the con-dition in Equation (31) is found (by a zero search). This isrepeated until the KKT conditions are jointly satisfied. Thisalgorithm is similar to the Algorithm 1 proposed in Refer-ence [8] for ergodic capacity maximization. Unfortunately,this gives only the local optimum and nothing can be saidabout the globally optimum yet.

4. NUMERICAL SIMULATIONS

4.1. Outage probability over rate R

In Figure (1), we show the outage probability overtransmission rateR for different number of transmitantennasnT and correlationλ at SNR of 10 dB. Alsoshown are the bounds for which the outage probability is

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1MISO n × 1 @ 10 dB

Rate R [bit/s]

Ou

tag

e P

rob

abili

ty

n=2, λT=[0.6, 0.4]

n=2, λT=[0.9, 0.1]

n=3, λT=[0.4, 0.35, 0.25]

n=3, λT=[0.7, 0.2, 0.1]

R = log(1 + ρ)

R = log(1 + 2 ρ)

Figure 1. Outage probability over transmission rateR for differentnumber of transmit antennasn and correlationλT at SNR of 10 dB.

Copyright© 2006 AEIT Euro. Trans. Telecomms. (in press)

OUTAGE PROBABILITY IN MULTIPLE ANTENNA SYSTEMS

Schur-convex and Schur-concave, respectively. The inter-section point between the different outage probabilities liesalways between these two bounds, that is in this examplebetween

R = log(1+ ρ) ≈ 3.4594 [bit/s]

and

R = log(1+ 2 × ρ) ≈ 4.3923 [bit/s]

The outage probability in Figure (2) was computedusing the alternative representation that is derived bya decomposition of product into partial fractions given

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 110

0

101

102

Rate R=1 [bit/s]

SNR ρ

Nu

mb

er o

f ac

tive

an

ten

nas

d

2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 310

0

101

102

Rate R=2 [bit/s]

SNR ρ

Nu

mb

er o

f ac

tive

an

ten

nas

d

Figure 2. Number of active antennas over linear SNRρ fortransmission rateR = {1, 2} [bit/s].

as

Pout(ρ, R, λT , p) =

1

2π

∞∫−∞

1

jω

nT∏l=1

1

1 + plλTl jω

ejωzdω + 1

2

=nT∑l=1

nT∏

j=1j =l

plλTl

plλTl − pjλ

Tj

(

1 − e− 2R−1

ρplλTl

)(33)

4.2. Optimal number of active antennas over SNR ρ

In Figure (2), the question about the SNR range betweenρ

andρ is answered. The points at which the number of activeantennas‡ is increased accumulate on the right borderρ. Thecloser you get to the right border the more active antennasyou have. The upper SNR bounds in Figure (2) are

ρR=1 = 1 ρR=2 = 3.

This fits the results in Figure (2) well. The lower SNR rangein which single-antenna processing is optimal is given by

ρR=1 = 0.7957 ρR=2 = 2.3874.

5. OUTAGE PROBABILITY IN MIMO SYSTEMS

Let us consider the case in which the transmitter is unin-formed and the channel is spatially correlated. Then theoutage probability in MIMO systems depends on the SNRρ, the transmission rateR, the transmitRT and receiveRR

correlation, and the transmit strategyQ is given in Equation(5). Since the transmitter is uninformed about the instanta-neous channel realisations as well as the channel statistics,it performs equal power allocation.

In order to analyse the functionf it is necessary to knowthe probability density function of the determinant in theRHS of Equation (5). Unfortunately, even for the simplestcase in whichRT = I andRR = I, there is no closed formexpression for this pdf which can be used for further an-alytical optimisation. The matrixWWH is called Wishartmatrix [24, 25]. If the dimension of the Wishart grows with-out bound, but the ratio of rows and columns stays fixed, the

‡ The number of active antennas is only of theoretical interest in this figure,because 100 transmit antennas are seldom to find in reality.

