Performance Analysis of Transmit Diversity Systems With

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Performance Analysis of Transmit DiversitySystems With Antenna Replacement

Seyeong Choi, Member, IEEE,Hong-Chuan Yang, Senior Member, IEEE,and Young-Chai Ko, Senior Member, IEEE

Abstract—We propose a new closed-loop transmit diversity scheme formultiple-input–multiple-output (MIMO) diversity systems based on or-thogonal space–time block coding (OSTBC). The receiver of the proposedscheme checks the output signal-to-noise ratio (SNR) of the space–timedecoder against an output threshold and requests the transmitter toreplace the transmit antenna resulting in the poorest path with an unusedantenna if the output SNR is below the threshold. We provide someinteresting statistical analysis and obtain closed-form expressions for thecumulative distribution function (cdf), the probability density function(pdf), and the moment-generating function of the received SNR. We showthrough numerical examples that the proposed scheme offers a signifi-cant performance gain with a very minimal feedback load over existingopen-loop MIMO diversity systems, and for a properly chosen threshold,its performance is commensurate with a more complicated generalized-selection-combining-based transmit diversity system while requiring amuch smaller feedback load.

Index Terms—Diversity techniques, fading channels, multiple-input–multiple-output (MIMO), performance analysis, switched diversity,transmit diversity.

I. INTRODUCTION

Future wireless communication systems should support not onlyhigh spectral efficiency but good link reliability as well. Antennadiversity systems with multiple transmit and/or receive antennas cansignificantly increase the reliability of wireless fading channels [1].Well-known receive diversity combining techniques include maximalratio combining (MRC), equal-gain combining, selection combining(SC), and switch-and-stay combining [2], [3]. Meanwhile, the mainadvantage of transmit diversity is that diversity gain can be obtainedfor downlink transmission without implementing multiple antennas atthe mobile station. Transmit diversity systems based on an orthogonalspace–time block code (OSTBC) have received considerable interest[4]–[7]. Its two-antenna special case, i.e., the so-called Alamoutischeme [8], has been incorporated into third-generation standards.

In general, OSTBC systems can achieve full diversity gain withsimple linear processing at the receiver and without any knowledge ofthe fading channels at the transmitter side. However, when the numberof transmit antennas is greater than two, the OSTBC will suffer a rateloss. Moreover, the more the transmit antennas are used, the larger thesignal-to-noise ratio (SNR) loss occurs due to power spreading over

Manuscript received August 17, 2006; revised February 7, 2007, March 20,2007, and March 21, 2007. This work was supported by the Ministry ofKnowledge Economy, Korea, under the Information Technology ResearchCenter (ITRC) support program supervised by the Institute of InformationTechnology Advancement (IITA) under Grant IITA-2008-C1090-0801-0037.This paper was presented in part at the IEEE International Symposium onWireless Pervasive Computing, Phuket, Thailand, January 2006. The reviewof this paper was coordinated by Prof. N. Arumugam.

S. Choi is with the Department of Electrical Engineering, Texas A&MUniversity at Qatar, Doha, Qatar (e-mail: [email protected]).

H.-C. Yang is with the Department of Electrical and Computer Engineering,University of Victoria, BC V8W 3P6, Canada (e-mail: [email protected]).

Y.-C. Ko is with the School of Electrical Engineering, Korea University,Seoul 136-701, Korea (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2007.912957

0018-9545/$25.00 © 2008 IEEE

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 4, JULY 2008 2589

Fig. 1. System model of the proposed transmit diversity systems.

multiple antennas. A viable solution to these issues is antenna selectionat the transmitter side. In particular, the performance of multiple-input–multiple-output (MIMO) systems with single antenna selectionat the transmitter and traditional MRC at the receiver was studied in[9]–[11]. Molisch et al. [12] applied the idea of generalized selectioncombining (GSC) [13]–[17], where a fixed number of best antennasare selected, to multiple transmit antenna selection in MIMO systems.Antenna subset selection was also considered using different criteriain [18]–[20]. Finally, the performance of the OSTBC, with emphasison the Alamouti scheme, with antenna subset selection in variouspractical scenarios was investigated in [21]–[23]. In summary, whilerequiring a certain amount of closed-loop feedback, these antennaselection schemes incur much less of a rate/power loss compared toconventional space–time block code (STBC)-based transmit diversitysystems.

