IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 8, AUGUST 2008 3079

Dynamic Transmit Power Allocation inSpace-Time Trellis Coded Systems

Agus Santoso, Yonghui Li, Member, IEEE, J. Kingsley A. Allan, Student Member, IEEE,and Branka Vucetic, Fellow, IEEE

Abstract—Space-time trellis codes (STTCs) have been shown toefficiently use transmit diversity to improve error performance.In existing STTCs, the transmit power is equally distributedacross all transmit antennas. In fact, when feedback informationis available at the transmitter, the equal power allocation strategyis not optimum regarding the error performance. In this paper,we propose a new scheme, referred to as space-time trelliscodes with dynamic transmit power allocation (STTCs/DTPA),when partial channel state information (CSI) is available atthe transmitter. A closed-form pairwise error probability (PEP)upper bound is derived. The optimum power allocation schemeand the code design criteria, based on the PEP upper bound,are proposed. Theoretical and simulation results show thatthe proposed scheme performs much better in terms of errorprobability than the general open-loop STTCs and other closed-loop transmit diversity schemes. For example, the new 16-stateSTTCs/DTPA scheme is 5.6 dB better than the standard 16-stateChen/Yuan/Vucetic code for three transmit antennas.

Index Terms—Antenna selection, diversity, multiple-inputmultiple-output (MIMO), space-time trellis codes, weighting.

I. INTRODUCTION

I INFORMATION theoretic studies have shown that amultiple-input-multiple-output (MIMO) system can sig-

nificantly increase the system capacity [1], [2]. Space-timetrellis codes (STTCs), which jointly take into account chan-nel coding, modulation and transmit diversity, have drawnparticular attention due to their ability to exploit the MIMOcapacity and improve error performance [3]. In [3], the rankand determinant criteria were proposed as the design criteriafor STTCs over quasi-static flat Rayleigh fading channels.Subsequent computer searches were carried out in [4], [5]to find STTCs with improved error performance based onthe rank and determinant criteria. Instead of following thetraditional design rules [3], different design criteria wereproposed in [6]–[9], where, in particular, the role of Euclideandistance was emphasized. In [10]–[13], Ionescu et al. andJafarkhani et al. proposed a modified determinant criterionusing super-orthogonal STTCs, independently.

In transmit and receive diversity, large numbers of transmitand receive antennas are required to achieve a high diversityorder. However, as discussed in [3], the memory order hasto be very large for STTCs with high numbers of transmit

Manuscript received January 10, 2007; revised March 5, 2007; acceptedApril 15, 2007. The associate editor coordinating the review of this paper andapproving it for publication was N. Arumugam. This work was supported inpart by the Australia Research Council under Grant DP0559263.

The authors are with the Telecommunications Lab, School of Electrical &Information Engineering, University of Sydney, Sydney, NSW 2006, Australia(e-mail: {asantos, lyh, branka, kallan}@ee.usyd.edu.au).

Digital Object Identifier 10.1109/TWC.2008.070031.

antennas to achieve a full diversity order. For example, 64and 1024 states is the minimum memory requirement for the4-PSK STTCs with four and six transmit antennas, respec-tively, not to mention higher order constellations. Furthermore,multiple transmit antenna systems require the same numberof radio-frequency (RF) chains as the number of antennas,which presents a hardware challenge in terms of cost. Thus,antenna selection was introduced in [14], [15] to decreasethe number of RF chains but keep the benefits of multipleantennas. The antenna selection can achieve the full diversityorder regardless of the memory order of the baseline STTCsover quasi-static flat fading channels. In [14] and [15], antennaselection at the transmitter for low-rank matrix channels andantenna selection at the receiver are considered, respectively.Due to the size and power limitations, most current hand-helddevices can accommodate one or at most two antennas.

In existing STTCs with transmit antenna selection [16],two transmit antennas, which maximise the signal-to-noiseratio (SNR) at the receiver, are chosen and weighted equallyto transmit the baseline STTCs designed for two transmitantennas. In fact, when feedback information is available atthe transmitter, this power allocation strategy is not opti-mum regarding the error performance. Many different powerallocation strategies have been discussed only for uncodedsystems [17]–[23] and space-time block codes [24]–[28]. Thenatural extension of these results is to apply the proposedpower allocation strategies to STTCs. Following the sameapproach of capacity optimization as in [1], an optimumpower allocation strategy can be derived for systems with fullchannel state information (CSI) available at the transmitter.However, if only partial CSI is provided at the transmitter, theimplementation of the power allocation strategies for STTCsis not straightforward and presents a number of challenges.

In [29], we have proposed a weighted space time trelliscoding scheme for the system with two antennas when partialCSI available at the transmitter. In the proposed scheme, thetransmitter dynamically allocates the transmit power betweentwo antennas. The power allocation coefficients have beenfound based on theoretical analysis in [29]. However the powerallocation solutions found in [29] are not always optimal,especially for a high number of states, because of the loosenessof the bound derived in [29].

In this paper, we will consider a general MIMO system withan arbitrary number of antennas and design space-time trelliscodes with dynamic transmit power allocation (STTCs/DTPA),when partial CSI is available at the transmitter. More specifi-cally, the transmitter only needs to know which two transmit

1536-1276/08$25.00 c© 2008 IEEE

3080 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 8, AUGUST 2008

antennas are best among nT transmit antennas. Based on thisinformation, the transmitter selects two out of nT antennas toperform space time trellis coding and dynamically allocatestransmit power among two selected antennas. A new and moreaccurate closed-form pairwise error probability (PEP) upperbound is derived. The optimum power allocation scheme andthe code design criteria based on the new PEP upper boundare also proposed. Simulation results validate that the optimalpower allocation coefficients are consistent with the simulationresults. The STTCs/DTPA scheme is a good approach toachieving a high diversity order, and optimum coding andweighting gains, while maintaining a low decoding complexityat the receiver. We demonstrate that the proposed schemeperforms much better in error probability than the generalopen-loop STTCs [3], [4], [6], [11], [30] and the other closed-loop transmit diversity schemes [16], [31], [32].

