998 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 2, FEBRUARY 2009

CorrespondencePower Control for MIMO Diversity Systems With

Nonidentical Rayleigh Fading

Le Cao, Student Member, IEEE, Meixia Tao, Member, IEEE, andPooi Yuen Kam, Senior Member, IEEE

Abstract—We investigate the performance limits and associatedpower-allocation problems in a multiple-antenna diversity system withstatistical channel information available at the transmitter. The channelsare assumed to be independent and nonidentically distributed Rayleighfading. By studying both ergodic mutual information and informationoutage, we derive two simple and near-optimal power-allocation schemes.Specifically, for ergodic information rate maximization in a system withmultiple transmit antennas and a single receive antenna, the proposedpower control only depends on the ratios between channel gains and isindependent of the total transmit signal-to-noise ratio (SNR). Meanwhile,for information outage minimization in a system with an arbitrary numberof transmit and receive antennas, the proposed power allocation follows thewater-filling principle and pours more power into the transmit antennawith a larger geometric mean of channel gains.

Index Terms—Multiple-input–multiple-output (MIMO), mutual infor-mation, nonidentical fading, outage probability, power allocation.

I. INTRODUCTION

Multiple-input–multiple-output (MIMO) technology offers signif-icant increases in data throughput and link reliability without ad-ditional bandwidth or transmit power in wireless communications.These advantages are well represented in two forms of gains from theinformation-theoretic perspective, namely, diversity and multiplexing.In particular, the diversity advantage is built upon the transmission ofthe same message over multiple independently faded spatial branches.It can be accomplished by using space–time block codes (STBCs)or other spreading codes together with appropriate combining at thereceiver. Tremendous work has been done on the design and analysisof space–time diversity techniques, such as [1]–[3] and referencestherein.

Recently, MIMO diversity schemes over nonidentical fading chan-nels have attracted great attention because of their applications incooperative communications and distributed antenna systems. Indecode-and-forward (DF) cooperative communications systems [4], atransmission from a source to a destination is facilitated with the helpof a set of relays. In the second stage, when those relays have knownthe transmitted signal, the subsequent transmission can be performedby employing space–time coding in a distributed manner, resulting ina nonidentical multiple-input–single-output (MISO) fading channel.In distributed antenna systems [5], multiple antennas that are distrib-uted at different radio ports and connected through coaxial cableswork together to simulcast signals. A nonidentical MIMO channelis actually formed, which enhances signal quality, increases system

Manuscript received November 1, 2007; revised March 19, 2008 andApril 27, 2008. First published June 6, 2008; current version publishedFebruary 17, 2009. The review of this paper was coordinated by Dr. A. Ghrayeb.

L. Cao and P. Y. Kam are with the Department of Electrical and ComputerEngineering, National University of Singapore, Singapore 117576 (e-mail:caole@nus.edu.sg; elekampy@nus.edu.sg).

M. Tao is with the Department of Electronic Engineering, Shanghai JiaoTong University, Shanghai 200240, China (e-mail: mxtao@ieee.org; mxtao@sjtu.edu.cn).

Digital Object Identifier 10.1109/TVT.2008.927040

capacity, and improves coverage. In the aforementioned cases, thechannels on different transmit–receive antenna pairs can be modeled asindependent but not necessarily identically distributed (i.n.d.) fading.

Previous studies on the effects of i.n.d. fading for MIMO diversitysystems have focused on the bit error analysis. The bit error perfor-mances over i.n.d. Rayleigh/Ricean and Nakagami fading channelsusing STBCs are studied in [6] and [7], respectively. In [8], the biterror performance over semiidentical Rayleigh fading channels usingdifferential STBCs with noncoherent receivers is analyzed. In thispaper, we are interested in the performance limits and associatedpower-allocation problems in a MIMO diversity system given thenonidentical fading statistics available at the transmitter. It is wellknown that equal power allocation (EPA) is optimal for traditionalMIMO channels with identical fading distribution [9] (assuming noinstantaneous channel state information at the transmitter). However,it is no longer optimal for nonidentical MIMO channels. In [10],the mutual information outage of a transmit diversity system with asingle receiver over nonidentical Rayleigh fading channels is studied.Therein, a heuristic power control scheme, named EPA with channelselection, is proposed and is shown to be near optimal in minimizingthe outage probability. For ergodic mutual information maximizationin a MIMO diversity system, to the best of our knowledge, no closed-form power control is available in the literature.