Copyright© 2006 AEIT Euro. Trans. Telecomms. (in press)

E. A. JORSWIECK AND H. BOCHE

empirical distribution of the eigenvalues of the Wishart ma-trix converges to a deterministic distribution [26, 27]. Thisimportant result has been used frequently in the literature,for example in the context of CDMA systems [28, 29] andin the context of MIMO systems [30–33]. In Reference [7],the asymptotic eigenvalue distribution for correlated MIMOchannels was analysed using the Stieltjes transform.

There are approaches in which the pdf of the determinantin Equation (5) is bounded, for example [34] fornT > nR

nT∑k=nT−nR+1

log(1+ ρχ22k) < log det

(I + ρWWH

)

<

nT∑k=1

log(1+ ρχ22nR,k) (34)

with χ22k are independent chi-square distributed random

variables with 2k degrees of freedom andχ2nR,k are inde-pendent chi-square variates each with 2nR degrees of free-dom. The inequality in Equation (34) is in probability.§

In Reference [35], asymptotic results were derived forconvergence in distribution of the instantaneous mutual in-formation in uncorrelated MIMO channels for four cases:largenT and fixednR, fixednT and largenR, largenT andlarge nR with low or high SNR. Always the mutual in-formation converges in distribution to a Gaussian distribu-tion. This convergence is shown to be very fast. Even withtwo transmit and receive antennas, the Gaussian approxi-mation with the same mean and variance as the mutual in-formation fits the real outage probability well. The Gaussianapproximation of the mutual information is also studied inReference [36] where it is shown that the pdf of the mu-tual information can be well approximated by a Gaussiandistribution. The mean and variance of the Gaussian ap-proximation are computed in Reference [36].

The results from the outage probability analysis for MISOsystems from the last section carry over to the high and lowSNR MIMO analysis. Assume the transmitter is uninformedand performs equal power allocation. Fix the transmissionrateR.

1. For small SNR values, the outage probability is Schur-concave with respect to transmit correlation.

2. For high SNR values, the outage probability is Schur-convex with respect to transmit correlation.

Remark. By symmetry, these statements hold with respectto receive correlation, too.

§ One says thatx < y in probability if Pr(x < t) < Pr(y < t).

To argue plastic, consider the instantaneous mutual in-formation as a function of transmit correlation with uncor-related receive antennas.

Pout(�T) = log det(I + ρ�TWHW

)(35)

Now, for SNR approaching zero, we can approximate (35)by

fρ→0(�T) ∼ log(1 + ρtr

(�TWHW

))∼ log

(1 + ρ

nT∑k=1

λTk χ2nR,k

)(36)

The expression in Equation (36) can be written as the sumof weighted standard exponential random variables by

log

(1 + ρ

nT∑k=1

λTk χ2nR,k

)= log

(1 + ρ

nT∑k=1

nR∑l=1

λTk sk,l

)

with iid standard exponentially distributedsk,l for 1 ≤ k ≤nT and 1≤ l ≤ nR. This expression corresponds to the out-age probability of a MISO system and the analysis in the lastsection has proved that the outage probability of a MISOsystem is Schur-concave at low SNR values.

For high SNR values, assume that�T has fullrank. Then we can approximatef (�T) = log

∏nTk=1(1 +

ρ eigk(�TWHW)) with eigk(A) denoting thek-th eigen-value ofA by

fρ→∞(�T) ∼ lognT∏

k=1

(ρ eigk

(�TWHW

))= log det

(�TρWHW

)= log det(�T) + log det

(ρWHW

)The summand log det(�T) depends on the transmit corre-lation. The function

∏nTk=1 λT

k is Schur-concave. Hence, theoutage capacity is Schur-concave, too, and the outage prob-ability is Schur-convex for high SNR values.

