energy-efficient full diversity collaborative unitary space-time block code designs via unique...

1678 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 3, MARCH 2013

Energy-Efficient Full Diversity CollaborativeUnitary Space-Time Block Code Designs via

Unique Factorization of SignalsDong Xia, Jian-Kang Zhang, Senior Member, IEEE, and Sorina Dumitrescu, Member, IEEE

Abstract—In this paper, a novel concept called a uniquely fac-torable constellation pair (UFCP) is proposed for the systematicdesign of a noncoherent full diversity collaborative unitary space-time block code by normalizing two Alamouti codes for a wire-less communication system having two transmitter antennas anda single receiver antenna. It is proved that such a unitary UFCPcode assures the unique identification of both channel coefficientsand transmitted signals in a noise-free case as well as full diversityfor the noncoherent maximum likelihood receiver in a noise case.To further improve error performance, an optimal unitary UFCPcode is designed by appropriately and uniquely factorizing a pairof energy-efficient cross quadrature amplitude modulation (QAM)constellations to maximize the coding gain subject to a transmis-sion bit rate constraint. After a deep investigation of the fractionalcoding gain function, a technical approach developed in this paperto maximizing the coding gain is to carefully design an energy scaleto compress the first three largest energy points in the corner ofthe QAM constellations in the denominator of the objective as wellas carefully design a constellation triple forming two UFCPs, withone collaborating with the other two so as to make the accumulatedminimum Euclidean distance along the two transmitter antennasin the numerator of the objective as large as possible, and at thesame time, to avoid as many corner points of the QAM constella-tions with the largest energy as possible to achieve the minimumof the numerator. In other words, the optimal coding gain is at-tained by intelligent constellations collaboration and efficient en-ergy compression. Computer simulations demonstrate that errorperformance of the optimal unitary UFCP code presented in thispaper outperforms those of the differential code and the signal-to-noise-ratio-efficient training code.

Index Terms—Coding gain, constellations collaboration, crossquadrature amplitude modulation (QAM) constellations, energyscale, full diversity, noncoherent ML receiver, uniquely factoriz-able constellation pair and unitary space-time block code.

I. INTRODUCTION

A T present, the technology intelligently combining mul-tiple antennas [1]–[3] with space-time block coding

[4]–[36] has been well developed to improve the spectralefficiency of a coherent wireless communication system. In this

Manuscript received August 23, 2011; revised June 02, 2012; accepted Au-gust 24, 2012. Date of publication November 16, 2012; date of current versionFebruary 12, 2013. J.-K. Zhang and S. Dumitrescu were supported in part bythe Natural Sciences and Engineering Research Council of Canada Award.The authors are with the Department of Electrical and Computer Engineering,

McMaster University, Hamilton, ON L8S 4L8, Canada (e-mail: [email protected]; [email protected]; [email protected]).Communicated by J.-C. Belfiore, Associate Editor for Coding Theory.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TIT.2012.2227453

paper, we are specifically interested in a flat-fading wirelesscommunication system with two transmitter antennas anda single receiver antenna. This system is often encounteredin mobile down-link communications for which the mobilereceiver may not be able to deploy multiple antennas. Forsuch a system, if the exact knowledge of the channel coeffi-cients is available at the receiver, the orthogonal Alamouti [4]space-time block code is particularly appealing, since it enablesthe coherent maximum likelihood (ML) receiver to extract fulldiversity not only with linear processing complexity, but withinformation losslessness as well [37].Unfortunately, perfect channel state information at the

receiver, in practice, is not easily obtainable. If the channelchanges slowly, then the transmitter may have sufficiently longchannel coherence time and send training signals enabling thechannel coefficients to be estimated accurately. However, thefading coefficients in mobile wireless communications mayvary rapidly and the coherence time may be too short to allowfor reliable estimation of the coefficients. Therefore, the timecost on sending training signals cannot be ignored since moretraining signals need sending for the accurate estimation of thechannel [38]–[40]. In this paper, we consider the communi-cation scenario where channel fading changes very promptly,assuming that the channel gains are completely unknown atboth the transmitter and the receiver, but remain unchangedwithin the four transmission time slots, after which they changeto new independent values that are fixed for next four timeslots, and so on. Such fast varying flat-fading channel for asingle transmitter and single receiver antenna was first consid-ered for determining the capacity-achieving input distribution[41]–[45].In order to make communication as reliable as possible under

this severe environment and avoid sending the training signalsfor estimation of the channel, using differential space-timeblock coding [46]–[56] is one of the possible solutions. Un-fortunately, this approach results in an approximate loss of 3dB in performance compared to coherent detection. Recently,some techniques of blind signal processing such as the sub-space method based on the second-order statistics have beenutilized to blindly identify the space-time block coded channel[57]–[61]. However, phase ambiguity incurs the channel notbeing able to be identified uniquely, even in the noise-freecase. In addition, even if there were no phase ambiguity, thesubspace method could not be successfully applicable to ourcase, since the four coherence time slots are too short to allowthe second-order statistics to be estimated accurately.