In this paper, we consider the power control for both ergodic mutualinformation maximization and information outage minimization inMIMO diversity systems over nonidentical Rayleigh fading channels.We first derive explicit and closed-form expressions for ergodic mutualinformation and information outage probability at any given powerallocation and with any number of transmit and receive antennas. Asuboptimal power allocation in simple and analytical form is thenproposed for ergodic mutual information maximization when there isonly one receive antenna (i.e., MISO). Numerical results show that itis near-capacity achieving. For outage minimization, a water-filling-based power allocation is proposed. The derivation is based on theChernoff bound and is different from [10]. Moreover, it generalizesthe case with multiple receive antennas.

II. SYSTEM MODEL

Consider a narrow-band system with Nt > 1 transmit antennas andNr ≥ 1 receive antenna(s). The channel is Rayleigh fading with addi-tive white Gaussian noise of power spectral density N0. The entity hij

of channel matrix H is the channel coefficient between the ith receiveantenna and the jth transmit antenna, and {hij}Nr,Nt

i=1, j=1 is a set ofi.n.d. zero-mean complex Gaussian variables, each with variance σ2

ij .To achieve the antenna diversity gain, the transmit information spreadsacross all the transmit antennas, and the receiver is equipped with amaximum ratio combiner. Let pj denote the power radiated from thejth transmit antenna subject to a normalized total power constraint∑Nt

j=1pj = 1. The instantaneous postdetection signal-to-noise ratio

(SNR) of the diversity system can then be expressed as

γe =1

N0

Nr∑i=1

Nt∑j=1

pj |hij |2. (1)

The conditional mutual information for a given power vectorp = (p1, . . . , pNt) and a channel realization H = (h1, . . . ,hNt) is

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 2, FEBRUARY 2009 999

given by

I(p,H) = log2(1 + γe). (2)

III. ERGODIC MUTUAL INFORMATION AND POWER ALLOCATION

A. Ergodic Mutual Information Analysis

The ergodic mutual information of the considered system isgiven by

I(p) = E [log2(1 + γe)] (3)

where the expectation E[·] is taken over all realizations of H. From(1), it is clear that γe is a weighted sum of NtNr independentand normalized exponential random variables with weights given bypjσ

2ij/N0. Its probability distribution function can thus be expressed

as [11, eq. (1)]

p(γe) =∑

k

−N0e−x/λk

(−λk)mk

g(mk−1)k (0, x)

(mk − 1)!

∣∣∣∣∣x=N0γe

(4)

where {λk}N′−1k=0 represent the N ′ (≤ NtNr) distinct values, each

with multiplicity mk, as a result of reordering and grouping theNtNr values of pjσ

2ij ; αkl is defined as αkl = 1 − λl/λk; and

g(mk−1)k (s, x) denotes the (mk − 1)th derivative of gk(s, x) given in

(5) with respect to s, that is

gk(s, x) = e−sx∏l �=k

1

(αkl − λls)ml. (5)

In the case where pjσ2ij’s are all distinct, (4) can be reduced to

p(γe) = N0

Nr∑i=1

Nt∑j=1

Bij

pjσ2ij

exp

(−N0γe

pjσ2ij

)(6)

where

Bij =∏

{m,n}�={i,j}

pjσ2ij

pjσ2ij − pnσ2

mn

. (7)

Applying (6) to (3) and using [12, eq. (4.331.2)], we obtain the closed-form expression of the ergodic mutual information over independentand distinctly distributed (i.d.d.) channels as

I(p) =1

ln 2

Nr∑i=1

Nt∑j=1

Bij exp

(N0

pjσ2ij

)E1

(N0

pjσ2ij

)(8)

where E1(·) is the exponential integral function, which is defined asE1(x) =

∫∞x

e−t/t dt, for x > 0.In the following, we study the power allocation p that can maximize

the ergodic mutual information I(p). The expression of Bij in (7)makes it difficult to directly maximize I(p) with respect to p. To makethe problem more tractable, we only consider the MISO case. First,we study the simplest case, where there are only two transmit anten-nas (Nt = 2). After that, we propose a suboptimal power-allocationscheme for Nt > 2 transmit antennas.

