1948 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO....

1948 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 11, NOVEMBER 2006

Constellation Design for Trellis-Coded UnitarySpace–Time Modulation Systems

Yi Wu, Member, IEEE, Vincent K. N. Lau, Senior Member, IEEE, and Matthias Pätzold, Senior Member, IEEE

Abstract—We consider the problem of creating signal constel-lations for trellis-coded unitary space–time communication links,where neither the transmitter nor the receiver knows the fadinggains of the channel. Our study includes constellation-design tech-niques for trellis-coded schemes with and without parallel paths,which allows us to find a tradeoff between low complexity andhigh performance. We present a new formulation of the constella-tion design problem for trellis-coded unitary space–time modula-tion (TCUSTM) schemes. The two key differences in our approachagainst those of other authors are that we not only combine the con-stellation design and mapping by set partitioning into one step, butwe also use directly the Chernoff bound of the pairwise error prob-ability as a design metric. By novelly employing a theorem for theClarke subdifferential of the sum of the k largest singular valuesof the unitary matrix, we also present a numerical optimizationprocedure for finding signal constellations resulting in high-perfor-mance communications systems. To demonstrate the advantages ofour new design method, we report the best constellations found forTCUSTM systems. Simulation results show that these constella-tions achieve a 1-dB coding gain at a bit-error rate of 10�4 againstusually used constellations.

Index Terms—Constellation design, multiple antennas, non-coherent communications, trellis-coded modulation (TCM).

I. INTRODUCTION

CONSIDERING the rapidly increasing demand for highdata rates and reliable wireless communications, spectrally

efficient transmission schemes are of great importance for fu-ture wireless systems. Recent information-theoretic works byFoschini and Gans [1], as well as Telatar [2], indicate that usingmultiple antennas at both the transmitter and the receiver candramatically increase the spectral efficiency. Several codingand modulation schemes have been proposed to exploit thispotential increase in spectral efficiency through space diversity(see, e.g., [3]–[7], references therein, and for an overview, [8]).All these results are derived under the assumption that the re-ceiver has perfect channel state information (CSI). However,

Paper approved by X. Dong, the Editor for Modulation and Signal Designof the IEEE Communications Society. Manuscript received March 1, 2005; re-vised January 14, 2006. This paper was presented in part at the IEEE GlobalTelecommunications Conference, St. Louis, MO, November 2005.

Y. Wu was with the Department of Electrical and Electronic Engineering,Hong Kong University of Science and Technology, Clear Water Bay, HongKong. He is now with the Department of Information and CommunicationTechnology, Agder University College, NO-4876 Grimstad, Norway (e-mail:[email protected]).

V. K. N. Lau is with the Department of Electrical and Electronic Engineering,Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong(e-mail: [email protected]).

M. Pätzold is with the Department of Information and CommunicationTechnology, Agder University College, NO-4876 Grimstad, Norway (e-mail:[email protected]).

Digital Object Identifier 10.1109/TCOMM.2006.884828

the known-channel assumption may not be realistic in rapidlychanging fading environments or when a large number of an-tennas is used. Motivated by these considerations, the authors of[9]and[10]studied thecapacityofnoncoherentmultiple-antennachannels,where neither the transmitternor the receiver knows thefading gains of the channel. They showed that in the noncoherentcase, the channel capacity still increases with the number oftransmit and receive antennas, but quite as expected, the increasein capacity is somewhat smaller than with perfect CSI. A class ofunitary space–time signals that are well-tailored for noncoherentchannels was initiated by Hochwald and Marzetta in [11]. Sincethen, numerous methods have been proposed to construct unitaryspace–time signals operating in the absence of CSI [12], [13].All these designs, however, focus mainly on uncoded systems.

To exploit the diversity and to boost coding gains, a power-and bandwidth-efficient trellis-coded unitary space–time modu-lation (TCUSTM) system was suggested in [14]–[17]. There, itwas shown that TCUSTM systems perform significantly betterthan uncoded unitary space–time modulation (USTM) systemswith the same spectral efficiency. Nonetheless, they all directlyuse the unitary space–time signal constellations designed for un-coded systems. As we will see in this paper, this is not optimalfor trellis-coded systems with multiple antennas, even thougha general set-partitioning method, based on an idea stemmingfrom conventional set partitioning for phase-shift keying (PSK)-or quadrature amplitude modulation (QAM)-type signal con-stellations, has been employed.

The primary purpose of this paper is to design trellis-matchedunitary space–time signal constellations for TCUSTM systemswith an arbitrary number of transmit antennas. For such sys-tems, we give a new formulation of the constellation designproblem. There are two key differences in our approach againstthose of other authors. The first one is that we combine the con-stellation design and mapping by set partitioning into one step,and the second is that we use directly the Chernoff bound ofthe pairwise error probability (PEP) as a design metric, whichis a better figure of merit than the previously used chordal dis-tance in unitary constellation design, as suggested in [12] and[13]. By employing a theorem for the Clarke’s subdifferentialof the sum of the largest singular values of the unitary ma-trix, we present in this paper a numerical optimization proce-dure for finding good constellations using the Chernoff boundof the PEP as a design metric. We also carried out simulationsand found that with the proposed new constellations, one canachieve a coding gain of about 1 dB at a bit-error rate (BER)of , compared with the previously used constellations inTCUSTM schemes, such as those in [12] and [13].

The rest of the paper is organized as follows. In Section II,we describe our channel model and some preliminaries of

0090-6778/$20.00 © 2006 IEEE

WU et al.: CONSTELLATION DESIGN FOR TRELLIS-CODED UNITARY SPACE–TIME MODULATION SYSTEMS 1949

Fig. 1. Block diagram of a TCUSTM system.

