On the Channel Capacity of Multiantenna Systems With Nakagami Fading

Hindawi Publishing CorporationEURASIP Journal on Applied Signal ProcessingVolume 2006, Article ID 39436, Pages 1–11DOI 10.1155/ASP/2006/39436

On the Channel Capacity of Multiantenna Systemswith Nakagami Fading

Feng Zheng1 and Thomas Kaiser2

1 Department of Electronic and Computer Engineering, University of Limerick, Limerick, Ireland2 Department of Communication Systems, Faculty of Electrical Engineering, University Duisburg-Essen,47048 Duisburg, Germany

Received 12 September 2005; Revised 13 January 2006; Accepted 9 March 2006

Recommended for Publication by Christoph Mecklenbrauker

We discuss the channel capacity of multiantenna systems with the Nakagami fading channel. Analytic expressions for the ergodicchannel capacity or its lower bound are given for SISO, SIMO, and MISO cases. Formulae for the outage probability of the capacityare presented. It is shown that the channel capacity could be increased logarithmically with the number of receive antennas forSIMO case; while employing 3–5 transmit antennas (irrespective of all other parameters considered herein) can approach the bestadvantage of the multiple transmit antenna systems as far as channel capacity is concerned for MISO case. We have shown that fora given SNR, the outage probability decreases considerably with the number of receive antennas for SIMO case, while for MISOcase, the upper bound of the outage probability decreases with the number of transmit antennas when the transmission rate islower than some value, but increases instead when the transmission rate is higher than another value. A critical transmission rateis identified.

Copyright © 2006 F. Zheng and T. Kaiser. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

1. INTRODUCTION

Since Foschini and Gans [1] and Telatar [2] established thatthe capacity of a Rayleigh distributed flat fading channel willincrease almost linearly with the minimum of the number oftransmit and receive antennas when the receiver has accessto perfect channel state information but not the transmitter,multiple transmit and receive antenna (MIMO) systems andspacetime coding have received great attention as a meansof providing substantial performance improvement againstchannel fading in wireless communication systems. In [3],Jayaweera and Poor extended the capacity result to the caseof Rician fading channel, considering that Rician fading isa better model for some fading environments. For example,when there is a direct line of sight (LOS) path in additionto the multiple scattering paths, the natural fading model isRician.

In this paper, we will investigate how the capacity ofMIMO systems changes in a Nakagami fading [4] environ-ment. The main reason that motivated our study is thatin some communication scenarios such as ultra-wideband(UWB) wireless communications, which has become a very

hot topic recently, the Nakagami fading gives a better fit-ting for the channel model [5]. Another reason is that theNakagami fading is an extension to the Rayleigh fading, andtherefore the results to be presented in this paper will be ageneralization of previous MIMO results. On the other hand,there are no reports in the literature on the study of the chan-nel capacity of MIMO systems with the Nakagami fading tothe best of the authors’ knowledge.

This paper is organized as follows. Section 2 describes themodel we are considering. The ergodic channel capacity forthe case of single transmit antenna and single receive antennais discussed in Section 3. Then the MIMO case is studied inSection 4. The outage probability about the capacity is dis-cussed in Section 5. In Section 6, numerical results are pro-vided to demonstrate the dependence of the channel capacityon various kinds of channel parameters. Finally, concludingremarks are given in Section 7.

Notation 1. The notation in this paper is fairly standard. I isan identity matrix whose dimension is either implied by con-text or indicated by its subscript when necessary, Pr denotesthe probability of an event, PA(x) represents the cumulative

2 EURASIP Journal on Applied Signal Processing

distribution function of a random variableA, that is, PA(x) =Pr{A ≤ x}, pA(x) stands for the probability density func-tion of a random variable A, that is, pA(x) = Pr{x ≤ A <x + dx}/dx, ϕA(ν) represents the characteristic function of arandom variable A, EA( f (A)) stands for the expectation ofa function of a random variable A, taking expectation overthe statistics of A, and tr represents the trace of a square ma-trix. Throughout this paper, the function log is understoodas the natural logarithm of its argument. Hence the unit ofthe channel capacity is nat.

2. MODEL DESCRIPTION

Consider a single user communications link in which thetransmitter and receiver are equipped with mX and mY an-tennas, respectively. The received signal in such a system canbe written in vector form as

Y(t) = A(t)X(t) + N(t), (1)

where X(t) ∈ RmX and Y(t) ∈ RmY are the transmitted andreceived signals, respectively, A(t) = [anm(t)]mY×mX is a ran-dom matrix characterizing the amplitude fading of the chan-nel, and N(t) ∈ RmY is the receiver noise. Note that all thesignals considered in this paper are in real spaces, in accor-dance with some communication scenarios such as UWB.

Throughout this paper we will assume that all the ran-dom processes are blockwise stationary. Therefore the nota-tion of time will be omitted for briefness.

To make the analysis tractable, the following assumptionsare needed.