B. Power Allocation for Two-Transmit One-Receive Antenna Systems

For the sake of brevity, in the rest of this section, we omit the receiveantenna subindex i in both Bij and σ2

ij as only Nr = 1 is considered.

For a MISO diversity system with two transmit antennas, the problemof maximizing I(p) is equivalent to

maxp1

{B1e

N0p1σ2

1 E1

(N0

p1σ21

)+B2e

N0(1−p1)σ2

2 E1

(N0

(1−p1)σ22

)}.

(9)

It is shown in Appendix A that the second derivative of the objectivefunction in (9) with respect to p1 is nonpositive for 0 ≤ p1 ≤ 1. Hence,the optimization problem is convex. It can be solved by letting its firstderivative with respect to p1 be zero. Note that when there does notexist such a p1 ∈ [0, 1], which can make the first derivative zero, theobjective function degenerates to a monotonic function of p1. In otherwords, one of the two ends of p1’s range should be the optimum value.Consequently, one antenna should be turned off. Hence, two differentcases with regard to power allocation are analyzed.

In the first case, both antennas are active. Assume that the power onthe first antenna is 0 < p1 < 1, whereas on the second antenna, it isp2 = 1 − p1. Taking the first derivative of the objective function in (9)with respect to p1 and equating it to zero, after applying a high-SNRassumption (N0 → 0), we obtain

ln(1 − p1)σ

22

p1σ21

=(σ2

1 + σ22) (σ2

2 − p1 (σ22 + σ2

1))

σ21σ2

2

. (10)

The details are also shown in Appendix A. Although the solution to(10) cannot be obtained in closed form, we can still point it out byfinding the cross point of the two curves specified by the left andright sides of (10), respectively. We prove in Appendix B that therealways exist one tangent point and, at most, one cross point for thetwo curves. The tangent point is p1 = σ2

2/(σ21 + σ2

2), but it is notvalid to be the optimal solution since it violates the assumption ofdistinct distribution made in (6), i.e., p1σ

21 �= p2σ

22 . By inspection,

we can find that p1 > 1/2 when σ21 > σ2

2 and p1 increases when theratio ξ12 = σ2

1/σ22 increases. Moreover, there does not exist any cross

point anymore when the difference between channel conditions is largeenough. In other words, a valid solution 0 < p1 < 1 to (10) does notexist for highly unbalanced channels. This leads to the second case,where only one antenna is active.

When only one antenna is active, the problem of maximizing I(p) istrivial. It is clear that the total power should be assigned to the antennawith a larger channel variance.

Based on the aforementioned discussion, the asymptotic powerallocation scheme at a high SNR only depends on the channel ratioξ12 = σ2

1/σ22 ≥ 1. In Fig. 1, we plot the optimal solution (obtained by

a graphic method) of p1 as a function of ξ12 based on (10). It is shownthat p1 can be well approximated by

p1 = f(ξ12) =

{1 − 1

2exp(−ξ12 + 1), 1 ≤ ξ12 ≤ ξT

1, ξ12 > ξT(11)

where the ratio threshold ξT can be chosen larger than 10. Therefore,(11) is chosen as the power-allocation function.

C. Power Allocation for Multiple-Transmit One-ReceiveAntenna Systems

For multiple-transmit antenna systems, it is difficult to directlyoptimize (8) with respect to p. Motivated by the results obtained fortwo-transmit antenna systems, i.e., more power on a better channel,we propose a suboptimal but simple power-allocation scheme thatcan provide near-optimal performance. It is assumed without loss ofgenerality that σ2

1 ≥ σ22 · · · ≥ σ2

Nt. The power on each antenna is

sequentially assigned. In other words, p1 is first computed; afterward,

1000 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 2, FEBRUARY 2009

Fig. 1. Power functions for a diversity system with two transmit antennas andone receive antenna.

