Bounded Delay-Tolerant Space-Time Codes for Distributed Antenna Systems

4644 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 8, AUGUST 2014

Bounded Delay-Tolerant Space-Time Codes forDistributed Antenna Systems

Yun Liu, Student Member, IEEE, Wei Zhang, Senior Member, IEEE, Soung Chang Liew, Fellow, IEEE,and Pak-Chung Ching, Fellow, IEEE

Abstract—Distributed space-time codes (STCs) can achieve co-operative diversity in distributed antenna systems. But the pathdelay difference may cause diversity loss. In this paper, a familyof asynchronous STC is proposed to achieve full diversity, giventhat the path delay difference is within a tolerance bound. Theproposed code structure allows the overall decoding process tobe decomposed into several independent sub-processes, which canbe tackled by low-complexity maximum-likelihood decoders. Twocode designs based on the Alamouti code and the Golden code aregiven for a system with two distributed antennas. Moreover, twodesigns based on orthogonal STC and fast group decodable STCare introduced for a system with four distributed antennas, witheach transmitter having two antennas. Full diversity gain is provedfor all code designs and their associated decoding complexitiesare analyzed. Simulations of the proposed codes confirm ourtheoretical results on full diversity gain.

Index Terms—Asynchronous cooperative communication,space-time codes, full diversity.

I. INTRODUCTION

S PACE-TIME codes (STC) [1]–[4] were originally pro-posed to achieve antenna diversity gain in multiple-input

multiple-output (MIMO) systems. Cooperative communication[5]–[8], viewed as a distributed MIMO, offers cooperativediversity gain that can be achieved by distributed STC. Dueto the distributed nature of the cooperative communication,symbols sent from distributed transmitters may not arrive atthe receiver simultaneously. The asynchronous communicationmay lead to full rank deficiency of the STC matrix, therebycausing diversity loss [9], [10]. Much recent research has beendevoted to the study of asynchronous STC that can achieve fulldiversity [11]–[26].

Manuscript received November 18, 2013; revised April 11, 2014; acceptedApril 17, 2014. Date of publication May 5, 2014; date of current version August8, 2014. The work of S. C. Liew is partially supported by the General ResearchFunds (Project No. 414713) established under the University Grant Committeeof the Hong Kong Special Administrative Region and the China NSFC grant(Project No. 61271277). Part of this work was presented at the IEEE Inter-national Conference on Acoustics, Speech and Signal Processing (ICASSP),Vancouver, Canada, May 26–31, 2013. The associate editor coordinating thereview of this paper and approving it for publication was A. Ghrayeb.

Y. Liu and P.-C. Ching are with the Department of Electronic Engineer-ing, The Chinese University of Hong Kong, Shatin, Hong Kong (e-mail:[email protected]; [email protected]).

W. Zhang is with the School of Electrical Engineering and Telecommuni-cations, The University of New South Wales, Sydney, NSW 2052, Australia(e-mail: [email protected]).

S. C. Liew is with the Department of Information Engineering, The ChineseUniversity of Hong Kong, Shatin, Hong Kong (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/TWC.2014.2321382

Asynchronous STC designs can generally be grouped intotwo categories: frequency-domain approaches [12], [13] andtime-domain approaches [14]–[26]. For frequency-domain ap-proaches, the use of OFDM can avoid diversity loss in asyn-chronous transmissions. Such approaches are not applicable tonon-OFDM systems.

Time-domain approaches exploit the linear dependenceamong the signal vectors transmitted by relays to maintainfull diversity gain. Some designs have been proposed to main-tain full diversity, including delay-tolerant distributed threadedalgebraic space-time codes (D-TAST) [16] and delay-tolerantlinear dispersion codes (DT-LDC) [24]. Besides achieving fulldiversity, [17] further reduces the decoding complexity ofdelay-tolerant distributed STC by minimizing memory length.However, these designs work for short sequences made outof two or four different symbols only, limiting their usabilityin many real applications. In [19], distributed linear convolu-tive STC (DLC-STC) was presented to achieve full diversitywithout restriction on sequence length. DLC-STC applies alinear independent vector on each symbol and expands thevector over the entire sequence under the linear dispersioncode structure [3], [4]. Some other works extended DLC-STCto different scenarios like frequency-selective channels [20],analog network coding [27] and full-duplex relays [25]. In [22],the authors proposed a simple time-division multiple accessSTC (TDMA-STC) transmission scheme without restriction onsequence length. This scheme adds zero padding in front ofthe signals transmitted by each relay to maintain full diversityat the receiver. An advantage of this method is the simpleimplementation at relays. However, these designs [19], [22]require highly complex ML decoding. First, their ML decodingrequires exhaustive search over all these symbols. Second, fordifferent delay differences, the received signal structures aredifferent, resulting in different ML decoders. As a result, therespective ML decoding complexity essentially increases expo-nentially with the sequence length and grows linearly with thedelay difference bound. This motivates us to design a family ofasynchronous STC amenable to low-complexity ML decoding.

In this paper, we propose a family of bounded delay-toleranttime interleave reversal space-time codes (BDT-TIR STC).Given the bound Δτm in delay difference, BDT-TIR STCcan decompose a large signal block into multiple independentsignal sub-blocks, and each small signal sub-block can be ML-decoded with low complexity. For two-transmit-antenna scenar-ios, two STC designs are given to achieve full diversity ordersof 2 and 4. If a Viterbi decoder is used, then their ML decodingcomplexities are O(Δτm|S|2) and O(Δτm|S|4), respectively,

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LIU et al.: BOUNDED DELAY-TOLERANT SPACE-TIME CODES FOR DISTRIBUTED ANTENNA SYSTEMS 4645

Fig. 1. An asynchronous distributed antenna system with two transmitters andone receiver.

where |S| is the cardinality of the signal constellation. Forfour-transmit-antenna scenarios, two STC designs are givento achieve full diversity orders of 4 and 8. With a Viterbidecoder, their ML decoding complexities are O(Δτm|S|3) andO(Δτm|S|9), respectively. Full diversity proof and decodingcomplexity analysis are given for all the proposed code designs.

The rest of this paper is organized as follows. In Section II,we introduce the system model of distributed antenna systems.In Section III, we propose STC designs for systems withtwo distributed transmit antennas. We further propose STCdesigns for systems with four distributed transmit antennas inSection IV. Section V shows the simulation results. Finally,Section VI concludes this paper.

II. SYSTEM MODEL

We consider a distributed antenna system as shown in Fig. 1.The system comprises two transmitters T1 and T2 and onereceiver D. T1 and T2 transmit the same data to D. Cooperationbetween T1 and T2 is utilized to provide diversity gain in datatransmission. The channel and path delay coefficients from Ti

to D are denoted by (hi, τi), i = 1, 2, where τi is normalized bythe symbol period. The channel and path delay coefficients areassumed to be known at D only and invariant during each blocktransmission. The delay difference is Δτ = τ2 − τ1, which isbounded by |Δτ | ≤ Δτm, where Δτm is assumed to be knownat T1 and T2 before transmission. For simplicity, we assumeΔτ is a positive integer. The fractional part of Δτ can beregarded as multipath effects addressed by equalizers usingoversampling [14].