TCUSTM. In Section III, the unitary space–time constellationdesign is treated, including both the design rules and the algo-rithm. Various simulation results and related discussions aregiven in Section IV. Finally, Section V contains the conclusions.

The following notation is used throughout the paper: ,, denotes conjugate, transpose, and conjugate transpose,

respectively. The symbol denotes the expectation operator,is the real part of the complex variate , represents

the Frobenius norm, denotes the trace of the matrix ,and represents the identity matrix. The multivariatecircularly symmetric complex Gaussian distribution with mean

and covariance matrix is denoted by .

II. CHANNEL MODEL AND PRELIMINARIES OF TCUSTM

A. Channel Model

We consider a point-to-point noncoherent wireless communi-cation link with transmit and receive antennas, operatingin a Rayleigh flat-fading environment. We assume that each re-ceive antenna responds to each transmit antenna through a sta-tistically independent fading coefficient. We also assume thatthe fading is quasi-static, i.e., the channel coefficients remainconstant for a coherence interval of symbol periods, and maychange to new independent realizations from one symbol pe-riod to the next symbol period. Using the complex basebandrepresentation, the received signals over one coherence intervalcan be written in matrix form as [11]

(1)

where is the complex matrix of the received signals,is the complex matrix of the transmitted signals,

is the complex matrix of the Rayleigh fading channelcoefficients, is the complex matrix of the additivereceiver noise components, and is the expected signal-to-noiseratio (SNR) at each receive antenna. Note that the th column of

represents the signal transmitted over the th transmit antennaas a function of time. Furthermore, we assume that each entry in

and is independent, identically distributed (i.i.d.) complexGaussian . The transmitted symbols are normalized toobey

(2)

where is the signal sent by the th transmit antenna at time. This means that the mean signal power, averaged over the

transmit antennas, is kept constant for each channel use. Forthe case that the realizations of are unknown to the receiverand transmitter, it was shown in [9] and [10] that a class of ca-pacity-achieving signals may be constructed from a constella-tion of signals consisting of complex matrices ,

, obeying . Basedon the results in [9], a new class of signaling scheme namedUSTM has been developed in [11]. In the USTM schemes, theused constellation of signals are unitary space–time signalswhich can be represented in matrix notation as

.

B. Review of TCUSTM

The block diagram of the TCUSTM system under investiga-tion is shown in Fig. 1. Assume the number of information bitsto be transmitted during each block interval of symbol pe-riod is , i.e., the uncoded system has a constellation of size

. From the -bit input vector ,bits are expanded by a rate- convolutional encoderinto coded bits. These bits are then used to select one of

subsets of a redundant -ary signal set. The remaininguncoded bits determine which of the signals in the

subset is to be transmitted. Hence, one redundant bit is addedevery input bits, which amounts to expanding the constella-tion size to . If , parallel transitions are al-lowed in the trellis; if , no parallel transitions exist. Theselected signals are then transmitted through a wireless channelwith transmit antennas and receive antennas, as stated inthe previous section. Note that the transmitted signals here arethe elements of the scaled two-dimensional unitary matrix set

.At the receiver, the Viterbi algorithm (VA) is employed for

noncoherent demodulation. Suppose a coded signal sequenceis transmitted, where is the time index of

each signal block (or signal matrix) andis the index of the transmitted signal relate to the time index .With the assumption that the fading is independent from blockto block, the conditional probability density function of the re-ceived signal sequence , given the transmittedcoded signal sequence , is [16]

(3)


Hence, after omitting the constant factor and irrelevant term inthe metric calculation, the maximum-likelihood Viterbi decoderis described by

(4)

III. UNITARY SPACE–TIME CONSTELLATION DESIGN

In this section, we consider the problem of constellation de-sign for TCUSTM schemes. We will start this section by brieflyreviewing the performance analysis of a TCUSTM schemeand the existing constellation-design method. Then, we willpresent a new formulation of the constellation-design problemfor TCUSTM schemes with and without parallel paths. Finally,a numerical optimization algorithm to solve the problem willbe described.

A. Performance Analysis and Existing Design Methods

To perform the constellation design, it would be useful toinvestigate the PEP of TCUSTM schemes. Suppose a coded

sequence is transmitted, while another sequenceis chosen by the Viterbi decoder. An upper bound

on the PEP of mistaking for is given by [14]

(5)

where is the set of for which , and isthe Chernoff upper bound on the PEP of mistaking fordefined as [11]

(6)

with denoting the th singular value of the correla-tion matrix . For notational convenience, in the follow-ing, we may refer to as the product distancealong the path of error events. Another notational convenience isto denote as . Using the union bound, theupper bound on the average error-event probability of the Viterbidecoder can then be obtained by averaging the upper bound onthe PEP over all the possible error events. At sufficiently highSNRs, the upper bound on the average error-event probabilityis dominated by the largest term, i.e., the maximum of the ex-pression on the right-hand side of (5) over all the error events.Let denote the size of , which equals the Hamming distance

between and . It can easily be found from (5) that theerror events with the minimum Hamming distance will leadto the maximum PEP over all the error events, and thus deter-mine the average error-event probability of the Viterbi decoder.It should also be noticed that among all the error events with theminimum Hamming distance , the one with the maximumproduct distance along the corresponding path of error eventsplays the main role when evaluating the average error-event

probability. Therefore, the primary TCUSTM design criteria forquasi-static fading channels at high SNRs is to maximize theminimum Hamming distance defined over all error events,whereas the second TCUSTM design criteria is to minimizethe maximum product distance along the path of error eventshaving the minimum Hamming distance . While the code-design problem of maximizing the minimum Hamming distancein TCUSTM schemes is the same as that in trellis-coded mod-ulation (TCM) schemes for single-input single-output (SISO)Rayleigh fading channels, which has well been solved (see, e.g.,[17]–[21]), there is no satisfying method to construct TCUSTMschemes such that the maximum product distance along the pathof error events with Hamming distance is minimized. In pa-pers dealing with this problem [14]–[17], the authors simply usethe signal constellations designed for uncoded systems, and thenthey apply the “mapping by set partitioning” rules similar tothose in [22] to minimize the maximum product distance alongthe path of error events having the minimum Hamming distance