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XIA et al.: ENERGY-EFFICIENT FULL DIVERSITY COLLABORATIVE UNITARY SPACE-TIME BLOCK CODE DESIGNS 1679

Therefore, in order to attain a more satisfactory solution, non-coherent space-time block coding techniques [46], [62]–[66]have been developed. It has been proved that either at highsignal-to-noise ratio (SNR) or for long coherence time, the uni-tary code is optimal [40], [46], [67], [68]. Hence, most of thenoncoherent space-time block code designs have been primarilyconcentrated on unitary designs [46], [51], [55], [62]–[66], [69],[70]. Particularly, we would like here to mention the first non-coherent full diversity unitary code proposed by Tarokh andKim [62]. This code design is attractive because of the fact thatits encoding and ML decoding complexity is linear. More re-cently, the systematic design of nocoherent unitary space-timeblock codes with full diversity and a high transmission ratefor an arbitrary number of the transmitter antennas and the re-ceiver antennas has been established by using a pair of coprimephase-shift-keying (PSK) constellations and the QR decompo-sition [71]. Particularly, for the system with two transmitter an-tennas and a single receiver antenna, the phase ambiguity andfull diversity issue for the noncoherent Alamouti space-timeblock code has been completely resolved [72], [73].However, the PSK constellation except binary and 4PSK con-

stellations is not as energy efficient as the quadrature ampli-tude modulation (QAM) constellation. In addition, the strate-gies developed in [73] and [74] for the theoretic analysis of theunique identification and full diversity using a pair of coprimePSK constellations cannot be applied to the QAM constellation.Therefore, our primary target in this paper is to design a full di-versity unitary space-time block code for the system by usingthe two Alamouti codes and the energy-efficient cross QAMconstellations such that the noncoherent coding gain is maxi-mized subject a transmission bit rate constraint. Despite the factthat recent research on coherent multiple-input multiple-output(MIMO) communications has told us that the Alamouti codeenables coherent full diversity for any constellations and withany receivers, this is no longer true for noncoherent commu-nications, even if the commonly used QAM constellations aretransmitted and even if the noncoherent ML receiver is em-ployed, since the likelihood function is invariant under certainrotation of some QAM constellation points [61], [74]. Hence,signals must be carefully designed to combat against fading. Inthe noncoherent wireless communication scenario, the unknownof the fading channel at both the transmitter and the receiver re-quires that the transmitted signals emitted from different timeslots must be more correlated than in the coherent environmentso that reliable communications with noncoherent full diver-sity are made possible under a maximum allowable transmis-sion data rate.All the aforementioned factors greatly motivate us to pro-

pose a novel concept, a uniquely factorable constellation pair(UFCP), for the systematic design of an optimal unitary con-stellation for the system. The main idea of the UFCP designessentially comprises the following two major steps.1) Intelligent constellations collaboration: From a pair of theenergy-efficient cross QAM constellations, a constellationtriple constituting two UFCPs will be carefully designed,with the one collaboratively shared with the other twothrough the two transmitter antennas, so that the minimum

of the numerator in the fractional objective function ismade as large as possible and at the same time, as many asthe possible largest energy points of the QAM constella-tions are avoided to reach the minimum of the numerator.

2) Efficient energy compression: An energy scale will be care-fully designed to compress the first three largest energypoints of the QAM constellations in the denominator of theobjective.

Notation: Most notations used throughout this paper are stan-dard: column vectors and matrices are boldface lowercase anduppercase letters, respectively; the matrix transpose, the com-plex conjugate, the Hermitian are denoted by , , ,respectively; denotes the identity matrix; Notation

denotes the trace of an matrix , i.e.,, whereas notation denotes the determinant

of ; Notation denotes the Frobenius norm of ; No-tation denotes ; denotesan empty set.

II. CHANNEL MODEL AND UNITARY SPACE-TIME BLOCKCODING

In this section, we first briefly review the channel model inwhich we are interested in this paper. Then, we propose ourtransmission scheme and unitary code structure.

A. Channel Model

Let us consider a wireless communication system having twotransmitter antennas and a single receiver antenna. The trans-mitted symbols from the two transmitter antennas arrive at thereceiver via two different channels and . Then, the discretebaseband received signal can be represented as

(1)

Throughout this paper, we assume and are samples ofindependent circularly symmetric zero-mean complex whiteGaussian random variables with unit variances and remainconstant for the first four time slots, after which they changeto new independent values that are fixed for the next fourtime slots, and so on, the explanation for which will be givenin the ensuing subsection. and are two correspondingtransmitted symbols from these two antennas, and is a circu-larly symmetric complex Gaussian noise with zero mean andvariance .

B. Unitary Space-Time Block Codes

Let , , and be three constellations to be designed.Then, our unitary space-time block code for the channel model(1) is basically generated by normalizing two Alamouti codesand is described as follows: First, randomly, independently, andequally likely choose three symbols , and

and then transmit their normalized version from thetwo transmitter antennas within four time slots. During the firsttime slot, we transmit the signalsand in (1) at the same time from the respective Antennas1 and 2. During the second time slot, the signals and


are simultaneously transmittedfrom Antennas 1 and 2, respectively. Collecting these two re-ceived signals yields

(2a)

where , , , and. During the remaining two time slots, the second

and third symbols are transmitted using the Alamouti codingscheme, i.e.,

(2b)

where , and

By stacking the above four received signals (2a) and (2b), therelationship between the transmitted and received signals withinthe four time slots can be represented in a more compact matrixform as

(3a)

where , and

(3b)

for . We would like to make thefollowing comments on the unitary space-time block codedchannel model (3).1) Why do we need four time slots? For a noncoherent MIMOsystem with transmitter antennas, it has been proved[68] that a necessary condition for a space-time block codeto enable full diversity for the noncoherent ML receiver iscoherence time . Particularly, for , weshould have at least four time slots. Actually, in this paper,we consider the shortest coherence time which enables theunique identification and full diversity. See more details inSection V-B.