p2 is computed until pNt . At stage j for computing pj , we splitthe Nt − j + 1 antennas, which have not been assigned powers, intotwo groups. The channel gain with the first group is σ2

j , and with

the second group, they are {σ2j+1, . . . , σ

2Nt

}. Define σ2(e)j+1 as the

equivalent channel gain of the second group, whose exact definition isto be shortly determined. Based on (11), a proposed suboptimal powerallocation scheme is

pj =

⎧⎪⎪⎨⎪⎪⎩

(1 −

j−1∑k=1

pk

)f(σ2

j /σ2(e)j+1

), σ2

j ≥ σ2(e)j+1(

1 −j−1∑k=1

pk

)[1 − f

(σ

2(e)j+1/σ2

j

)], σ

2(e)j+1 > σ2

j

(12)

for j = 1, . . . , Nt.We now propose a novel definition for σ

2(e)j+1 . First, we compare each

channel gain of {σ2j+1, . . . , σ

2Nt

} with σ2j to form the set Sj+1, in

which each gain is larger than or equal to σ2j /ξT , i.e.,{

σ2k ∈ Sj+1|σ2

j /σ2k ≤ ξT , k ∈ [j + 1, Nt]

}. (13)

Assume that there are Kj+1 ∈ [0, Nt − j] elements in Sj+1. Wesimply regard those Kj+1 associated antennas as one single antennaand discard the rest of the antennas whose channel gains are smallenough compared with σ2

j . Then, we define the equivalent channel gain

σ2(e)j+1 as the norm of the vector [σ2

j+1, σ2j+2, . . . , σ

2j+Kj+1

], i.e.,

σ2(e)j+1 =

√√√√j+Kj+1∑k=j+1

(σ2k)2. (14)

Although we cannot rigorously prove it, the proposed equivalent chan-nel gain in (14), together with (12), provides near-capacity-achievingperformance, as will be shown in Section V.

IV. OUTAGE PROBABILITY AND POWER ALLOCATION

Given the instantaneous mutual information I(p,H) defined in(2) and the outage mutual information Iout, the outage probability isdefined as

Pout(p) = P (I(p,H) < Iout) = P (γe < γout) (15)

where γout = 2Iout − 1. Hence, the outage probability is the sameas the cumulative distribution function of γe, which is expressed as[11, eq. (32)]

Pout(p) = 1 +∑

k

e−x/λk g(mk−1)k (0, x)

(−λk)mk (mk − 1)!

∣∣∣∣∣x=γout

. (16)

Here, gk(s, x) is given by

gk(s, x) = −λke−sx∏

l

1

(αkl − λls)mkl(17)

with αkkΔ= −1, mkl = ml for l �= k, and mkk = 1. In the case where

pjσ2ij’s are all distinct (i.e., i.d.d. channels), the outage probability in

(16) can be simplified to

Pout(p) =

Nr∑i=1

Nt∑j=1

Bij

(1 − exp

(−N0γout

pjσ2ij

)). (18)

In the special case, where Nr = 1, our result (18) is consistent with[10, eq. (2)]. In [13], the authors derived the outage probability for DFcooperative communications. When assuming the source-relay link tobe error free, we can verify that [13, eq. (10a)] reduces to (18) withNr = 1.

In the following, we derive a suboptimal power allocation schemethat can minimize an upper bound on the outage probability at anygiven Iout. By applying the Chernoff bound, the outage probability in(15) can be upper bounded by

Pout(p) ≤ E[eu(γout−γe)

]=euγout

Nr∏i=1

Nt∏j=1

1

1 + upjσ2ij/N0

(19)

where u is a nonnegative constant and can be chosen to optimizethe tightness of the bound. Nevertheless, we choose u = NtNr/γout

for simplicity. This bound can be minimized with respect to pj’s bymaximizing the objective function

ϕ(pj) =

Nr∏i=1

Nt∏j=1

(1 + upjσ

2ij/N0

). (20)

By applying the inequality (1 + Nr

√∏Nr

i=1xi)

Nr ≤∏Nr

i=1(1 + xi)

[14, eq. (25)], (20) can be lower bounded by

ϕ(pj) ≥Nt∏j=1

⎛⎝1 +

upj

N0

Nr

√√√√Nr∏i=1

σ2ij

⎞⎠

Nr

. (21)

Using the Lagrange method, we maximize the lower bound in (21)of the objective function ϕ(pj) and obtain the water-filling-basedsuboptimal power allocation

pj =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

1

v− N0γout

NtNrNr

√Nr∏i=1

σ2ij

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

+

(22)

where {a}+ denotes max{0, a}, and v is a constant and determinedby the constraint

∑Nt

j=1pj = 1. According to the properties of wa-

ter filling, at a high transmit SNR, the power tends to be equally

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 2, FEBRUARY 2009 1001

Fig. 2. Topology of MIMO diversity systems with distributed antenna de-ployment. (Solid lines) System 1, (dashed lines) system 2, and (both solid anddashed lines) system 3.

allocated among all transmit antennas, whereas at a low SNR, someof the antennas with the geometric mean of channel variances (i.e.,Nr

√∏Nr

i=1σ2

ij) significantly lower than the others may be turned off.