The remainder of this paper is organized as follows. Sec-tion II describes the system and channel model. In Section III,we investigate the optimum power allocation strategy andderive the received SNR gain for the STTCs/DTPA. Wealso present the code construction and design criteria forthe STTCs/DTPA based on PEP bound in quasi-static flatRayleigh fading channels in Section III. In Section IV, thesimulation results are presented. Finally, conclusion is drawnin Section V.

II. SYSTEM AND CHANNEL MODEL

We consider a wireless communications system with nT

transmit antennas and a single receive antenna. It is assumedthat the channel state information (CSI) is perfectly availableat the receiver side and the total transmitted power is the sameas for one transmit antenna. The channel is modelled by annT × 1 vector h. The elements of h, denoted by hi, 1 ≤ i ≤nT , represent the fading coefficients between transmit antennai and the single receive antenna. The fading coefficients hi areconstant during one frame and vary independently from oneframe to another. They are modelled as independent samplesof complex Gaussian random variables with a zero mean andvariance of 0.5 per dimension.

In the STTCs/DTPA scheme, the receiver estimates thechannel h and informs the transmitter of which two antennasare the best in terms of the channel gain. Two out of nT

transmit antennas, which maximize the received SNR, arechosen for transmission and the power is optimally allocatedto each. We refer to this scheme as the (nT , 2) STTCs/DTPA.

Let h(1) and h(2) represent the fading coefficients for thebest antenna and the second best antenna, respectively. Letc(1)t and c

(2)t be the modulated signal transmitted at time t via

the best antenna and the second best antenna, respectively.Let w(1) and w(2) be the weighting coefficients for thebest antenna and the second best antenna, respectively. Thetransmitter transmits a fraction w2

(1) of the total power via thebest antenna and a fraction w2

(2) of the total power via thesecond best antenna. Here 0 ≤ w(1), w(2) ≤ 1 are parametersto be optimized and they should satisfy the normalizationconstraint

w2(1) + w2

(2) = 1. (1)

The received signal at time t, denoted by rt, t = 1, 2, · · · , l,is given by

rt =√

Es

(h(1)w(1)c

(1)t + h(2)w(2)c

(2)t

)+ nt, (2)

where l is the length of the frame, Es is the average energy ofthe signal transmitted, and nt is the additive white Gaussiannoise (AWGN) of the receive antenna at time t with the one-sided power spectral density N0.

III. PERFORMANCE ANALYSIS OF THE SPACE-TIME

TRELLIS CODES WITH DYNAMIC TRANSMIT POWER

ALLOCATION (STTCS/DTPA)

In this section, the PEP, and the code design criteria for theSTTCs/DTPA are presented.

A. The Pairwise Error Probability for STTCs/DTPA

In this subsection, the PEP for the STTCs/DTPA in quasi-static flat Rayleigh fading channels is derived. Two out ofnT transmit antennas, which maximize the received SNR, arechosen for transmission.

Assume that c is a transmitted code sequence matrix givenby

c =

(c(1)1 c

(1)2 · · · c

(1)l

c(2)1 c

(2)2 · · · c

(2)l

)(3)

where l is the frame length, c(1)t and c

(2)t represent the

transmitted modulated signal from the best antenna and thesecond best antenna, respectively, at time t. A maximum-likelihood (ML) receiver might decide erroneously in favorof another codeword matrix

e =

(e(1)1 e

(1)2 · · · e

(1)l

e(2)1 e

(2)2 · · · e

(2)l

)(4)

where e(1)t and e

(2)t represent another codeword, that was

decided in the receiver, from the best antenna and the secondbest antenna, respectively, at time t.

Let r (r ≤ 2) denote the rank of the 2× l difference matrixB(c, e) defined as

B(c, e) = e − c =

(e(1)1 − c

(1)1 · · · e

(1)l − c

(1)l

e(2)1 − c

(2)1 · · · e

(2)l − c

(2)l

)(5)

and λi be the eigenvalues of the distance matrix

A(c, e) = B(c, e)BH(c, e). (6)

Since A(c, e) is Hermitian, we have A(c, e) = VDVH , whereV is a unitary matrix with rows given by the eigenvectors ofA(c, e), VH denotes the transpose conjugate of V, and D isa real diagonal matrix whose elements are the eigenvalues ofA(c, e).

The conditional PEP is upper bounded as [3]

P (c → e | h) ≤ 12

exp(−d2(c, e)Es/4N0

)(7)

where d2(c, e) is the squared Euclidean distance between thetwo space-time codewords, c and e.