It should be pointed out that this system model is applicableto several well-known scenarios. For example, the system canbe regarded as a part of relay network, where relays transmitsymbols to a destination. The system may also be considered asuser cooperation [28], [29]. After sharing data symbols betweentwo mobile nodes T1 and T2, they cooperatively transmit to thedestination. The system may further be regarded as a simplifiedversion of distributed MIMO, where T1 and T2 are at differentgeographic locations with a backhaul link connecting them fordata sharing.

Given the above model, we aim to design an asynchronouscommunication strategy amenable to low-complexity ML de-coding.

III. STC DESIGNS WITH TWO DISTRIBUTED

TRANSMIT ANTENNAS

In this section, we propose two STC designs for the dis-tributed antenna system. The first code design is referred to as

the bounded delay-tolerant time interleave reversal Alamouticode (BDT-TIR AC) for a 2 × 1 system, where each transmit-ter has only one antenna. BDT-TIR AC has a low decodingcomplexity and a rate of one symbol per channel use. Thesecond code design is referred to as the bounded delay-toleranttime interleave reversal Golden code (BDT-TIR GC) for a 2 ×2 system, where each transmitter has one antenna but thereceiver has two antennas. Its code rate is 2 symbols per channeluse at the expense of a higher decoding complexity than theBDT-TIR AC.

A. BDT-TIR AC for a 2 × 1 System

For a 2 × 1 system, the well-known Alamouti code [1] hasthe following structure:

XA =

[s1 s2−s∗2 s∗1

],

where s1 and s2 are two data symbols. Let � ≥ Δτm, � ∈ Z+.To transmit a sequence s of 4� symbols, the codeword matrixA(s) of BDT-TIR AC is constructed as

A(s) =

[c0 c1 c2 c3

−←−c ∗2 −←−c ∗

3←−c ∗

0←−c ∗

1

],

where

ci = [si�+1, si�+2, . . . , si�+�], i = 0, 1, 2, 3

←−c ∗i =

[s∗i�+�, s

∗i�+�−1, . . . , s

∗i�+1

], i = 0, 1, 2, 3.

Time interleave reversal (TIR), contains two levels of struc-tural meanings. In the first level, “reversal,” represented by“←,” indicates the decreasing order of the index within eachci in the second row, i = 0, 1, 2, 3. This idea originates fromTR-STC [11]. Aided by “reversal,” the decoding computationburden is significantly reduced. In the second level, “timeinterleave” stands for the order among c0, c1, c2, and c3 inthe second row.

Example 1: When � = 2, assuming Δτ = 0, the codewordmatrix of BDT-TIR AC is[

s1 s2 s3 s4 s5 s6 s7 s8

−s∗6 −s∗5 −s∗8 −s∗7 s∗2 s∗1 s∗4 s∗3

].

When Δτ = 0, namely, a synchronized case, BDT-TIR ACcan be explicitly divided into several Alamouti codes. Thus,the performance of BDT-TIR AC in the synchronized case isequivalent to that of an Alamouti code.

Example 2: When � = 2, assuming Δτ = 1, the codewordmatrix of BDT-TIR AC becomes[

s1 s2 s3 s4 s5 s6 s7 s8 0

0 −s∗6 −s∗5 −s∗8 −s∗7 s∗2 s∗1 s∗4 s∗3

], (1)

where s2 and s6 form an Alamouti code, and s4 and s8 formanother Alamouti code. That is,[

s2 s6−s∗6 s∗2

]and

[s4 s8−s∗8 s∗4

]. (2)


The remaining columns are[s1 s3 s5 s7 00 −s∗5 −s∗7 s∗1 s∗3

]. (3)

The original eight-symbol matrix is split into a four-symbolmatrix and two Alamouti code matrices, thereby significantlyfacilitating an efficient decoding procedure.

Example 3: When � = 2, assuming Δτ = 2, the codewordmatrix of BDT-TIR AC becomes[

s1 s2 s3 s4 s5 s6 s7 s8 0 00 0 −s∗6 −s∗5 −s∗8 −s∗7 s∗2 s∗1 s∗4 s∗3

],

(4)

which can be decomposed into[s1 s4 s5 s8 00 −s∗5 −s∗8 s∗1 s∗4

],

[s2 s3 s6 s7 00 −s∗6 −s∗7 s∗2 s∗3

],

(5)

which have the identical code structure as (3), indicating thedecoder structure can be shared for different delay difference.

Theorem 1: In a 2 × 1 asynchronous communication systemwith delay difference bound Δτm, BDT-TIR AC of sequencelength 4� achieves full diversity, that is, diversity order 2, if� ≥ Δτm.

Proof: See Appendix A. �Theorem 2: In a 2 × 1 system with delay difference bound

Δτm, for PAM, PSK, and square QAM constellations, BDT-TIR AC of sequence length 4� (� ≥ Δτm) can achieve fulldiversity with linear decoders.

Proof: See Appendix B. �Remark 1 (Code Rate Analysis): At receiver D, 4� symbols

are received within 4�+|τ2−τ1| time slots. The code rate R is

R =4�

4�+ |τ2 − τ1|=

1

1 + |τ2−τ1|4�

.

When 4� is large, R approaches one symbol per channel use.Remark 2 (Decoding Complexity): As (1) is decomposed

into (2) and (3), a “divide-and-conquer” strategy can be adoptedto facilitate the decoding procedure. The codeword matrix AΔτ

like (1) can be “divided” into two parts: AΔτ1 like (2) with

Alamouti code structure and AΔτ2 that can be further decom-

posed into AΔτ2,i like (3) with only 4 symbols inside, i = 1, 2,

. . . , |Δτ |. For simplicity, the general forms of AΔτ , AΔτ1 ,

AΔτ2 , and AΔτ

2,i are given by (9)–(12) in Appendix A. In the“conquer” part, we deal with AΔτ

1 and AΔτ2,i separately. For

AΔτ1 with the Alamouti code structure, the computation burden

is O(4(l −Δτ)|S|), where |S| is the cardinality of the signalconstellation S . For each AΔτ

2,i , the computation burden of anexhaustive ML, Viterbi decoder, and belief propagation (BP)decoder [30] is O(|S|4), O(|S|2), and O(|S|2), respectively.We remark that as an equivalent form, the Viterbi decoderoutputs the codeword-wise ML result as the exhaustive ML,and BP outputs the symbol-wise ML result. Since |Δτ | =Δτm and 4� = 4Δτm in the worst case, the overall complexityof exhaustive ML, Viterbi decoder, and BP is O(Δτm|S|4),O(Δτm|S|2), and O(Δτm|S|2), respectively.