We conclude this section by some numerical illustrations.In Figure (3), the outage probability over transmissionrate R of a two times two MIMO system at 10 dB and−10 dB SNR is shown for different transmit correlations. InFigure (3), the predicted behaviour of the outage probabilityas a function of transmit and receive correlation for highSNR values can be observed. For high SNR values, transmitor receive correlation increases the outage probability for

Copyright© 2006 AEIT Euro. Trans. Telecomms. (in press)

OUTAGE PROBABILITY IN MULTIPLE ANTENNA SYSTEMS

1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1MIMO 2x2, SNR = 10 dB

R0

Pro

b[f

(ρ,I,

RT,R

T)<

R0]

RT = R

R = I

RT = [1.9 0.1], R

R = I

RT = R

R = [1.9 0.1]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1MIMO 2x2, SNR = −10 dB

Pro

b[f

(ρ,I,

RT,R

T)<

R0]

RT = R

R = I

RT = [1.9 0.1], R

R = I

RT = R

R = [1.9 0.1]

R0

Figure 3. Outage probability over transmission rateR for twotimes two MIMO system at−10 and 10 dB SNR for differenttransmit correlations and equal power allocation.

all transmission rates. In Figure (3), the behaviour of theoutage probability in dependence of transmit and receivecorrelation is shown at−10 dB. For lower rates, correlationincrease the outage probability, while for higher rates,transmit as well as receive correlation decrease the outageprobability and are therefore helpful. This correspondsto the result from the MISO analysis. In Figure (3), theupper and lower MISO boundsρ and ρ are shown forcomparison. In Figure (3) they are equal to

ρ = 0.4854 and ρ = 0.2630

From a practical point of view, the outage probability be-haves like the average mutual information, that is correlation

harms with uninformed transmitter. However, the completecharacterisation of the outage probability with transmit andreceive correlation for finite number of transmit and receiveantennas and for finite SNR larger than zero is still an openresearch problem.

6. CONCLUSION

We have studied the outage probability of MISO systems.In the first scenario, the channel is iid Rayleigh fading andwe ask about the optimal transmission strategy without CSIat the transmitter. In contrast to the equal power allocationwhich is optimal for the ergodic capacity, the optimalpower allocation turns out to use only a fraction of allavailable transmit antennas with equal power. The numberof active antennas increases with the SNR beginning fromone active antenna up to equal power allocation over allantennas. The SNR range in which one antenna is optimalis given byρ < (2R − 1)/2. The SNR range in which allantennas are active isρ > 2R − 1.

In the second scenario, we choose the equal powerallocation transmit strategy and study the impact ofspatial correlation on the outage probability. We showthat the outage probability for all SNR values smallerthan or equal toρ is Schur-concave, that is correla-tion reduces the outage probability. Furthermore, for allSNR values greater or equal toρ the outage probabil-ity is Schur-convex, that is correlation raises the outageprobability.

The results that were presented in Theorem 1 and The-orem 2 are only valid for the weighted sum of chi-square distributed random variables. It is not possible tocarry the results over to other positive distributed randomvariables.

The outage probability occurs also in the ‘good-put’ ormaximum throughput expression [37]

maxR≥0

R (1 − Pout(ρ, R, λ, p)) (37)

The maximum throughput in Equation (37) is defined asthe maximum of the transmission rateR times the proba-bility of successful transmission. The results for the out-age probability derived in the paper at hand cannot be di-rectly applied to the expression in Equation (37) due to themaximisation.

Finally, the covariance feedback case and the MIMO sce-nario are shortly discussed.

Copyright© 2006 AEIT Euro. Trans. Telecomms. (in press)

E. A. JORSWIECK AND H. BOCHE

APPENDIX A: A LEMMA AND ITS PROOF

Lemma 3. Let g a function with g(τ) ≥ 0 and∫∞0 g(τ)dτ = 1. Then the function

G(ξ, z) = 1

ξexp

(−z

ξ

) z∫0

exp

(τ

ξ

)g(τ)dτ (A.1)

for all z ∈ [0, 1] strict monotonic decreasing with respectto ξ.

Proof.