SANTOSO et al.: DYNAMIC TRANSMIT POWER ALLOCATION IN SPACE-TIME TRELLIS CODED SYSTEMS 3081

The PEP can be derived by averaging (7) with respect to thechannel coefficients under condition of |h(1)|2 > |h(2)|2

P (c → e) ≤∫ ∫

|h(1)|2>|h(2)|2P (c → e | h) (8)

fH(1)H(2)(h(1), h(2))dh(1)dh(2)

where fH(1)H(2)(h(1), h(2)) is the joint pdf of H(1) and H(2)

for any number of transmit antennas.Let us define

h(i) = riejθi (9)

Aij = |Aij |ejϕ. (10)

Then from (8), we have

P (c → e) ≤∫ 2π

0

∫ 2π

0

∫ ∞

0

∫ ∞

r2

P (c → e | r, θ) (11)

fR1R2(r, θ) |J(r, θ)| dr1dr2dθ1dθ2

where |J(r, θ)| = r1r2 is the determinant of the Jacobiantransformation matrix and the joint pdf of R1 and R2 for nT

transmit antennas is given by [33]

fR1R2(r, θ) =nT !

(nT − 2)!FnT −2(r2)f(r1)f(r2)

=nT (nT − 1)

π2(12){

nT −2∑i=0

(−1)i

(nT − 2

i

)(e−r2

2

)i}

e−r21e−r2

2

Let us define

α =Es

4N0w2

(1)A11 + 1 (13)

β =Es

4N0w2

(2)A22 + 1 (14)

γ =Es

4N0w(1)w(2)|A12|. (15)

By solving (11) analytically (detailed calculation is carried outin Appendix A), the final expressions for the upper bound ofthe PEP are1) For γ = 0

P (c → e) ≤nT (nT − 1)

2

{nT −2∑i=0

(−1)i

(nT − 2

i

)1

α2 + αβ + αi

}

= X1(c → e) (16)

2) For γ �= 0 and α−β−i �= 0 for any i ∈ {0, 1, · · · , nT −2}

P (c → e) ≤ nT (nT − 1)2

nT−2∑i=0

(−1)i

(nT − 2

i

)

1γ(α − β − i)

{1√

κ21 − 1

− 1√κ2

2 − 1

}

= X2(c → e) (17)

3) For γ �= 0 and and α−β−i = 0 for any i ∈ {0, · · · , nT−2}

P (c → e) ≤ nT (nT − 1)2

{κ2

2

(κ22 − 1)

32

}

nT −2∑i=0

(−1)i

(nT − 2

i

)1

2γ(β + i)

= X3(c → e) (18)

where

κ1 =α + β + i

2γ

κ2 =α

γ.

In the STTCs/DTPA scheme, two out of nT transmit anten-nas are chosen for transmission and we refer to it as a (nT ,2)STTCs/DTPA system. In Appendix B, by using Eqs. (16)-(18)we calculated the PEP upper bound for (3,2) STTCs/DTPAsystems. The results clearly show that a full diversity order 3can be achieved for the above two systems.

From (16) - (18), it is clear that the performance of theSTTCs/DTPA scheme depends on the weighting coefficients(w(1), w(2)) and the matrix A(c, e). Let us define

χ(c → e) = Max (X1(c → e), X2(c → e), X3(c → e)) (19)

where χ(c → e) is the global maximum. In order to optimizethe frame error rate (FER) performance, we should minimizethe maximum χ(c → e). For any numbers of transmit anten-nas, i.e. (nT , 2) case, the optimum STTC and the optimumpower allocation can be found from (16) - (18) by a computersearch. We will give more details in Section IV.

B. Design Criterion for STTCs/DTPA

Based on the derived PEP upper bound in (19), we canformulate the following Design Criterion for STTCs/DTPA.We consider a system with nT transmit antennas, where thetwo transmit antennas which maximize the received SNR atthe receiver are selected out of all the available transmit anten-nas to perform space-time trellis coding, and the transmittedsignals are weighted. If we define the total gain, includingthe weighting and coding gains, as the gain of the weightedcoded system over an uncoded system with no weighting andthe same diversity order, then the total gain from (19) isdetermined by the global maximum. In order to maximize thetotal gain, we should minimize the global maximum χ(c → e).The computer search is employed to obtain the best codewith joint optimum power allocation that minimizes the globalmaximum χ(c → e).

IV. SIMULATION RESULTS

In this section, we illustrate by simulation the perfor-mance of the STTCs/DTPA in quasi-static flat Rayleigh fadingchannels. In the simulation, each frame consists of 130 4-PSK information symbols. The performance curves are plot-ted against SNR. The standard Tarokh/Seshadri/Calderbank(TSC) codes [3], Baro/Bauch/Hansmann (BBH) codes [4], andChen/Yuan/Vucetic (CYV) codes [6] are considered as refer-ences. A single receive antenna is assumed. In the simulation,

3082 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 8, AUGUST 2008

TABLE ITHE GENERATOR SEQUENCES FOR THE STTCS/DTPA CODES

Code ν Generator Sequences χ %

STTCs/DTPA-TSC 2 g1 = [ (0,2) , (2,0) ] g2 = [ (0,1) , (1,0) ] 0.25 100%STTCs/DTPA-BBH 2 g1 = [ (2,2) , (1,0) ] g2 = [ (0,2) , (3,1) ] 0.25 100%STTCs/DTPA-CYV 2 g1 = [ (0,2) , (1,2) ] g2 = [ (2,3) , (2,0) ] 0.25 100%STTCs/DTPA-NEW 2 g1 = [ (2,3) , (2,2) ] g2 = [ (3,2) , (3,1) ] 0.06 100%

STTCs/DTPA-TSC 4 g1 = [ (0,2) , (2,0) , (0,2) ]g2 = [ (0,1) , (1,2) , (2,0) ]

0.22 80%

STTCs/DTPA-BBH 4 g1 = [ (0,2) , (2,0) , (0,2) ]g2 = [ (2,1) , (1,2) , (2,0) ]

0.24 80%

STTCs/DTPA-CYV 4 g1 = [ (1,2) , (1,3) , (3,2) ]g2 = [ (2,0) , (2,2) , (2,0) ]