In this paper, we propose a new closed-loop transmit diversityscheme for OSTBC-based systems. With the proposed scheme, allbut one transmit antenna are used for OSTBC transmission, andthe remaining transmit antenna is reserved for the purpose of an-tenna replacement. Whenever the combined SNR at the receiver afterspace–time decoding is below a certain threshold, the transmitterreplaces the antenna resulting in the weakest path with the unusedantenna. This transmit diversity with the antenna replacement schemecan be viewed as a suboptimal antenna subset selection scheme. Itrequires not only a smaller number of pilot channels for channelestimation but also a much lower feedback load. The proposed schemeincludes one of the switched transmit diversity schemes for the 2 × 1Alamouti’s OSTBC studied in [24] and [25] as a special case. It canbe easily implemented in the future generation of cellular systemswith a minimal change of the standard and a minimal increment ofcomplexity.

This paper extends its corresponding conference version [26] byproviding more analytical details on the statistical derivation, nu-merically illustrating the existence of optimal switching thresholds,and accurately quantifying the average feedback rate of the proposedscheme. We carry out a thorough performance analysis of the proposedsystem. In particular, we derive the statistics of the receiver outputSNR, including the probability density function (pdf), the cumula-

tive distribution function (cdf), and the moment-generating function(MGF). We then present the exact closed-form expressions for theperformance measures, such as the outage probability and the averageprobability of error. We also consider the optimization of the proposedsystem in terms of the output threshold and the number of transmitantennas. In addition, we investigate the average feedback load ofthe proposed scheme in comparison with existing antenna selectionschemes. We show that the proposed system can offer nearly the sameperformance as GSC-based antenna selection systems but with a muchsmaller feedback load. While our presentation and analysis are basedon a single receive antenna over independent identically distributed(i.i.d.) Rayleigh fading channels, the extension to multiple receiveantennas and to generalized fading environments is straightforward.To ensure the correctness of the analytical results in this paper,we have double-checked our derivation and verified the closed-formexpressions through Monte Carlo simulations.

II. SYSTEM MODEL

A. System and Channel Models

We consider the closed-loop transmit diversity systems based on theOSTBC in Fig. 1. The transmitter is equipped with N + 1 antennas,where N ≥ 2, whereas the receiver possesses one antenna. The gener-alization to the multireceive-antenna case is straightforward. At anytime instant, N out of the total N + 1 transmit antennas are usedfor orthogonal space–time transmission, and the remaining antennais used for the antenna replacement purpose. It is worth noting thatif more antennas are used for transmission with the OSTBC, thesystem will suffer both rate loss and power spread while exploringadditional diversity paths. As will be shown later in this paper, theproposed setup will provide better performance than the scenariowhere all N + 1 antennas are used under the same total transmit powerconstraint.

Let γi denote the instantaneous received SNR for the ith transmitantenna, i = 1, 2, . . . , N + 1. Then, the total received SNR afterspace–time decoding is given by

∑N

i=1γi, where we assume that the

first N antennas are currently being used without loss of generality.We assume that the signals from N + 1 antennas experience i.i.d.


Rayleigh fading. As such, the faded SNR γi, i = 1, 2, . . . , N + 1follows the same exponential distribution, with the common pdf andcdf given by

pγi(x) =

1

γexp

(−x

γ

), x ≥ 0 (1)

Pγi(x) = 1 − exp

(−x

γ

), x ≥ 0 (2)

respectively, where γ is the common average faded SNR.

B. Mode of Operation and Feedback Analysis

With the proposed transmit diversity scheme, the receiver willperiodically, usually at the rate of channel coherence time, comparethe total received SNR

∑N

i=1γi with a certain fixed threshold, which

is denoted by γT . If∑N

i=1γi is greater than or equal to γT , then no

antenna replacement happens. Whenever∑N

i=1γi is smaller than γT ,

the receiver will request the transmitter to replace the worst transmitantenna, i.e., the one resulting in the smallest received SNR, withthe unused (N + 1)th antenna. Based on the aforementioned mode ofoperation, we can see that the final combined SNR, which is denotedby γt, is mathematically given by

γt =

∑N

i=1γi,

∑N

i=1γi≥γT∑N

i=1γi−min{γi, i = 1, 2, . . . , N}

+γN+1,∑N

i=1γi <γT .