Assumption 2. It is assumed that all anm, n = 1, . . . ,mY , m =1, . . . ,mX , are independent and identically distributed.

Assumption 3. The noise N is zero-mean Gaussian with co-variance matrix σ2

NImY .

Assumption 4. The power of the transmitted signal isbounded by S, that is, E(XTX) ≤ S.

Assumption 5. The receiver possesses complete knowledge ofthe instantaneous channel parameters, while the transmitteris not aware of the information about the channel parame-ters.

In the following, we will describe the statistical propertyof the matrix A. For this purpose, we will generically use a todenote each entry of A. We suppose that the magnitude of a,denoted as |a|, takes a Nakagami distribution, whose generalform of the probability density function (pdf) is as follows:

p|a|(x) =

⎧⎪⎪⎨

⎪⎪⎩

2mmx2m−1

Γ(m)Ωme−mx2/Ω when x ≥ 0,

0 when x < 0,m ≥ 1

2,

(2)

where Γ denotes the Gamma function, Ω = E(a2), and m =[E(a2)]2/Var[a2]. In this paper, we substitutemwith another

parameter κ by simply defining κ = 2m. Hence it is clear thatκ ≥ 1. By doing so the pdf of |a| in (2) can be rewritten as

p|a|(x)

=

⎧⎪⎪⎨

⎪⎪⎩

2(κ

2Ω

)κ/2 1Γ(κ/2)

xκ−1e−κx2/2Ω when x ≥ 0,

0 when x < 0,

κ ≥ 1.

(3)

Note that we should specify the statistics of the sign of a todescribe completely the fading of a. However, for the purposeof this paper, we do not need it.

Define η = a2. It is easy to get the pdf of η as follows:

pη(x) =

⎧⎪⎪⎨

⎪⎪⎩

(κ

2Ω

)κ/2 1Γ(κ/2)

xκ/2−1e−κx/2Ω when x ≥ 0,

0 when x < 0.(4)

In the sequel development, we need the characteristic func-tion of the random variable η. First, we calculate the momentgenerating function of η through which the characteristicfunction of η can be easily obtained. The moment generat-ing function of η is given by

ψη(s) =∫ +∞

−∞esx pη(x)dx

=∫ +∞

0esx

1(2Ω/κ)κ/2Γ(κ/2)

xκ/2−1e−κx/2Ωdx.

(5)

Substituting the integral variable x with y = (κ/2Ω− s)x andusing the definition of the Gamma function, we obtain

ψη(s) = 1(2Ω/κ)κ/2Γ(κ/2)(κ/2Ω− s)κ/2

∫ +∞

0yκ/2−1e−ydy

= 1(2Ω/κ)κ/2Γ(κ/2)(κ/2Ω− s)κ/2 Γ

(κ

2

)

= 1(1− (2Ω/κ)s)κ/2

.

(6)

Thus according to the relationship between moment gener-ating function and characteristic function [6], the latter isgiven by

ϕη(ν) = ψη(jν) = 1(1− j(2Ω/κ)ν)κ/2

. (7)

Notice that the distribution defined by pdf (4) can be re-garded as a modified χ2 distribution with κ degree of freedom(we can write η = (Ω/κ)χ2).

Now we are ready to discuss the channel capacity.

F. Zheng and T. Kaiser 3

3. THE CASE OF SINGLE TRANSMIT ANDRECEIVE ANTENNA (SISO)

First we study the SISO case. For this case, the input-outputrelation is simplified to

Y(t) = a(t)X(t) +N(t), (8)

where X , Y , and N become scalars, a assumes the distribu-tion described by (3), and the noiseN is zero-mean Gaussianand white with variance σ2

N .In this and next sections we will discuss ergodic chan-

nel capacity. So in the following we will assume that the fad-ing process is ergodic, so that averaging the classical channelcapacity over the amplitude fading is of operational signifi-cance.

The channel capacity for an AWGN channel with a givenfading amplitude a is given by

C|a =WX log

(

1 +a2S

σ2N

)

, (9)

where WX is the bandwidth of the channel. So the ergodicchannel capacity, denoted as Ce, turns out to be

Ce = Ea(C|a

) =WX

∫∞

−∞log

(

1 +a2S

σ2N

)

pa(a)da

=WX

∫∞

−∞log

(

1 +ηS

σ2N

)

pη(η)dη

=WX

∫∞

0log

(

1 +xS

σ2N

)(κ

2Ω

)κ/2

× 1Γ(κ/2)

xκ/2−1e−κx/2Ωdx,

(10)

where η = a2 and the distribution of η is given by (4). Substi-tuting the variable x with x = (2Ω/κ)u in the above integralyields

Ce = WX

Γ(κ/2)

∫∞

0log

(

1 +u

β

)

uκ/2−1e−udu, (11)

where

β := κσ2N

2ΩS= κ

2 SNR, (12)

SNR := ΩS

σ2N

, (13)

can be considered as the ratio of signal power (at the receiverside) to the noise power. Let us define

J(κ;β) :=∫∞

0log

(

1 +u

β

)

uκ/2−1e−udu. (14)

Integrating the above integral by parts, we obtain

J(κ;β) =∫∞

0

1u + β

uκ/2−1e−udu

+(κ

2− 1

)∫∞

0log

(

1 +u

β

)

u(κ−2)/2−1e−udu

=∫∞

0

1u + β

uκ/2−1e−udu +(κ

2− 1

)

J(κ− 2;β).