0.06 90%

STTCs/DTPA-NEW 4 g1 = [ (3,1) , (0,2) , (2,0) ]g2 = [ (2,2) , (1,0) , (1,1) ]

0.05 90%

50 60 70 80 90 100

10−1

% Power allocated to the Best Antenna

Fra

me

Err

or R

ate

(FE

R)

50 60 70 80 90 100

10−1

% Power allocated to the Best Antenna

Fra

me

Err

or R

ate

(FE

R)

50 60 70 80 90 100

10−1

50 60 70 80 90 100

10−1

Fra

me

Err

or R

ate

(FE

R)

New 4 StatesNew 16 States

BBH 4 StatesBBH 16 States

TSC 4 StatesTSC 16 States

CYV 4 StatesCYV 16 States

% Power allocated to the Best Antenna % Power allocated to the Best Antenna

Fra

me

Err

or R

ate

(FE

R)

Fig. 1. The (2,2) STTCs/DTPA performance for different power allocations, SNR=15dB.

all schemes have the same spectral efficiency of 2 bits/s/Hzand all performance comparisons are conducted at a frameerror rate (FER) of 10−3. For all simulated codes, the optimumweighting coefficients are allocated to the transmit antennas.

Since it is impossible to optimize the code and the powerallocation coefficients w(1), w(2) separately, we need to doa joint search for the optimum code as well as the optimumw(1) and w(2). That is, for each code pair c and e, we need tovary the value of w2

(1) from 0.5 to 1 (w2(2) = 1−w2

(1)). In thecomputer search, we quantized the values of w2

(1) from 0.5 to1 with a step size of 0.1.

The code search can be conducted in the following steps.

(a) Choose a code generator polynomial (CGP).(b) By using the chosen CGP, randomly generate two

distinct codewords c and e.(c) Based on Eqs. (16) - (18), calculate their χ(c → e) with

various discrete values of w2(1) from 0.5 to 1. Go to step (b)

until the maximum χ(c → e) is found.(d) Go to step (a) until all the codes in an exhaustive code

search or all the chosen codes for a limited code search havebeen searched.

(e) Find the codes corresponding to the minimum maximumχ(c → e). They are the optimum codes. Through a joint searchfor the code and w2

(1), we can also find the optimum powerallocation coefficients w(1) and w(2) for the optimum code.

SANTOSO et al.: DYNAMIC TRANSMIT POWER ALLOCATION IN SPACE-TIME TRELLIS CODED SYSTEMS 3083

TABLE IIWEIGHTING GAIN VS. CODING GAIN FOR THE STTCS/DTPA

Code No Total Weighting Codingof Gain Gain Gain

States (dB) (dB) (dB)

Standard TSC 4 0.4 0 0.4Standard TSC 16 1.9 0 1.9

STTCs/DTPA - TSC 4 2 1.76 0.2STTCs/DTPA - TSC 16 2.7 1.14 1.56

Standard BBH 4 0.3 0 0.3Standard BBH 16 1.9 0 1.9

STTCs/DTPA - BBH 4 1.5 1.76 -0.3STTCs/DTPA - BBH 16 2.4 1.14 1.26

Standard CYV (Baseline) 4 0 0 0Standard CYV 16 1.4 0 1.4

STTCs/DTPA - CYV 4 2.5 1.76 0.7STTCs/DTPA - CYV 16 2.8 1.46 1.34

STTCs/DTPA - New Code 4 3 1.76 1.2STTCs/DTPA - New Code 16 3.6 1.46 2.14

50 60 70 80 90 1000

0.5

1

1.5

2

2.5

% Power allocated to the best antenna

Gai

n (d

B)

The Gain for the 4−State TSC Code with Two Transmit Antennas

50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3

% Power allocated to the best antenna

Gai

n (d

B)

The Gain for the 16−State TSC Code with Two Transmit Antennas

Weighting GainRelative Coding GainTotal Gain

Weighting GainRelative Coding GainTotal Gain

Fig. 2. Gain for different power allocations for the (2, 2) TSC STTCs/DTPA codes.

Following the above procedure, the computer code searchfor the STTCs/DTPA is employed to obtain the best param-eters for 4-state and 16-state codes. The generator sequencesfor the STTCs/DTPA codes are given in Table I, where χis the global maximum, % is the optimum percentage ofpower allocated to the best antenna obtained from theoreticalanalysis, gi is the generator matrix for transmit antenna i, ν isthe memory order and the total number of states is 2ν . Fig. 1shows the FER performance for the different power allocationsfor 4-state and 16-state codes in the case of two total transmitantennas. By comparing Table I and Fig. 1, we can see that thetheoretical optimum power allocation in Table I agrees with

the simulation results in Fig. 1.We define the weighting gain as the power gain of the

weighted system over the system with no weighting at thesame error probability, denoted by gw. It is given by [29]

gw =(

32w2

(1) +12w2

(2)

). (20)

Table II shows the total gain, weighting gain, and codinggain for the STTCs/DTPA. The total gain for any code isobtained directly from the simulation and is defined as thetotal gain relative to the 4-state 4-PSK CYV code with noweighting. The weighting gain is calculated directly from

3084 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 8, AUGUST 2008

(20). The coding gain for any code is obtained by subtractingthe weighting gain from the total gain of the code. For anycode, the STTCs/DTPA has the best FER performance sinceSTTCs/DTPA maximize the total gain, defined as the sumof the coding and weighting gains. The weighting gain isindependent of the code but depends only on the weightingcoefficients. However, the coding gain depends on manyadditional factors such as the weighting coefficients, type ofcode and the number of states.