(3)

The proposed scheme has several complexity advantages in compari-son with existing antenna selection schemes, such as the (N + 1)/NGSC-based scheme. Note that (N + 1)/N GSC corresponds to theantenna selection scheme where the N strongest antennas amongthe N + 1 available ones are selected. First, our proposed schemerequires N pilot channels for channel estimation, whereas the GSC-based scheme requires N + 1 pilot channels. As such, our schemecan be implemented in the future generation of cellular systems with aminimal change to the standard and a minimal increase of complexity.Another advantage of the proposed scheme is the reduction of thefeedback load. In particular, with the proposed scheme, the receiverfeedbacks one bit of information in the case of

∑N

i=1γi ≥ γT to indi-

cate that there is no need for antenna replacement or �1 + log2 N� bitsof information in the case of

∑N

i=1γi < γT to inform the transmitter

that the antenna replacement is required and which transmit antennais to be replaced, where �x� denotes the smallest integer not smallerthan x. Note that rather than constant, the feedback load is dependenton the channel conditions and the predetermined threshold. Hence,we can quantify the average feedback load of the proposed systemin terms of the average number of feedback bits, which is denotedby NA, as

NA =1 · Pr

[N∑

i=1

γi≥γT

]+�1+log2 N� · Pr

[N∑

i=1

γi <γT

]. (4)

For an i.i.d. Rayleigh fading scenario, it can be shown that (4) spe-cializes to

NA = 1 + �log2 N� ·

(1 − e

− γTγ

N−1∑i=0

(γTγ

)i

i!

). (5)

Note that the amount of feedback required by the (N + 1)/N GSC-based antenna selection scheme is always �log2(N + 1)� bits, con-

Fig. 2. Average number of feedback bits versus the output threshold γT ofthe proposed systems and (N + 1)/N GSC for the various values of N overi.i.d. Rayleigh fading channels with γ = 10 dB.

taining the index of the worst antenna. In general, the amount offeedback bits required to select L antennas out of N + 1 antennas is�log2

(N+1

L

)�, which becomes the smallest when L = 1 or N .

Fig. 2 shows the average number of feedback bits versus theoutput threshold γT of the proposed scheme and the (N + 1)/N GSCscheme with γ = 10 dB. Also, the optimum switching thresholds thatminimize the average bit error rate (BER) for chosen parameters areindicated. These optimum switching thresholds can be obtained usingthe numerical method and will be intensively investigated later on.From this figure, we can see that as the output threshold increases,the average number of feedback bits of the proposed system increasesfrom 1 to �1 + log2 N�, as expected. Also, we can verify that thenumber of feedback bits required at the optimum threshold is less thanthat of the GSC scheme. This saving in feedback comes, of course,with a certain amount of performance loss. As we will show in thefollowing accurate performance analysis, the proposed system onlyexperiences a slight loss of performance in comparison with GSC-based systems.

III. STATISTICS OF THE COMBINED SNR

In this section, we derive the statistics of the final combined SNR γt.In particular, closed-form expressions of the pdf, the cdf, and the MGFof γt are obtained over i.i.d. Rayleigh fading channels. These resultsare then applied to the performance analysis of the proposed transmitdiversity systems in Section IV.