. This method is suboptimal in the sense that the constella-tion design and the set-partitioning step are separated. Actually,the spatial dimension of multiple transmit antennas allows us toconstruct better constellations according to the used trellis struc-ture. In the following subsection, we will introduce a new con-stellation design method, which combines constellation designand “mapping by set partitioning” into one step, to minimize themaximum product distance along the path of error events withHamming distance . A new formulation of the constellationdesign problem for TCUSTM schemes will be given.

B. Constellation Design Criteria

Since the problem of trellis-code design for TCUSTMschemes has well been solved, as discussed above, we will nowfocus on creating constellations that match properly with thosewell-designed trellis codes. In general, a TCUSTM schemeshould choose a trellis code with the minimum Hammingdistance as large as the computational complexity willallow. This means that parallel paths are not allowed, andthe minimum number of trellis states is . Whenthe data rate is high, the TCUSTM schemes with no parallelpaths may lead to prohibitive complexity. Therefore, we needto consider TCUSTM schemes with parallel paths, as well.Although TCUSTM schemes with parallel paths are inferior toTCUSTM schemes with no parallel paths, we emphasize thatit is applicable for complex sensitive applications. Thus, theconstellation design problem for TCUSTM schemes with andwithout parallel paths will be discussed in the following.

1) Constellation Design for TCUSTM Schemes With ParallelPaths: We first consider the constellation design problem forTCUSTM schemes with parallel paths. In this case, it can easilybe found that , and this occurs between two consec-utive states. Therefore, as argued in the preceding subsection,a good constellation should minimize the maximum productdistance along the path of error events having a Hamming dis-tance . It can be found from (5) that the product dis-tance along the path of error events having a Hamming distance

is , i.e., the Chernoff bound on the PEP ofmistaking for , as defined in (6). We will use this productdistance as a design metric which characterizes the performance


of TCUSTM schemes as a function of the constellation. Notethat the design criteria of TCUSTM schemes, as stated in theprevious subsection, are quite close to the well-known designcriteria of TCM schemes for SISO Rayleigh fading channelsand AWGN channels. Thus, we will employ a technique sim-ilar to Ungerboeck’s mapping by set partitioning to constructsignal constellations resulting in high-performance TCUSTMschemes. While the conventional mapping by set partitioningtechnique is used to improve the performance of TCM schemeswith the given constellation, we will combine the constellationdesign and mapping by set partitioning into one step to improvethe performance of TCUSTM schemes.

For the given trellis code, the first step of signal constellationdesign is to assign the indexes of the signal points in the constel-lation to the paths in the trellis diagram. This can be done in asimilar way as is the conventional approach, known as mappingby set partitioning, for PSK- or QAM-type signal constellations.Let us consider the indexes of the whole signal set as the zerothlevel. These indexes are split arbitrarily into two subsets at thefirst level. The index partition can be done arbitrarily, becausewe have no further information about the constellation exceptthe size. At the second level, each index subset from the firstlevel is partitioned arbitrarily into two smaller subsets. This pro-cedure continues for the succeeding levels and stops at a desiredlevel. Since the index partition is arbitrary, any simple permuta-tion rules may also apply. Actually, we can simply use the sameindex partitioning results, as it is for the SISO TCM schemeswith PSK- or QAM-type signal constellations of the same size.After index partitioning, we can assign the indexes of points tothe trellis branches by applying Ungerboeck’s rules as follows.

• Parallel paths should be assigned with points from thehighest level subset.

• Paths originating from or merging into the same state mustbe assigned with points from the second highest levelsubset.

• All points are used in the trellis diagram with equal fre-quency.

Then, using the idea in mapping by set partitioning for PSK- orQAM-type constellations, we propose to optimize the constel-lation according to the following rules, with decreasing priority.

• First, minimize the maximum metric of the subset that isassigned to parallel transitions.

• Next, minimize the maximum metric between any twohighest level subsets originating or ending in the samestate.

• Finally, minimize the maximum metric between all of thehighest level subsets except those originating or ending inthe same state.

With the design techniques elaborated above, we can generatetrellis-matched unitary space–time constellations resulting inhigh-performance TCUSTM systems. In the following, an ex-ample will be given to illustrate the design procedure.

Example 1: In Fig. 2(a), we depict a rate four-statetrellis diagram with parallel paths. The first step of the constel-lation design is to perform the index partitioning following theprocedures described above. Here, we use the integer ,

, to represent the index of the signal point in the con-stellation. These indexes can be split into four subsets, denoted

Fig. 2. Two trellis diagrams of the TCUSTM systems with parallel paths.

by , , , and ,where each of the subsets contains two signal indexes. Then, weassign the indexes of the signal points to the trellis branches fol-lowing the rules stated above. The results of the assignment aredepicted in Fig. 2(a). Finally, following the rules for constella-tion optimization, we introduce an objective function as

(7)

where , , , ,are three weighting factors, and is the Chernoff upperbound of the PEP, as defined in (6).

