Báo cáo hóa học: " On the Channel Capacity of Multiantenna Systems with Nakagami Fading" pot

Analytic expressions for the ergodic channel capacity or its lower bound are given for SISO, SIMO, and MISO cases.. It is shown that the channel capacity could be increased logarithmical

Volume 2006, Article ID 39436, Pages 1 11

DOI 10.1155/ASP/2006/39436

On the Channel Capacity of Multiantenna Systems

with Nakagami Fading

Feng Zheng 1 and Thomas Kaiser 2

1 Department of Electronic and Computer Engineering, University of Limerick, Limerick, Ireland

2 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 ergodic channel capacity or its lower bound are given for SISO, SIMO, and MISO cases Formulae for the outage probability of the capacity are presented It is shown that the channel capacity could be increased logarithmically with the number of receive antennas for SIMO case; while employing 3–5 transmit antennas (irrespective of all other parameters considered herein) can approach the best advantage of the multiple transmit antenna systems as far as channel capacity is concerned for MISO case We have shown that for

a given SNR, the outage probability decreases considerably with the number of receive antennas for SIMO case, while for MISO

case, the upper bound of the outage probability decreases with the number of transmit antennas when the transmission rate is lower than some value, but increases instead when the transmission rate is higher than another value A critical transmission rate

is identified

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

1 INTRODUCTION

the capacity of a Rayleigh distributed flat fading channel will

increase almost linearly with the minimum of the number of

transmit and receive antennas when the receiver has access

to perfect channel state information but not the transmitter,

multiple transmit and receive antenna (MIMO) systems and

spacetime coding have received great attention as a means

of providing substantial performance improvement against

Jayaweera and Poor extended the capacity result to the case

of Rician fading channel, considering that Rician fading is

a better model for some fading environments For example,

when there is a direct line of sight (LOS) path in addition

to the multiple scattering paths, the natural fading model is

Rician

In this paper, we will investigate how the capacity of

environ-ment The main reason that motivated our study is that

in some communication scenarios such as ultra-wideband

(UWB) wireless communications, which has become a very

hot topic recently, the Nakagami fading gives a better

Nakagami fading is an extension to the Rayleigh fading, and therefore the results to be presented in this paper will be a generalization 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 to the best of the authors’ knowledge

model we are considering The ergodic channel capacity for the case of single transmit antenna and single receive antenna

Section 4 The outage probability about the capacity is

pro-vided to demonstrate the dependence of the channel capacity

on various kinds of channel parameters Finally, concluding

Notation 1 The notation in this paper is fairly standard I is

an identity matrix whose dimension is either implied by con-text or indicated by its subscript when necessary, Pr denotes

distribution function of a random variableA, that is, P A(x) =

Pr{ A ≤ x }, p A(x) stands for the probability density

x + dx } /dx, ϕ A(ν) represents the characteristic function of a

ma-trix Throughout this paper, the function log is understood

as the natural logarithm of its argument Hence the unit of

the channel capacity is nat

2 MODEL DESCRIPTION

Consider a single user communications link in which the

an-tennas, respectively The received signal in such a system can

be written in vector form as

ran-dom matrix characterizing the amplitude fading of the

signals 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 assumptions

are needed

Assumption 2 It is assumed that all a nm,n =1, , m Y,m =

1, , m X, are independent and identically distributed

Assumption 3 The noise N is zero-mean Gaussian with

N Im Y

Assumption 4 The power of the transmitted signal is

Assumption 5 The receiver possesses complete knowledge of

the instantaneous channel parameters, while the transmitter

is not aware of the information about the channel

parame-ters

In the following, we will describe the statistical property

form of the probability density function (pdf) is as follows:

p | a |(x) =

⎧

⎪

⎪

m ≥1

(2)

[E(a2)]2/ Var[a2] In this paper, we substitutem with another

p | a |(x)

=

⎧

⎪

⎪ 2



κ

2Ω

κ/2 1

2/2Ω whenx ≥0,

κ ≥1.

