IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 1, JANUARY 2003 317
Hotelling’s Generalized Distribution andPerformance of 2D-RAKE Receivers
Ming Kang, Student Member, IEEE,andMohamed-Slim Alouini, Member, IEEE
Abstract—Two-dimensional (2-D) RAKE (2D-RAKE) receivers processsignals in both time and space to improve the performance of code-divisionmultiple-access (CDMA) systems. In this correspondence, we establishthe connection between the 2D-RAKE signal-to-interference-ratio (SIR)output statistics and the generalized Hotelling’s (in the complex case)distribution. We study this distribution and offer new simple explicitclosed-form expressions (in terms of hypergeometric functions) for itsprobability density function (PDF) and cumulative distribution function(CDF). Based on that, we derive the outage probability of dual-branch2D-RAKE receivers over Rayleigh-fading channels. Some numericalexamples for particular cases of interests are plotted and discussed.
Two-dimensional (2-D) RAKE (2D-RAKE) receivers processsignals not only in time to take advantage of the multipath diversitybut also in space to reduce the level of cochannel interference [1]–[3].The significant capacity and coverage gain promised by these hybridreceivers, which are typically implemented in the form of an arrayof antenna elements each followed by a traditional RAKE receiver,make them one of the key tools for future generations of wirelesscode-division multiple-access (CDMA) systems. Unfortunately, thecapabilities of these receivers will be limited by both complexityand cost considerations. This is particularly true for small andlightweight hand-held/portable terminals for which the size of apractical adaptive array will typically be restricted to two antennaelements. In this correspondence, we focus on the performance ofdual-branch 2D-RAKE receivers operating in interference-limitedsystems and over a frequency-selective Rayleigh-fading channel witha flat power-delay profile.
Outage probability (which is related to the coverage area of a wire-less system) is one of the most important criteria for evaluating theperformance of wireless communication systems. In the context of in-terference-limited systems, this performance criterion is nothing elsebut the cumulative distribution function (CDF) of the output signal-to-interference ratio (SIR) evaluated at a predetermined threshold. Re-cently, in a statistical multivariate framework, Shah and Haimovich[4] showed that the statistics of the output SIR of adaptive arrays withoptimum combining in a flat Rayleigh-fading environment is relatedto the Hotelling’sT 2
0 distribution. They then capitalized on that resultto derive a closed-form expression for the outage probability of theseadaptive arrays. In this correspondence, we show that when each an-tenna element of these arrays is followed by a traditional RAKE re-ceiver (to constitute a spatio–temporal 2D-RAKE receiver) then theoutput SIR follows a generalized Hotelling’sT 2
0 distribution. Because
Manuscript received April 10, 2001; revised April 9, 2002. This work wassupported in part by a Grant-in-Aid of Research from the Office of the VicePresident for Research and Dean of the Graduate School of the University ofMinnesota and in part by the National Science Foundation under Grant CCR-9983462.
The authors are with the Department of Electrical and Computer Engi-neering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail:[email protected]; [email protected]).
Communicated by D. N. C. Tse, Associate Editor for Communications.Digital Object Identifier 10.1109/TIT.2002.806151
of the contribution of this statistic to the wartime multivariate qualitycontrol problems, a great deal of theoretical investigations were carriedout on it since then (e.g., [5]–[14]). However, to the best of our knowl-edge, the particular form that arises in the context of 2D-RAKE recep-tion has not been investigated yet. In this correspondence, we study thisparticular form of interest and offer new simple explicit closed-formexpressions (in terms of hypergeometric functions) for the probabilitydensity function (PDF) and CDF (and, therefore, the outage proba-bility) of the SIR output. Some numerical examples for particular casesof interest are also plotted and discussed.
The remainder of this correspondence is organized as follows. Thenext section states the problem and then derives the new PDF andCDF results of the generalized Hotelling’sT 2
0 (in the complex case).Section III establishes the connection between the 2D-RAKE SIRoutput statistics and the generalized Hotelling’sT 2
0 (in the complexcase) distribution then offers some numerical examples validatingthe analysis and showing the effect of various system parameterson the performance of dual-branch 2D-RAKE receivers. Section IVconcludes the correspondence.