B. BDT-TIR GC for a 2 × 2 System

For a 2 × 2 system, the Golden code was proposed in [31]with the following structure:

XG(a1, b1, a2, b2) =

[α(a1 + b1θ) α(a2 + b2θ)jα(a2 + b2θ) α(a1 + b1θ)

],

where a1, b1, a2, b2 are symbols, j =√−1, α = 1 + j((1−√

5)/2), α = 1 + j((1 +√5)/2), θ = (1 +

√5)/2, and θ =

(1−√5)/2 are constants defined in [31]. The equivalent ma-

trix of XG is defined as[c1 c2jc2 c1

]Δ=

[α(a1 + b1θ) α(a2 + b2θ)jα(a2 + b2θ) α(a1 + b1θ)

].

Let � ≥ Δτm, � ∈ Z+. To transmit a sequence s of 8� sym-bols, the codeword matrix G(s) of BDT-TIR GC is

G =

[c0 c1 c2 c3j←−c 2 j

←−c 3

←−c 0

←−c 1

], (6)

where

ci = [α(ai�+1 + bi�+1θ), α(ai�+2 + bi�+2θ),

. . . , α(ai�+� + bi�+�θ)] ,←−c i =

[α(ai�+� + bi�+�θ), α(ai�+�−1 + bi�+�−1θ),

. . . , α(ai�+1 + bi�+1θ)],

i =0, 1, 2, 3.

Theorem 3: In a 2 × 2 asynchronous communication systemwith a delay difference bound Δτm, BDT-TIR GC of sequencelength 8� achieves full diversity, that is, diversity order 4, if� ≥ Δτm.

Proof: See Appendix C. �Remark 3 (Code Rate Analysis): At receiver D, 8� symbols


R =8�

4�+ |τ2 − τ1|=

2

1 + |τ2−τ1|4�

.

When 8� is large, R approaches two symbols per channel use.Remark 4 (Decoding Complexity): The “divide-and-

conquer” strategy is adopted to facilitate the decodingprocess. The overall decoding complexities of exhaustive ML,Viterbi decoder, and BP are O(Δτm|S|8), O(Δτm|S|4), andO(Δτm|S|4), respectively.

C. Comparisons Among Different Codes

Table I compares BDT-TIR AC, DLC-STC [19] and TDMA-STC [22]. The total number of symbols transmitted is L. AsTable I shows, the proposed BDT-TIR AC has the highest coderate and the lowest ML decoding complexity.

IV. STC DESIGNS WITH FOUR DISTRIBUTED

TRANSMIT ANTENNAS

In this section, we propose two STC designs with fourdistributed transmit antennas, with each transmitter having twoantennas. The first code design is referred to as the bounded


TABLE ICOMPARISON BETWEEN BDT-TIR AC, DLC-STC [19] AND TDMA-STC

[22] FOR 2 × 1 SYSTEMS WITH L AS THE TOTAL NUMBER OF

SYMBOLS TRANSMITTED, USING VITERBI DECODER

AS AN EQUIVALENT FORM OF ML DECODING

delay-tolerant time interleave reversal complex orthogonal de-sign STC (BDT-TIR COD4) for a 4 × 1 system, where eachtransmitter has two antennas. BDT-TIR COD4 has low de-coding complexity and its code rate is 3/4. To increase coderate to 2, we propose another code design, referred to as thebounded delay-tolerant time interleave reversal fast group de-codable STC (BDT-TIR FGD4) for a 4 × 2 system, where eachtransmitter has two antennas and the receiver has two antennas.

A. BDT-TIR COD4 for a 4 × 1 System

In a four-transmit-antenna scenario, COD4 was proposed in[32], [33] with following structure:

Xcod =

⎡⎢⎣s1 −s∗2 −s∗3 0s2 s∗1 0 −s∗3s3 0 s∗1 s∗20 s3 −s2 s1

⎤⎥⎦ .

Let � ≥ Δτm, � ∈ Z+. To transmit a sequence s of 6� symbolsfrom the two distributed transmitters, each having two antennas,the codeword matrix C(s) of BDT-TIR COD4 is designed as

C=

⎡⎢⎢⎣

c0 c1 −c∗2 −c∗3 −c∗4 −c∗5 0l 0l

c2 c3 c∗0 c∗1 0l 0l −c∗4 −c∗5←−c 4

←−c 5 0l 0l←−c ∗

0←−c ∗

1←−c ∗

2←−c ∗

3

0l 0l←−c 4

←−c 5 −←−c 2 −←−c 3←−c 0

←−c 1

⎤⎥⎥⎦ ,

where

ci = [si�+1, si�+2, . . . , si�+�], i = 0, 1, 2, 3, 4, 5;

←−c i = [si�+�, si�+�−1, . . . , si�+1], i = 0, 1, 2, 3, 4, 5.

The first two rows of C are transmitted by one transmitter, andthe last two rows of C are transmitted by the other transmitter.

Theorem 4: In a 4 × 1 asynchronous communication sys-tem with delay difference bound Δτm, BDT-TIR COD4 ofsequence length 6� achieves full transmit diversity, that is,diversity order 4, if � ≥ Δτm.

Proof: See Appendix D. �Remark 5 (Code Rate Analysis): At receiver D, 6� symbols


R =6�

8�+ |τ2 − τ1|=

3

4 + |τ2−τ1|2�

.

When 6� is large, R approaches 3/4 symbol per channel use.Remark 6 (Decoding Complexity): The “divide-and-conquer”

strategy is utilized to facilitate the decoding. The overall decod-ing complexity of exhaustive ML, Viterbi decoder, and BP isO(Δτm|S|6), O(Δτm|S|3) and O(Δτm|S|3), respectively.

B. BDT-TIR FGD4 for a 4 × 2 System

BDT-TIR COD4 can achieve full diversity with code rate 3/4.To increase code rate, we propose BDT-TIR FGD4. Fast groupdecodable STC (FGD) [34] in a 4 × 2 system can be expressedas follows:

XF =

⎡⎢⎣

s0+jks4 −s1+jks5 −s2+jks6 jks3+s7s1+jks5 s0−jks4 jks3+s7 s2−jks6s2+jks6 −jks3−s7 s0−jks4 −s1+jks5−jks3−s7 −s2−jks6 s1+jks5 s0+jks4

⎤⎥⎦ ,

where si ∈ R, j =√−1 and k =

√(3/5). We define xi,j

Δ=

si + jksj and the equivalent matrix of XF as⎡⎢⎣

x0,4 −x∗1,5 −x∗

2,6 x7,3

x1,5 x∗0,4 x7,3 x∗

2,6

x2,6 −x7,3 x∗0,4 −x∗

1,5

−x7,3 −x2,6 x1,5 x0,4

⎤⎥⎦

Δ=

⎡⎢⎣

s0 + jks4 −s1 + jks5 −s2 + jks6 jks3 + s7s1 + jks5 s0 − jks4 jks3 + s7 s2 − jks6s2 + jks6 −jks3 − s7 s0 − jks4 −s1 + jks5−jks3 − s7 −s2 − jks6 s1 + jks5 s0 + jks4

⎤⎥⎦.