Note that the three terms in (7) correspond to three opti-mization rules. A different priority can be assigned to eachrule simply by adjusting corresponding weighting factors. Theproblem of finding proper weighting factors is similar tothe problem of determining the step size in steepest descentmethods. It is interesting to note that if we set weighting factors

equal to 1 for , then the objective function willbe the same as that for the uncoded system in [23]. If we furtherreplace the metric in (7) by the sum of squared singularvalues of , i.e., the chordal distance, then the objectivefunction will be the same as that for the uncoded system in[12]. Therefore, our constellation design method for TCUSTMsystems includes the conventional constellation design methodfor uncoded systems as a special case. Now, we can clearlysee why the signal constellations designed for uncoded USTMsystems are not optimal for TCUSTM schemes: they can notmatch well with the trellis structure. This is due to the fact thatequal priority is assigned to the three terms in the objectivefunction (7). Actually, the signal constellations designed foruncoded systems in [23] and [12] have almost uniform profilesconcerning the product distance for any . Inthis sense, systematically designed unitary constellations foruncoded USTM systems [13] with nonuniform profiles of theproduct distance are more suitable for TCUSTM schemes. Aswe will see in Section IV, systematically designed unitary con-stellations [13] for uncoded systems are optimal for TCUSTMschemes in some cases. Notice also that in [14], [16], and [18],this kind of nonuniform constellation is used for the research inTCUSTM schemes.


Fig. 3. Two trellis diagrams of the TCUSTM systems with no parallel paths.

Adopting the objective function (7) as an optimality criterion,the signal-constellation design problem can be formulated as thefollowing optimization problem:

(8)

2) Constellation Design for TCUSTM Schemes With No Par-allel Paths: Now, we consider the signal-constellation designproblem for TCUSTM schemes with no parallel paths. In thiscase, it can easily be found that . As a result, the asymp-totic product distance with a Hamming distance can be ex-pressed as the product of terms, where each term is givenby (6). This makes the problem of signal constellation design,using the product distance along the path of error events with aHamming distance as design metric, very complex, if notintractable. Therefore, we propose to simply use the same metricas that in the case of parallel paths, i.e., we use the Chernoffupper bound on the PEP of mistaking for , de-fined in (6) as the design metric. Although it is suboptimal, weshall see later that significant coding gains can still be achieved,compared with existing constellations.

Similar to the case with parallel paths, we first assign the in-dexes of the signal points in the constellation to each path in thetrellisdiagram. Inparticular,weconsider the indexesof thewholesignal set as the zeroth level. These indexes are split arbitrarilyinto two subsets at the first level. Since there are no parallel paths,two subsets are sufficient, so that we do not need to perform fur-ther partitions. After the index partitioning, we can assign the in-dexes of points to the trellis branches following Ungerboeck’srules:

• transitions originating from or merging into the same statemust be assigned with signal points from the first-levelsubset;

• all signal points are used in the trellis diagram with equalfrequency.

Then, using the idea in mapping by set partitioning for PSK-or QAM-type constellations again, we propose to optimize theconstellation according to the following rules, with decreasingpriority:

• first, minimize the maximum metric of each first-levelsubset that is assigned to transitions originating or endingin the same state;

• next, minimize the maximum metric between two subsetsoriginating or ending in different states.

Note that there is a difference between the constellation de-sign problem for trellis diagrams with parallel paths and thosewithout parallel paths. In the following, we will present an ex-ample to illustrate the design procedure.

Example 2: In Fig. 3(a), we depict a rate16-state trellis diagram with no parallel paths. The first stepof constellation design is to perform the index partitioningfollowing the procedures described above. We use the integer ,

, to represent signal point in the constellation.The indexes of the signal points in this example can be splitinto two subsets, denoted byand , with each of the subsetscontaining eight signal elements. Then, we assign the indexesof the signal points to the trellis branches, following the rulesstated above. The results of the assignment are depicted inFig. 3(a). Finally, following the rules for constellation opti-mization, we can formulate our objective function as

(9)

where , , are two weighting factors, and isthe Chernoff upper bound of the PEP as defined in (6).


Note that the two terms in (9) correspond to two different op-timization rules. Different priority can be assigned to each rulesimply by adjusting the corresponding weighting factors. Again,these parameters can be determined using a method which iscommonly used to determine the step size in the steepest de-scent algorithm. If we set the weighting factors equal to 1 for

, we can easily find that the above constellation designmethod for TCUSTM systems includes the previous constella-tion design method for an uncoded system as a special case.Adopting the objective function (9) as optimality criterion, theconstellation design problem can be formulated as the followingoptimization problem:

(10)

A detailed optimization algorithm will be described in thenext subsection.

C. Constellation Design Algorithm

To find the solution of the above problem, a key step is tocompute the partial derivatives of the objective function [see,e.g., (8) and (10)] with respect to the elements of the matrix .Below, we will first give a brief review about the parameteriza-tion of the unitary matrix. Then, some implementation detailsabout the constellation design algorithm, including the partialderivatives of the objective function (8) with respect to the ele-ments of the matrix , will be given.

First, let us consider the parameterization of the unitary ma-trix. There are several ways to parameterize the matrix . Asdiscussed in [12], a more practical way to parameterize isto overparameterize the space, which will simplify the deriva-tives of the objective function with respect to its parameters.This over-parameterization is built by expressing any rectan-gular matrix as , where is aunitary matrix parameterized by . The square unitary matrix

can, in turn, be expressed as the product of a diagonalmatrix and some basic unitary matrices , i.e.,

(11)

where , and the element in the th rowand th column of the unitary matrix with

and , is given by

if andif andif andif andotherwise.