Based on the mode of antenna replacement operation summarizedin (3), the cdf of γt, i.e., Pγt(x), can be written as

Pγt(x) = Pr[γt < x]

= Pr

[γT ≤

N∑i=1

γi < x

]

+ Pr

[N∑

i=1

γi < γT &

N∑i=1

γi

− min{γi, i=1, 2, . . . , N} + γN+1 <x

]. (6)


To simplify the notation, let us denote∑N

i=1γi − min{γi, i =

1, 2, . . . , N} by ΓN−1. Note that ΓN−1 is the sum of the first N − 1order statistics from a sample of size N . Now, we can rewrite (6) as (7),shown at the bottom of the page. Since γi, i = 1, 2, . . . , N + 1, areindependent random variables, the probability Pr[ΓN−1 + γN+1 <

x|∑N

i=1γi < γT ] can be calculated as the probability that the sum of

two independent random variables, i.e., γN+1 and ΓN−1|∑N

i=1γi <

γT , is less than x, i.e.,

Pr

[ΓN−1 + γN+1 < x|

N∑i=1

γi < γT

]

= Pr

[γs ≡ γN+1 +

(ΓN−1|

N∑i=1

γi < γT

)< x

](8)

where we have defined a new random variable γs = γN+1 +

(ΓN−1|∑N

i=1γi < γT ).

Jointly considering (7) and (8), we can write the pdf of the combinedSNR, i.e., pγt(x), as

pγt(x)=

p∑N

i=1γi

(x)+pγs(x)γT∫0

p∑N

i=1γi

(z)dz, x ≥ γT

pγs(x)γT∫0

p∑N

i=1γi

(z)dz, 0 ≤ x < γT

(9)

where p∑N

i=1γi

(x) is the pdf of the sum of N i.i.d. fading SNRs,

which is available in closed form for most fading scenarios, and pγs(x)

denotes the pdf of the sum γN+1 + (ΓN−1|∑N

i=1γi < γT ), which

can be calculated as the convolution of two individual pdfs as

pγs(x)=

x∫0

pγN+1(x−z) pΓN−1|

∑N

i=1γi<γT

(z)dz, x≥0. (10)

Noting that pγN+1(x) is readily available, we now derive the con-ditional pdf p

ΓN−1|∑N

i=1γi<γT

(x). If we define γi:N as the ith

largest among γi, i = 1, 2, . . . , N , then we can rewrite∑N

i=1γi =

ΓN−1 + γN :N . Hence, the conditional pdf can be calculated from thejoint pdf of ΓN−1 and γN :N , which is denoted by pΓN−1,γN:N (x, y)[27, eq. (3)]. In particular, it can be shown that p

ΓN−1|∑N

i=1γi<γT

(x)

is given by

pΓN−1|

∑N

i=1γi<γT

(x)

= pΓN−1|ΓN−1+γN:N <γT(x)

=1

Pr[∑N

i=1γi < γT ]

min{γT −x, xN−1}∫

0

pΓN−1,γN:N (x, y)dy

0 ≤ x < γT (11)

where the integration upper limit is based on the condition x + y < γT

and the support of the joint pdf, 0 ≤ y ≤ x/(N − 1).For the i.i.d. Rayleigh fading environment, the joint pdf

pΓN−1,γN:N (x, y) is given by [27, eq. (9)]

pΓN−1,γN:N (x, y)=N !

(N−1)!(N−2)!γN[x−(N−1)y](N−2) e

− x+y

γ

y ≥ 0, x ≥ (N − 1)y. (12)

After carrying out the integration, the conditional pdf ofΓN−1|

∑N

i=1γi < γT can be obtained as (13), shown at the

bottom of the page.

Pγt(x) =

{Pr

[γT ≤

∑N

i=1γi < x

]+ Pr

[∑N

i=1γi < γT & ΓN−1 + γN+1 < x

], x ≥ γT

Pr[∑N

i=1γi < γT & ΓN−1 + γN+1 < x

], 0 ≤ x < γT

=

Pr[γT ≤

∑N

i=1γi < x

]+ Pr

[ΓN−1 + γN+1 < x|

∑N

i=1γi < γT

]×Pr

[∑N

i=1γi < γT

], x ≥ γT

Pr[ΓN−1 + γN+1 < x|

∑N

i=1γi < γT

]× Pr

[∑N

i=1γi < γT

], 0 ≤ x < γT

(7)

pΓN−1|

∑N

i=1γi<γT

(x) =1

Pr[∑N

i=1γi < γT ]

{∫ x/(N−1)

0pΓN−1,γN:N (x, y)dy, 0 ≤ x < N−1

NγT∫ γT −x

0pΓN−1,γN:N (x, y)dy, N−1

NγT ≤ x < γT

=1

Pr[∑N

i=1γi < γT ]

Ne− x

γ

(N − 2)!γN

N−2∑k=0

(N − 2

k

)(1 − N)kk!γk+1xN−k−2

×

(1 − e

− xγ(N−1)

∑k

p=0

(x

γ(N−1)

)p

p!