It is shown in Fig. 2 that for a lower number of states (i.e.,4-state codes), the optimal power allocation is the same as inpure antenna selection, where only the best transmit antennais selected. This is due to the fact that the coding gain isvery marginal and the major component of the overall gain isthe weighting gain. For example, for the 4-state 4-PSK TSCcode, pure antenna selection can provide about 2.1 dB overallgain. It consists of 0.3 dB coding gain and 1.76 dB weightinggain. When 80% power allocation is employed, although itcan increase the coding gain to 0.6 dB, it also reduces theweighting gain to 1.1 dB. Thus, for the 4-state 4-PSK TSCcode, 80% power allocation can only provide 1.7 dB overallgain, that is 0.4 dB less than pure antenna selection. Therefore,pure antenna selection has the best performance for the 4-stateTSC code.

When the number of states increases, the coding gain alsoincreases and has a significant contribution to the overallperformance. Thus, the optimum power allocation attemptsto achieve a high coding gain by properly allocating thepower to the two antennas, as the number of states increases.In other words, both antennas should be used at highernumbers of states to obtain the optimum performance sincethe contribution of the coding gain is significant. Fig. 2 showsthat for the 16-state 4-PSK TSC code, the coding gain hasa more significant contribution on the overall gain comparedto the weighting gain. As shown in Fig. 2, the coding gainfor the 16-state TSC code decreases but the weighting gainincreases, as more power is allocated to the best antenna.Thus, STTCs/DTPA will allocate the power to optimize bothcoding and weighting gain. For example, for the 16-state TSCcode, 80% power allocation provides the best performance. Itprovides the coding gain and weighting gain of 1.6 dB and1.1 dB, respectively. Compared to pure antenna selection, itsweighting gain reduces by 0.6 dB, but its coding gain increasesby 0.9 dB. Thus, the overall gain is better by 0.3 dB. Comparedto the equal power allocation, its coding gain reduces by 0.3dB, but its weighting gain increases by 1.1 dB, which meansthat the total gain is better by 0.8 dB.

By summarizing Table 2, Figs. 1 and 2, we can find thatas the number of states increases, the coding gain providedby STTC will play a more important role than the weightinggain. Since the optimum coding gain is achieved at w2

(1) =w2

(2), the optimum power allocation solutions tend to balancethe power allocated to two selected antennas by decreasingw2

(1) and increasing w2(2).

Figs. 3 and 4 show the performance comparison between thestandard STTCs and STTCs/DTPA for the (2,2) case for the4-state and 16-state codes, respectively. In the simulation, theperformance of the new codes is compared to the TSC, BBH,CYV and Ionescu et al. (IMYL) codes. The STTCs/DTPA

16 17 18 19 20 21 22 23 24 25 26 2710

−3

10−2

10−1

Performance Comparison of the 4−State QPSK Codes with Two Transmit Antennas

SNR (dB)

Fra

me

Err

or R

ate

(FE

R)

Standard TSC Code, Two Tx antennas Standard BBH Code, Two Tx antennasStandard CYV Code, Two Tx antennasSTTCs/DTPA using TSC Code, (2, 2)STTCs/DTPA using BBH Code, (2, 2)STTCs/DTPA using CYV Code, (2, 2)STTCs/DTPA using New Code, (2, 2)

Fig. 3. FER performance of the 4-state 4-PSK (2, 2) STTCs/DTPA andstandard STTCs with total two transmit antennas.

16 17 18 19 20 21 22 23 24 2510

−3

10−2

Performance Comparison of the 16−State QPSK Codes with Two Transmit Antennas

SNR (dB)

Standard TSC Code, Two Tx antennasStandard BBH Code, Two Tx antennasStandard CYV Code, Two Tx antennasSTTCs/DTPA using TSC Code, (2, 2)STTCs/DTPA using BBH Code, (2, 2)STTCs/DTPA using CYV Code, (2, 2)STTCs/DTPA using New Code, (2, 2)IMYL Code [11], Two Tx antennas

Fra

me

Err

or R

ate

(FE

R)

Fig. 4. FER performance of the 16-state 4-PSK (2, 2) STTCs/DTPA andstandard STTCs with two transmit antennas.

scheme and the standard STTCs achieve the same diversitygain. Figs. 3 and 4 show that the STTCs/DTPA significantlyoutperform the standard STTCs for any number of states be-cause of the contribution of the weighting gain. Furthermore,the performance of the STTCs/DTPA using the new codes isbetter than the STTCs/DTPA using other existing codes. Forexample, the 16-state STTCs/DTPA using the new code andthe 16-state STTCs/DTPA using the TSC code can providea 1.8 dB and a 0.9 dB improvement, respectively, over thestandard TSC code with the same number of states. Fig. 4also shows that the 16-state STTCs/DTPA scheme using thenew code outperforms the IMYL code [11] by 0.5 dB.