G(ξ, z)

∂ξ= − 1

ξ2exp

(−z

ξ

) z∫0

exp

(τ

ξ

)g(τ)dτ

+ z

ξ2exp

(−z

ξ

) z∫0

exp

(τ

ξ

)g(τ)dτ

− 1

ξ2exp

(−z

ξ

) z∫0

τ exp

(τ

ξ

)g(τ)dτ

(A.2)

Study the sign of

−z∫

0

exp

(τ

ξ

)g(τ)dτ + z

z∫0

exp

(τ

ξ

)g(τ)dτ

−z∫

0

τ exp

(τ

ξ

)g(τ)dτ

= (z − 1)

z∫

0

exp

(τ

ξ

)g(τ)dτ

−z∫

0

τ exp

(τ

ξ

)g(τ)dτ

and note that it is negative for allz ∈ [0, 1]. This completesthe proof. �

ACKNOWLEDGEMENT

The authors thank the reviewers for their detailed and insightfulcomments, which significantly enhanced the quality and readabil-ity of the paper. The second author presented the results in part

at University of California at Berkeley and wishes to thank Prof.David Tse for stimulating discussions. The Work of H. Boche issupported by the Deutsche Forschungsgemeinschaft (DFG).

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AUTHORS’ BIOGRAPHIES

Eduard A. Jorswieck did his Diplom-Ingenieur (MS) degree and received Doktor-Ingenieur (PhD), both in Electrical Engineeringand Computer Science from the Technische Universität Berlin, Germany, in 2000 and 2004, respectively. He has been with theFraunhofer Institute for Telecommunications, Heinrich-Hertz-Institut (HHI) Berlin, in the Broadband Mobile Communication NetworksDepartment since 2001. He joined the Department of Signals, Sensors and Systems at the Royal Institute of Technology (KTH) in 2006as post-doc. His research activities comprise performance and capacity analysis of wireless systems, optimal transmission strategiesfor single- and multi-user multiple antenna systems, and cross-layer optimization.

Holger Boche did his MSc and received his PhD in Electrical Engineering from the Technische Universität Dresden, Germany, in1990 and 1994, respectively. In 1992 he graduated in Mathematics from the Technische Universität Dresden. From 1994 to 1997 he didpostgraduate studies in mathematics at the Friedrich-Schiller Universität Jena, Germany. In 1997 Dr. Boche joined the Heinrich-Hertz-Institut (HHI) für Nachrichtentechnik Berlin. In 1998 he received his PhD in pure mathematics from the Technische Universität Berlin.He is the head of the Broadband Mobile Communication Networks Department at HHI. Since 2002 he is a full professor for MobileCommunication Networks at the Technische Universität Berlin at the Institute for Communications Systems. Since 2003 Dr. Boche isdirector of the Fraunhofer German-Sino Lab for Mobile Communications, Berlin, Germany. In October 2003 he received the ResearchAward ‘Technische Kommunikation’ from the Alcatel SEL Foundation. Dr. Boche was visiting professor at the ETH Zurich duringwinter term 2003/2004 and at KTH Stockholm during summer term 2005.

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33. Jorswieck EA, Wunder G, Jungnickel V, Haustein T. Inverse eigen-value statistics for rayleigh and ricean MIMO channels.IEE Seminaron MIMO: Communications Systems from Concept to Implementa-tions, pp. 3/1–3/6, December 2001.

34. Foschini GJ, Gans MJ. On limits of wireless communications in afading environment when using multiple antennas.Wireless PersonalCommunications 1998;6:311–335.

35. Hochwald BM, Marzetta TL, Tarokh V. Multi-antenna channel-hardening and its implications for rate feedback and scheduling.IEEETransaction on Information Theory 2004;50(9):1893–1909.

36. Whang Z, Giannakis GB. Outage mutual information of space-timemimo channels.IEEE Transaction on Information Theory 2004;50(4):657–662.

37. Jorswieck EA, Boche H, Sezgin A. Delay-limited capacity and maxi-mum throughput of spatially correlated multiple antenna systems un-der average and peak-power constraints.Proceedings of IEEE Infor-mation Theory Workshop, 2004.

Copyright© 2006 AEIT Euro. Trans. Telecomms. (in press)