Now, we will use the objective function in (8) as an exampleto present some implementation details about the constellationsearch algorithm. Note that the parameters involved in the pa-rameterization of the unitary matrix lie in a compactdifferential manifold, and is a continuous function of

its parameters. Therefore, it achieves at least one global min-imum. However, we also note that may have many localminima that are far away from the global minimum. In fact,

is not very smooth, and it is not even differentiable ev-erywhere. These facts prevent us from the direct use of gra-dient-based search algorithms. Consequently, as in [12], fol-lowing [24], we choose a family of surrogates for

to overcome the difficulties posed by the undesirableproperties of the objective function. The strategy is to minimize

for small values of , while mimicking lesswell, and then track this minimum while increases. The func-tion closely tracks , and converges toas tends to infinity. An example of such a family is [12]

(12)

In order to use the steepest descent algorithm to get the high-performance unitary space–time constellation, we need to com-pute the partial derivatives of the objective functionwith respect to the elements of the matrix , while assumingthat all other matrices , , are fixed. To get these deriva-tives, a key step is to compute the partial derivatives ofwith respect to the parameters that determine the matrix . Theadvantage of matrix overparameterization arises from the factthat we can perturb the matrix by premultiplying with theunitary matrix for , since . This allowsan efficient computation of the partial derivatives of byjust using the entries in and the derivatives of at

. Specifically, for , we obtain

(13)

with


To get in (13), we will introduce a theorem re-garding the derivative of the sum of the largest singular valuesof a matrix with respect to each entry. We start with some helpfuldefinitions. Let the singular values of an complex ma-trix be , where and

. According to the singularvalue decomposition, there exist an unitary matrix andan unitary matrix such that

(14)

The sum of the largest singular values is a function mappingcomplex matrix to real numbers , defined by

(15)

where . We also note that the matrix inequalitymeans that is positive semidefinite, and

denotes the spaces of all by real symmetric matrices. Withthese definitions, we can state the theorem about the derivativeof the sum of the largest singular values of a matrix with respectto each entry as follows.

Theorem 1: (Sum Derivative) Let be a matrix-valuedfunction mapping from to . Suppose that the singularvalues of are

, and let

(16)

If is locally Lipschitzian and -differentiable at ,then is locally Lipschitzian and semiregular at , and theClarke subdifferential is given by

where, and consist of the first columns of and ,

respectively, and consist of the next columns of and, respectively, and the matrix is given by

(17)

Furthermore, if , then is -differentiable at , and

(18)

Proof: First, we note that in the case of real matrices, thetheorem has been proved in [25]. The extension to the complexcase uses a standard trick in matrix algebra, in which there existsa one-to-one correspondence between complex matricesand real matrices. See Appendix A for a review of thedetailed proof.

By applying Theorem 1, we can readily obtainas follows. Define an complex matrix

, and suppose that the singular values of are

. By thesingular value decomposition, there exist an unitary

matrix and an unitary matrix suchthat

. Also define and consistingof the first columns of and , respectively,

and consisting of the next columns ofand , respectively. With these definitions and

Theorem 1, we can easily get as

where , andis given by

(19)

Furthermore, if , then is -differentiable at, and

In (19), can easily be calculated using (11).With the techniques and algorithms described above, we

can summarize our procedure of constellation optimizationas follows [12]. The search procedure starts with a randomlygenerated signal set and a relatively small value of , say

. We then use the steepest descent algorithm to iterativelyupdate the set , and finally find a set such thatthe value of is (nearly) locally minimized.At each iteration, we update, in a fixed order, each element of

individually, i.e., we calculate the partial derivatives ofthe objective function with respect to the parameters of onlyone matrix and update that matrix suitably. After isfound, we slightly increase to , and start from the set

to find a new set such that the value ofis (nearly) locally minimized. We continue

in this manner, each time increasing the value of slightlyand tracking the minimium of . For very largevalues of , would be essentially equivalent to

, and minimizing will also essentiallyminimize .

Note that the main bottleneck of the above constellationsearching algorithm is the computation of the partial derivativesof the objective function with respect to the parameters that de-termine the matrix . Compared with the searching algorithmin [12], the derivatives of our algorithm are more complicatedto compute. In a sense, we get high-performance constellationsat the price of increased searching complexity. The good newsis that this design procedure can be performed offline. Thus,although the computing power needed for the constellationsearch is substantial, the transmission scheme can efficientlybe implemented. The time it takes to complete such an offline


TABLE ICOMPARISON OF THE MAXIMUM CHORDAL DISTANCE AND THE MAXIMUM

PEP AMONG A CONSTELLATION WITH SNR � = 20 dB FOR OUR NEW

CONSTELLATIONS AND OTHER WELL-KNOWN CONSTELLATIONS

procedure increases with the size of the signal constellation andthe number of transmit antennas. For moderate values of ,

, and , a solution can usually be obtained in a reasonableamount of time. For large values of , , and , the authorsare currently investigating ways to modify the design metricpresented here to reduce the algorithm complexity and toincrease the search speed.

IV. NUMERICAL RESULTS

In this section, we use the constellation design rules andthe algorithm developed in the previous section to generateconstellations for TCUSTM schemes and to examine theirperformance. We shall see that our method leads to thehigh-performance constellations of unitary space–time signalsfor TCUSTM schemes. We look specifically at systems with

transmit antennas and receive antenna. Wechoose typical parameters of with the spectral effi-ciency b/channel use. Thus, we require a constellation of atleast signals, each described by an matrix.

Table I lists the maximum chordal distances[12], [13] and the max-

imum PEPs for the new generated

constellations, as well as for two other types of constella-tions, which are named the structured constellation [13] andchordal-distance constellation [12]. From this table, we observethat in all three cases with different rates, our new constellationshave neither a minimum nor a minimum , comparedwith the other two types of constellations. However, as we willsee in the following simulation results, our new constellationshave a better performance against the other two types of con-stellations in TCUSTM schemes. These findings are in linewith the analysis presented in the previous section: instead ofpursuing the minimum [12] or the minimum [23] asin the constellation design for the uncoded case, we should tryto generate a constellation which matches well with the trellisstructure for the coded case.

