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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 7, JULY 2003 1093

Further Results on DifferentialSpaceTime Modulations

Eugenio Chiavaccini, Member, IEEE,and Giorgio M. Vitetta, Senior Member, IEEE

AbstractIn this paper, some novel results on the spacetimedifferential modulation schemes described by Hochwald andSweldens, and Tarokh and Jafarkhani, are derived under theassumption of a quasi-static multiple-antenna channel. First, theoptimality (in the maximum-likelihood sense) of the one-shotdetection algorithms of Hochwald and Sweldens, and Tarokh andJafarkhani, is proved. Then the expression of the pairwise errorprobability for these detectors is derived, and, as a particularcase, the bit-error rate is computed for the binary phase-shiftkeyed scheme of Tarokh and Jafarkhani. Finally, a per-survivorprocessing (PSP) detection algorithm for this class of modulationsis illustrated. Numerical results evidence both the superiorityof the PSP strategy over the one-shot spacetime differentialdetectors and the accuracy of the above-mentioned bit-error rate

formula.

Index TermsMaximum-likelihood detection, per-survivor pro-cessing (PSP), spacetime coding, Viterbi algorithm.

I. INTRODUCTION

I N THE LAST few years, it has been shown that the informa-tion capacity of wireless communication systems canbe sub-stantially increased by employing multiple transmit and receive

antennas [3]. A practical solution to approach the enlarged ca-

pacity, i.e., to increase the data rate in wireless communications,

is the use of coding schemes for multiple antennas, namely,

spacetime codes (STCs) [4], which can provide receiver diver-

sity and coding gain without bandwidth expansion.Recently, different spacetime block codes (STBCs) [5][7]

and spacetime trellis codes (STTCs) [8], [9] have been pro-

posed. All these ST block-coding schemes have been designed

under the assumption that ideal channel state information (CSI)

is available at the receiver side, so that coherent detection algo-

rithms can be employed. With multiple antennas and in a time-

varying scenario, however, channel estimation may represent a

serious problem in the implementation of these coding tech-

niques. This consideration has motivated the search for novel

STCs which can be detected in a noncoherent fashion. The first

example of an ST block-coding scheme admitting differential

detection has been proposed by Tarokh and Jafarkhani [2]. A

deeper insight into the problem has been provided by Hochwald

Paper approved by V. A. Aalo, the Editor for Diversity and Fading ChannelTheory of the IEEE Communications Society. Manuscript received November9, 2001; revised November 15, 2002. This work has been supported in part bythe MIUR, Italy, within the framework of the national project MC-CDMA: ARadio Interface for the Fourth Generation of Mobile Wireless Systems. Thispaper was presented in part at the IEEE International Conference on Commu-nications, New York, NY, April 2002.

The authors are with the Department of Information Engineering, Universityof Modena and Reggio Emilia, 41100 Modena, Italy (e-mail: chiavaccini.eu-genio@unimo.it, giorgio.vitetta@unimo.it).

Digital Object Identifier 10.1109/TCOMM.2003.814211

and Sweldens [1], [10] and Hughes [11], [12], who have devel-

oped a general approach, based on finite group theory, to dif-

ferential modulation for multiple antennas. In all these papers,

bounds on error performance and optimal differential block-by-

block (i.e., one-shot) detectors are derived under the hypothesis

that the multiple-antenna channel can be assumed static over at

least two consecutive blocks. More recently, the use of multiple-

symbol differential detection [23] and decision-feedback differ-

ential detection [24] has been proposed as a possible solution to

avoid the substantial loss of conventional differential detectors

in moderately fast fading [22]. In this paper, we investigate the

detection of the differential ST modulation schemes proposed

by HochwaldSweldens (HS) [1] and TarokhJafarkhani (TJ)

[2], assuming a quasi-static (QS) communication channel. This

means that, both in the derivation of optimal detection algo-

rithms and in performance analysis, channel variations within

each block are deemed negligible, whereas changes from block

to block are taken into account.

The purpose of this paper is threefold. First, we show that

the block-by-block differential detectors of [1] and [2]1 are still

optimal in the maximum-likelihood (ML) sense under the QS

hypothesis. Then, a closed-form solution for the pairwise error

probability (PEP) of the block-by-block differential detector for

HS codes [1] is evaluated. This result is exploited to derive the

bit-error probability of the TJ scheme when binary phase-shiftkeying (BPSK) symbols feed the ST encoder. Finally, an ap-

proximate ML receiver for sequence detection is derived. It is

based on the Viterbi algorithm and incorporates per-survivor

processing (PSP) techniques for channel estimation [13].

The paper is organized as follows. Signal and channel models

are illustrated in Section II and are employed in Section III to

derive one-shot ML differential detectors for both HS and TJ

schemes. The PEP of the block-by-block differential detector

for HS codes is derived in Section IV. The PEP expression is

also employed to derive the bit-error probability of the BPSK

TJ code. An approximate ML sequence detector for differential

ST modulations is illustrated in Section V. Numerical results

are illustrated in Section VI. Finally, Section VII offers someconclusions.

II. SIGNAL ANDCHANNELMODELS

In this paper, we focus on a STBC system employing

transmit and receive antennas [4]. The set of channel symbols

transmitted during the th block2 is denoted by the

1The differential detector of [2] has been heuristically derived without anyclaim of optimality.