Fig. 5 compares the STTCs/DTPA scheme using the newcodes with other closed-loop transmit diversity schemes for the(3, 2) case. These adaptive schemes are transmit antenna selec-tion with space-time trellis codes (TAS/STTCs) [16], transmitantenna selection with maximal-ratio combining (TAS/MRC)[31], and transmit antenna selection with space-time blockcodes (TAS/STBCs) [32]. In the TAS/MRC scheme, a single

SANTOSO et al.: DYNAMIC TRANSMIT POWER ALLOCATION IN SPACE-TIME TRELLIS CODED SYSTEMS 3085

12 13 14 15 16 17 18 19 20 21 22 2310

−3

10−2

10−1

Comparison FER Performance of of the New STTCs/DTPA with Three Transmit Antennas

SNR (dB)

Fra

me

Err

or R

ate

(FE

R)

TAS/STTCs using CYV Code, 4−State, (3, 2)TAS/STTCs using CYV Code, 16−State, (3, 2)TAS/STBCs using Alamouti Code, (3, 2)TAS/MRC, (3, 1)Standard CYV Code, 16−State, 3 Tx antennasSTTCs/DTPA using New Code, 4−state, (3, 2)STTCs/DTPA using New Code, 16−state, (3, 2)

Fig. 5. Comparison FER performance of the (3, 2) STTCs/DTPA schemeusing the new codes and other closed-loop transmit diversity schemes for twoout of three transmit antennas.

transmit antenna, which maximizes the SNR at the outputof the maximal-ratio combiner, is chosen for uncoded trans-mission. In the TAS/STBCs and TAS/STTCs schemes, twoantennas are chosen out of three transmit antennas and theAlamouti [34] and STTC coded signals are transmitted fromthese two chosen antennas. TAS/STTCs in [16] is a specialcase of the STTCs/DTPA. In TAS/STTCs scheme, the transmitpower is equally distributed across the two transmit antennasout of nT , that maximize the received SNR.

Fig. 5 shows that the 4-state and 16-state (3, 2)STTCs/DTPA schemes using the new code outperform otherclosed-loop transmit diversity schemes. For example, the new16-state (3, 2) STTCs/DTPA scheme outperforms by about2 dB, 2.5 dB and 3.2 dB the TAS/MRC, the 16-state CYVTAS/STTCs scheme, and the TAS/STBCs, respectively. Fig. 5also shows that the new 16-state (3, 2) STTCs/DTPA schemeis 5.6 dB better than the standard 16-state CYV code for threetransmit antennas. The proposed scheme also achieves a fulldiversity order of 3 as if all the transmit antennas were used.

In this paper, we have only considered a (nT , 2)STTC/DPTA scheme. The calculation of a general closed-form expression for the (nT , m) STTC/DPTA scheme is nottractable. Actually it can be noted that the STTC/DPTA systemwith m > 2 will significantly increase the system complexity,but provide only limited coding gain. The m = 2 STTC/DPTAscheme benefits from a simplicity of antenna selection andsignificant coding gain of STTC and thus achieving the besttradeoff of performance and complexity.

V. CONCLUSION

In this paper, we present the optimum power allocation andcode design for the STTCs/DTPA scheme, when only partialCSI feedback is available at the transmitter. The weightinggain depends only on the weighting coefficients, while thecoding gain depends on the weighting coefficients, the codesand the number of states. The STTCs/DTPA scheme opti-mizes both the coding and weighting gain. It significantly

outperforms the standard STTCs scheme and the other closed-loop transmit diversity methods. For example, the new 16-state(3, 2) STTCs/DTPA scheme is 5.6 dB better than the standard16-state CYV code for three transmit antennas. We also showthat the proposed scheme achieves a full diversity order as ifall the transmit antennas were used. These results are validprovided that the partial CSI is available at the transmitterand the fading channels corresponding to different transmitantennas are independent.

APPENDIX ATHE CALCULATION OF THE PEP UPPER BOUND IN EQS.

(16) - (18)

Consider (11)

P (c → e) ≤∫ 2π

0

∫ 2π

0

∫ ∞

0

∫ ∞

r2

P (c → e | r, θ)

fR1R2(r, θ)|J(r, θ)| dr1dr2dθ1dθ2 (21)

where

fR1R2(r, θ) =nT (nT − 1)

π2{nT −2∑i=0

(−1)i

(nT − 2

i

)(e−r2

2

)i}

× e−r21e−r2

2 (22)

|J(r, θ)| = r1r2. (23)

P (c → e | h) ≤ 12

exp(−d2(c, e)Es/4N0

)(24)

and

d2(c, e) =l∑

t=1

∣∣∣∣∣2∑

i=1

h(i)w(i)(c(i)t − e

(i)t )

∣∣∣∣∣2

= r21w

2(1)A11 + r2

2w2(2)A22

+ 2r1r2w(1)w(2)|A12| cos(θ1 − θ2 + ϕ) (25)

By substituting (25) into (24), P (c → e | h) can be furtherwritten as in (26) shown at the top of the next page.

If we substitute (22) - (26) into (21) and integrate theequation with respect to r1, we can obtain (27) shown at thetop of the next page.

By using the definition of α, β and γ in (13), (14) and(15), (27) can be rewritten in (28) at the top of the next page,where

C0 =nT (nT − 1)

2π2

nT −2∑i=0

(−1)i

(nT − 2

i

)(29)

L11 = r2

exp{−r2

2 (α + β + i + 2γ cos(θ1 − θ2 + ϕ))}

2α(30)

L12 = r22

(√πγ cos(θ1 − θ2 + ϕ)

2α1.5

)

exp{−r2

2

(β + i − γ2 cos2(θ1 − θ2 + ϕ)

α

)}(31)

L13 = L12Erf

{r2

γ cos(θ1 − θ2 + ϕ) + α√α

}(32)

3086 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 8, AUGUST 2008

P (c → e | r, θ) ≤ 12

exp{− Es

4N0

(r21w

2(1)A11 + r2

2w2(2)A22 + 2r1r2w(1)w(2)|A12| cos(θ1 − θ2 + ϕ)

)}(26)

L1 =∫ ∞

r2

P (c → e | r, θ) fR1R2(r, θ) |J(r, θ)| dr1

=nT (nT − 1)