Let � ≥ Δτm, � ∈ Z+. To transmit a sequence s of 16� symbolsfrom the two distributed transmitters, each having two antennas,the codeword matrix F(s) of BDT-TIR FGD4 is in (7), shown atthe bottom of the page. The first two rows of F are transmittedby one transmitter, and the third and fourth rows of F aretransmitted by the other transmitter.

F =

⎡⎢⎢⎣

x0,4 x8,12 −x∗1,5 −x∗

9,13 −x∗2,6 −x∗

10,14 x7,3 x15,11

x1,5 x9,13 x∗0,4 x∗

8,12 x7,3 x15,11 x∗2,6 x∗

10,14

←−x 2,6←−x 10,14 −←−x 7,3 −←−x 15,11

←−x ∗0,4

←−x ∗8,12 −←−x ∗

1,5 −←−x ∗9,13

−←−x 7,3 −←−x 15,11 −←−x 2,6 −←−x 10,14←−x 1,5

←−x 9,13←−x 0,4

←−x 8,12

⎤⎥⎥⎦ (7)

xi,j = [xi�+1,jl+1, xi�+2,jl+2, . . . , xi�+�,jl+�],

←−x i,j = [xi�+�,j�+�, xi�+�−1,j�+�−1, . . . , xi�+1,j�+1],

(i, j) = (0, 4), (1, 5), (2, 6), (7, 3), (8, 12), (9, 13), (10, 14), (15, 11)


TABLE IICOMPARISON BETWEEN BDT-TIR COD4 AND BDT-TIR FGD4 USING

VITERBI DECODER AS AN EQUIVALENT FORM OF ML DECODING

Theorem 5: In a 4 × 2 asynchronous communication systemwith delay difference bound Δτm, BDT-TIR FGD4 of sequencelength 16� achieves full diversity, that is, diversity order 8, if� ≥ Δτm.

Proof: See Appendix E. �Remark 7 (Code Rate Analysis): At receiver D, 16� symbols

are received in 8�+ |τ2 − τ1| time slots. The code rate R is

R =16�

8�+ |τ2 − τ1|=

2

1 + |τ2−τ1|8�

. (8)

When 16� is large, R approaches two symbols per channel use.Remark 8 (Decoding Complexity): We use the “divide-and-

conquer” strategy to facilitate the decoding process. The com-plexity of exhaustive ML is O(Δτm|S|16). By taking advantageof the fast group decodable property [34], the complexity canbe reduced to O(Δτm|S|9).

The general form of the codeword matrix is FΔτ , whichcan be decomposed into FΔτ

1 and FΔτ2,i . FΔτ

1 follows thestructure of XF , which can be easily decoded. For FΔτ

2,i , theQR decomposition of its equivalent channel matrix results inan upper triangular matrix R:

R =

[A B0 C

]

where A is an 8 × 8 diagonal matrix referring to the eightgroups and C is an upper triangular matrix of size 8 × 8. Wecan determine the eight symbols of the columns in B and C andsearch each column of A independently. Therefore, the overallML decoding complexity can be reduced to O(Δτm|S|9).

C. Comparisons Among Different Codes

Table II compares BDT-TIR COD4 and BDT-TIR FGD4.BDT-TIR COD4 has the lower decoding complexity than BDT-TIR FGD4, but its code rate is only 3/4. BDT-TIR FGD4increases the code rate to 2 with an increased ML decodingcomplexity of O(Δτm|S|9).

V. SIMULATION RESULTS

This section examines the performance of the proposed BDT-TIR STC over Rayleigh fading channels based on simulationresults.

Fig. 2 shows the bit error rate (BER) performance of theproposed BDT-TIR AC under different Δτ as Es/N0 varies,where Es is total transmit power for each symbol. The synchro-nized case, that is, Δτ = 0, can be regarded as the performanceof the Alamouti code, which serves as a lower bound. Fig. 2illustrates the performance of BDT-TIR AC with Δτ = 1,Δτ = 2 and Δτ = 10, which are barely distinguishable from

Fig. 2. Performance of BDT-TIR AC (� = Δτm) in a 2 × 1 system withQPSK.

Fig. 3. Performance of BDT-TIR AC (� = Δτm) in a 2 × 1 system withΔτ = 1 using QPSK.

the synchronized case. Identical slope of the four lines in thehigh Es/N0 region demonstrates the full diversity gain.

Fig. 3 compares the BER performance among different de-coders of BDT-TIR AC. Viterbi decoder and exhaustive MLoutput the same result. Linear decoders such as Zero-Forcingand Minimum Mean Square Error Decision Feedback (MMSE-DF) have the same BER slope as the exhaustive ML in highSNR region, demonstrating full diversity gain. This result sup-ports our analytical results in Theorem 2.

Fig. 4 compares the BER performance among the proposedBDT-TIR AC, DT-LDC [24], and Alamouti code [1]. As ex-pected, the performance of BDT-TIR AC in the synchronizedcase is identical to that of the Alamouti code. For the caseconsidered, the SNR gap between BDT-TIR AC (Δτ = 1)and the other two codes is approximately 0.3 dB and 1.4 dB,respectively.

Fig. 5 shows the BER performance of the proposed BDT-TIRGC in a 2 × 2 system. The synchronized case, that is, Δτ = 0,can be regarded as the performance of the Golden code [31].The performance of BDT-TIR GC with Δτ = 1, Δτ = 2, and


Fig. 4. BER comparison between the proposed BDT-TIR AC, DT-LDC [24],and Alamouti code [1] in a 2 × 1 system with QPSK and Δτm = 2.

Fig. 5. BER performance of BDT-TIR GC (� = Δτm) in a 2 × 2 systemwith QPSK.

Δτ = 10 is presented. These four lines have the same slope inhigh SNR region, verifying full diversity gain.

Fig. 6 demonstrates the BER performance of the proposedBDT-TIR COD4 in a 4 × 1 system. The synchronized case,that is, Δτ = 0, can be regarded as the performance of COD[32], [33]. The performance of BDT-TIR COD4 with Δτ = 1,Δτ = 2, and Δτ = 10 is presented. The identical slope ofthe four lines in the high Es/N0 region demonstrates the fulldiversity gain. As shown in the subfigure, the gaps betweencases of Δτ �= 0 and the synchronized one are around 0.2 dB,when Es/N0 = 20 dB.

Fig. 7 demonstrates the BER performance of the proposedBDT-TIR FGD4 in a 4 × 2 system. The synchronized case, thatis, Δτ=0, can be regarded as the performance of FGD [34].The performance of BDT-TIR FGD4 with Δτ=1, Δτ=2, andΔτ=10 is presented. The identical slope of the four lines inthe high Es/N0 region demonstrates the full diversity gain. Asshown in the subfigure, the gaps between cases of Δτ �=0 andthe synchronized one are within 1 dB, when Es/N0=24 dB.