II. HOTELLING’S GENERALIZED DISTRIBUTION
A. Problem Statement
Let XXXp�m andYYY p�n (p � n) be independent matrices whosecolumns are independent and identically distributed (i.i.d.)p-variateGaussian vectors, i.e.,XXX; YYY � N (0; �), thenSSS1 = XXXXXX
TTT whenp � m andSSS0 = YYY YYY
TTT , where[ � ]T denotes the transpose oper-ator, are Wishart distributed matrices, i.e.,SSS0 � Wp(n; �) andSSS1 �Wp(m; �) [15]. Lawley [16] and Hotelling [17], [18] introduced thestatistic
u = trace(SSS�10 SSS1) :=T 2
0
n:
It can be easily shown thatu is the sum of the roots of the characteristicequation
jSSS�10 SSS1 � �IIIj = 0 (1)
or equivalently
jSSS1 � �SSS0j = 0 (2)
where in (1) and (2),j � j denotes the determinant operator. Whenp = 1,m andn = 1; 2 . . ., andXXX andYYY arereal (complex) Gaussian vari-ates,u is obviously proportional toF distributed random variables withm andn (2m and2n) degrees of freedom. On the other hand, whenm = 1, p = 1; 2; . . . ; n � p, andXXX andYYY are real (complex)Gaussian variates,u equalsXXXH(YYY YYY H)�1XXX, which is in this case ascalar, and is the well-known Hotelling’sT 2
0 statistic [19] whose PDFcan be obtained as a special case of [20, eq. (72)] for the real caseand [20, eq. (105)] for the complex case. The complex case of theHotelling’s T 2
0 distribution is the one used by Shah and Haimovich[4] to derive the statistics of the output SIR of adaptive arrays with op-timum combining in a flat Rayleigh-fading environment. Note that inabove cases,SSS�11 SSS0 is of rank1 (with probability1). Whenp = 2,m � 2, andXXX andYYY arereal Gaussian matrix variates,u is knownas the Hotelling’s generalizedT 2
318 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 1, JANUARY 2003
where Bx( � ; � ) is the incomplete beta function [21, Sec. 6.6]andU(�) is the unit step function.. Using the relation between theincomplete beta function and the Gaussian hypergeometric function2F1( : ; : ; : ; : ) [22, eq. (8.391)], as well as the identity [22, eq.(9.131)], (3) can also be written in the following equivalent form [5,p. 223]:
fu(u) =�(m+ n� 1)
2�(m)�(n� 1)
(u=2)m�1
(1 + u=2)m+n
� 2F1 1;n+m
2;m+ 1
2;
u2
(2 + u)2U(u) (4)
which corrects a minor typo (factor of1=2 missing) in [5], as pointedout by Phillips in [14].
B. Hotelling’s GeneralizedT 20 Distribution: Complex Case
To the best of our knowledge, the statistics ofu whenp = 2,m � 2,andXXX , YYY arecomplexGaussian matrix variates has not been inves-tigated so far (even in the specialized statistics literature [5]–[14]).However, as we will see later, this is the statistics needed to derive theperformance of dual-branch 2D-RAKE receivers over Rayleigh-fadingchannels under consideration in Section III. In the following section,we study this special case and offer new simple explicit closed-formexpressions for the PDF and CDF ofu corresponding to this scenario.
Theorem 1: Let XXX be a2 � m (m � 2) matrix whose columnsare i.i.d. bi-variate complex Gaussian vectors with zero mean and co-variance matrix�, i.e.,XXX � CN (0; �). LetYYY be a2 � n (n � 2)matrix independent ofXXX whose columns are i.i.d. bi-variate complexGaussian distribution with zero mean and covariance matrix�, i.e.,YYY � CN (0; �), then
u =trace XXXXXXH(YYY YYY H)�1
=trace XXXH(YYY YYY H)�1XXX � 0
is distributed as
fu(u) =
p� �(n+m)�(n+m� 1)
�(n)�(n� 1)�(m)� m+ 1
2
� (u=2)2m�1
(1 + u=2)2m+2n
� 2F13
2; m+ n;
2m+ 1
2;
u2
(2 + u)2U(u): (5)
It is noted that, contrary to the cases whenm = 1 orp = 1, the newlyderived complex Hotelling’s generalizedT 2
0 distribution (5) cannot beobtained by simply scalingn andm by 2 in the real Hotelling’s gen-eralizedT 2
0 distribution given in (4).Proof: See Appendixes A and B.