2π2

nT −2∑i=0

(−1)i

(nT − 2

i

)∫ ∞

r2

exp{− (r2

1 + (i + 1)r22

)}

exp{− Es

4N0

(r21w

2(1)A11 + r2

2w2(2)A22 + 2r1r2w(1)w(2)|A12| cos(θ1 − θ2 + ϕ)

)}r1r2dr1 (27)

L1 = C0

∫ ∞

r2

exp{−αr2

1 − 2γ cos(θ1 − θ2 + ϕ)r1r2 − (β + i)r22

}r1r2dr1

= C0 (L11 − L12 + L13) (28)

L22 =∫ ∞

0

L14dr2

=γ

8αγ2

⎧⎨⎩ cos(θ1 − θ2 + ϕ)

cos2(θ1 − θ2 + ϕ) +(

3α2γ+α(β+i)γ2αγ2

)cos(θ1 − θ2 + ϕ) +

(α3+α2(β+i)

2αγ2

)⎫⎬⎭ . (39)

and Erf(·) is the error function.The complementary error function, denoted Erfc(·), is a

function defined by [35]

Erf(z) = 1 − Erfc(z). (33)

Thus, (28) can be written as

L1 = C0(L11 − L14) (34)

where

L14 = L12Erfc

{r2

γ cos(θ1 − θ2 + ϕ) + α√α

}. (35)

Let us integrate (34) with respect to r2.

L2 =∫ ∞

0

L1dr2

= C0

∫ ∞

0

L11dr2 − C0

∫ ∞

0

L14dr2

= C0(L21 − L22). (36)

Note that for large values of z, we can have

Erfc(z) <exp(−z2)

z√

π. (37)

Thus from [36], we can have

L21 =∫ ∞

0

L11dr2

=1

4α2 + 4α(β + i) + 8αγ cos(θ1 − θ2 + ϕ)(38)

and L22 can be calculated in (39) at the top of the page.

Let us integrate (36) with respect to dθ1 and dθ2.

L3 =∫ 2π

0

∫ 2π

0

L2dθ1dθ2

= C0

∫ 2π

0

∫ 2π

0

L21dθ1dθ2 − C0

∫ 2π

0

∫ 2π

0

L22dθ1dθ2

= C0(L31 − L32) (40)

We have the following integral relationship∫ π

0

11 + a cos(z)

dz =π√

1 − a2(41)

where a is a constant.Therefore

L31 =∫ 2π

0

∫ 2π

0

L21dθ1dθ2

=π2√

α4 + α2(β + i)2 + 2α3(β + i) − 4α2γ2(42)

1) For γ = 0. The following condition applies: L12 =L13 = L14 = 0. Thus, L22 = L32 = 0.Therefore, we can obtain

P (c → e)

≤ nT (nT − 1)2

{nT −2∑i=0

(−1)i

(nT − 2

i

)1

α2 + αβ + αi

}

= X1(c → e) (43)

as shown in (16).2) For γ �= 0 and α−β− i �= 0 for any i ∈ {0, 1, · · · , nT −

2}.

SANTOSO et al.: DYNAMIC TRANSMIT POWER ALLOCATION IN SPACE-TIME TRELLIS CODED SYSTEMS 3087

P (c → e) ≤ nT (nT − 1)2π2

nT −2∑i=0

(−1)i

(nT − 2

i

){

π2√α4 + α2(β + i)2 + 2α3(β + i) − 4α2γ2

− π2

2αγ

[κ3√

κ21 − 1

+κ4√

κ22 − 1

]}(46)

Let us recall (39). We can rewrite (39) as

L22 =γ

8αγ2[κ3

cos(θ1 − θ2 + ϕ) + κ1+

κ4

cos(θ1 − θ2 + ϕ) + κ2

](44)

where

κ1 =α + β + i

2γ

κ2 =α

γ

κ3 =α + β + i

β + i − α

κ4 =2α

α − β − i.

Using the relationship in (41), we have

L32 =∫ 2π

0

∫ 2π

0

L22dθ1dθ2

L32 =π2

2αγ

[κ3√

κ21 − 1

+κ4√

κ22 − 1

](45)

If we substitute (42) and (45) into (40), P (c → e) can becalculated in (46) at the top of the next page.Thus,

P (c → e) ≤ nT (nT − 1)2

nT−2∑i=0

(−1)i

(nT − 2

i

)

1γ(α − β − i)

{1√

κ21 − 1

− 1√κ2

2 − 1

}= X2(c → e) (47)

as shown in (17).3) For γ �= 0 and and α − β − i = 0 for any i ∈

{0, 1, · · · , nT − 2}.Let us recall (39). When α − β − i = 0, we can rewrite (39)as

L22 =γ

8αγ2

[cos(θ1 − θ2 + ϕ)

cos2(θ1 − θ2 + ϕ) + 2κ2 cos(θ1 − θ2 + ϕ) + κ22

]

=γ

8αγ2

[cos(θ1 − θ2 + ϕ)

(cos(θ1 − θ2 + ϕ) + κ2)2

](48)

We have this following integral relationship∫ π

0

cos(z)(a + b cos(z))2

dz =π

2(a + b)√

a2 + b2

[ −2b

a − b

](49)

where a and b are constants and a2 > b2.