These conclusions with Nr = 1 are consistent with the heuristic powercontrol scheme for the MISO channel in [10].

V. NUMERICAL RESULTS

To quantify the nonidentical fading channels, we consider distrib-uted antenna deployment, whose layout is sketched in Fig. 2. Here,the three transmit ports {T1, T2, T3} and receive port R1 are alignedon a straight line spaced with an equal distance, whereas receive portR2 is located at its perpendicular bisector. Suppose receiver R1 isactive and receiver R2 is disabled. This scenario is referred to assystem 1 and is indicated by solid lines in the figure. By adoptingthe well-known path loss model and assuming the path loss exponentequal to 3, the channel variances can be obtained as σ2

1 = 648/251,σ2

2 = 81/251, and σ23 = 24/251. Alternatively, in system 2, receiver

R1 is disabled, and receiver R2 is active, as denoted by the dashed linesin Fig. 2. In this case, the channel variances become σ2

1 = 117/100,σ2

2 = 117/100, and σ23 = 66/100. The sum of channel variances in

both the aforementioned systems is normalized to satisfy the constraintNtNr = 3 with Nr = 1. Either of the systems may be equivalent toa DF cooperating system with one source (T3), two relays (T2 andT1), and one destination (R1 or R2) by assuming error-free decodingat the relays. When both receivers are active, we obtain system 3, forwhich the channel gain parameters are obtained as σ2

11 = 648/251,σ2

12 = 81/251, σ213 = 24/251, σ2

21 = 117/100, σ222 = 117/100, and

σ223 = 66/100, which sum to NtNr = 6.To illustrate the results for ergodic mutual information, we consider

only the 3 by 1 systems (systems 1 and 2). In Fig. 3, the ergodic mutualinformation using different power allocations is presented. The resultof the optimal power allocation is obtained by using a 2-D exhaustivesearch. It is shown that the proposed suboptimal scheme (12) performsalmost the same as the optimal scheme and, hence, is near-capacityachieving. In addition, a 3-dB SNR gain is achieved over EPA insystem 1. Fig. 4 compares the power values assigned to T1 (p1) andT3 (p3) by using the criterion (12) (for simplicity, p2 is not shownbut can be straightforwardly obtained). Note that the power values areall constant at different transmit SNRs since the power allocation (12)only depends on the ratio of channel variances. The results show thatfor system 1 at the cooperative transmission scenario, only relay II isneeded for forwarding signals without the cooperation from the source

Fig. 3. Ergodic mutual information with different power allocations insystems 1 and 2.

Fig. 4. Power values assigned on T1 and T3 for maximizing ergodic mutualinformation in systems 1 and 2.

and relay I. This is expected as the channels are highly unbalanced.On the other hand, for slightly unbalanced channels encountered insystem 2, the source and the two relays all need to be active, but morepower is given to the node with a larger channel gain.

The outage probability for a given Iout = 2 bits per channel useis shown in Fig. 5. It is shown that the proposed power allocation(22) provides performance very close to the optimal scheme (via anexhaustive search). The outage probability after using the proposedpower allocation is smaller than that of EPA. This improvement ismore significant in systems 1 and 3 in low and moderate SNR regions.In Fig. 6, the power values assigned to T1 and T3 by using the water-filling principle (22) are compared. An interesting finding is that for thesame transmitter, the power value assigned in system 3 lies between thevalues assigned in systems 1 and 2.

VI. CONCLUSION

We analyzed the mutual information of MIMO diversity sys-tems with i.n.d. Rayleigh fading. Closed-form expressions of ergodic

1002 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 2, FEBRUARY 2009

Fig. 5. Outage probability with different power allocations in systems 1–3.