Corollary 1: The CDF of the Hotelling’s generalizedT 20 distribu-
tion in the complex case is given by
Fu(uth) =u
0
fu(u)du
=
p� �(n+m)�(n+m� 1)
�(n)�(n� 1)�(m)� m+ 1
2
�2n�1
k=0
22n�k�(2n)�(2m)
�(2n� k)�(2m+ k + 1)
u2m+kth
(2 + uth)2m+2n�1
� 3F23
2; m+ n; m;
2m+ k + 1
2;2m+ k + 2
2;
u2th(uth + 2)2
(6)
Fig. 1. Block diagram of a dual-branch (two antennas) 2D-RAKE receiverwith -finger RAKES.
where3F2( � ; � ; � ; � ; � ; � ) is the generalized hypergeometric functionpFq(a1; . . . ; ap; b1; . . . ; bq; z)with the parametersp = 3 andq = 2[22, Sec. 9.14].
Proof: See Appendix C.
III. PERFORMANCE OF2� L 2D-RAKE RECEIVERS
A. System and Channel Model
We consider a direct-sequence (DS) CDMA system in whichNI+1
mobile units are equipped with dual-branch 2D-RAKE devices con-sisting of an array of two antennas, each followed by anL-finger RAKEreceiver, as shown in Fig. 1. The transmitted signal of theith user canbe modeled as
si(t) =p2P ci(t� �i)bi(t� �i) exp(j�i) (7)
whereP , ci(t), bi(t), �i, and�i denote the transmitted power, binaryspreading sequence, data waveform, phase shift, and delay of theithuser, respectively. We assume that�i are i.i.d. uniformly distributedover(0; 2�) and�i are i.i.d. uniformly distributed over(0; Tb), whereTb denotes the bit duration. Without loss of generality, the phase shiftand the delay of the desired user are assumed to be zero.
The equivalent impulse response from the base station to theith user2D-RAKE receiver over a frequency-selective multipath channel maybe modeled as
hhhi(t) =
L
l=1
aaali � �(t� l � Tc) (8)
whereLp (Lp � L) is the number of multipath,aaali = [ali; 1 ali; 2]T
denote the channel complex gain vector for the two antennas of theith user over thelth path andTc is the chip duration. We assume thatall users are subject to frequency-selective Rayleigh fading with a flatpower-delay profile and independent multipaths. As such, the channelcomplex gain vectors for the desired user andNI interferers are allassumed to be i.i.d. complex Gaussian bivariates with zero mean andnormalized covariance matrix�, i.e.,aaali � CN (0; �). We also as-sume that the system is interference limited and thus the thermal noise
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 1, JANUARY 2003 319
can be neglected. Hence, the low-pass equivalent received signal at theinput of the desired user’s 2D-RAKE receiver is given by
rrr(t) =
L
l=1
p2D aaalD � cD(t� �D � l � Tc)
� bD(t� �D � l � Tc) � exp(j�D)
+
N
i=1
L
l=1
p2I aaali � ci(t� �i � l � Tc)
� bi(t� �i � l � Tc) � exp(j�i) (9)
whereaaalD (with unit variance/power) is the channel complex gainvector for the desired user over thelth path,aaali (with unit vari-ance/power) is the channel complex gain vector for theith interfererover thelth path,D is the average received fading power of thedesired user (assumed to be equal for theLp multipaths), andI isthe average received fading power of the interferers (assumed to beequal for theLp multipaths of theNI interferers).
The weights of the 2D-RAKE receiverWl; i, (l = 1; . . .L,i = 1; 2) are set adaptively to maximize the output SIR. The optimalweighting vectorWWW l = [Wl; 1; Wl; 2]
T for thelth path is known to begiven by [3, eq. (12)]
WWW l = RRR�1aaalD (10)
whereRRR is given by [3]
RRR =NILpI
3ND� (11)
with N denoting the processing gain. When the weights are set as per(10), the maximum output SIR� is achieved and is given by [3, eq.(14)]
� =
L
l=1
aaaHlD �RRR�1 � aaalD : (12)
We assume that the covariance matrix� is estimated by the samplecovariance matrix̂� given by
�̂ =1
Lp �NI
L
i=1
N
j=1
aaaijaaaHij (13)
where[ � ]H denotes the conjugate transpose operator. Thus, (11) canbe rewritten as
RRR =I
3ND
L
i=1
N
j=1
aaaijaaaHij : (14)
Therefore, the maximum output SIR in (12) can be expressed as
� =3ND
I
L
l=1
aaaHlD �L
i=1
N
j=1
aaaijaaaHij
�1
� aaalD = � � � (15)
where� = 3N
, � is given by
� = trace aaaHD(aaaIaaaHI )�1aaaD (16)
and the2� L (we only combineL � Lp out ofLp paths) matrixaaaDand2 � LpNI matrixaaaI are given by
SinceaaaI andaaaD are i.i.d. complex Gaussian bivariates with zeromean, we see from Theorem 1 that the normalized SIR� as given in(16) is a Hotelling’s generalizedT 2
0 random variable with parametersn = LpNI andm = L. Hence, using (5) and the transformation ofrandom variables� = ���, the PDF of the output SIR� can be deducedas
f�(�) =
p� �2L N �(LpNI + L)�(LpNI + L� 1)
�(LpNI)�(LpNI � 1)�(L)� L+ 1
2
� 2F13
2; LpNI + L;
2L+ 1
2;
�2
(2� + �)2
� (�=2)2L�1
(� + �=2)2L+2L NU(�): (19)
Fig. 2 shows the effect of the number of combined diversity multipathsL on the PDF of�, whenNI = 5,D = 5 I ,N = 15, andLp = 10.As expected, these curves indicate that the output SIR� tends to takeon larger values (which implies better performance) whenL increases.Correspondingly, Fig. 3 shows that the output SIR� has a better chanceof taking on larger values as the number of interferersNI decreases.