Fig. 6. BER performance of BDT-TIR COD4 (� = Δτm) in a 4 × 1 systemwith QPSK.

Fig. 7. BER performance of BDT-TIR FGD4 (� = Δτm) in a 4 × 2 systemwith size-4 real constellation.

VI. CONCLUSION

We have proposed a family of bounded delay-tolerant timeinterleave reversal space time codes for distributed antennasystems. The proposed codes achieve full diversity when de-lay difference is bounded. The unique code structure allowsbreaking down the decoding into several independent parts,greatly reducing the overall decoding complexity and facilitat-ing parallel decoding processes. We give full diversity proof anddecoding complexity analysis for all code designs. Simulationsverify the full diversity gain.

APPENDIX APROOF OF THEOREM 1

The codeword matrix of s is AΔτ (s) given in (9), shownat the bottom of the next page. Let s be another sequencewith codeword matrix AΔτ (s), such that s �= s. The codeworddifference matrix ΔAΔτ is defined as

ΔAΔτ Δ= AΔτ (s)−AΔτ (s).


To prove the existence of full diversity gain, we need to show

that ΔAΔτ is always full rank for any s �= s, where ΔsjΔ=

sj − sj , j = 1, 2, . . . , 4�.Let us analyze the structure of AΔτ (s). AΔτ (s) can be

separated into two parts: AΔτ1 given in (10), shown at the

bottom of the page, and AΔτ2 given in (11), shown at the bottom

of the page. AΔτ1 includes the columns with Alamouti code

structure, and the remaining columns are included in AΔτ2 .

Furthermore, AΔτ2 can be separated intoΔτ matricesAΔτ

2,i as[si s�+Δτ+1−i s2�+i s3�+Δτ+1−i 00 −s∗2�+i −s∗3�+Δτ+1−i s∗i s∗�+Δτ+1−i

],

(12)

where i = 1, 2, . . . ,Δτ .Similar to ΔAΔτ , we define the difference matrix ΔAΔτ

1

and ΔAΔτ2,i as

ΔAΔτ1

Δ= AΔτ

1 (s)−AΔτ1 (s),

ΔAΔτ2,i

Δ= AΔτ

2,i (s)−AΔτ2,i (s).

ΔAΔτ2,i is given in (13), shown at the bottom of the page.

• For Δsi ∈ ΔAΔτ1 , when Δsi �= 0, AΔτ

1 is full rank be-cause of the Alamouti code structure.

• For Δsi ∈ ΔAΔτ2,i , we analyze each symbol in ΔAΔτ

2,i

accordingly.

1) If Δsi �= 0, the first and fourth columns of ΔAΔτ2,i

in (13) form a rank-two matrix as[Δsi Δs3�+Δτ+1−i

0 Δs∗i

],

which means that ΔAΔτ2,i is full rank.

2) For the case in which Δsi = 0 and Δs3�+Δτ+1−i �=0, and the case in which [Δsi,Δs3�+Δτ+1−i] =[0, 0] and Δs2�+i �= 0, the same rank analysis can beapplied to ΔAΔτ

2,i .3) Finally, if [Δsi,Δs3�+Δτ+1−i,Δs2�+i] = [0, 0, 0]

and Δs�+Δτ+1−i �= 0, the second and fifth columnsof ΔAΔτ

2,i form a rank two matrix as[Δs�+Δτ+1−i 0

0 Δs∗�+Δτ+1−i

],

which means that ΔAΔτ2,i is full rank.

Therefore, ∀ j ∈ [1, 4�], if Δsj �= 0, ΔAΔτ is full rank.

APPENDIX BPROOF OF THEOREM 2

As a part of AΔ has the Alamouti code structure, we focus onthe remaining part AΔτ

2,i as shown in (12). The received signalvector of transmitting AΔτ

2,i is

yi =√

PT [h1, h2]AΔτ2,i + ni,

which can be reformulated in an equivalent channel form as

yeq =

[{yTi

}�{yTi

} ]=

√PT H

[{sTi

}�{sTi

} ]+

[{nTi

}�{nTi

} ] ,(14)

where si = [si, s�+Δτ+1−i, s2�+i, s3�+Δτ+1−i], {x} showsthe real part of x, �{x} shows the imaginary part of x, and Hin (15), shown at the bottom of the next page, is the equivalentchannel matrix.

AΔτ (s) =

[s1 . . . sΔτ sΔτ+1 . . . s� s�+1 . . . s�+Δτ s�+Δτ+1 . . . s2� s2�+1

0 . . . 0 −s∗3� . . . −s∗2�+Δτ+1 −s∗2�+Δτ . . . −s∗2�+1 −s∗4� . . . −s∗3�+Δτ+1 −s∗3�+Δτ

. . . s2�+Δτ s2�+Δτ+1 . . . s3� s3�+1 . . . s3�+Δτ s3�+Δτ+1 . . . s4� 0 . . . 0

. . . −s∗3�+1 s∗� . . . s∗Δτ+1 s∗Δτ . . . s∗1 s∗2� . . . s∗�+Δτ+1 s∗�+Δτ . . . s∗�+1

](9)

AΔτ1 =

[sΔτ+1 . . . s� s�+Δτ+1 . . . s2� s2�+Δτ+1 . . . s3� s3�+Δτ+1 . . . s4�−s∗3� . . . −s∗2�+Δτ+1 −s∗4� . . . −s∗3�+Δτ+1 s∗� . . . s∗Δτ+1 s∗2� . . . s∗�+Δτ+1

](10)

AΔτ2 =

[s1 . . . sΔτ s�+1 . . . s�+Δτ s2�+1 . . . s2�+Δτ s3�+1 . . . s3�+Δτ 0 . . . 00 . . . 0 −s∗2�+Δτ . . . −s∗2�+1 −s∗3�+Δτ . . . −s∗3�+1 s∗Δτ . . . s∗1 s∗�+Δτ . . . s∗�+1

](11)

[Δsi Δs�+Δτ+1−i Δs2�+i Δs3�+Δτ+1−i 00 −Δs∗2�+i −Δs∗3�+Δτ+1−i Δs∗i Δs∗�+Δτ+1−i

](13)


According to Theorem 1 in [35], we need to prove that

‖H‖ ≤ c1‖h‖, det(HHH) ≥ c2‖h‖2k, (16)

where c1 and c2 are positive constants independent of h, andk = 8 for H. ‖ · ‖, (·)H , and det(·) denote the Frobeniusnorm, the Hermitian transpose, and the determinant of a matrix,respectively.

Given (15), we have

‖H‖ = 2√2√|h1|2 + |h2|2, ‖h‖ =

√|h1|2 + |h2|2.

Thus,

‖H‖ ≤ 2√2‖h‖.

Similarly, we can derive

det(HHH) ≥ 1

36‖h‖16.

As (16) is fulfilled, the full diversity of BDT-TIR AC can beachieved using linear decoders.

APPENDIX CPROOF OF THEOREM 3

To prove the full diversity gain, we need to show that ΔGΔτ

is always full rank for any [c0, c1, c2, c3] �= [c0, c1, c2, c3].