(15)

From [7, page 319], one sees that

∫∞

0

1u + β

uκ/2−1e−udu = eββ(κ−2)/2Γ(κ

2

)

Γ(

1− κ

2,β)

,

(16)

where it is required that κ ≥ 1 to guarantee the integral toconverge and Γ(α, z) denotes the incomplete Gamma func-tion, defined by (see [7, page 940])

Γ(α, z) =∫∞

ze−uuα−1du. (17)

Thus we have

J(κ;β) = eββ(κ−2)/2Γ(κ

2

)

Γ(

1− κ

2,β)

+(κ

2− 1

)

J(κ− 2;β).

(18)

To use the recursive formula (18) to calculate J(κ;β), we needto know J(1;β) and J(2;β), respectively. By definition, wehave

J(1;β) =∫∞

0

log(1 + u/β)e−u√u

du

= π3/2 erfi(√

β)

− (γ + 2 log 2 + logβ)√π − 2

√πβ

·2 F2

(

[1, 1],[

2,32

]

,β)

= √π[

π erfi(√

β)

− γ − 2 log 2− logβ − 2β

·2 F2

(

[1, 1],[

2,32

]

,β)]

,

(19)

where γ ≈ 0.5772 is the Euler’s constant, erfi(z) and 2F2([α1,α2], [α3,α4], z) are the imaginary error function and general-ized hypergeometric function, respectively, which are definedby (cf. [7, page 1045])

erfi(z) = 2√π

∫ z

0eu

2du,

2F2([α1,α2

],[α3,α4

], z) =

∞∑

k=0

(α1)k(α2)k(α3)k(α4)k

zk

k!,

(20)

where (α)k = α(α+ 1) · · · (α+k−1) = Γ(α + k)/Γ(α). Whileusing the definition of the incomplete Gamma function, weobtain

J(2;β) =∫∞

0log

(

1 +u

β

)

e−udu =∫∞

0

e−u

u + βdu

= eβ∫∞

β

e−v

vdv = eβΓ(0,β).

(21)

Finally, the ergodic channel capacity can be calculated ac-cording to

Ce = WX

Γ(κ/2)J(κ;β). (22)


From (22), (12), and (13), it is interesting to observe that thechannel capacity depends only on parameters WX , κ, and β,and that parameter Ω plays the same role as S. This is an ex-pected result since Ω is proportional to the power of a, whichcan be seen from the fact that E(a2) = Ω.

4. THE CASE OF MULTIPLE TRANSMITAND RECEIVE ANTENNAS

In this case, the input-output relation (channel model) is de-scribed by (1). The mutual information between X and Y fora given A is

I(X; Y | A) =H(Y | A)−H(Y | X, A)

=H(Y | A)−H(N),(23)

where H denotes the entropy of a random variable, whosedefinition can be found in [8] for the case of continuous ran-dom variables. It is well known that if X is constrained tohave covariance Q, the choice of X that maximizes I(X; Y |A) is a Gaussian random variable with covariance Q. Thusthe channel capacity for a given fading matrix A turns out tobe [2, 8]

C|A = maxpX(x)

I(X; Y | A)

=WX log det(σ2

NImY + AQAT)−WX log det(σ2

NImY )

=WX log det(

ImY +1σ2

NAQAT

)

,

(24)

where AT represents the transpose of matrix A. Let us define

Ψ(Q) = EA

[

log det(

ImY +1σ2

NAQAT

)]

. (25)

Then the ergodic channel capacity is given by

Ce =WX maxtr(Q)≤S

Ψ(Q). (26)

The optimization problem described by (25) and (26) isdifficult to solve. In the following we will solve the subopti-mal problem described by (25) and

C¯ e=WX max

tr(Q)≤SQ is diagonal

Ψ(Q). (27)

The constraint in (27) says that the transmitted signalsamong all antennas are uncorrelated. As is well known, a niceproperty for the case of the (complex) Gaussian fading chan-nel is that the optimal solution of Q for problem (25)-(26) isa diagonal matrix, but for our problem, whether or not Q isdiagonal is still an open problem. In principle, a nondiagonalQ may yield a greater maximum mutual information than adiagonal Q for general fading matrix A. Therefore, we willgenerally have Ce ≥ C

¯ e. In some cases, we will see Ce = C

¯ e.