C. Outage Probability
The outage probability is an important statistical measure to assessthe quality of service provided by the system. It is defined as the prob-ability of failing to achieve a specified SIR value�th sufficient for sat-isfactory reception. Mathematically speaking, the outage probabilityPout is defined by
Pout = P[� < �th] = P � <�th�
=� =�
0
f�(�)d�: (20)
Therefore, the outage probability is simply the CDF of� evaluated at� = �th=�. Therefore,Pout can be obtained by substitutingn =LpNI , m = L, anduth = �
�in Corollary 1. Fig. 4 plots the
outage probability versus the normalized “average” SIR �
withthe number of combined diversity multipathsL as a parameter whenNI = 6, Lp = 8, andN = 15. This figure shows the good matchbetween the outage probability obtained via the analytical formula andthe one obtained via Monte Carlo simulations. Furthermore, from thisfigure, we see that significant improvement in the outage probabilitycan be obtained by going fromL = 1 to L = 2 but diminishing re-turns are obtained asL increases.
IV. CONCLUSION
We extended the generalized Hotelling’sT 20 distribution to the com-
plex case and derived new simple explicit closed-form expressions forits PDF and CDF. We then established the connection between this dis-tribution and the statistics of the output SIR of optimized 2D-RAKE
320 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 1, JANUARY 2003
Fig. 2. Effect of the number of combined diversity multipathson the PDF of the output SIR when = 5, = 5 , = 15, and = 10.
Fig. 3. Effect of the number of interferers on the PDF of the output SIR when = 2, = 5 , = 15, and = 10.
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 1, JANUARY 2003 321
Fig. 4. Outage probability versus normalized threshold in decibels with the number of combined diversity multipathsas a parameter and = 6,= 8, and = 15.
receivers. Based on this, we obtained the outage probability of dual-branch 2D-RAKE receivers over Rayleigh-fading channels.
APPENDIX AJOINT PDFOF THE CHARACTERISTIC ROOTS
In this appendix, we present the joint PDF of the characteristic rootsof a certain quadratic form of complex Gaussian matrix variates whichwe use to derive the PDF of the Hotelling’s generalizedT 2
0 distributionin the complex case.