Define ΔcjΔ= α(aj + bjθ)− α(aj + bjθ) and Δcj

Δ= α(aj +

bj θ)− α(aj + bj θ), j = 1, 2, . . . , 4�. The asynchronous ver-sion of (6) can also be decomposed, similar to the procedure

in decoupling (9). We can obtain GΔτ2,i as in (17), shown at the

bottom of the page, similar as AΔτ2,i in (12).

We now prove that GΔτ2,i has full diversity gain. For two

symbol sequences, the difference matrix ΔGΔτ2,i is given in

(18), shown at the bottom of page.If Δci �= 0, we have Δci �= 0. The first and fourth columns

of ΔGΔτ2,i form a rank-two matrix as[

Δci Δc3�+Δτ+1−i

0 Δci

].

If Δci = 0 and Δc3�+Δτ+1−i �= 0, the third and fourthcolumns form a rank-two matrix as[

Δc2�+i Δc3�+Δτ+1−i

jΔc3�+Δτ+1−i 0

].

The same rank analysis can be applied to the case inwhich [Δci,Δc3�+Δτ+1−i] = [0, 0] and Δc2�+i �= 0, or thecase in which [Δci,Δc3�+Δτ+1−i,Δc2�+i] = [0, 0, 0] andΔc�+Δτ+1−i �= 0. Therefore, the full rank is maintained.

APPENDIX DPROOF OF THEOREM 4

The asynchronous version of C is CΔτ in (19), shown onthe next page, which can be decomposed into CΔτ

1 in (20),shown on the next page, and CΔτ

2 in (21), shown on the nextpage. CΔτ

2 can be further divided into CΔτ2,i in (22), shown

on the next page, where i = 1, 2, . . . ,Δτ , and i = Δτ + 1− i.As CΔτ

1 holds the COD structure, we focus on CΔτ2,i for the

H =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

{h1} 0 0 0 −�{h1} 0 0 0

0 {h1} −{h2} 0 0 −�{h1} −�{h2} 0

0 0 {h1} −{h2} 0 0 −�{h1} −�{h2}{h2} 0 0 {h1} �{h2} 0 0 −�{h1}

0 {h2} 0 0 0 �{h2} 0 0

�{h1} 0 0 0 {h1} 0 0 0

0 �{h1} −�{h2} 0 0 {h1} {h2} 0

0 0 �{h1} −�{h2} 0 0 {h1} {h2}�{h2} 0 0 �{h1} −{h2} 0 0 {h1}

0 �{h2} 0 0 0 −{h2} 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(15)

GΔτ2,i =

[ci c�+Δτ+1−i c2�+i c3�+Δτ+1−i 00 jc2�+i jc3�+Δτ+1−i ci c�+Δτ+1−i

](17)

[Δci Δc�+Δτ+1−i Δc2�+i Δc3�+Δτ+1−i 00 jΔc2�+i jΔc3�+Δτ+1−i Δci Δc�+Δτ+1−i

](18)


CΔτ =

⎡⎢⎢⎢⎣

s1 . . . sΔτ sΔτ+1 . . . s� s�+1 . . . s�+Δτ s�+Δτ+1 . . . s2� −s∗2�+1 . . .

s2�+1 . . . s2�+Δτ s2�+Δτ+1 . . . s3� s3�+1 . . . s3�+Δτ s3�+Δτ+1 . . . s4� s∗1 . . .

0 . . . 0 s5� . . . s4�+Δτ+1 s4�+Δτ . . . s4�+1 s6� . . . s5�+Δτ+1 s5�+Δτ . . .

0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . .

−s∗2�+Δτ −s∗2�+Δτ+1 . . . −s∗3� −s∗3�+1 . . . −s∗3�+Δτ −s∗3�+Δτ+1 . . . −s∗4� −s∗4�+1 . . . −s∗4�+Δτ

s∗Δτ s∗Δτ+1 . . . s∗� s∗�+1 . . . s∗�+Δτ s∗�+Δτ+1 . . . s∗2� 0 . . . 0

s5�+1 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0

0 s5� . . . s4�+Δτ+1 s4�+Δτ . . . s4�+1 s6� . . . s5�+Δτ+1 s5�+Δτ . . . s5�+1

−s∗4�+Δτ+1 . . . −s∗5� −s∗5�+1 . . . −s∗5�+Δτ −s∗5�+Δτ+1 . . . −s∗6� 0 . . . 0

0 . . . 0 0 . . . 0 0 . . . 0 −s∗4�+1 . . . −s∗4�+Δτ

s∗� . . . s∗Δτ+1 s∗Δτ . . . s∗1 s∗2� . . . s∗�+Δτ+1 s∗�+Δτ . . . s∗�+1

−s3� . . . −s2�+Δτ+1 −s2�+Δτ . . . −s2�+1 −s4� . . . −s3�+Δτ+1 −s3�+Δτ . . . −s3�+1

0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0

−s∗4�+Δτ+1 . . . −s∗5� −s∗5�+1 . . . −s∗5�+Δτ −s∗5�+Δτ+1 . . . −s∗6� 0 . . . 0

s∗3� . . . s∗2�+Δτ+1 s∗2�+Δτ . . . s∗2�+1 s∗4� . . . s∗3�+Δτ+1 s∗3�+Δτ . . . s∗3�+1

s� . . . sΔτ+1 sΔτ . . . s1 s2� . . . s�+Δτ+1 s�+Δτ . . . s�+1

⎤⎥⎥⎥⎦ (19)

CΔτ1 =

⎡⎢⎢⎢⎣

sΔτ+1 . . . s� s�+Δτ+1 . . . s2� −s∗2�+Δτ+1 . . . −s∗3� −s∗3�+Δτ+1 . . . −s∗4�s2�+Δτ+1 . . . s3� s3�+Δτ+1 . . . s4� s∗Δτ+1 . . . s∗� s∗�+Δτ+1 . . . s∗2�

s5� . . . s4�+Δτ+1 s6� . . . s5�+Δτ+1 0 . . . 0 0 . . . 0

0 . . . 0 0 . . . 0 s5� . . . s4�+Δτ+1 s6� . . . s5�+Δτ+1

−s∗4�+Δτ+1 . . . −s∗5� −s∗5�+Δτ+1 . . . −s∗6� 0 . . . 0 0 . . . 0

0 . . . 0 0 . . . 0 −s∗4�+Δτ+1 . . . −s∗5� −s∗5�+Δτ+1 . . . −s∗6�s∗� . . . s∗Δτ+1 s∗2� . . . s∗�+Δτ+1 s∗3� . . . s∗2�+Δτ+1 s∗4� . . . s∗3�+Δτ+1

−s3� . . . −s2�+Δτ+1 −s4� . . . −s3�+Δτ+1 s� . . . sΔτ+1 s2� . . . s�+Δτ+1

⎤⎥⎥⎥⎦

(20)