2) Why is the symbol rate per channel use reasonable?The answer to this question is mainly motivated from thefollowing observation: For a noncoherent MIMO commu-nication system with transmitter antennas and re-ceiver antennas, Zheng and Tse [40] have proved that in ahigh-SNR regime and for the Rayleigh-faded channel, theaverage channel capacity is given by

(4)

where and is the coherencetime. This benchmark result tells us that the original non-coherent MIMO system can be asymptotically regarded as

parallel spatial channels and thus, the number

is the total number of degrees of freedom tocommunication. The result also suggests us that the symbolrate of a space-time block code for the noncoherent MIMOchannel should be . Especially for the nonco-herent system with , , and , the symbolrate should be .

C. Problem Statement

To formally state our design problem, we make the followingassumptions throughout this paper.1) The channel coefficients and are samples of inde-pendent circularly symmetric white Gaussian random vari-ables with zero mean and unit variances, and remain con-stant for the first four time slots, after which they changeto new independent values that are fixed for the next fourtime slots, and so on.

2) The elements of are circularly symmetric zero-meancomplex Gaussian samples with covariance matrix .

3) During consecutive four observable time slots, the space-time block coding matrix is transmitted with , andbeing independently and equally likely chosen from the

respective constellations , , and .4) Channel state information is not available at either thetransmitter or the receiver.

Under the above assumptions, our primary purpose in this paperis to solve the following problem.Problem 1: Design the constellation triple , , and for

the unitary space-time block coded channel (3) such that1) in the noise-free case, for any given nonzero receivedsignal vector , the equation reduced from (3a)

(5)

with respect to the transmitted symbol variables , and, and the channel vector has a unique solution, and

2) in the noisy environment, full diversity and the optimalcoding gain are enabled for the noncoherent ML receiver.

Here, a natural question is: Why is the study of the noise-freecase so important when all realistic communication systems arenoisy? Themain reason for this can be explained as follows: Thediversity gain is to measure how quickly the error probabilitydecreases as SNR increases [5], [68], and thus, quantitativelycharacterizes how accurate the estimate of the transmitted signalis in a noise environment, particularly in a high-SNR scenario.Therefore, a full diversity noncoherent code must have ability toallow the channel as well as the transmitted signal to be uniquelyidentified in a noise-free case. In other words, if a signal designis not able to provide the unique identification of the channel andthe transmitted signals in the noise-free case, then the reliableestimation of the signal will not be guaranteed, even in highSNR. In fact, it will be proved in Theorems 1 and 2 that theunique identification of both the channel and the transmittedsignal in the noise-free case is equivalent to full diversity inthe noise case. In addition, the study of the noise-free case willsignificantly facilitates the analysis of the transmitted signals.


III. UNIQUE IDENTIFICATION AND FULL DIVERSITY

To solve Problem 1, in this section, we derive a necessaryand sufficient condition for the code generated from (3b) andthe constellation triple , , and to not only enable theunique identification of both the channel coefficients and thetransmitted signals in the noise-free case, but full diversity inthe noise case as well.

A. Unique Identification

The following theorem, which is the first main result of thispaper, characterizes the relationship among the three constel-lations , , and for the unique identification of both thechannel and the transmitted signals in the noise-free case.

Theorem 1: Let . Then, for any given nonzeroreceived signal vector without noise, i.e., , both thechannel coefficients and the transmitted signals are uniquelyidentified if and only if the constellation triple , , andsatisfies a so-called unique factorization property: If there exist

and such that and, then we have , , and .

Proof: Using the code structure (3b), we have

(6a)

(6b)

Since , we have and as a result, the matrix isinvertible. Eliminating from (6) yields

(7)

Notice that

is the Alamouti codeword matrix. Therefore, (7) can berewritten as

(8)

where , , and isgiven by

It can be verified that is unitary up to a scale. In addition,since and is invertible, we haveand thus, , which is equivalent to the fact that isinvertible. Therefore, from (8), we obtain

(9)

That is

(10a)

(10b)

Now, it can be seen clearly that once quotients forhave been determined, then and themselves

are able to be uniquely determined if and only if theredo not exist andwith such that and

, which is equivalent to the unique factoriza-tion property described in Theorem 1. Moreover, after wehave determined and , the channel vector can beuniquely determined by

. This completes the proof ofTheorem 1.

Some observations on Theorem 1 are made as follows.1) Theorem 1 reveals that using the code with the unique fac-torization property, both the channel coefficients and thetransmitted signals can be uniquely identified by just pro-cessing the four received signals. Theorem 1 is very desir-able in the design of noncoherent constellations, albeit allthe realistic communication channels are noisy, since if asignal design is not able to provide the unique identificationof the channel and the transmitted signals in the noise-freecase, then it is impossible to reliably estimate the signal,even in a high-SNR environment.