Lemma 1 (Khatri, 1965 [23]):Let XXXp�m andYYY p�n (p � n,p � m) be independent matrix variates whose columns are i.i.d.p-variate complex Gaussian vectors with zero mean and covariancematrix �, i.e., XXX; YYY � CN (0; �). Then the joint distributionf� ; ...;� (�1; . . . ; �p) of the characteristic roots of thep� p matrix(YYY YYY HHH)�1XXXXXX
HHH , 0 � �1 � �2 � � � � � �p < 1, is given by [23,eq. (7.1.3)]
f� ; ...;� (�1; . . . ; �p)
=
p
j=1
�(n+ q + j � 1)
�(n+ j � 1)�(p� j + 1)�(q� j + 1)
�p
j=1
�q�pj
(1 + �j)n+q�
p�1
k=1
p
i=k+1
(�k � �i)2
: (21)
Using Lemma 1 for the special case whenp = 2, we can write thejoint distribution of�1 and�2, f� ; � (�1; �2), as
f� ; � (�1; �2) =�(n+m)�(n+m� 1)
�(n)�(n� 1)�(m)�(m� 1)
� (�1�2)m�2(�1 � �2)
2
((1 + �1)(1 + �2))n+m: (22)
APPENDIX BDERIVATION OF THE PDFOF THE HOTELLING’S GENERALIZED T 2
0
DISTRIBUTION IN THE COMPLEX CASE
In this appendix, we prove Theorem 1 by deriving the PDF of theHotelling’s generalizedT 2
0 distribution in the complex case.Making the transformationu = �1 + �2 andv = �1�2, yields the
JacobianJ = �2 � �1. Based on the joint distribution of the charac-teristic roots given by (22), the joint distribution ofu andv is given by
fu; v(u; v) =�(n+m)�(n+m� 1)
�(n)�(n� 1)�(m)�(m� 1)
� vm�2
(1 + u+ v)n+m
pu2 � 4v: (23)
322 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 1, JANUARY 2003
Integratingfu; v(u; v) with respect tov over the interval[0; u
4] yields
the PDF ofu as
fu(u) =2�(n+m)�(n+m� 1)
�(n)�(n� 1)�(m)�(m� 1)
�0
vm�2(1 + u+ v)�(n+m) u2
4� v dv: (24)
Using the identity [22, eq. (3.197.8)], the integral in (24) can be evalu-ated in terms of the Gaussian hypergeometric function as
fu(u) =2�(n+m)�(n+m� 1)
�(n)�(n� 1)�(m)�(m� 1)
� 2F1 m+ n; m� 1;2m+ 1
2; � u2
4(1 + u)
�B3
2; m� 1
u
4
m�
(1 + u)m+nU(u): (25)
Using the identity [22, eq. (9.131.1)] and after simplification, (25) re-duces to (5), which completes the proof of Theorem 1.
APPENDIX CDERIVATION OF THE CDF OF THE HOTELLING’S GENERALIZED T 2
0
DISTRIBUTION IN THE COMPLEX CASE
In this appendix, we prove Corollary 1 by deriving the CDF of theHotelling’s generalizedT 2
0 distribution in the complex case. Based on(5), the CDF can be evaluated as
Fu(uth) =u
0
fu(u)du
=C �u
0
(u=2)2m�1
(1 + u=2)2m+2n
� 2F13
2; m+ n;
m+ 1
2;
u2
(2 + u)2du (26)
where the constant
C =
p� �(n+m)�(n+m� 1)
�(n)�(n� 1)�(m)�(m+ 12):
Expressing the Gaussian hypergeometric function in its series expan-sion [22, eq. (9.100)] and interchanging the order of integration and thesummation, (26) can be rewritten as
Fu(uth) = C �1
k=0
32 k
(m+ n)k
m+ 12 k
k!
1
22m+2k�1
�u
0
u2m+2k�1
(1 + u=2)2m+2n+2kdu
= C �1
k=0
32 k
(m+ n)k
m+ 12 k
k!
1
22m+2k�1
u2m+2kth
2m+ 2k
� 2F1 2m+2n+2k; 2m+2k; 2m+2k+1;�uth2
(27)
where
3
2k
=3
2
3
2+ 1 � � � 3
2+ k � 1 :
Using the identity [22, eq. (9.131.1)], we can express (27) as
Fu(uth) =C �1
k=0
32 k
(m+ n)k
m+ 12 k
k!
� 1
22m+2k�1
u2m+2kth
2m+ 2k
2
uth + 2
2m+2n+2k�1
� 2F1 1� 2n; 1; 2m+ 2k + 1;�uth2
: (28)
Since1�2n is a negative integer, the series expansion of the Gaussianhypergeometric function reduces to be a finite sum. Therefore, (28) canbe rewritten as
Fu(uth) =C �1
k=0
32 k
(m+ n)k
m+ 12 k
k!
� 1
22m+2k�1
u2m+2kth
2m+ 2k
2
uth + 2
2m+2n+2k�1
�2n�1
l=0
(1� 2n)l(2m+ 2k + 1)l
� (�uth=2)l: (29)
Interchanging the order of summations and after some manipulations,the CDF ofu can be evaluated in terms of the3F2( � ; � ; � ; � ; � ; � )function [22, Sec. 9.14] as
Fu(uth) =
p� �(n+m)�(n+m� 1)
�(n)�(n� 1)�(m)� m+ 12
�2n�1
k=0
22n�k�(2n)�(2m)
�(2n�k)�(2m+k+1)
u2m+kth
(2+uth)2m+2n�1
� 3F23
2; m+ n; m;
2m+ k + 1
2;2m+ k + 2
2;
u2th(uth + 2)2
(30)
which completes the proof of Corollary 1.