Now following the same argument as that in [2], we showthat the optimal solution of Q for problem (25) and (27) is

Qopt = S

mXI. (28)

Suppose that Q is any given nonnegative diagonal matrix sat-isfying tr(Q) ≤ S and Π is any permutation matrix. ConsiderQΠ := ΠQΠT and AΠ := AΠT . Since AΠ is obtained by in-terchanging two corresponding columns, it can be inferredfrom the independence of the elements in A that pA(Z) =pAΠ(Z), where Z is a matrix with the same dimension as A.Therefore, we have

Ψ(Q) = EA

[

log det(

ImY +1σ2

NAΠT(ΠQΠT)ΠAT

)]

= EAΠ

[

log det(

ImY +1σ2

NAΠQΠ

(AΠ)T)]

= Ψ(

QΠ).

(29)

Let Q̃ = (1/mX !)∑

Π QΠ. It is well known [8] that the map-ping Q �→ Ψ(Q) is convex ∩ (in the convention of [8])over the set of positive definite matrices. Thus it follows thatΨ(Q̃) ≥ Ψ(Q). Notice that Q̃ is simply a multiple of the iden-tity matrix and tr(Q̃) = tr(Q). Thus C

¯ eis achieved by let-

ting Q = αI. Applying the trace constraint to Q yields thatα = S/mX. Therefore, we arrive at

Ce ≥ C¯ e=WXEA

[

log det(

ImY +S

mXσ2N

AAT)]

. (30)

Equation (30) provides a lower bound for the channel capac-ity of the Nakagami fading channels. The conservativeness ofthe lower bound comes from the diagonal assumption on Q.If, on the other hand, Q is nondiagonal, some kind of knowl-edge, either statistical property on or the exact value of thefading matrix should be provided to the transmitter. Consid-ering Assumption 5, we can conclude that the lower bounddescribed by (30) is a useful performance measure for thewireless systems with the Nakagami fading.

To use (30), we need to know the distribution of det(ImY +(S/mXσ

2N)AAT) or that of the eigenvalues of matrix AAT . Un-

fortunately, these distributions are known only when A pos-sesses some special distribution (typically normal distribu-tion) if both mX > 1 and mY > 1, see, for example, [9].Therefore, we will consider some special cases in the follow-ing.

Here we would like to point out that, in the above deriva-tion, we have used the property that the distribution of A isinvariant under permutation transformations, but this prop-erty does not hold for A under general unitary transforma-tions, such as the case of normal distribution discussed in[2].

4.1. Single transmit and multiple receive antennas

In this case, mX = 1 and mY > 1. We denote A = [a1, . . . ,amY ]T . First notice the fact that for any two matrices M1 andM2 with compatible dimensions, we have

det(

I + M1M2) = det

(I + M2M1

). (31)


Note also that in this case the matrix Q reduces to a scalar.Applying these two facts to (30), one sees that

Ce = C¯ e=WXEA

[

log(

1 +S

σ2N

ATA)]

. (32)

Let

Υ := ATA =mY∑

l=1

a2l :=

mY∑

l=1

Υl, (33)

where Υl := a2l , l = 1, . . . ,mY. According to (7) and noticing

the fact that {Υl, l = 1, . . . ,mY} are independent, we can seethat the characteristic function of Υ is given by

ϕΥ(ν) =[

1(1− j(2Ω/κ)ν)κ/2

]mY

= 1(1− j(2Ω/κ)ν)(κ/2)mY

.

(34)

From (34) we can see (cf. [6, page 148]) that ϕΥ(ν) is thecharacteristic function of the Gamma distribution. Thus Υhas the following pdf:

pΥ(x)

=

⎧⎪⎪⎨

⎪⎪⎩

1(2Ω/κ)(κ/2)mYΓ((κ/2)mY )

x(κ/2)mY−1e−κx/2Ω when x ≥ 0,

0 when x < 0.(35)

Therefore, the ergodic channel capacity is given by

Ce =WXEΥ

(

log

(

1 +1σ2

NSΥ

))

=WX

∫∞

0log

(

1 +S

σ2Nx

)

× 1(2Ω/κ)(κ/2)mYΓ((κ/2)mY )

x(κ/2)mY−1e−κx/2Ωdx

=WX

∫∞

0log

(

1 +2ΩSκσ2

Ny

)1

Γ((κ/2)mY )y(κ/2)mY−1e−ydy

= WX

Γ((κ/2)mY )J(mYκ;β

).

(36)

4.2. Multiple transmit and single receive antennas

In this case,mX > 1 andmY = 1. Thus AAT is a scalar. DefineΥ = AAT . It is clear that Υ has the following distribution:

pΥ(x)

=

⎧⎪⎪⎨

⎪⎪⎩

1(2Ω/κ)(κ/2)mXΓ((κ/2)mX)

x(κ/2)mX−1e−κx/2Ω when x ≥ 0,

0 when x < 0.(37)

Therefore, from (30), we have

Ce ≥ C¯ e=WXEΥ

[

log(

1 +S

mXσ2NΥ)]

=WX

∫∞

0log

(

1 +S

mXσ2Nx

)

× 1(2Ω/κ)(κ/2)mXΓ((κ/2)mX)

x(κ/2)mX−1e−κx/2Ωdx

= WX

Γ((κ/2)mX)J(mXκ;mXβ

).