CΔτ2 =

⎡⎢⎢⎢⎣

s1 . . . sΔτ s�+1 . . . s�+Δτ −s∗2�+1 . . . −s∗2�+Δτ −s∗3�+1 . . . −s∗3�+Δτ −s∗4�+1 . . . −s∗4�+Δτ

s2�+1 . . . s2�+Δτ s3�+1 . . . s3�+Δτ s∗1 . . . s∗Δτ s∗�+1 . . . s∗�+Δτ 0 . . . 0

0 . . . 0 s4�+Δτ . . . s4�+1 s5�+Δτ . . . s5�+1 0 . . . 0 0 . . . 0

0 . . . 0 0 . . . 0 0 . . . 0 s4�+Δτ . . . s4�+1 s5�+Δτ . . . s5�+1

−s∗5�+1 . . . −s∗5�+Δτ 0 . . . 0 0 . . . 0 0 . . . 0

0 . . . 0 −s∗4�+1 . . . −s∗4�+Δτ −s∗5�+1 . . . −s∗5�+Δτ 0 . . . 0

s∗Δτ . . . s∗1 s∗�+Δτ . . . s∗�+1 s∗2�+Δτ . . . s∗2�+1 s∗3�+Δτ . . . s∗3�+1

−s2�+Δτ . . . −s2�+1 −s3�+Δτ . . . −s3�+1 sΔτ . . . s1 s�+Δτ . . . s�+1

⎤⎥⎥⎥⎦ (21)

⎡⎢⎢⎢⎣

si s�+i −s∗2�+i −s∗3�+i

−s∗4�+i −s∗5�+i

0 0 0

s2�+i s3�+i s∗i s∗�+i

0 0 −s∗4�+i −s∗5�+i

0

0 s4�+i s5�+i 0 0 s∗i s∗�+i

s∗2�+i s∗3�+i

0 0 0 s4�+i s5�+i −s2�+i −s3�+i si s�+i

⎤⎥⎥⎥⎦ (22)


⎡⎢⎢⎣

Δsi Δs�+i −Δs∗2�+i −Δs∗3�+i

−Δs∗4�+i 0 0 0 0

Δs2�+i Δs3�+i Δs∗i Δs∗�+i

0 0 −Δs∗4�+i 0 0

0 Δs4�+i 0 0 0 s∗i Δs∗�+i

Δs∗2�+i s∗3�+i

0 0 0 Δs4�+i 0 −Δs2�+i −Δs3�+i Δsi Δs�+i

⎤⎥⎥⎦ (32)

⎡⎢⎢⎣

xi,4�+i x8�+i,12�+i −x∗�+i,5�+i −x∗

9�+i,13�+i−x∗

2�+i,6�+i −x∗10�+i,14�+i

x7�+i,3�+i x15�+i,11�+i 0

x�+i,5�+i x9�+i,13�+i x∗i,4�+i x∗

8�+i,12�+ix7�+i,3�+i x15�+i,11�+i x∗

2�+i,6�+i x∗10�+i,14�+i

0

0 x2�+i,6�+i x10�+i,14�+i −x7�+i,3�+i −x15�+i,11�+i x∗i,4�+i x∗

8�+i,12�+i−x∗

�+i,5�+i −x∗9�+i,13�+i

0 −x7�+i,3�+i −x15�+i,11�+i −x2�+i,6�+i −x10�+i,14�+i x�+i,5�+i x9�+i,13�+i xi,4�+i x8�+i,12�+i

⎤⎥⎥⎦(33)

full diversity proof. For two 6�-symbol sequences, s and s, we

define the difference matrix of CΔτ2,i

Δ= CΔτ

2,i (s)−CΔτ2,i (s).

• Assume that Δs5�+i �= 0 and Δs4�+i �= 0. If ΔCΔτ2,i

is not full rank, ∃ α, β, γ, which are not zeroes,satisfy that ΔCΔτ

2,i [1, :] = αΔCΔτ2,i [2, :] + βΔCΔτ

2,i [3, :] +

γΔCΔτ2,i [4, :]. A contradiction can be derived from this

equation.

Δs�+i = αΔs3�+i + βΔs4�+i (23)

−Δs∗3�+i

= αΔs∗�+i

+ γΔs4�+i (24)

0 = − αΔs∗4�+i + βΔs∗�+i

− γΔs3�+i (25)

0 = βΔs∗3�+i

+ γΔs�+i. (26)

From (26), we have γΔs�+i = −βΔs∗3�+i

. Based on (26),(23)–(25) become

αγΔs3�+i+βγΔs4�+i=− βΔs∗3�+i

(27)

−αβ∗Δs3�+i+|γ|2Δs4�+i=− γ∗Δs∗3�+i

(28)

γ∗αΔs∗4�+i +(|γ|2+|β|2

)Δs3�+i=0. (29)

Replacing Δs∗3�+i

in (28) with (27), we obtain

α(|β|2 + |γ|2

)s3�+i = 0 (30)

−(|β|2 + |γ|2

)s3�+i = αγ∗Δs∗4�+i. (31)

Therefore, Δs4�+i = 0, which contradicts with theassumption.

• When Δs5�+i = 0 and Δs4�+i �= 0, the remaining struc-ture will be (32), shown on top of the page. The first,third, sixth, and eighth columns of (32) will contain twoAlamouti code structure that lead to full rank if Δsi �=0 or Δs2�+i �= 0 or Δs�+i �= 0 or Δs3�+i �= 0. For theremaining five columns as in (34), as Δs4�+i �= 0, the first,third, and fourth rows are linearly independent.⎡

⎢⎢⎣Δs�+i −Δs∗

3�+i−Δs∗4�+i 0 0

Δs3�+i Δs∗�+i

0 −Δs∗4�+i 0

Δs4�+i 0 0 Δs∗�+i

s∗3�+i

0 Δs4�+i 0 −Δs3�+i Δs�+i

⎤⎥⎥⎦ (34)

Assuming that the remaining five columns are not fullrank, ∃ α, β, where α, β are not zeroes, α[Δs4�+i, 0, 0,Δs∗

�+i, Δs∗

3�+i] + β[0, Δs4�+i, 0, −Δs3�+i, Δs�+i] =

[Δs3�+i,Δs∗�+i

, 0,−Δs∗4�+i, 0], that is,⎧⎪⎪⎨⎪⎪⎩

αΔs4�+i = Δs3�+i,βΔs4�+i = Δs∗

�+i,

αΔs∗�+i

+ β(−Δs3�+i) = −Δs∗4�+i,

αΔs∗3�+i

+ βΔs�+i = 0.

The solution is Δs4�+i = 0, which contradicts the as-sumption. Therefore, the matrix in (32) is full rank.

• When Δs5�+i �= 0 and Δs4�+i = 0, the full rank can bederived in the same way.

• When Δs5�+i = 0 and Δs4�+i = 0, because of the exis-tence of the Alamouti code in the remaining structure, fulldiversity is assured.