Fig. 6. Power values assigned on T1 and T3 for minimizing outage insystems 1–3.

mutual information and outage probability over i.d.d. channels wereobtained with an arbitrary number of transmit and receive antennas.We then derived two suboptimal power-allocation schemes exploitingthe nonidentical channel statistics for ergodic mutual informationmaximization and outage minimization, respectively. With a singlereceive antenna, the power-allocation scheme for maximizing theergodic mutual information is particularly novel. It assigns more powerto antennas with larger channel variances. To minimize the outageprobability at an arbitrary number of receive antennas, the powerallocation depends on the geometric mean of channel variances andfollows the water-filling principle. Numerical results show that theproposed power controls are near optimal and considerably outperformEPA when the channels are highly unbalanced.

APPENDIX A

We derive the first derivative of (9) with respect to p1. Based on theidentity E1(x) = −Ei(−x) = −

∫ −x

−∞(et/t)dt, x > 0, let A = B1 ×exp(N0/(p1σ

21)), B = −Ei(−N0/(p1σ

21)), C = B2 exp(N0/((1 −

p1)σ22)), and D = −Ei(−N0/((1 − p1)σ

22)). Taking the first deriva-

tive with respect to p1, respectively, we get

A′=−σ2

1σ22eN0/(p1σ2

1)p1−N0eN0/(p1σ2

1)(p1σ21−(1−p1)σ

22)

p1(p1σ21−(1−p1)σ2

2)2

B′= e−N0/(p1σ21)/p1

C′=eN0/((1−p1)σ2

2)((1−p1)σ21σ2

2)+N0((1−p1)σ22−p1σ

21))

((1−p1)σ22−p1σ2

1)2 (1−p1)

D′= −e−N0/((1−p1)σ22)/(1−p1).

Note that limx→0 −xEi(−x) = 0 and limN0→0 E1(N0/(1 −p1)σ

22) − E1(N0/p1σ

21) = ln((1 − p1)σ

22/p1σ

21). For a high-SNR

assumption, the first derivative of objective function in (9) becomes

σ21σ2

2

(ln

(1−p1)σ22

p1σ21

)((1 − p1)σ2

2 − p1σ21)2

− σ21 + σ2

2

(1 − p1)σ22 − p1σ2

1

. (23)

By letting (23) be zero, after simple manipulations, (10) can beobtained.

Differentiating (23) with respect to p1, and multiplying the result bya positive term {[(1 − p1)σ

22 − p1σ

21 ]2p1(1 − p1)}−1, we obtain the

weighted second derivative

y = −σ21σ2

2 +2σ2

1σ22 (σ2

1 + σ22) p1(1 − p1)

(1 − p1)σ22 − p1σ2

1

× ln(1 − p1)σ

22

p1σ21

− p1(1 − p1)(σ2

1 + σ22

)2. (24)

Without loss of generality, by assuming ξ = σ21/σ2

2 ≥ 1 and the sumof channel gains to be Nt (in this case, Nt = 2), one has σ2

1 =2ξ/(ξ + 1) and σ2

2 = 2/(ξ + 1). Hence, (24) can be expressed by afunction of ξ and p1, i.e.,

y=−4ξ

(1 + ξ)2+

4 × 2ξp1(1 − p1)

(ξ + 1)(1 − p1 − ξp1)ln

1 − p1

ξp1

− 4p1(1 − p1).

(25)

Now, we need to show that y is nonpositive for 0 ≤ p1 ≤ 1. To doso, we consider three different cases. First, in the case of x = (1 −p1)/(ξp1) = 1, the term ln((1 − p1)/εp1)/(1 − p1 − εp1) as part ofthe second term in (25) can be rewritten as ln x/(ξp1x − ξp1). Byapplying the L’Hospital principle, it is easy to show that y = 0. In thesecond case of x=(1−p1)/(ξp1)>1, we substitute p1 =1/(xξ+1)into (25) and get

y =−4ξ

(ξ + 1)2+

4 × 2ξx ln(x)

(ξ + 1)(x − 1)(ξx + 1)− 4ξx

(ξx + 1)2. (26)

Multiplying the two sides of (26) by the positive term (1 + ξ)(ξx +1)(x − 1)/(4ξx), we get

y1 =−(ξx + 1)(x − 1)

(ξ + 1)x− (x − 1)(ξ + 1)

ξx + 1+ 2 ln x. (27)

By differentiating (27) with respect to x and multiplying the resultby the positive term x2(1 + ξ)(ξx + 1)2, y1 comes to y2 = −(x −1)2(x2ξ3+1). Considering the property of quadratic function (ξ≥1)and only one real root x = 1 of y2, we can find that (27) is a monotonic

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 2, FEBRUARY 2009 1003

decreasing function of x. Hence, y1 = 0 is global maximum atpoint x = 1.