2) From the proof of Theorem 1, i.e., (10), we can observethat if either and , or and satisfy the uniquefactorization property, then both the channel coefficientsand the transmitted signals can be still uniquely identified.In addition, (10a) divided by (10b) gives us

Now, it becomes clear that if and satisfy the uniquefactorization property, then and, thus, can beuniquely determined. As a consequence, the channel coef-ficients can also be uniquely determined. In other words,that any two of the constellation triple , , andsatisfy the unique factorization property sufficiently as-sures that both the channel coefficients and the transmittedsignals can be uniquely identified.

B. Full Diversity

In order to analyze the full diversity of the code (3b), letus first consider a general space-time block coded noncoherentMIMO system with transmitter antennas, receiver an-tennas and flat-fading channels as follows:

(11)

where denotes a received signal matrix, denotesan channel matrix, is a codeword matrix,and denotes a noise matrix. We assume that the ele-ments and of the channel matrix and the noisymatrixare samples of independent circularly symmetric zero-mean

complex Gaussian random variables with the respective vari-ance one and . Under these assumptions, the probability den-sity function of the received signal matrix conditioned on thetransmitted signal matrix is the Gaussian distribution, i.e.,


and thus, its likelihood is given by

Then, the ML receiver for the noncoherent MIMO system isequivalent to solving the following optimization problem:

, where

and .To avoid estimating the variance of noise, consider the condi-tional probability density function of the received signal matrixgiven the channel matrix and the transmitted signal matrix, i.e.,

and thus, its likelihood is given by

The generalized likelihood ratio test (GLRT) receiver for thejoint estimation of and is to maximize the likelihood, whichis essentially equivalent to solving the following nonlinear leastsquare error optimization problem [68], [75], [76]:

(12)

Its solution can be obtained by first estimating the transmittedsignal matrix as

(13)

and then estimating the channel matrix as. Particularly for any unitary code, the ML

receiver and the GLRT receiver for the optimal estima-tion of the transmitted signal matrix are equivalent, i.e.,

.In addition, Brehler and Varanasi [68] analyzed the asymp-

totic performance of the GLRT detector for the noncoherentMIMO system and proved the following lemma.

Lemma 1: Let a matrix be defined as. If each matrix has full rank for all pairs

of distinct codewords and , then the resulting space-timeblock code provides full diversity for the GLRT receiver, andmoreover, the pairwise error probability oftransmitting and deciding in favor of has the followingasymptotic formula:

We like to make the following two comments on Lemma 1.1) In spite of the fact that Proposition 6 in [68] did not ex-plicitly state that full rank for each matrix is a neces-sary condition for full diversity, its proof actually implic-itly infers that it is indeed a necessary condition. Therefore,Lemma 1 and this comment together essentially tell us that

the full rank of every matrix for all the distinct code-words and is a necessary and sufficient condition forfull diversity.

2) Particularly for a unitary code, i.e., for all , weobtain

This gives us, where it is assumed that

are all the singular values of matrix. Therefore, an optimal code in terms of error perfor-

mance should maximize the minimum product, which isexactly the same design criterion proposed by Hochwaldand Marzetta [46] to optimize the worst case dominantterm of the Chernoff bound in [46, eq. (18)] when SNRis large.

In addition, it is not difficult to prove that a necessary condi-tion for to have full rank is . Hence, in this paper,we consider the case of the shortest coherence time when it ispossible for the code design (3b) to enable the unique identifica-tion of both the channel coefficients and the transmitted signalsas well as full diversity. Now, we are in a position to state thesecond main result in this paper.

Theorem 2: Let . Then, the code designedby (3b) enables full diversity for the noncoherent ML receiverif and only if the constellation triple , and satisfies theunique factorization property defined in Theorem 1.

Proof: We first note that since each codeword matrix (3b)is unitary, we have . By Lemma1 and its comment, we only need to prove the statement that thecode enables full diversity if and only if is invertible forany pair of distinct and . Since is a square matrix

and , proving that the matrix isinvertible is equivalent to proving that the matrix isinvertible. Notice that

By some algebraic manipulations, we can obtain that the de-

terminant of is given by

. There-

fore, we have if and only if there do not exist, and with

such that and . In other words,the unique factorization property defined in Theorem 1 is in-deed a necessary and sufficient condition for full diversity. Thiscompletes the proof of Theorem 2.

Theorems 1 and 2 together uncover the fact that it is theunique factorization property of the constellation triple , ,and that enables the unique identification of both the channelcoefficients and the transmitted signals in the noise-free case as


well as full diversity in the noisy case. In addition, Comment2 on Theorem 1 is also true for Theorem 2. That being said, ifany two of the three constellations , , and satisfies theunique factorization property, then full diversity is achieved.

IV. UNIQUELY FACTORABLE CONSTELLATION PAIR

As we have observed from Theorems 1 and 2, the uniquefactorization property plays a crucial role in solving Problem1. This greatly motivates us to deeply and systematically studysuch a constellation triple , , and . Therefore, in this sec-tion, we propose a novel concept called UFCP.