(3)

of this paper, we do not need it

p η(x) =

⎧

⎪

⎪



κ

κ/2 1

(4)

In the sequel development, we need the characteristic

ψ η(s) =

+∞

−∞esx p η(x)dx

=

+∞

0 esx 1

(5)

using the definition of the Gamma function, we obtain

+∞

0 y κ/2 −1e− y dy

2



(6)

Thus according to the relationship between moment

given by

ϕ η(ν) = ψ η(jν) = 1

Now we are ready to discuss the channel capacity

3 THE CASE OF SINGLE TRANSMIT AND

RECEIVE ANTENNA (SISO)

First we study the SISO case For this case, the input-output

relation is simplified to

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 averagfad-ing the classical channel

capacity over the amplitude fading is of operational

signifi-cance

The channel capacity for an AWGN channel with a given

C |a = W Xlog

σ N2

C e =Ea

C |a= W X

∞

−∞log



σ N2



p a(a)da

= W X

∞

−∞log

σ2

N p η(η)dη

= W X

∞

0

log

σ2

N

κ

2Ω

κ/2

(10)

yields

C e = W X

Γ(κ/2)

∞

0 log



β



u κ/2 −1e− u du, (11) where

β : = κσ N2

σ N2

can be considered as the ratio of signal power (at the receiver

side) to the noise power Let us define

J(κ; β) : =

∞

0

log



β



u κ/2 −1e− u du. (14) Integrating the above integral by parts, we obtain

J(κ; β) =

∞

0

1

u + β u

κ/2 −1e− u du

+



κ

 ∞

0 log



β



u(κ −2)/2 −1e− u du

=

∞

0

1

u + β u

κ/2 −1e− u du +



κ



J(κ −2;β).

(15)

∞

0

1

u + β u

κ/2 −1e− u du =eβ β(κ −2)/2Γκ

2



 , (16)

∞

z e− u u α −1du. (17) Thus we have

J(κ; β) =eβ β(κ −2)/2Γκ

2



 +



κ



J(κ −2;β).

(18)

have

J(1; β) =

∞

0

√

= π3/2erfi β

−(γ + 2 log 2 + log β) √

π −2√

πβ

·2F2

 [1, 1],



2





= √ π



π erfi β

·2F2

 [1, 1],



2





,

(19)

α2], [α3,α4],z) are the imaginary error function and

general-ized hypergeometric function, respectively, which are defined

π

z

0eu2du,

2F2



α1,α2



α3,α4



=

∞



k =0

z k

k!,

(20)

using the definition of the incomplete Gamma function, we obtain

J(2; β) =

∞

0 log



β



∞

0

u + β du

=eβ

∞

β

(21)

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

C e = W X

From (22), (12), and (13), it is interesting to observe that the

4 THE CASE OF MULTIPLE TRANSMIT

AND RECEIVE ANTENNAS

In this case, the input-output relation (channel model) is

a given A is

ran-dom variables It is well known that if X is constrained to

A) is a Gaussian random variable with covariance Q Thus

the channel capacity for a given fading matrix A turns out to

C |A=max

σ2

N Im Y)



Im Y+ 1

σN2 AQAT

 ,

(24)



Im Y+ 1

σ2

N AQAT



Then the ergodic channel capacity is given by

C e = W X max

tr(Q)≤ S Ψ(Q). (26)

difficult to solve In the following we will solve the

C

tr(Q)≤ S

Q is diagonal

among all antennas are uncorrelated As is well known, a nice

property for the case of the (complex) Gaussian fading

a diagonal matrix, but for our problem, whether or not Q is

diagonal is still an open problem In principle, a nondiagonal

Q may yield a greater maximum mutual information than a

diagonal Q for general fading matrix A Therefore, we will

Qopt= S

m X

Suppose that Q is any given nonnegative diagonal matrix

in-terchanging two corresponding columns, it can be inferred

Therefore, we have

 log det



Im Y+ 1

σ2

N

A ΠT

ΠQΠT

ΠAT



=EAΠ

 log det



Im Y+ 1

σN2

A ΠQΠ

AΠT

=ΨQ Π

.

(29)

over the set of positive definite matrices Thus it follows that

α = S/m X Therefore, we arrive at

C e ≥C

 log det



Im Y+ S

m X σ2

N

AAT



capac-ity of the Nakagami fading channels The conservativeness of

the 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 the fading matrix should be provided to the transmitter

wireless systems with the Nakagami fading

Un-fortunately, these distributions are known only when A

pos-sesses some special distribution (typically normal

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 is

invariant 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

4.1 Single transmit and multiple receive antennas

I + M1M2

=det

I + M2M1

Note also that in this case the matrix Q reduces to a scalar.

C e =C

 log



σ2

N

ATA



Let

m Y



l =1

a2

l :=

m Y



l =1

ϕΥ(ν) =



1

m Y

(34)

has the following pdf:

pΥ(x)

=

⎧

⎪

⎪

1

(35) Therefore, the ergodic channel capacity is given by

C e = W XEΥ

log

σ2

N

SΥ

= W X

∞

0 log

σ2

N

x

(2Ω/κ)(κ/2)m Y Γ((κ/2)m Y)x(κ/2)m Y −1e− κx/2Ω dx

= W X

∞

0 log

m Y κ; β

.