APPENDIX B

In this Appendix, we show that there always exist one tangentpoint and, at most, one cross point for the two curves in (10).First, we denote f1(p1) = ln((1 − p1)σ

22/p1σ

21) and f2(p1) = (σ2

1 +σ2

2)(σ22 − p1(σ

22 + σ2

1))/σ21σ2

2 , respectively. We can show that whenp1 > 1/2, the function f1(p1) is convex, whereas when p1 < 1/2, thefunction f1(p1) is concave. Since f2(p1) is a linear function, thereare at most three values of p1, which can make f1(p1) = f2(p1). Inour case, two of the three values are always equal. In other words,there is one tangent point between f1(p1) and f2(p1), and it lies atp1 = σ2

2/(σ21 + σ2

2). This point is the tangent point between the twocurves since

f ′i

(p1 =

σ22

σ21 + σ2

2

)= − (σ2

1 + σ22)

2

σ21σ2

2

, i = 1, 2 (28)

and since f1(p1) = f2(p1) holds when p1 = σ22/(σ2

1 + σ22). How-

ever, the tangent point p1 = σ22/σ2

1 + σ22 is not the valid solution

since p1σ21 = p2σ

22 when p1 = σ2

2/σ21 + σ2

2 , which conflicts with theassumption made when obtaining (6).

REFERENCES

[1] S. Sandhu and A. Paulraj, “Space–time block codes: A capacity perspec-tive,” IEEE Commun. Lett., vol. 4, no. 12, pp. 384–386, Dec. 2000.

[2] G. Ganesan and P. Stoica, “Space–time block codes: A maximum SNRapproach,” IEEE Trans. Inf. Theory, vol. 47, no. 5, pp. 1650–1656,May 2001.

[3] H. Zhang and T. A. Gulliver, “Capacity and error probability analysis fororthogonal space–time block codes over fading channels,” IEEE Trans.Wireless Commun., vol. 4, no. 2, pp. 808–819, Mar. 2005.

[4] J. N. Laneman and G. W. Wornell, “Distributed space–time-coded protocols for exploiting cooperative diversity in wirelessnetworks,” IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2415–2425,Oct. 2003.

[5] M. V. Clark, T. M. Willis, III, L. J. Greenstein, A. J. Rustako, Jr., V. Erceg,and R. S. Roman, “Distributed versus centralized antenna arrays in broad-band wireless networks,” in Proc. IEEE Veh. Technol. Conf.—Spring,2001, vol. 1, pp. 33–37.

[6] J. He and P. Y. Kam, “On the performance of orthogonal space–time blockcodes over independent, nonidentical Rayleigh/Ricean fading channels,”in Proc. IEEE GLOBECOM, Nov. 2006, pp. 1–5.

[7] H. Zhao, Y. Gong, Y. L. Guan, C. L. Law, and Y. Tang, “Space–time blockcodes in Nakagami fading channels with non-identical m-distributions,”in Proc. WCNC, Mar. 2007, pp. 536–540.

[8] M. Tao and P. Y. Kam, “Analysis of differential orthogonal space–timeblock codes over semi-identical MIMO fading channels,” IEEE Trans.Commun., vol. 55, no. 2, pp. 282–291, Feb. 2007.

[9] E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans.Telecommun., vol. 10, no. 6, pp. 585–595, 1999.

[10] J. Luo, R. S. Blum, L. Cimini, L. Greenstein, and A. Haimovich, “Powerallocation in a transmit diversity system with mean channel gain informa-tion,” IEEE Commun. Lett., vol. 9, no. 7, pp. 616–618, Jul. 2005.