A. Uniquely Factorable Constellation Pair

First, the unique factorization property in Theorems 1 and 2naturally leads us to formally introducing the following defini-tion for a pair of constellations:

Definition 1: A pair of constellations and is said to be aUFCP, which is denoted by , if there exist and

such that , then , .

The following two remarks on Definition 1 are made for clari-fying the relationship between the unique factorization propertyon the constellation triple defined in Theorem 1 and the uniquefactorization property for a pair of constellations defined in Def-inition 1.1) As we have explained in Comment 2 on Theorem 1, if anytwo of the constellation triple , , and forms a UFCP,i.e., , or , or , then the constella-tion triple , , and satisfies the unique factorizationproperty in Theorem 1. Unfortunately, the necessary condi-tion is not true, i.e., that the constellation triple , , andsatisfies the unique factorization property does not nec-

essarily implies that any two of them constitutes a UFCP.For example, , ,and . It can be verified by calcu-lation that such a constellation triple satisfies the uniquefactorization property given in Theorem 1. However, anytwo of them cannot form a UFCP.

2) Despite the fact that the aforementioned necessary condi-tion is not true, the sufficient condition, i.e., UFCP, helpsus significantly facilitate the systematic construction ofthe constellation triple satisfying the unique factorizationproperty in Theorems 1 and 2 as well as of good unitarycodes. See more explanations in Section V.

The following example provides us with a trivial UFCP.

Example 1: For any set , if we take , then andform a UFCP.

Example 1 tells us that the constellation pair used in thetraining transmission scheme naturally forms a UFCP. Here isa nontrivial example:

Example 2: Let and be given by

Then, by Definition 1, it can be verified that such a pair ofand constitutes a UFCP.

In this paper, we are interested in the systematic design ofnontrivial UFCPs each element of which is a complex integer.To do that, we need to develop a necessary condition which

a UFCP must satisfy.

Proposition 1: Let and form a UFCP. If , then.

From Proposition 1 and Definition 1, we can immediatelyobtain the following proposition:

Proposition 2: For a pair of given constellations andwith each having finite size and , if a new constellationis defined as

then such a pair of and constitutes a UFCP if and only if

(14)

Proposition 2 tells us that a UFCP can be constructedby factorizing a constellation in such a way that each frac-tion is unique. For notation simplicity, this kind of constructionis specifically denoted by . In general, when the sizeof is large, it is not easy to find a nontrivial unique factor-ization. However, for some special constellations such as thesquare QAM constellation, which satisfy rotation-invariance,i.e., every element in the constellation multiplied by stillbelongs to the constellation, we can utilize this property to sys-tematically construct a UFCP. In general, we always have thefollowing property, which can be verified directly by the defini-tion and thus, whose proof is omitted.

Proposition 3: Let be rotation-invariant with respect toand , where is real. Then, the following

statements are true:1) If we let

then such a pair of and constitutes a UFCP and.

2) If we let

then such a pair of and forms another UFCP and.

In addition, it is noticed that for a given UFCP , whenthe symbol is fixed, all the fractions of the formcan be chosen only from a certain subset of , which will beemployed in the design of the constellation triple , andin Section V. Therefore, we particularly give a formal definitionas follows:


Definition 2: Given a UFCP and a fixed , a setgenerated from , denoted by ,

(15)

is called a Group- .

Here, we make an abuse of the terminology “Group” justfor the purpose of discussion conveniences. Specifically,throughout this paper wherever we use the term “Group- ”, italways means a set generated from in terms of Definition 2,but it does not mean “the algebraic group.” The groups havesome interesting properties.

Proposition 4: Let . Then, the following three state-ments are true:1) Nonintersection: For any , there is nointersection between Group- and Group- , i.e.,

(16)

2) Decomposition: The union of all the groups is equal to theoriginal constellation , i.e.,

(17)

3) The number of groups is equal to , for anyand .

Proof: Statement 2 can be derived directly from the defi-nition of the Group. Here, we only examine Statements 1 and3. For any and , suppose that there existssome belonging to . Then, and ,and thus, there exist such that , which im-plies that . Since and form a UFCP, we have

and , which contradicts with the assumptionthat . Hence, , i.e., Statement 1 is true.Using Statements 1 and 2, we obtain

(18)

By the definition of the group, we have . Substitutingthis into (18) yields . This completes the proof ofStatement 3 and thus, of Proposition 4.

B. Unique Factorizations of the Modified Cross QAMConstellations

In spite of the fact that Proposition 2 tells us that a UFCPcan be constructed by factorizing a constellation ,

in general, the UFCP so derived from the given isnot unique. In other words, the same constellation can gen-erate two different UFCPs. For instance, one 16-QAM constel-lation can generate two UFCPs: ,

, and ,. Now, a natural question is:

for a given constellation , which pair of constellations andgenerated by is better? A general answer to this question

is hard to be given. However, in this paper, we are interestedin the UFCPs which are derived from the cross -ary QAMconstellation [77] and every element of which is still a complexinteger. This implicitly requires that each element of must

be of unit-norm, i.e., or . Specifically, we focus on sucha UFCP generated from the cross QAM constellation that theminimum distance of is as large as possible. Since the con-ventional 8-QAM constellation does not satisfy the rotation-in-variant property, we need to modify it into a new 8-QAM con-stellation so as to be rotation-invariant under . For discus-sion self-containment, a formal definition of a -ary modifiedcross QAM constellation is provided here.