Now let us look at the BER performance of the constella-tions presented in Table I. Fig. 4 illustrates the performanceof the new constellation, the structured constellation, and thechordal-distance constellation for the TCUSTM scheme shownin Fig. 2(a) with a rate of 0.5 b/s/Hz. It can be seen that thenew constellation achieves a coding gain of 3 dB against thechordal-distance constellation at a BER of . However, we

Fig. 4. Comparison of the BER performance based on the constellations forthe TCUSTM scheme depicted in Fig. 2(a) with a rate of 0.5 b/s/Hz.

Fig. 5. Comparison of the BER performance based on the constellations forthe TCUSTM scheme depicted in Fig. 2(b) with a rate of 0.75 b/s/Hz.

also observe that the structured constellation has the same per-formance as that of the new constellation. This may look sur-prising, but actually, it is not. It can easily be checked that eachof the subsets of the structured constellation, which are assignedto two parallel transitions, contains two orthogonal unitary ma-trices. This means that the maximum metric of each subset isminimized already. Furthermore, we can also confirm the factthat the maximum metric between any two subsets originatingor ending in the same state is minimized. Hence, the structuredconstellation is optimal, in the sense that it satisfies all constel-lation design rules for TCUSTM schemes with parallel paths.Thus, this previously used constellation is one of the optimalsolutions with closed-form expressions. As our numerical re-sults suggest, there are many other optimal solutions withoutclosed-form expressions.

Fig. 5 shows the BER performance of the new constellation,the structured constellation, and the chordal-distance constella-tion for the TCUSTM scheme shown in Fig. 2(b) with a rateof 0.75 b/s/Hz. We see that the new constellation has an im-provement in SNR of about 2.5 dB over the chordal-distance


Fig. 6. Comparison of the BER performance based on the constellations forthe TCUSTM scheme depicted in Fig. 3(a) with a rate of 0.75 b/s/Hz.

Fig. 7. Comparison of the BER performance based on the constellations forthe TCUSTM schemes depicted in Fig. 3(b) with a rate of 1 b/s/Hz.

constellation at a BER of . Compared with the structuredconstellation, the new constellation has an improvement in SNRof about 1 dB at a BER of . This gain is due to the factthat each subset in the new constellation has the same metric(distance), while the other constellations have different metriceswithin each subset where those pairs with the larger metric leadto worse performance.

The same relative performance found for TCUSTM schemeswith parallel paths also extends to TCUSTM schemes with noparallel paths. Fig. 6 shows the performance of the constella-tions for the TCUSTM scheme shown in Fig. 3(a) with a rateof 0.75 b/s/Hz. The new constellation gained about 1.3 dB inperformance at a BER of relative to both the structuredand chordal-distance constellations. If the rate is increased to1 b/s/Hz and the TCUSTM scheme in Fig. 3(b) is used, thenthe new constellation achieves a 1.5 dB performance gain at aBER of against both the structured and the chordal-dis-tance constellations, as shown in Fig. 7.

V. CONCLUSIONS

We proposed a novel method for creating constellationsof unitary space–time signals for TCUSTM schemes, whereneither the transmitter nor the receiver knows the fading coef-ficients of the channel. TCUSTM schemes with and withoutparallel paths were considered. By combining the constellationdesign and the principle of mapping by set partitioning into onestep, we gave a new formulation of the signal-design problemin TCUSTM schemes. We also presented a numerical opti-mization algorithm for finding good constellations. Simulationresults have shown that these new constellations improve theperformance of TCUSTM schemes by 1 dB at a BER ofagainst other well-known constellations.

APPENDIX

A REVIEW OF THE PROOF OF Theorem 1

To compute the subdifferential (Michel–Penot subdifferen-tial) of the sum of the largest singular values of a matrixwith respect to each entry, we need to introduce an appropriateEuclidean structure on the set of complex matrices. The innerproduct of two complex matrices is given by

(20)

We define

(21)

where . As in [25], we also define the absolutetrace by setting

the absolute sum by

and the maximum row-column of by

where and denote the th row and the th column ofthe matrix , respectively. For notational convenience, we use

and to denote the spaces of all by real matricesand all by real symmetric matrices, respectively.

Assume the singular values of be .By the singular value decomposition for every complex matrix,there are unitary matrices and such that

(22)

where , and all the off-diagonal elements of arezero. Let us denote the sum of the largest singular values by

(23)


Our proof will consist of the following three steps. First, wegive some properties about the bounds on the singular values.Then, we will discuss and analyse a max characterization of thesum of the largest singular values of a matrix. We finally getour main theorem in the third step. We now present Lemma 1about the bounds on the singular values.

Lemma 1: For every :(a) ;(b) .(c) If , then

(1) , if or .(2) , if and

.Proof: The proof of statements (a) and (b) are exactly as

the proofs in [25, Prop. 2.1, Cor. 2.2, and the first part of Th.2.3]. There the proofs are provided for real matrices; but theyapply for complex matrices as well.

The proof of the last statement requires a slight modificationin the complex case. Assume that and that

. Let us multiply the matrix with a unitary matrix ,where except

The multiplication of matrix by an unitary matrix does not alterthe singular values. Hence

by part (b). However, the absolute trace of the matrixis

which is a contradiction. The case and the part (2) areproved in the same manner.

A key step is a max characterization of the sum of the largestsingular values of a matrix. Let us define the set of matrices

Clearly, is a compact set, since the functions are con-tinuous. By the previous theorem, part (a), we have the fol-lowing.

Corollary 1: For any , and if forsome and , then , .

Then, we have following lemma.Lemma 2: For every

is a convex function.Proof: Let be the singular value decomposi-

tion. Therefore, for every

where , since singular values are invariantunder unitary transformations.