), 0 ≤ x < N−1

NγT(

1 − e− γT −x

γ∑k

p=0

(γT −x

γ

)p

p!

), N−1

NγT ≤ x < γT

(13)


After successive substitution of (13) into (10) and (10) into (9) andintegration, while noting that for Rayleigh fading, pγN+1(x) is givenin (1), and p∑N

i=1γi

(x) is given by

p∑N

i=1γi

(x) =e− x

γ

γ

N−1∑w=0

(xγ

)w − w(

xγ

)w−1

w!(14)

we finally obtain a closed-form expression for the pdf of the combinedSNR, i.e., γt, of the proposed transmit diversity systems over the i.i.d.Rayleigh fading environment in (15)–(17), shown at the bottom ofthe page.

The closed-form expressions for the cdf and MGF of γt can be rou-tinely obtained, after some manipulations, in (18)–(22), respectively,shown at the top of the next page. Note that for the particular case ofN = 2, (15), (18), and (22), respectively, reduce to the previous resultsfor a dual-branch case [24, eqs. (4)–(6)]. Also, by letting γT = 0,we can obtain the statistics for the conventional N -fold MRC, whichfurther verify our closed-form results.

IV. PERFORMANCE ANALYSIS

In this section, we apply the closed-form results from the previoussection to the performance analysis of the proposed transmit diversitysystems over Rayleigh fading channels. We examine the performancemeasures of the outage probability and the average error rate.

A. Outage Probability

The outage probability is the probability that the received SNR fallsbelow a threshold value γth. It can be calculated by replacing x withγth in (18), i.e.,

Pout = Pγt(γth). (23)

The optimum output threshold γ∗T for the minimum outage probability

can be shown as

γ∗T = γth. (24)

As such, from (18), we can obtain the optimum outage probabilityP ∗

out, which is actually the outage probability of (N + 1)/N GSC[15]. Intuitively, with the optimal choice of outage threshold γ∗

T =γth, the proposed system experiences outage if and only if the totalreceived SNR

∑N

i=1γi is less than γ∗

T , the antenna replacementoccurs, and the final combined SNR is still less than γth. Under thesame condition, the combined SNR of (N + 1)/N GSC is also smallerthan γth.

Fig. 3 shows the outage performance of the proposed transmitdiversity systems. For each value of N , we plot the outage probabilityof the proposed scheme with the fixed output threshold γT = 0 dB,the optimal output threshold γth, and N -fold MRC. Note that forall cases, the proposed systems always have better performance than

pγt(x) =

ζ(γ, γT , N, x), 0 ≤ x < N−1N

γT

ζ(γ, γT , N, N−1

NγT

)+ ξ(γ, γT , N, x) − ξ

(γ, γT , N, N−1

NγT

), N−1

NγT ≤ x < γT

ζ(γ, γT , N, N−1

NγT

)+ ξ(γ, γT , N, γT ) − ξ

(γ, γT , N, N−1

NγT

)+ e

− xγ

γ

∑N−1

w=0

(xγ

)w−w

(xγ

)w−1

w!, x ≥ γT

(15)

ζ(γ, γT , N, α) =

α∫0

pγN+1(x − z)

z/(N−1)∫

0

pΓN−1,γN:N (z, y)dy

dz

=Ne

− xγ

(N − 2)!γN

N−2∑k=0

(N − 2

k

)(γ(1 − N))k k!

×

(αN−k−1

N − k − 1− (γ(N − 1))N−k−1

k∑p=0

(N − k + p − 2)!

p!

(1 − e

− αγ(N−1)

N−k+p−2∑t=0

(α

γ(N−1)

)t

t!