Definition 3: Amodified -ary cross QAM constellationis defined as follows:1) If is even, is the standard square -ary QAM con-stellation, i.e.,

2) If , is a new 8-QAM constellation modified fromthe conventional 8-ary QAM constellation, i.e.,

3) If is an odd number exceeding 3, is the union of ahorizontal rectangular QAM constellation and a verticalrectangular QAM constellation, i.e.,

Proposition 5: Let be the given modified -cross QAMconstellation. Then, subject to with a fixedsize of greater than one, one solution to the following opti-mization problem:

(19)

is given as follows:1) If , then

a) For

b) For

c) For

d) For , the optimal is determined as follows:


i) When is even

ii) When is an odd number exceeding 5, isgiven by (20), at the bottom of the page.

2) If and , then

a) For

b) For

c) For , the optimal can be determined asfollows:i) When is even

ii) When is an odd number exceeding 5, isgiven by (21), at the bottom of the page.

The proof of Proposition 5 is given in Appendix A. In principle,the optimal constellations and in Proposition 5 canbe attained by starting at any corner point in with the largestenergy, and then, for , successively selecting all nearestneighbors of the previously already selected points along the di-

agonal lines and for , successively selecting all everyother points of the previously already selected points along thehorizontal and vertical lines. Figures 1 and 2 visually demon-strate how the optimal constellations and in Proposi-tion 5 have been obtained by starting at the corner point in thefirst quadrant with the largest energy via the diagonal line or thehorizontal and vertical lines search over the cross QAM con-stellations when is either even or odd. However, we shouldclearly point out here that the way in obtaining the optimal con-stellation in Proposition 5 for is exactly thesame as the one in partitioning a large constellation into twosmall subconstellations in the trellis coded modulation (TCM)proposed by Urgerboeck in [78] and [79]. The major differencebetween these two partition methods is that the partition of aconstellation in the TCM is based on a union operation, i.e., theunion of the two partitioned subconstellations is equal to theoriginal constellation, whereas the partition of a constellationin the UFCP is based on multiplication. As an application ofProposition 5, we give the following example.Example 3: . By Proposition 5, we have that1) if , then

2) If , then

V. COLLABORATIVE UNITARY UFCP SPACE-TIMEBLOCK CODES

In this section, we take advantage of UFCPs established inSection IV to systematically and optimally design the constella-tion triple , , and so that the resulting unitary space-time

(20)

(21)


Fig. 1. Optimal UFCP selections from the square 64QAM and cross 128QAMconstellations for in Proposition 5. (a) Diagonal line search over thesquare QAM constellation for the optimal and even : . (b) Diagonalsearch over the cross QAM constellation for the optimal and odd : .

block code assures full diversity and the unique identification ofboth the channel coefficients and the transmitted signals as wellas the optimal coding gain.

A. Unitary UFCP Space-Time Block Codes

By Theorems 1 and 2, in order for the code (3b) to enablethe unique identification of both the channel coefficients andthe transmitted signals as well as full diversity, the three con-stellations , , and must work cooperatively such that theunique factorization property is satisfied. From the commentsof Theorem 1, we know that there may exist several cooper-ative agreements to make these constellations work together.Different collaborative ways will produce different codes, but

Fig. 2. Optimal UFCP selections from the square 64QAM and cross 128QAMconstellations for in Proposition 5. (a) Horizontal and vertical linessearch over the square QAM constellation for the optimal and even :. (b) Horizontal and vertical lines search over the cross QAM constellation

for the optimal and odd : .

in this paper, we require that , , and cooperate in such away that and and and constitute two pairs of UFCPs.In other words, the same constellation collaborates with boththe constellations and , since the same constellation col-laborating with the other two constellations and not onlyenables the unique identification and full diversity (see Theo-rems 1 and 2), but also provides an opportunity for the twotransmitter antennas to accumulate their minimum Euclideandistances such that the coding gain is increased. See more de-tails in Section V-B. To make the selection of these constella-tions more clear and more specific, we change notations , ,


and into the respective , , and . Correspondingly, thecode now takes the following form:

(22)

where and constitute two UFCPs. Such acode is called a collaborative unitary UFCP space-time blockcode. Using Theorems 1, 2 and Lemma 1 immediately yields thefollowing corollary, whose proof, therefore, is omitted.

Corollary 1: Let and constitute twoUFCPs. Then, the collaborative unitary UFCP code gener-ated from (22) enables the unique identification of both thechannel and the transmitted signals as well as full diversity withthe GLRT receiver. Moreover, the pairwise error probability

of transmitting and deciding in favor ofhas the following asymptotic formula:

where .