(38)

Remark 6. Notice that when κ = 2, the fading model foreach element of A reduces to Rayleigh distribution, whichcorresponds to the classic narrowband wireless communica-tion channel. So we expect that the results obtained for thisspecific κ also recover the results obtained in [2]. Substitutingκ = 2 into (36) and (38), respectively, readily reveals that (36)and (38) indeed reduce to (12) and (13) in [2], respectively.

5. CAPACITY VERSUS OUTAGE PROBABILITY

The results we have obtained in the previous sections apply tothe case where the fading matrix is ergodic and there are noconstraints on the decoding delay on the receiver. In practicalcommunication systems, we often run into the case where thefading matrix is generated or chosen randomly at the begin-ning of the transmission, while no significant channel vari-ability occurs during the whole transmission. In this case, thefading matrix is clearly not ergodic. We suppose that the fad-ing matrix still has the distribution defined in the previoussections. In this case it is more important to investigate thechannel capacity in the sense of outage probability. An out-age is defined as the event where the communication channeldoes not support a target data rate. Thus, according to [10],outage probability, denoted by Pout(R), is defined as follows.With a given rate R, we associate a set ΘR in the space of thefading matrix A. The set is the largest possible set for whichCΘ, the capacity of the compound channel with parameterA ∈ ΘR, satisfies CΘ ≥ R. The outage probability is then de-fined as Pout(R) = Pr{A /∈ ΘR}. Thus it is clear that

Pout(R) = Pr{

A /∈ ΘR} = Pr

{C|A < R

} = Pr{C|A ≤ R

},

(39)

that is, the outage probability can be actually viewed as thecumulative distribution function (cdf) of the conditionalShannon capacity. Notice that the last equality of the aboveequation follows from the fact that C(X; Y|A) is a continuousfunction of the continuous random matrix A.

Based on the above discussion, we can evaluate the out-age probability for the following three cases.


(i) SISO case

Let us define η = a2. Recall that η has the pdf defined by (4).Thus its cdf is as follows:

Pη(x) = Pr{η ≤ x}

=∫ x

0

(κ

2Ω

)κ/2 1Γ(κ/2)

yκ/2−1e−κy/2Ωdy

= 1Γ(κ/2)

γ(κ

2,κ

2Ωx)

,

(40)

where γ(α, z) is the incomplete Gamma function, defined by(cf. [7, page 940])

γ(α, z) =∫ z

0e−xxα−1dx. (41)

Therefore, from (9) it follows that

Pout(R) = Pr

{

WX log

(

1 +a2S

σ2N

)

≤ R

}

= Pr

{

η ≤ σ2N

S

(eR/WX − 1

)}

= Pη

(σ2N

S

(eR/WX − 1

))

= 1Γ(κ/2)

γ(κ

2,β(eR/WX − 1

))

.

(42)

(ii) SIMO case

Recalling the definition of Υ and its pdf (35), we can obtainits cdf as follows:

PΥ(x) = Pr{Υ ≤ x}

=∫ x

0

1(2Ω/κ)(κ/2)mYΓ((κ/2)mY )

y(κ/2)mY−1e−κy/2Ωdy

= 1Γ((κ/2)mY )

γ(κ

2mY ,

κ

2Ωx)

.

(43)

Following the same argument as (32), the conditional capac-ity can be derived as

C|A =WX log

(

1 +1σ2

NSATA

)

. (44)

Thus the outage probability turns out to be

Pout(R) = Pr

{

WX log

(

1 +1σ2

NSATA

)

≤ R

}

= Pr

{

Υ ≤ σ2N

S

(eR/WX − 1

)}

= PΥ

(σ2

N

S

(eR/WX − 1

))

= 1Γ((κ/2)mY )

γ(κ

2mY ,β

(eR/WX − 1

))

.

(45)

(iii) MISO case

First, we have

PΥ(x) = 1Γ((κ/2)mX)

γ(κ

2mX ,

κ

2Ωx)

. (46)

Then according to (38), we have

Pout(R) = Pr{C|A ≤ R

}

≤ Pr

{

WX log

(

1 +S

mXσ2N

AAT

)

≤ R

}

= Pr

{

Υ ≤ mXσ2N

S

(eR/WX − 1

)}

= PΥ

(mXσ

2N

S

(eR/WX − 1)

)

= 1Γ((κ/2)mX

)γ(κ

2mX ,mXβ

(eR/WX − 1

))

= Pout(R).

(47)

Pout, as defined in (47), provides an upper bound for the con-cerned outage probability.

6. NUMERICAL RESULTS

In this section, we will investigate the variation of channel ca-pacity with respect to various kinds of parameters. It is foundfrom (12), (22), (36), and (38) that the ergodic channel ca-pacity depends only on channel bandwidth WX , the numberκ, mY , mX , and the signal-to-noise power ratio SNR in thesense defined by (13), respectively, and it depends on WX

linearly, so we let WX = 1 and only focus our attention onthe variation of Ce with respect to κ, SNR, mY , and mX , re-spectively.