Using the relationship in (49), we have

L32 =∫ 2π

0

∫ 2π

0

L22dθ1dθ2

L32 = − π2

2αγ

[1

(κ22 − 1)

32

](50)

If we substitute (42) and (50) into (40) and expand theequation, we can obtain

P (c → e) ≤ nT (nT − 1)2

{κ2

2

(κ22 − 1)

32

}

nT −2∑i=0

(−1)i

(nT − 2

i

)1

2γ(β + i)

= X3(c → e) (51)

as shown in (18).It completes the proof.

APPENDIX BPEP UPPER BOUND FOR SELECTION OF TWO OUT OF

THREE ANTENNAS

For a system with three transmit antennas, i.e. two out ofthree antennas are chosen for transmission, referred to as the(3, 2) STTCs/DTPA scheme, the upper bound of PEP in (16)- (18) becomes

P (c → e) ≤ X1(c → e)

=3

α(α2 + β2 + 2αβ + α + β)(52)

P (c → e) ≤ X2(c → e)

= 3(a2 − a1)α + (a1 − a2)β + a1

γ(α2 + β2 − 2αβ − α + β)(53)

P (c → e) ≤ X3(c → e) =32

a3

γβ2 + γβ(54)

where a1, a2, and a3 are constants given by

a1 =1√(

αγ

)2

− 1

− 1√(α+β2γ

)2

− 1

(55)

a2 =1√(

αγ

)2

− 1

− 1√(α+β+1

2γ

)2

− 1

(56)

a3 =κ2

2

(κ22 − 1)

32

(57)

3088 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 8, AUGUST 2008

For high SNR and w(1) �= 0, a1 � a2 and inequalities (52) -(54) can be approximated by

P (c → e) ≤ X1(c → e)

=3(

Es

4N0

)−3

w6(1)A

311 + w2

(1)w4(2)A11A2

22 + 2w4(1)w

2(2)A

211A22

(58)

P (c → e) ≤ X2(c → e)

=3(

Es

4N0

)−3

w2(1)A11 + w(2)A22

2w2

(1)A11

(59)

P (c → e) ≤ X3(c → e)

=32a3

(Es

4N0

)−3

w(1)w5(2)A

222|A12| (60)

which clearly shows that a full diversity order of 3 is achieved.

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[33] H. A. David, Order Statistics. John Wiley & Sons, 1970.[34] S. M. Alamouti, “A simple transmit diversity technique for wireless

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[35] S. Haykin, Communication Systems, 4th ed. Wiley, 2001.[36] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and

Products, 6th ed. Academic Press, 2000.

Agus Santoso was born in Indonesia in 1976. He re-ceived the B. Eng. Sc. degree (with 1st class Honors)and the M. Eng. Sc. (Special Commendation) degreein Electrical Engineering from the University ofAdelaide, Australia in 2000 and 2004, respectively,and the Ph.D. degree in Electrical Engineering fromthe University of Sydney, Australia, in 2006. In2004, he received the Max Simon’s IEEE New SouthWales Award for the Best Student Paper. Dr. Santosois also the recipient of the Australian Postgradu-ate Awards (APA) and the University Postgraduate

Awards (UPA). His research interests include wireless communications,multiple-input-multiple-output (MIMO) systems, space-time coding, spreadspectrums, and equalizations.

SANTOSO et al.: DYNAMIC TRANSMIT POWER ALLOCATION IN SPACE-TIME TRELLIS CODED SYSTEMS 3089

Yonghui Li (M’04) received his PhD degree inElectronic Engineering in November 2002 fromBeijing University of Aeronautics and Astronauticsand his PhD thesis has been awarded the best PhDthesis award. From 1999 - 2003, he was affiliatedwith Linkair Communication Inc., where he held theposition of project manager with responsibility forthe design of physical layer solutions for the LAS-CDMA system. Since 2003, he has been with theTelecommunication Lab, the University of Sydney,Australia. He is now a senior lecturer in the School

of Electrical and Information Engineering, the University of Sydney. Cur-rently, he is also the Australian Queen Elizabeth II fellow.

His current research interest is in the area of wireless communications,with a particular focus on MIMO, multiple user communications, codingtechniques, and wireless sensor networks. He holds a number of patentsgranted and pending in these fields. He serves as Associate Editor for theEURASIP JOURNAL ON WIRELESS COMMUNICATIONS AND NETWORK-ING and JOURNAL OF NETWORKS. He has also been involved in the technicalcommittees of several international conferences, such as ICC, Globecom,PIMRC, etc.

J. Kingsley A. Allan (S’04) received his B. Eng(Computer) (Honours) degree from the Universityof Western Sydney, Australia, in 2004, receiving theDean’s Medal and University Medal on graduation.He is currently working towards a PhD degree inthe Telecommunications Laboratory in the Schoolof Electrical and Information Engineering at TheUniversity of Sydney, Australia. His current researchis in wireless communications, specifically the areasof space-time coding and wireless sensor networks.

Branka Vucetic (M’83-SM’00-F’03) received theB.S.E.E., M.S.E.E., and Ph.D. degrees in 1972,1978, and 1982, respectively, in electrical engineer-ing, from The University of Belgrade, Belgrade,Yugoslavia.

During her career she has held various researchand academic positions in Yugoslavia, Australia, andthe UK. Since 1986, she has been with the Syd-ney University School of Electrical and InformationEngineering in Sydney, Australia. She is currentlyDirector of The Telecommunications Laboratory at

Sydney University. Her research interests include wireless communications,digital communication theory, coding, and multi-user detection.

In the past decade she has been working on a number of industry sponsoredprojects in wireless communications and mobile Internet. She has taught awide range of undergraduate, postgraduate, and continuing education coursesworldwide. Prof. Vucetic co-authored four books and more than two hundredpapers in telecommunications journals and conference proceedings.