Therefore, full rank is maintained.

APPENDIX EPROOF OF THEOREM 5

The asynchronous version of (7) can also be decomposed.

We can obtain FΔτ2,i in (33). For simplicity, we denote i

Δ= Δτ +

1− i.We define the difference matrix of FΔτ

2,i as ΔFΔτ2,i

Δ=

FΔτ2,i (s)− FΔτ

2,i (s).• When ΔsiΔs4�+iΔs�+iΔs5�+i �= 0 and Δs8�+iΔs12�+i

Δs9�+iΔs13�+i �= 0, we can derive that ΔFΔτ2,i forms a

full rank matrix.• When Δsi = 0, Δs4�+iΔs�+iΔs5�+i �= 0 and Δs8�+i

Δs12�+iΔs9�+iΔs13�+i �= 0, ΔFΔτ2,i is still a full rank

matrix.• Similarly, we can derive that as long as s �= s, ΔFΔτ

2,i is afull rank matrix.

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Yun Liu (S’12) received the B.Eng. degree intelecommunication engineering from The BeijingUniversity of Posts and Telecommunications, Bei-jing, China. He is currently working toward the Ph.D.degree with the Department of Electronic Engineer-ing, The Chinese University of Hong Kong, Shatin,Hong Kong.

In 2011, he was a visiting Ph.D. student at theUniversity of Missouri, Columbia, MO, USA. In2014, awarded an Australian Endeavour Fellowship,he was a visiting Ph.D. student at the University of

New South Wales, Sydney, Australia. His research interests include cooperativecommunications, space-time coding and physical-layer network coding.

Wei Zhang (S’01–M’06–SM’11) received the Ph.D.degree in electronic engineering from The Chi-nese University of Hong Kong, Shatin, Hong Kong,in 2005.

He was a Research Fellow with the Depart-ment of Electronic and Computer Engineering, TheHong Kong University of Science and Technology,Kowloon, Hong Kong, during 2006–2007. He hasbeen with the School of Electrical Engineering andTelecommunications, The University of New SouthWales, Sydney, N.S.W., Australia, since 2008, where

he is an Associate Professor. His current research interests include cognitiveradio, cooperative communications, space-time coding, and multiuser MIMO.

Dr. Zhang received the Best Paper Award at the 50th IEEE Global Commu-nications Conference, Washington DC, USA, in 2007 and the IEEE Communi-cations Society Asia-Pacific Outstanding Young Researcher Award in 2009. Hewas TPC Co-Chair for the Communications Theory Symposium of the IEEEInternational Conference on Communications, Kyoto, Japan, in 2011. He iscurrently serving as a TPC Chair of the Symposium on Signal Processing forCognitive Radios and Networks of the 2nd IEEE Global Conference on Signaland Information Processing, Atlanta, GA, USA, 2014. He is an Editor of theIEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS and an Editor ofthe IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS (CognitiveRadio Series).


Soung Chang Liew (F’12) received the S.B., S.M.,E.E., and Ph.D. degrees from the Massachusetts In-stitute of Technology, Cambridge, MA, USA.

From 1984 to 1988, he was at the MIT Laboratoryfor Information and Decision Systems, where heinvestigated fiber-optic communications networks.From March 1988 to July 1993, he was at Bell-core (now Telcordia), NJ, USA, where he engagedin Broadband Network Research. He has been aProfessor with the Department of Information En-gineering, the Chinese University of Hong Kong,

Shatin, Hong Kong, since 1993. He is an Adjunct Professor at PekingUniversity, Shenzhen, China and Southeast University, Nanjing, China. Hisresearch interests include wireless networks, Internet protocols, multimediacommunications, and packet switch design. His research group won best paperawards at IEEE MASS in 2004 and IEEE WLN in 2004. Separately, TCPVeno, a version of TCP to improve its performance over wireless networksproposed by his research group, has been incorporated into a recent releaseof Linux OS. In addition, he initiated and built the first inter-universityATM network testbed in Hong Kong in 1993. More recently, his researchgroup pioneered the concept of physical-layer network coding (PNC). Besidesacademic activities, he is active in the industry. He co-founded two technol-ogy start-ups in Internet software and has been serving as a consultant tomany companies and industrial organizations. He is the holder of nine U.S.patents and a Fellow of the IEEE, IET, and HKIE. He currently serves asEditor for IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS andAd Hoc and Sensor Wireless Networks. He is the recipient of the first Vice-Chancellor Exemplary Teaching Award in 2000 and the Research ExcellenceAward in 2013 from the Chinese University of Hong Kong. Publications ofDr. Liew can be found at www.ie.cuhk.edu.hk/soung.

Pak-Chung Ching (F’11) received the B. Eng. (1stClass Honors) and Ph.D. degrees from the Universityof Liverpool, Liverpool, UK, in 1977 and 1981,respectively.

From 1981 to 1982 he was Research Officer at theUniversity of Bath, Bath, UK. In 1982, he returned toHong Kong and joined the Hong Kong PolytechnicUniversity, Kowloon, Hong Kong, as a Lecturer.Since 1984, he has been with the Department ofElectronic Engineering, the Chinese University ofHong Kong (CUHK), Shatin, Hong Kong, where he

is currently Professor of Electronic Engineering. He was Department Chairmanfrom 1995 to 1997, Dean of Engineering from 1997 to 2003, and Head of ShawCollege, Shatin, Hong Kong, from 2004 to 2008. He became Director of theShun Hing Institute of Advanced Engineering in 2004. On August 1, 2006, heassumed his new responsibility as Pro-Vice-Chancellor, CUHK. His researchinterests include adaptive digital signal processing, time delay estimation andtarget localization, blind signal estimation and separation, automatic speechrecognition, speaker identification/verification and speech synthesis, and ad-vanced signal processing techniques for wireless communications. She is veryactive in promoting professional activities, both in Hong Kong and overseas.He was a Council Member of the Institution of Electrical Engineers (IEE), pastchairman of the IEEE Hong Kong Section, an Associate Editor of the IEEETRANSACTIONS ON SIGNAL PROCESSING from 1997 to 2000 and IEEESIGNAL PROCESSING LETTERS from 2001 to 2003. He was also a memberof the Technical Committee of the IEEE Signal Processing Society from 1996to 2004. He was appointed Editor-in-Chief of the HKIE Transactions between2001 and 2004. He is also an Honorary Member of the editorial committee forthe Journal of Data Acquisition and Processing.

Dr. Ching has been instrumental in organizing many international confer-ences in Hong Kong including the 1997 IEEE International Symposium onCircuits and Systems where he was the Vice-Chairman and the 2003 IEEEInternational Conference on Acoustics, Speech and Signal Processing wherehe was the Technical Program Chair. He is a Fellow of the IEEE, IEE, HKIE,and HKAES. He was awarded the IEEE Third Millennium Medal in 2000 andthe HKIE Hall of Fame in 2010.

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Bounded Delay-Tolerant Space-Time Codes for Distributed Antenna Systems