Figure 1 depicts the variation of channel capacity Ce withrespect to the number κ for SISO case. We can see from thisfigure that even though the channel capacity increases withthe number κ, the quantities increased are not large com-pared to the base case (κ = 1), especially when κ ≥ 10. Forexample, for the case of SNR = 0 dB, when κ is increasedfrom 1 to 10, Ce increases (0.6695−0.5335)/0.5335 = 25.5%,while when κ is increased from 10 to 30, Ce increases only by2.31%.

Figure 2 demonstrates the relationship between channelcapacity and SNR for SISO case, which shows that when SNRbecomes large, Ce is approximately a logarithmic function ofSNR. This is a result coinciding with our expectation.

Figure 3 shows the relationship between the channel ca-pacity and the number of receive antennas for SIMO case.It can be seen from this figure that C

¯ eincreases with mY al-

most logarithmically. This phenomenon is similar to the cor-responding one in the case of the Rayleigh fading channels(cf. [2, Example 3]).

Figure 4 shows the relationship between the lower boundof the channel capacity and the number of transmit anten-nas for MISO case. It is interesting to observe from this figure


0 5 10 15 20 25 30

κ

0

0.5

1

1.5

2

2.5

Ch

ann

elca

paci

tyCe

SNR = −10 dBSNR = 0 dBSNR = 10 dB

Figure 1: Variation of ergodic channel capacity Ce (in nats/s/Hz)with the number κ for SISO case.

−20 −15 −10 −5 0 5 10

SNR

0

0.5

1

1.5

2

2.5

Ch

ann

elca

paci

tyCe

κ = 2κ = 4κ = 6

Figure 2: Variation of ergodic channel capacity Ce (in nats/s/Hz)with the ratio SNR (in dB ) for SISO case.

that the capacity increases with mX rapidly when mX is small(mX ≤ 6), however, the increase is very slow when mX be-comes large (mX > 6). This phenomenon is different fromthe one in the case of the Rayleigh fading channel, see [2,Example 4], where it is found that Ce does not change withmX when mX ≥ 2. An important phenomenon can alsobe observed by comparing Figures 4(a) and 4(b), that is,when the signal-to-noise ratio is low, the benefit obtained by

0 2 4 6 8 10 12 14 16 18 20

mY

0

0.2

0.4

0.6

0.8

1

Ch

ann

elca

paci

tyCe

κ = 2κ = 4κ = 6

Figure 3: Variation of ergodic channel capacity Ce (in nats/s/Hz)with the number of receive antennas mY for SIMO case (SNR =−10 dB).

distributing the available power to different transmit anten-nas is very limited as far as the average capacity is concerned.

From Figures 3 and 4, we can see that increasing thenumber of receiver antennas can obtain more benefit inchannel capacity than increasing the number of transmit an-tennas. Principally, the channel capacity could be increasedinfinitely by employing a large number of receive antennas,but it appears to increase only logarithmically in this num-ber; while employing 3—5 receive antennas can approach thebest advantage of the multiple transmit antenna systems (forthe case of single receive antenna). The reason for this phe-nomenon is two fold. First, the power is constrained to bea constant, for different mX , among all the transmit anten-nas, while no such constraint is applied to receive antennas.Second, it is assumed that the receiver possesses the fullknowledge about the channel state.

The variations of outage probability Pout or Pout withrespect to the transmission rate (in nats/s/Hz), R/WX , areshown in Figures 5, 6, and 7 for SISO, SIMO, and MISO cases,respectively. From these figures, it can be observed that fora given SNR, the outage probability decreases considerablywith the number of receive antennas in the range of wholetransmission rate, while Pout(R) decreases with the numberof transmit antennas when R/WX is lower than some value(denoted as R1), but increases instead when R/WX is largerthan another value (denoted as R2). Notice that the outageprobability is so large when R/WX is larger than R2 that totransfer information at this rate is of little practical interest.Therefore, we can conclude that increasing the number oftransmit antennas is of some significance at a transmissionrate of practical communications with tolerable outage prob-ability.


0 5 10 15

mX

0.0915

0.092

0.0925

0.093

0.0935

0.094

0.0945

0.095

0.0955Lo

wer

bou

nd

ofch

ann

elca

paci

tyCe

κ = 2κ = 4κ = 6

(a) SNR = −10 dB

0 5 10 15

mX

2

2.05

2.1

2.15

2.2

2.25

2.3

2.35

2.4

Low

erbo

un

dof

chan

nel

capa

cityCe

κ = 2κ = 4κ = 6

(b) SNR = +10 dB

Figure 4: Variation of the lower bound of the ergodic channel ca-pacity Ce (in nats/s/Hz) with the number of transmit antennas mX

for MISO case.