2Throughout the paper, the parameter denotes the block index, whereas specifies the location of a channel symbol within each block.

0090-6778/03$17.00 2003 IEEE

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CHIAVACCINI AND VITETTA: FURTHER RESULTS ON DIFFERENTIAL SPACETIME MODULATIONS 1099

Fig. 4. BERperformanceof theTJC CR andVR withsingle diversity at the receiver and 1 .

Fig. 5. BERperformanceof theTJC CR andVR withsingle diversity at the receiver and 1 .

Figs. 4 and 5 illustrate the error performance of the CR and

the VR for the TJC with and ,

respectively, and single diversity at the receiver . In the

VR (corresponding to a four-state trellis), a prediction

length and a decision d elay have b een s elected.

Similarly, Figs. 6 and 7 compare the error performance of the

CRs with that of the VRs for the HS1 and the HS2 codes with

and , respectively, and singlediversity at the receiver . Here, and have

been chosen for the VRs. This means that an eight-state (16-

state) trellis decoder has been employed for the HSC1 (HSC2)

with a decision delay .

Figs. 47 show that VRs substantially outperform the CRs

for medium and high SNRs. For low SNRs, the small differ-

ence between the VRs and the CRs can be related to the poor

quality of the channel prediction evaluated in (53) on the basis

of the last symbols. The superiority of the VRs over the CRs

is also exemplified by Figs. 8 and 9, showing the error floor of

the CR and the VR for the TJC and the HSC2 with single di-

versity and for the TJC with double diversity ,

Fig. 6. BER performance of the HSC CR and VR with single diversity at the receiver and .

Fig. 7. BER performance of the HSC CR and VR with single diversity at the receiver and 1 .

respectively, and [the continuous lines

for the TJC-CR have been generated using (41)]. It can be easily

inferred that the VR has a substantially lower error floor than the

CR for , and the performance gap gets larger

as increases.

Finally, we note that the CR performs better with the TJC

than with HSC2 (see Fig. 8). The opposite occurs when the VR

is employed. The VR results can be interpreted as follows. WithHSCs all the matrices are diagonal, so that the th antenna

transmits only in the block-spaced symbol intervals ,

with . Therefore, in this case, the signal

model (2) and the structure of the matrix (45) hold, even

if the fading channel changes within each block. On the other

hand, these considerations do not apply to TJC, as (2) is ap-

proximate.

VII. CONCLUSIONS

In this paper, some new results on the detection of the dif-

ferential STBCs proposed in [1] and [2] have been illustrated.

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1100 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 51, NO. 7, JULY 2003

Fig. 8. Error floor of the TJC and HSC2 CR and VR with single diversity at the receiver.

Fig. 9. Error floor of the TJC CR and VR with double diversity atthe receiver.

In particular, the optimality of the detector suggested in [ 1] and

[2] has been proved and error-rate formulas have been derived

under the assumption of a QS scenario. Finally, a PSP-based re-

ceiver substantially outperforming conventional differential de-

tectors has been developed.

APPENDIX

In this Appendix, the derivation of the eigenvalues of the ma-

trix

(54)

is briefly described. Let undergo a similitude transformation

producing

(55)

where is an nonsingular matrix. Then the eigen-

values of are the same as [21]. Here, we choose

(56)

with

(57)

Substituting (56) and (57) into (55) yields the lower-triangular

block matrix

(58)

with

(59)

and . It is easy to verify that eigen-

values of the matrix (58) are null and that its remaining

eigenvalues are the roots of the characteristic polynomial

of . If we take into account the

property

(60)

(holding when the matrices and commute) and note that

[where is given by (38)], the polynomial

can be simplified as

(61)

Finally, let denote the eigenvalues of . Then, solving

produces (37).

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Eugenio Chiavaccini (S99M02) was born inLivorno, Italy, in 1973. He received the Dr. Ing. de-gree (cum laude) in telecommunication engineeringfrom the University of Pisa, Pisa, Italy, in 1998. In2002, he received the Ph.D. degree in informationengineering at the University of Modena and ReggioEmilia, Modena, Italy.

His main research interests include digital trans-mission theory, detection/equalization techniques,and coded modulations. Presently, he is workingon pure and applied electromagnetic compatibility

topics and on the application of digital signal processing techniques to EMissues.

Giorgio M. Vitetta(S89M91SM99) was bornin Reggio Calabria, Italy, in April 1966. He receivedthe Dr. Ing. degree in electronic engineering (cumlaude) in 1990 and the Ph.D. degree in 1994, bothfrom the University of Pisa, Pisa, Italy.

From 1992 to 1993 he was with the Universityof Canterbury, Christchurch, New Zealand, doingresearch for digital communications on fadingchannels. From 1995 to 1998, he was a ResearchFellow at the Department of Information Engi-neering, University of Pisa. From 1998 to 2001,

he was an Associate Professor of Telecommunications at the University ofModena and Reggio Emilia, Modena, Italy, where he is now a Full Professorof Telecommunications. His main research interests lie in the broad areaof communication theory, with particular emphasis on coded modulation,synchronization, and channel equalization. He is serving as an Editor of boththe IEEE TRANSACTIONS ONCOMMUNICATIONS and the IEEE TRANSACTIONS

ONWIRELESSCOMMUNICATIONS.