Because is diagonal, then

By Lemma 1 [part (c)], if , then. At the same time, Corollary 2 implies that .

Combining these, we obtain

The equality is valid for a matrix whose elements are

otherwise.

Hence we obtain that the sum of largest singular valuesis a maximum of convex func-

tions. By the convex analysis and the previous considerationswe obtain the lemma.

Assume that the singular values of A are

As in [25, Th. 3.4], the matrices achieves the maximumif and only if there exists a matrix such that

, and except:(i) for ;

(ii) for and;

(iii) for ;(iv) is Hermitian and positive

semidefinite and the spectral radius .In other words, the maximum is achieved by matrices from theset

, where the matrix inequality meansthat is positive semidefinite. Now by the convex analysis,the subdifferential of at is [26]

If the multiplicity of the th singular value is one, then isdifferentiable at and

Let us now consider a real-valued function , defined by

where is a smooth function, as in it is the casein our situation. Then is a regular and locally Lipschitz,and then it possesses the Clarke subdifferential. Using [26, Prop.6.2.1], the Michel–Penot and Clarke subdifferentials coincide,because is Gâteaux-differentiable (provided is smoothenough).

From the previous considerations and the theory of non-smooth optimization, we finally get our main results.


If is locally Lipschitz and -differentiable at , thenthe function is locally Lipschitz and regular at , and theClarke subdifferential is given by

where

where and consist of the first columns of and ,respectively, and and consist of the next columns ofand . The matrix is given by

Furthermore, if is a smooth function and is of multi-plicity 1, then is -differentiable at , and

Thus, we complete the proof of Theorem 1.

REFERENCES

[1] G. J. Foschini and M. J. Gans, “On limits of wireless comunicationsin a fading environment when using multiple antennas,” Wireless Pers.Commun., vol. 6, no. 3, pp. 311–335, Mar. 1998.

[2] I. Telatar, “Capacity of multi-antenna Gaussian channels,” Euro. Trans.Telecommun., vol. 10, no. 6, pp. 585–595, Nov. 1999.

[3] S. M. Alamouti, “A simple transmitter diversity scheme for wirelesscommunications,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp.1451–1458, Oct. 1998.

[4] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space–time codesfor high data rate wireless communications: Performance criterionand code construction,” IEEE Trans. Inf. Theory, vol. 44, no. 2, pp.744–765, Mar. 1998.

[5] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space–time blockcodes from orthogonal designs,” IEEE Trans. Inf. Theory, vol. 45, no.6, pp. 1456–1467, Jun. 1999.

[6] J.-C. Guey, M. P. Fitz, M. R. Bell, and W.-Y. Kuo, “Signal design fortransmitter diversity wireless communication systems over Rayleighfading channels,” IEEE Trans. Commun., vol. 47, no. 4, pp. 527–537,Apr. 1999.

[7] A. R. J. Hammons and H. El Gamal, “On the theory of space–timecodes for PSK modulation,” IEEE Trans. Inf. Theory, vol. 46, no. 2,pp. 524–542, Mar. 2000.

[8] B. Vucetic and J. Yuan, Space–Time Coding. New York: Wiley, 2003.[9] T. L. Marzetta and B. M. Hochwald, “Capacity of a mobile multiple-

antenna communication link in Rayleigh flat fading,” IEEE Trans. Inf.Theory, vol. 45, no. 1, pp. 139–157, Jan. 1999.

[10] L. Zheng and D. N. C. Tse, “Communication on the Grassmannmanifold: A geometric approach to the noncoherent multiple-antennachannel,” IEEE Trans. Inf. Theory, vol. 48, no. 2, pp. 359–383, Feb.2002.

[11] B. M. Hochwald and T. L. Marzetta, “Unitary space–time modulationfor multiple-antenna communications in Rayleigh flat fading,” IEEETrans. Inf. Theory, vol. 46, no. 2, pp. 543–563, Mar. 2000.

[12] D. Agrawal, T. J. Richardson, and R. L. Urbanke, “Multiple-antennasignal constellations for fading channels,” IEEE Trans. Inf. Theory, vol.47, no. 6, pp. 2618–2626, Sep. 2001.

[13] B. M. Hochwald, T. L. Marzetta, T. J. Richardson, W. Sweldens, andR. Urbanke, “Systematic design of unitary space–time constellations,”IEEE Trans. Inf. Theory, vol. 46, no. 6, pp. 1962–1973, Sep. 2000.

[14] Z. Sun and T. T. Tjhung, “On the performance analysis and design cri-teria for trellis coded unitary space–time modulation,” IEEE Commun.Lett., vol. 7, no. 4, pp. 156–158, Apr. 2003.

[15] ——, “Multiple-trellis-coded unitary space–time modulation inRayleigh flat fading,” IEEE Trans. Wireless Commun., vol. 3, no. 6,pp. 2335–2344, Nov. 2004.

[16] I. Bahceci and T. M. Duman, “Trellis-coded unitary space–time mod-ulation,” IEEE Trans. Wireless Commun., vol. 3, no. 6, pp. 2005–2012,Nov. 2004.

[17] W. Zhao, G. Leus, and G. B. Giannakis, “Orthogonal design of unitaryconstellations for uncoded and trellis-coded noncoherent space–timesystems,” IEEE Trans. Inf. Theory, vol. 50, no. 6, pp. 1319–1327, Jun.2004.

[18] M. Tao and R. S. Cheng, “Trellis-coded differential unitary space–timemodulation over flat fading channels,” IEEE Trans. Commun., vol. 51,no. 4, pp. 587–596, Apr. 2003.

[19] D. Divsalar and M. K. Simon, “The design of trellis coded MPSKfor fading channels: Set partitioning for optimum code design,” IEEETrans. Commun., vol. COM-36, no. 9, pp. 1013–1022, Sep. 1988.