It would be difficult to calculate R2 exactly. However, R1

can be calculated in the following way. Notice the fact thatthe total signal power S is equally distributed among transmitantennas for MISO case withmX transmit antennas. Thus thereceived power from the useful signals should be

SY =mX∑

k=1

a2k ·

S

mX=∑mX

k=1 a2k

mX· S. (48)

Therefore it is clear that the larger the number mX , thesmaller the variance of the received power from the usefulsignals. In the extreme case, when mX approaches infinity,we have

SY −→ ΩS with probability 1 as mX −→ ∞, (49)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

R/WX

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ou

tage

prob

abili

tyP

out

κ = 2κ = 4κ = 6

(a) SNR = −10 dB

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

R/WX

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ou

tage

prob

abili

tyP

out

κ = 2κ = 4κ = 6

(b) SNR = 0 dB

0 0.5 1 1.5 2 2.5 3 3.5 4

R/WX

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ou

tage

prob

abili

tyP

out

κ = 2κ = 4κ = 6

(c) SNR = +10 dB

Figure 5: Outage probability Pout versus transmission rate R/WX

(in nats/s/Hz) for various κ and SNR for SISO case.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

R/WX

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1O

uta

gepr

obab

ilityP

out

mY = 1mY = 2

mY = 4mY = 8

(a) SNR = −10 dB

0 0.5 1 1.5 2 2.5 3

R/WX

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ou

tage

prob

abili

tyP

out

mY = 1mY = 2

mY = 4mY = 8

(b) SNR = 0 dB

0 1 2 3 4 5 6

R/WX

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ou

tage

prob

abili

tyP

out

mY = 1mY = 2

mY = 4mY = 8

(c) SNR = +10 dB

Figure 6: Outage probability Pout versus transmission rate R/WX

(in nats/s/Hz) for various mY and SNR for SIMO case (κ = 4).

according to strong law of large numbers [6]. Therefore inthis extreme case, we obtain

R

WX= log

(

1 +ΩS

σ2N

)

= log(1 + SNR) := Rc. (50)

The above says that the channel capacity will approach aconstant Rc when the number of transmit antennas ap-proaches infinity. We call Rc the critical transmission rate.From Figure 7, we can see that R1 and R2 satisfy the relation-ships

R1 = Rc, R2 > Rc. (51)

The above analysis yields that Rc = 0.0953, 0.6931, and2.3979 for SNR being −10 dB, 0 dB, and +10 dB, respectivelyin the case of Figure 7. It is seen that R2 almost coincides withRc.

7. CONCLUDING REMARKS

In this paper, the analytic expression for the ergodic chan-nel capacity or its lower bound of wireless communicationsystems with the Nakagami fading is presented for three spe-cial cases: (i) single transmit antenna and single receive an-tenna, (ii) single transmit and multiple receive antennas, and(iii) multiple transmit and single receive antennas, respec-tively. Formulae on the outage probability about the channelcapacity are also presented. Numerical results are provided todemonstrate the dependence of the channel capacity on var-ious kinds of channel parameters. It is shown that increasingthe number of receive antennas can obtain more benefit inchannel capacity than increasing the number of transmit an-tennas. Principally, the channel capacity could be increasedinfinitely by employing a large number of receive antennas,but it appears to increase only logarithmically in this num-ber for SIMO case; while employing 3—5 transmit anten-nas can approach the best advantage of the multiple transmitantenna systems (irrespective of all other parameters consid-ered herein) as far as channel capacity is concerned for MISOcase. We have also observed that when the signal-to-noiseratio is low, the benefit in average capacity obtained by dis-tributing the available power to different transmit antennasis very limited. We have shown numerically that for a givensignal-to-noise ratio, the outage probability decreases consid-erably with the number of receive antennas for SIMO case,while for MISO case, the upper bound of the outage proba-bility decreases with the number of transmit antennas whenthe communication rate is lower than the critical transmis-sion rate (Rc), but increases when the rate is higher than an-other value (R2). The gap between R2 and Rc is not big for thecases considered here. Rc is determined by the fading powerand the signal-to-noise ratio of the system at the transmit-ter side. We can roughly say that it is not beneficial to usemultiple transmit antennas if the required transmission rate(normalized by system bandwidth) is higher than the criticaltransmission rate.

Due to the fact that the probability density function ofthe eigenvalues of nonnormal distributed random matrices is


0 0.05 0.1 0.15 0.2 0.25 0.3

R/WX

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1U

pper

bou

nd

ofou

tage

prob

abili

tyP

out

mX = 1mX = 2

mX = 4mX = 8

(a) SNR = −10 dB

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

R/WX

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Upp

erbo

un

dof

outa

gepr

obab

ilityP

out

mX = 1mX = 2

mX = 4mX = 8

(b) SNR = 0 dB

0 0.5 1 1.5 2 2.5 3 3.5 4

R/WX

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Upp

erbo

un

dof

outa

gepr

obab

ilityP

out

mX = 1mX = 2

mX = 4mX = 8

(c) SNR = +10 dB

Figure 7: The upper bound of the outage probability Pout versustransmission rate R/WX (in nats/s/Hz) for various mX and SNR forMISO case (κ = 4).

unknown yet, the problem about the calculation of the chan-nel capacity for the general MIMO case is still open.