))(16)

ξ(γ, γT , N, α) =

α∫0

pγN+1(x − z)

γT −z∫

0

pΓN−1,γN:N (z, y)dy

dz

=Ne

− xγ

(N − 2)!γN

N−2∑k=0

(N − 2

k

)(γ(1 − N))k k!

×

(αN−k−1

N−k−1− (−γ)N−k−1e

− γTγ

k∑p=0

p∑t=0

(N−k+p−t−2)!

p!

(p

t

)(γT

γ

)t(

1−eαγ

N−k+p−t−2∑w=0

(−α

γ

)w

w!

))(17)


Pγt(x) =

x∫0

pγt(z)dz

=

χ(γ, γT , N, x), 0 ≤ x < N−1N

γT

χ(γ, γT , N, N−1

NγT

)+ ψ(γ, γT , N, x) − ψ

(γ, γT , N, N−1

NγT

), N−1

NγT ≤ x < γT

χ(γ, γT , N, N−1

NγT

)+ ψ(γ, γT , N, γT )

−ψ(γ, γT , N, N−1

NγT

)+ µ(γ, γT , N, x), x ≥ γT

(18)

χ(γ, γT , N, α) =

α∫0

ζ(γ, γT , N, x)dx (19)

ψ(γ, γT , N, α) =

α∫0

[ζ(γ, γT , N,

N−1

NγT

)+ξ(γ, γT , N, x)−ξ

(γ, γT , N,

N−1

NγT

)]dx (20)

µ(γ, γT , N, α) =

α∫0

[ζ(γ, γT , N,

N − 1

NγT

)+ ξ(γ, γT , N, γT ) − ξ

(γ, γT , N,

N − 1

NγT

)+

e− x

γ

γ

N−1∑w=0

(xγ

)w − w(

xγ

)w−1

w!

]dx (21)

Mγt(s) =N

(N − 2)!γN(

1γ− s

) N−2∑k=0

(N − 2

k

)(γ(1 − N))k k!

×

{(N−k)!(

1γ−s

)N−k−1

(1−e

(s− 1

γ

)γT

N−k−1∑v=0

((1γ−s

)γT

)v

v!

)+

γN−k−1T e

(s− 1

γ

)γT

N−k−1−

k∑p=0

(N−k+p−2)!

p!(γ(N−1))N−k−1

×

[1+

N−k+p−2∑t=0

((N−1)(sγ−1)

(N−sγ(N−1))t+1×

(1−e

(N−1

Ns− 1

γ

)γT

t∑v=0

((1γ− N−1

Ns)

γT

)v

v!

)−

(γTNγ

)te

(N−1

Ns− 1

γ

)γT

t!

)]

+ (−1)N−k(γ)N−k−1e

(s− 1

γ

)γT

k∑p=0

p∑t=0

N−k+p−t−2∑w=0

(N − k + p − t − 2)!

p!

(γT

γ

)t (p

t

)

×

[(− γT

γ

)w(e−

γTN

s(

N−1N

)w − 1)

w!− sγ − 1

(sγ)w+1×

(e−

γTN

s

w∑v=0

(−N−1

NsγT

)v

v!−

w∑v=0

(−sγT )v

v!

)]}

+e

(s− 1

γ

)γT

γ

N−1∑w=0

1

(1 − sγ)w

(w∑

v=0

((1γ− s

)γT

)v(1γ− s

)v!

− γ

w−1∑v=0

((1γ− s

)γT

)v

v!

)(22)

N -fold MRC, particularly when the outage threshold γth is smallerthan γT . Therefore, the antenna replacement can significantly improvethe outage performance. As earlier mentioned, the outage performanceof the proposed systems with the optimal output threshold γ∗

T isactually the same as that of the (N + 1)/N GSC scheme. We cansee from Fig. 3 that as the number of antennas used for trans-mission, i.e., N , increases, the outage performance of the proposedscheme with the fixed switching threshold approaches that of the(N + 1)/N GSC scheme when the outage threshold γth is smallerthan γT .