(36)

4.2 Multiple transmit and single receive antennas

Υ=AAT It is clear that Υ has the following distribution:

pΥ(x)

=

⎧

⎪

⎪

1

(37)

C e ≥C

 log



m X σN2

= W X

∞

0 log

m X σ2

N

x

(2Ω/κ)(κ/2)m X Γ((κ/2)m X)x(κ/2)m X −1e− κx/2Ω dx

m X κ; m X β

.

(38)

Remark 6 Notice that when κ = 2, the fading model for

each element of A reduces to Rayleigh distribution, which

corresponds to the classic narrowband wireless communica-tion channel So we expect that the results obtained for this

5 CAPACITY VERSUS OUTAGE PROBABILITY

The results we have obtained in the previous sections apply to the case where the fading matrix is ergodic and there are no constraints on the decoding delay on the receiver In practical communication systems, we often run into the case where the fading 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, the fading matrix is clearly not ergodic We suppose that the fad-ing matrix still has the distribution defined in the previous sections In this case it is more important to investigate the channel capacity in the sense of outage probability An out-age is defined as the event where the communication channel

fading matrix A The set is the largest possible set for which

Pout(R) =Pr

A∈ / ΘR



=Pr

C |A< R

=Pr

C |A≤ R

, (39)

that is, the outage probability can be actually viewed as the cumulative distribution function (cdf) of the conditional Shannon capacity Notice that the last equality of the above

function 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

Thus its cdf is as follows:

P η(x) =Pr{ η ≤ x }

=

x

0



κ

κ/2 1



κ

κ

 ,

(40)

γ(α, z) =

z

0e− x x α −1dx. (41)

Pout(R) =Pr



W Xlog

σ N2

≤ R

=Pr



η ≤ σ N2

S

= P η

σ2

N

S

Γ(κ/2)γ



κ

.

(42)

(ii) SIMO case

its cdf as follows:

PΥ(x) =Pr{Υ≤ x }

=

x

0

1



κ



.

(43)

capac-ity can be derived as

C |A= W Xlog

Thus the outage probability turns out to be

Pout(R) =Pr



W Xlog

σ2

N

SA TA ≤ R

=Pr



S

= PΥ

σ2

N

S



κ

.

(45)

(iii) MISO case

First, we have



κ

2Ωx



Pout(R) =Pr

C |A≤ R

≤Pr



W Xlog

m X σN2

AAT ≤ R

=Pr



S

= PΥ

m X σ2

N

S

=Γ(κ/2)m1 X

γκ

= Pout(R).

(47)

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 found

re-spectively

Figure 1depicts the variation of channel capacityC ewith

figure that even though the channel capacity increases with

2.31%.

Figure 2demonstrates the relationship between channel capacity and SNR for SISO case, which shows that when SNR

SNR This is a result coinciding with our expectation

Figure 3shows 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

al-most logarithmically This phenomenon is similar to the cor-responding one in the case of the Rayleigh fading channels

Figure 4shows the relationship between the lower bound

of 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

C e

SNR= −10 dB

SNR=0 dB

SNR=10 dB

Figure 1: Variation of ergodic channel capacityC e(in nats/s/Hz)

with the numberκ for SISO case.

SNR 0

0.5

1

1.5

2

2.5

C e

Figure 2: Variation of ergodic channel capacityC e(in nats/s/Hz)

with the ratio SNR (in dB ) for SISO case

when the signal-to-noise ratio is low, the benefit obtained by

0

0.2

0.4

0.6

0.8

1

C e

Figure 3: Variation of ergodic channel capacityC e(in nats/s/Hz) with the number of receive antennasm Y for SIMO case (SNR =

−10 dB)

anten-nas is very limited as far as the average capacity is concerned

number of receiver antennas can obtain more benefit in channel capacity than increasing the number of transmit an-tennas Principally, the channel capacity could be increased infinitely 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 the best advantage of the multiple transmit antenna systems (for the case of single receive antenna) The reason for this phe-nomenon is two fold First, the power is constrained to be

anten-nas, while no such constraint is applied to receive antennas Second, it is assumed that the receiver possesses the full knowledge about the channel state

respectively From these figures, it can be observed that for

a given SNR, the outage probability decreases considerably with the number of receive antennas in the range of whole

transfer information at this rate is of little practical interest Therefore, we can conclude that increasing the number of transmit antennas is of some significance at a transmission rate of practical communications with tolerable outage prob-ability