[20] C. Schlegel and D. J. Costello, Jr., “Bandwidth-efficient coding forfading channels: Code construction and performance analysis,” IEEEJ. Sel. Areas Commun., vol. 7, no. 9, pp. 1356–1368, Dec. 1989.

[21] J. Du, B. Vucetic, and L. Zhang, “Construction of new MPSK trelliscodes for fading channels,” IEEE Trans. Commun., vol. 43, no. 2–4,pp. 776–784, Feb.–Apr. 1995.

[22] G. Ungerboeck, “Trellis-coded modulation with redundant signal sets,Part II: State of the art,” IEEE Commun. Mag., vol. 25, pp. 12–21, Feb.1987.

[23] Y. Wu, K. Ruotsalainen, and M. Juntti, “A novel design of unitaryspace–time constellations,” in Proc. IEEE Int. Conf. Commun., Seoul,Korea, May 2005, pp. 825–829.

[24] J. H. Conway, R. H. Hardin, and N. J. A. Sloane, “Packing lines, planes,etc.: Packings in Grassmannian spaces,” Exper. Math., vol. 5, no. 2, pp.139–159, 1996.

[25] L. Qi and R. S. Womersley, “On extreme singular values of matrixvalued functions,” J. Convex Anal., vol. 3, no. 1, pp. 463–472, 1996.

[26] J. Browein and A. Lewis, Convex Analysis and Nonlinear Optimiza-tion: Theory and Examples. New York: Springer, 2000.

Yi Wu (M’05) received the B.S. degree from Shanghai Jiaotong University,Shanghai, China, the M.S. degree from the Beijing University of Posts andTelecommunications, Beijing, China, and the Ph.D. degrees from Tsinghua Uni-versity, Beijing, China, all in electrical engineering, in 1991, 1997, and 2001,respectively.

He is currently with Agder University College, Grimstad, Norway, as a Re-search Staffer. His research interests are in the area of wireless communications,with a focus on space–time coding, wideband wireless communication systems,and multiuser detection.

Vincent K. N. Lau (M’97–SM’01) received theB.Eng. (Distinction 1st Hons) degree in 1992 fromthe University of Hong Kong, Hong Kong, and thePh.D. degree in mobile communications in 1995from the University of Cambridge, Cambridge, U.K.

He was with HK Telecom, Hong Kong, as ProjectEngineer, and later promoted to System Engineer. Hejoined Lucent Technologies, Bell Labs (ASIC De-partment), Whippany, NJ, as a Member of TechnicalStaff in 1997. In 2004, he joined the Department ofElectrical and Electronic Engineering, Hong Kong

University of Science and Technology, Hong Kong. At the same time, he isa Technology Advisor for HK-ASTRI on the research and development ofwireless LAN access infrastructure with smart antennas. His research interestsinclude adaptive modulation and channel coding, information theory with statefeedback, multiuser MIMO scheduling, cross-layer optimization, basebandSoC design (UMTS base station ASIC, 3G1x mobile ASIC, Wireless LANMIMO ASIC). He is the principal author of a book on MIMO technologies(to be published by John Wiley and Sons), as well as the chapter author oftwo books on wideband CDMA technologies. He has published more than 40papers in IEEE Transactions and journals, 47 papers in international conferenceproceedings, and 14 Bell Labs technical memos. He has eight U.S. patentspending.

Dr. Lau recieved the Sir Edward Youde Memorial Fellowship and theCroucher Foundation Fellowship in 1995, and has received two Best Paperawards.


Matthias Pätzold (M’94–SM’98) was born inEngelsbach, Germany, in 1958. He received theDipl.-Ing. and Dr.-Ing. degrees in electrical engi-neering from Ruhr-University Bochum, Bochum,Germany, in 1985 and 1989, respectively, and thehabil. degree in communications engineering fromthe Technical University of Hamburg-Harburg,Hamburg, Germany, in 1998.

From 1990 to 1992, he was with ANT Nachrich-tentechnik GmbH, Backnang, Germany, where hewas engaged in digital satellite communications.

From 1992 to 2001, he was with the Department of Digital Networks atthe Technical University Hamburg-Harburg. Since 2001, he has been a FullProfessor of mobile communications with Agder University College, Grimstad,Norway. He is author of the books “Mobile Radio Channels—Modelling,Analysis, and Simulation” (Wiesbaden, Germany: Vieweg, 1999, in German)and “Mobile Fading Channels” (Chichester, U.K.: Wiley, 2002). His current

research interests include mobile radio communications, especially multipathfading channel modelling, multi-input multi-output (MIMO) systems, channelparameter estimation, and coded-modulation techniques for fading channels.

Prof. Pätzold received the 1998 and 2002 “Neal Shepherd Memorial BestPropagation Paper Award” from the IEEE Vehicular Technology Society. Heis the recipient of the “2003 Excellent Paper Award” of the IEEE Interna-tional Symposium on Personal, Indoor and Mobile Radio Communications(PIMRC’03) in Beijing, China, as well as of the “Best Paper Award” of the 8thInternational Symposium on Wireless Personal Multimedia Communications(WPMC’05) in Aalborg, Denmark. He was the local Organizer of the confer-ence “Kommunikation in Verteilten Systemen (KiVS) 2001” and Organizer ofthe 2nd International Workshop on “Research Directions in Mobile Communi-cations and Services 2002.” He served as a member of the Technical ProgramCommittee (TPC) for IST’05, VTC’05-Fall, and WPMC’05. He also served asSession Chair for several reputed international conferences (VTC’04-Spring,NRS’04, PIMRC’04, IST’05, and WPMC’05).

Date post:	19-Jul-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

1948 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO....

Documents