B. Average Probability of Error

The MGF-based method for the evaluation of the average errorrate over fading channels can be used. For example, the averagesymbol error rate (SER) of M -ary phase-shift keying (PSK) signals isgiven by

PS(E) =1

π

(M−1)π/M∫0

Mγt

(− gpsk

sin2 φ

)dφ (25)


Fig. 3. Outage probability of the proposed systems and N -fold MRC for thevarious values of N over i.i.d. Rayleigh fading channels with γ = 0 dB.

Fig. 4. Average BER of binary PSK (BPSK) versus the output threshold γT

of the proposed systems, (N + 1)/N GSC, and N -fold MRC for the variousvalues of N over i.i.d. Rayleigh fading channels with γ = 0 dB.

where gpsk = sin2(π/M). Since the explicit closed-form solution ofthe optimum switching threshold γ∗

T for the minimum SER is notpossible to obtain, we rely on the numerical method to obtain thenumerical solution for γ∗

T in (25).We examine the error performance of the proposed transmit di-

versity systems in Figs. 4 and 5. In Fig. 4, the average BER of theproposed scheme is plotted as a function of the output threshold γT .As we can see, there is a general trend that the BER performance ofthe proposed scheme improves as γT increases. Note that althoughit is not easily visible from Fig. 4, there exists an optimal choice ofthe output threshold γT , which minimizes the average BER. Table Ipresents the numerical values of the average BER for the differentvalues of γT and N . From this table, we can see that there is a finite

Fig. 5. Average BER of BPSK versus the average SNR per branch γ of theproposed systems with the optimum switching threshold, (N + 1)/N GSC,and N -fold MRC for the various values of N over i.i.d. Rayleigh fadingchannels.

value of the optimal output threshold for the different values of N .In all cases, the proposed scheme provides considerable performanceimprovement over N -fold MRC for the relatively large values of γT ,and its performance approaches that of (N + 1)/N GSC for the largevalues of N . Note that the optimal values of γT are also indicatedin Fig. 2. If we jointly consider Figs. 2 and 4, we can concludethat the proposed scheme possesses the attractive feature of achievingnearly the same performance as that of the GSC-based scheme withlower complexity in terms of fewer pilot channels and much less ofa feedback load when N is large. The same conclusion can be drawnfrom Fig. 5, where we plot the average BER of the proposed schemewith the optimal values of γT in comparison with those of N -foldMRC and (N + 1)/N GSC. Again, when the number of transmitantennas N increases, the performance gap between the proposedscheme and the GSC-based scheme diminishes.

In Fig. 6, we compare the performance of the proposed transmitdiversity systems with that of the traditional Alamouti scheme andthe generalized OSTBC systems under the condition of common totaltransmit power. In particular, the average BERs of the three types ofsystems are plotted as a function of the total average received SNR.Note that if N antennas are used for transmission, then the averageSNR per path will be 1/N of the total average SNR. For the N = 2case, we can see that the proposed scheme considerably outperformsthe traditional Alamouti scheme. The proposed system can benefitfrom the third antenna through occasional antenna replacement. Whencomparing the performance of the proposed scheme with that of the(N + 1) × 1 orthogonal design, both of which are equipped withN + 1 antennas at the transmitter, we can see that the proposed schemeoffers about a 1-dB performance advantage. While a little surprising,this can be explained as the (N + 1) × 1 orthogonal STBC systemtries to take advantage of the additional antenna by further dividingthe transmit power, whereas the proposed system incurs less of a powerloss by exploiting the extra antenna through antenna replacement. Notealso that the (N + 1) × 1 OSTBC always suffers a larger rate loss thanthe N × 1 OSTBC. In conclusion, the proposed antenna replacementscheme is a viable closed-loop solution for OSTBC-based transmitdiversity systems.


TABLE IAVERAGE BER OF BPSK FOR THE VARIOUS VALUES OF THE OUTPUT THRESHOLD γT (IN DECIBELS) AND THE

NUMBER OF TRANSMIT ANTENNAS N WITH γ = 0 dB

Fig. 6. Average BER of BPSK versus the total received average SNR ofthe proposed systems with the optimum switching threshold, (N + 1) × 1orthogonal STBC, and pure Alamouti STBC for the various values of N overi.i.d. Rayleigh fading channels.

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