[11] D. Raphaeli, “Distribution of noncentral indefinite quadratic forms incomplex normal variables,” IEEE Trans. Inf. Theory, vol. 42, no. 3,pp. 1002–1007, May 1996.

[12] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, andProducts. Boston, MA: Academic, 1994.

[13] N. C. Beaulieu and J. Hu, “A closed-form expression for the outageprobability of decode-and-forward relaying in dissimilar Rayleigh fad-ing channels,” IEEE Commun. Lett., vol. 10, no. 12, pp. 813–815,Dec. 2006.

[14] M.-K. Byun and B. G. Lee, “New bounds of pairwise error probability forspace–time codes in Rayleigh fading channels,” IEEE Trans. Commun.,vol. 55, no. 8, pp. 1484–1493, Aug. 2007.

Maximizing the Average Spectral Efficiencyof Dual-Branch MIMO Systems With

Discrete-Rate Adaptation

Sébastien de la Kethulle de Ryhove, Geir E. Øien, and Frode Bøhagen

Abstract—The capacity of multiple-input–multiple-output (MIMO) sys-tems with perfect transmitter and receiver channel state information (CSI)can be attained by decoupling the MIMO channel into a set of independentsubchannels and distributing the power among these subchannels in accor-dance with the water-filling solution. Implementation of this scheme on atime-varying channel, however, requires continuous-rate adaptation andis not feasible in any such practical system. In this paper, we show howto maximize the average spectral efficiency (ASE) of dual-branch MIMOsystems (either two transmit or two receive antennas) with perfect trans-mitter and receiver CSI when using a fixed number of codes (discrete-rateadaptation). This maximum ASE is compared with the system’s ergodiccapacity and with the maximum ASE that can be attained if the availablepower is distributed among the different subchannels in accordance withthe water-filling solution although only discrete-rate adaptation is possible.We assume that capacity-achieving codes for additive white Gaussiannoise (AWGN) channels are available and that the power available to thetransmitter to transmit the ith symbol frame is fixed and independent ofthe frame index i.

Index Terms—Average spectral efficiency (ASE), capacity-achieving code, discrete-rate adaptation, dual-branch multiple-input–multiple-output (MIMO) system, nonlinear optimization, powerallocation, water filling.

I. INTRODUCTION

Information-theoretic results on the capacity of multiple-input–multiple-output (MIMO) systems with both transmitter and receiverchannel state information (CSI) suggest decoupling such channels intoindependent subchannels by means of linear precoding and distributingthe available power between these subchannels according to the water-filling solution [1]–[4]. Communication based on this scheme on atime-varying channel requires, however, continuous-rate adaptation(i.e., the ability to implement any rate within the continuum coveredby the statistical distribution of the channel realizations) and is un-fortunately not feasible in a practical system. Indeed, although it isrealistic to assume that capacity-achieving codes for additive whiteGaussian noise (AWGN) channels of any desired rate can be designeddue to advances in coding theory (mainly concerning turbo codes [5]and low-density parity-check codes [6]), only a finite number of suchcodes—and thus rates—will be available in any practical system dueto memory and complexity constraints. Recent publications addressingthe maximization of the average spectral efficiency (ASE) of single-input–single-output systems operating over fading channels under the

Manuscript received November 12, 2006; revised August 5, 2007, April 11,2008, and May 23, 2008. First published June 3, 2008; current version pub-lished February 17, 2009. This paper was presented in part at the Interna-tional Workshop on Signal Processing Advances for Wireless Communications(SPAWC’06), Cannes, France, July 2006. The review of this paper was coordi-nated by Dr. S. Vishwanath.

S. de la Kethulle de Ryhove was with the Department of Electronics andTelecommunications, Norwegian University of Science and Technology, 7491Trondheim, Norway. He is now with PricewaterhouseCoopers advisory Bergen,5835 Bergen, Norway (e-mail: sebastien.delakethulle@no.pwc.com).

G. E. Øien is with the Department of Electronics and Telecommunications,Norwegian University of Science and Technology, 7491 Trondheim, Norway(e-mail: oien@iet.ntnu.no).

F. Bøhagen is with Telenor Research and Innovation, 1331 Fornebu, Norway(e-mail: frode.bohagen@telenor.com).

Digital Object Identifier 10.1109/TVT.2008.926612

0018-9545/$25.00 © 2009 IEEE