B. Optimal Designs of Unitary UFCP Space-Time BlockCodes

Our main task in this section is to efficiently and effectivelyoptimize the coding gain for the unitary UFCP space-timeblock codes generated from the energy-efficient cross QAMconstellations.1) Problem Formulation: Corollary 1 tells us that the unitary

UFCP code designed by (22) enables the unique identificationof the channel coefficients and the transmitted signals as wellas full diversity for the noncoherent ML receiver. Therefore,the code design partially gives a solution to Problem 1. In orderto further optimize its error performance, we can see from theasymptotic formula of the pairwise error probability in Corol-lary 1 that when SNR is large, the error performance is domi-nated by the term . Hence,

following the way similar to coherent MIMO communications[5], we define the coding gain for the unitary code as

(23)

Theoretically speaking, we should maximize the coding gaindirectly among all two UFCPs and

. However, this optimization problem, in general, is too dif-ficult to be solved, since the optimal design of constellations isgenerally extremely difficult to be reformulated into a tractableoptimization problem, even for an additive white Gaussian noisechannel [77], [80]–[85]. To make the problem tractable, in thispaper, we restrict ourselves to using the energy-efficient crossQAM constellations to generate the two UFCPs and

with . Specifically, let andbe two given -ary and -ary cross QAM constellations,

respectively. Then, the three constellations , and areselected in such a way that and . In addition,it is not difficult to verify that if and , then

and for any positive . Therefore, afamily of UFCP codes resulting from the cross QAM constella-tions and an energy scale is characterized by (24), shown atthe bottom of the page. By employing the code structure (24)and performing some algebraic manipulations, expression (23)can be further simplified into (25) at the bottom of the page,where notation is defined as

(26)

which is called a coding gain function. Our design problem isnow formally stated as follows.Problem 2: Let be fixed. Find

an energy scale and three nonnegative integers and withand such that the coding gain

is maximized subject to a total transmission bitsconstraint: , i.e.,

(24)

(25)


2) Solution to Problem 2: In order to solve each individualoptimization problem in Problem 2, we first introduce some no-tations for discussion simplicity. Recall that denotes the mod-ified -ary cross QAM defined in Definition 3. Let denoteone of its corner points with the largest energy , and let anddenote the two nearest neighbors of this corner point, with

the respective energies and , where . For givenpositive integers , and satisfying with ,let and denote the respective modified -ary and -arycross QAM constellations; denotes one of the corner point inwith the largest energy ; its two nearest neighbors are de-

noted by and , respectively, with energies being and, where . Specifically, for a given positive integer, integers and are defined as follows:

(27)

Correspondingly, all notations , , and are defined inthe same way as , , and . As will be seen shortly inthe following discussions on the solution to Problem 2, the in-tegers , and are to be replaced by , and , respec-tively, and thus, is to be regarded as . Hence, the corre-sponding positive integers and and energy notation ,

, and are used directly without the need of redefini-tions. Some properties regarding these energies are collected asthe following Lemma.

Lemma 2: For the modified -ary cross QAM , the fol-lowing statements are true.1) If , then each corner point has only one nearestneighbor, and .

2) If is even, then and.

3) If is odd and greater than 3, then,

and .

Lemma 2 can be verified directly by the calculation and thus,its proof is omitted. Now, applying Lemma 2 to a pair of QAMconstellations and yields the following lemma.

Lemma 3: Let and denote the modified -ary and-ary QAM constellations, respectively, with . Then

(28a)

(28b)

The proof of Lemma 3 is provided in Appendix B.

Lemma 4: Let and denote the modified -ary and-ary QAM constellations, respectively, with .

Then, the following four inequalities hold:

(29a)

(29b)

(29c)

(29d)

The proof of Lemma 4 is postponed to Appendix C. We alsoneed the following lemma, whose proof is given in Appendix D.

Lemma 5: Let and represent the modified -ary and-ary QAM constellations with and defined in (27). Then

(30a)

(30b)

Now, it is time to start solving Problem 2. For discussion sim-plicity, we assume without loss of generality that ,i.e, . Let us consider the following three cases of all thepossible values of .[A]. . In this case, the constellation includes

only one element, i.e., , where . Sinceand matrix is unitary, we have

, withhaving the same statistical property as and

(31)

Therefore, we only need to consider the case where . Inthis situation, andis reduced to

(32)

Let us first consider a special case where . In this case,, and hence

(33)

It is very interesting to observe that for any fixed , and ,the objection function with respectto the variables , and is minimized when its numer-ator achieves the minimum and simultaneously, its denominatorachieves the maximum, both optimums being achieved whenis the nearest neighbor of and is the corner


point with the largest energy in . Therefore, the minimumin this case is given by

(34)

Now, we consider a general case where . In this case,notice (35), at the bottom of the page. Realizing the followingfacts is key to obtaining the minimum of the objective function(32) with respect to variables , and :a) Under the conditions that

and or , the numer-ator of the objection function (32) is lower-bounded by

, i.e.,

(36a)

where the equality holds when either is the nearestneighbor of , i.e., , and or isthe nearest neighbor of and . The denominatorof the objective function (32) is upper-bounded by

(36b)

where the equality holds when both and are thecorner points in with each having the largest energy. Therefore, both equalities in (36a) and (36b) hold si-

multaneously when both are the corner points inwith each having the largest energy and is the

nearest neighbor of .b) Under the conditions that ,

, and , the numerator of the objectivefunction (32) is lower-bounded by

(37a)

The denominator of the objective function (32) is upper-bounded by (37b) at the bottom of the page.

c) If and , then. Note that the numerator of the objection function (32)

is lower-bounded by

(38a)

where the equality holds when either is the nearestneighbor of and or is the nearest neighborof and . In addition, the denominator of theobjective function (32) is upper-bounded by

(38b)

where the equality holds when both and are thecorner points in with each having the largest energy, while one of and is the corner point in with

the largest energy , and the other is of the second largestenergy . In other words, the upper bound

is the second largest maximumof the denominator of the objection function (32). Veryinterestingly, under this condition, both equalities in (38a)and (38b) are able to be achieved at the same time when

is the corner point having the largest energyin , is the nearest neighbor of and of the largestenergy and has the second largest energy in.