0 5 10 15

0.092

0.093

0.094

0.095

C e

(a) SNR= −10 dB

2

2.05

2.1

2.15

2.2

2.25

2.3

2.35

2.4

C e

(b) SNR=+10 dB

Figure 4: Variation of the lower bound of the ergodic channel

ca-pacityC e(in nats/s/Hz) with the number of transmit antennasm X

for MISO case

can be calculated in the following way Notice the fact that

received power from the useful signals should be

S Y =

m X



k =1

a2

k · S

m X =

m X

k =1a2

k

smaller the variance of the received power from the useful

we have

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pout

(a) SNR= −10 dB

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pout

(b) SNR=0 dB

0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pout

(c) SNR=+10 dB

Figure 5: Outage probabilityPout versus transmission rateR/W X

(in nats/s/Hz) for variousκ 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

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pout

(a) SNR= −10 dB

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pout

(b) SNR=0 dB

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pout

(c) SNR=+10 dB

Figure 6: Outage probabilityPout versus transmission rateR/W X

(in nats/s/Hz) for variousm and SNR for SIMO case (κ =4)

this extreme case, we obtain

R

W X =log

σ N2

The above says that the channel capacity will approach a

FromFigure 7, we can see thatR1andR2satisfy the relation-ships

2.3979 for SNR being −10 dB, 0 dB, and +10 dB, respectively

R c

7 CONCLUDING REMARKS

In this paper, the analytic expression for the ergodic chan-nel capacity or its lower bound of wireless communication systems 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 channel capacity are also presented Numerical results are provided to demonstrate the dependence of the channel capacity on var-ious kinds of channel parameters It is shown that increasing the number of receive antennas can obtain more benefit in channel capacity than increasing the number of transmit an-tennas Principally, the channel capacity could be increased infinitely 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 transmit antenna systems (irrespective of all other parameters consid-ered herein) as far as channel capacity is concerned for MISO case We have also observed that when the signal-to-noise ratio is low, the benefit in average capacity obtained by

is very limited We have shown numerically that for a given

signal-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 when the communication rate is lower than the critical

and the signal-to-noise ratio of the system at the transmit-ter side We can roughly say that it is not beneficial to use multiple transmit antennas if the required transmission rate (normalized by system bandwidth) is higher than the critical transmission rate

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

0 0.05 0.1 0.15 0.2 0.25 0.3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pout

(a) SNR= −10 dB

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pout

(b) SNR=0 dB

0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pout

(c) SNR=+10 dB

Figure 7: The upper bound of the outage probabilityPoutversus

transmission rateR/W X(in nats/s/Hz) for variousm Xand SNR for

MISO 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 their helpful 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-antenna

systems in the presence of Rician fading,” in Proceedings of

IEEE 56th Vehicular Technology Conference (VTC ’02), vol 4,

pp 1963–1967, Vancouver, BC, Canada, September 2002 [4] M Nakagami, “Them-distribution - a general formula of

in-tensity distribution of rapid fading,” in Statistical Methods in

Radio 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-wide bandwidth 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, Academic

Press, San Diego, Calif, USA, 2nd edition, 1997

[7] I S Gradshteyn and I M Ryzhik, Table of Integrals, Series and

Products, Academic Press, New York, NY, USA, 1980, corrected

and 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, John

Wiley & Sons, New York, NY, USA, 1982

[10] E Biglieri, J Proakis, and S Shamai, “Fading channels:

information-theoretic and communications aspects,” IEEE

Transaction 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, both

in electrical engineering from Xidian Uni-versity, Xi’an, China, and the Ph.D degree

in automatic control in 1993 from Beijing University of Aeronautics and Astronautics, Beijing, China In the past years, he held Alexander-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 Space Science and Applied Research, Chinese Academy of Sciences, as

an 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, including the National Natural Science Award in 1999 from the Chinese gov-ernment, the Science and Technology Achievement Award in 1997

capac-ity of the Nakagami fading channels The conservativeness of

the lower bound comes from the diagonal assumption on Q.

If, on the other hand, Q is nondiagonal, some kind of. .. the outage probability can be actually viewed as the cumulative distribution function (cdf) of the conditional Shannon capacity Notice that the last equality of the above

function of. .. Formulae on the outage probability about the channel capacity are also presented Numerical results are provided to demonstrate the dependence of the channel capacity on var-ious kinds of channel