ACKNOWLEDGMENT

The authors wish to thank the anonymous referees for theirhelpful comments that have significantly improved the qual-ity of the paper.

REFERENCES

[1] G. J. Foschini and M. J. Gans, “On limits of wireless commu-nications in a fading environment when using multiple an-tennas,” Wireless Personal Communications, vol. 6, no. 3, pp.311–335, 1998.

[2] E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eu-ropean Transactions on Telecommunications, vol. 10, no. 6, pp.585–595, 1999, see also Tech. Rep., AT&T Bell Labs., 1995.

[3] S. Jayaweera and H. V. Poor, “On the capacity of multi-antennasystems in the presence of Rician fading,” in Proceedings ofIEEE 56th Vehicular Technology Conference (VTC ’02), vol. 4,pp. 1963–1967, Vancouver, BC, Canada, September 2002.

[4] M. Nakagami, “The m-distribution - a general formula of in-tensity distribution of rapid fading,” in Statistical Methods inRadio Wave Propagation, W. C. Hoffman, Ed., pp. 3–36, Perg-amon, Oxford, UK, 1960.

[5] D. Cassioli, M. Z. Win, and A. F. Molisch, “The ultra-widebandwidth indoor channel: from statistical model to simu-lations,” IEEE Journal on Selected Areas in Communications,vol. 20, no. 6, pp. 1247–1257, 2002.

[6] G. G. Roussas, A Course in Mathematical Statistics, AcademicPress, San Diego, Calif, USA, 2nd edition, 1997.

[7] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series andProducts, Academic Press, New York, NY, USA, 1980, correctedand enlarged edition, prepared by A. Jeffrey.

[8] R. G. Gallager, Information Theory and Reliable Communica-tion, John Wiley & Sons, New York, NY, USA, 1968.

[9] R. J. Muirhead, Aspects of Multivariate Statistical Theory, JohnWiley & Sons, New York, NY, USA, 1982.

[10] E. Biglieri, J. Proakis, and S. Shamai, “Fading channels:information-theoretic and communications aspects,” IEEETransaction on Information Theory, vol. 44, no. 6, pp. 2619–2692, 1998.

Feng Zheng received the B.S. and M.S. de-grees in 1984 and 1987, respectively, bothin electrical engineering from Xidian Uni-versity, Xi’an, China, and the Ph.D. degreein automatic control in 1993 from BeijingUniversity of Aeronautics and Astronautics,Beijing, China. In the past years, he heldAlexander-von-Humboldt Research Fellow-ship at the University of Duisburg and re-search positions at Tsinghua University, Na-tional University of Singapore, and the University Duisburg-Essen,respectively. From 1995 to 1998 he was with the Center for SpaceScience and Applied Research, Chinese Academy of Sciences, asan Associate Professor. Now he is with the Department of Elec-tronic & Computer Engineering, University of Limerick, as a Se-nior Researcher. He is a corecipient of several awards, includingthe National Natural Science Award in 1999 from the Chinese gov-ernment, the Science and Technology Achievement Award in 1997


from the State Education Commission of China, and the SICE BestPaper Award in 1994 from the Society of Instrument and ControlEngineering of Japan at the 33rd SICE Annual Conference, Tokyo,Japan. His research interests are in the areas of systems and control,signal processing, and wireless communications.

Thomas Kaiser received the Ph.D. degreein 1995 with distinction and the GermanHabilitation degree in 2000, both from Ger-hard-Mercator-University, Duisburg, and inelectrical engineering. From April 2000 toMarch 2001 he was the Head of the Depart-ment of Communication Systems at Ger-hard-Mercator-University, Duisburg, andfrom April 2001 to March 2002 he was theHead of the Department of Wireless Chips& Systems (WCS) at Fraunhofer Institute of Microelectronic Cir-cuits and Systems. In summer 2005 he joined Stanford’s Smart An-tenna Research Group (SARG) as a Visiting Professor. Now he iscoleader of the Smart Antenna Research Team (SmART) at the Uni-versity Duisburg-Essen. He has published more than 90 papers ininternational journals and at conferences, and he is the coeditorof three forthcoming books on ultra-wideband systems. He is thefounder of PLANET MIMO Ltd. and is a Member of the Edito-rial Board of EURASIP Journal on Applied Signal Processing andthe Advisory Board of a European multiantenna project. He is thefounding Editor-in-Chief of the IEEE Signal Processing Society e-letter. His current research interest focuses on applied signal pro-cessing with emphasis on multiantenna systems, especially its ap-plicability to ultra-wideband systems.

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On the Channel Capacity of Multiantenna Systems With Nakagami Fading