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Service Outage Based Power and Rate Allocation
Jianghong Luo, Student Member, IEEE,Lang Lin, Student Member, IEEE, Roy Yates, Member, IEEE, and
Predrag Spasojevic´, Member, IEEE
Abstract—This correspondence combines the concepts of ergodic ca-pacity and capacity versus outage for fading channels, and explores vari-able-rate transmissions under a service outage constraint in a block flat-fading channel model. A service outage occurs when the transmission rateis below a given basic rate. We solve the problem of maximizing the ex-pected rate subject to the average power constraint and the service outageprobability constraint. When the problem is feasible, the optimum powerpolicy is shown to be a combination of water filling and channel inversionallocation, where the outage occurs at a set of channel states below a certainthreshold. The service outage approach resolves the conflicting objectivesof high average rate and low outage probability.
Index Terms—Adaptive transmission, block-fading channel, ergodic ca-pacity, outage capacity, service outage.
Manuscript received June 5, 2001; revised June 21, 2002. The material in thiscorrespondence was presented in part at the Conference on Information Scienceand Systems 2001, Baltimore, MD, March 2001.
Communicated by D. N. C. Tse, Associate Editor for Communications.Digital Object Identifier 10.1109/TIT.2002.806150
I. INTRODUCTION
Wireless communication channels vary with time due to mobility ofusers and changes in the environment. For a time-varying channel, dy-namic allocation of resources such as power, rate, and bandwidth canyield improved performance over fixed allocation strategies. Indeed,adaptive techniques are employed in EDGE [1], GPRS [2], and HDR[3], and are proposed as standards for next-generation cellular systems.The system performance criterion is usually application specific; there-fore, different classes of applications will benefit from specific adaptivetransmission schemes. In order to differentiate real-time service fromnon-real-time service, three capacity measures have been defined inthe literature: ergodic capacity [4], delay-limited capacity [5], and ca-pacity versus outage probability [9]–[11]. A comprehensive survey ofthese concepts can be found in [6].
The ergodic capacity [4] was developed for non-real-time dataservices. It determines the maximum achievable rate averaged overall fading states. The corresponding optimum power allocation is thewell-known water filling allocation [7], [8]. The ergodic capacity maynot be relevant for real-time applications in a slow fading environment,where substantial delay can occur when averaging over all fadingstates. Delay-limited capacity [5] and the capacity versus outageprobability [9]–[11] were developed for constant-rate real-time appli-cations. The delay-limited capacity specifies the highest achievablerate with a decoding delay independent of fading correlation structures[5]. The outage capacity in the capacity versus outage probabilityproblem determines the�-achievable rate [12] of theM -block fadingchannel. The corresponding optimum power allocation was derivedin [10] for M parallel flat-fading blocks (frequency diversity or spacediversity), and in [11] forM consecutive flat-fading blocks (timediversity). The zero-outage capacity in [10], [11] is also referred to asthe delay-limited capacity.
Though the outage capacity studies the capacity for constant-ratereal-time applications, the constant-rate assumption may not be es-sential for many real-time applications. For applications with simul-taneous voice and data transmissions, for example, as soon as a basicratero for the voice service has been guaranteed, any excess rate canbe used to transmit data in a best effort fashion. For some video oraudio applications, the source rate can be adapted according to thefading channel conditions to provide multiple quality of service levels.Typically, these applications require a nonzero basic service ratero
to achieve a minimum acceptable service quality. Motivated by thesevariable-rate real-time applications, we study variable-rate transmis-sion schemes subject to a basic rate requirement in a slow-fading envi-ronment. By allowing variable-rate transmissions, the variation of thefading channel can be exploited to achieve an average rate higher thanthe outage capacity. By imposing a basic rate requirement, the systemcan be guaranteed to operate properly.
Since infinite average power is needed to achieve any nonzero rate atall times in a Rayleigh-fading channel, we impose a probabilistic basicservice rate requirement, that we call a service outage constraint. Theservice is said to be in an outage when the information rate is smallerthan the basic service ratero. Service quality is acceptable as long asthe probability of the service outage is less than�, a parameter indi-cating the outage tolerance of the application. Unlike the informationoutage in the capacity versus outage probability problem [10], [11], thebits transmitted during the service outage may still be valuable in thatthey will be transmitted reliably and will contribute to the average rate.
For variable-rate systems, the expected rate determines how muchrate can be transmitted on the average and is a meaningful measureof system performance. Therefore, in this correspondence, the alloca-tion problem is to find the power and rate allocation that maximize the