The above three observations reveal that for ,

(39a)

(39b)

(39c)

(35)

(37b)


When , and as a result, comparing(39a) with (39c) gives us

and thus, we have

(40a)

In addition, since , we can obtain from (39b) and(39c) that

(40b)

Combining (35) with (40) yields

(41)Since if , , the conclusion (41) in-cludes (34) as a special case. Therefore, no matter whether ornot , we always have (41).Notice that (41) can be rewritten as

(42a)

(42b)

where we have used the arithmetic mean and the geometricmean inequality from (42a) to (42b): , and theequality holds when . Thus,we have

(43)

where we have used Lemma 4 with , andand the inequality in (43) holds when and .

All the above discussions can be summarized as Property 1.Property 1: When and , the optimal solution to

Problem 2 is given as follows:(1) If is even, then .(2) If is odd, then , .

Once the optimal and have been determined, then, is the -ary QAM constellation , is the -ary

QAM constellation , and the optimal energy scale is deter-mined by

(44)

Moreover, the optimal coding gain is given by

In fact, Property 1 gives us an optimal design of the unitarytraining Alamouti code based on the cross QAM constellations.[B]. . In this case, there are in total six possibilities

in choosing the constellation , i.e.,

. In order to attain the optimal solution, we take thefollowing two steps.Step 1: Determine

, where

constellations and are derived from Proposition 5with , i.e., and . To dothat, we first represent as

(45)

which leads us to individually considering the following twooptimization problems.a) . In this case, the objective function in (25) is sim-plified into

(46)

since . Following the discussion very similarto the case when , we can obtain (47) at the bottomof the next page.

b) . Under this condition, we further splitthe feasible domain of the optimization problem:

,


into four disjoint subdomains as follows:

Therefore, we have

(48)

Now, let us first consider each inner minimization problem.a) When and ,

and as a result, the numerator of

is lower-bounded by

(49)

Under the same condition, the denominator ofis upper-bounded by

(50)

Combining (49) with (50) results in

(51)

b) When , we first notice that the nu-merator of the objective is lower-bounded by

(52)

since and . In addition, thedenominator of the objective is upper-bounded by

(53)

Now, let us argue that both equalities can be achievedsimultaneously. Since , without loss ofgenerality, we can always assume that and .Recall the definition and properties of the Group- whichwe have discussed in Section IV. If we use andto denote two groups generated by for ,then we have and . Also,recall that notation denotes one of the corner points inwith the largest energy , and notation and

denote its two nearest neighbors in , i.e.,, with the respective energies and .

Then, for , since and. Hence, if we set and ,

then both the equalities in (52) and (53) hold at the sametime, and thus

(54)

c) When , we need to consider twopossibilities: and . If , thenfollowing the discussion similar to Situation a, we obtain

(55)

If , then and .Following the same discussion as Situation b andchoosing , we have

(56)

If we make the convention that whenis the 4-QAM constellation, then (54) includes (56) as aspecial case. To compare (51) with (54), we need to provethat the following inequality:

(57)

if or or and

if

if is even and greater than 2 and

if is an odd integer exceeding 5

(47)


is true for any positive . To do that, we first establish anenergy inequality:

(58)

To show this, let us discuss the following four possibilities.a) . In this special case, it can be verified bycalculation that

and . Thus, the inequality (58) istrue in this particular situation.

b) . Since , we have

(59a)

c) , In this case, note that ,. Hence, we attain

(59b)

d) . Then, and , andas a result

(59c)

Thus, the proof of (58) is complete. Now, (58) implies

(60)

Hence, the inequality (57) is true. Now, comparing (51) with(54) and using (57) lead to

if , which is equivalent to the fact that

(61)

Using a similar argument, we can derive that

(62)

Combining (48) with (61) and (62) altogether tells us that

(63)

Substituting (47) and (63) into (45) yields

(64)

Step 2: Establish the achievable upper bound of the codinggain for any fixed UFCP, i.e.,

(65)

for any positive . As we have mentioned before, when ,there are totally six candidates regarding the constellation ,i.e., , , , ,

, . Here, we only consider the casewhere , since the discussions for the other casesare very similar. Now, we examine the following possibilities:Case 1: . In this case, since includes the 4-QAM

constellation as a subset, there are two points and insuch that and , and thus

(66a)

Case 2: . There are totally eight corner points inwith the largest energy and includes four of them. If two ofthe four points, say and , are in the same quadrant, i.e.,

, then we have

(66b)

Case 3: and .(a) . Then, we have

(66c)


(b) . Similarly, we can obtain

(66d)

(c) . Then, and as a conse-quence, we arrive at

(66e)

Case 4: and . We consider the followingpossibilities:(a) If either or , then following case 3-(a),

we can have

(66f)

where and .(b) If either or , then similar to Case

3-(b), we can obtain

(66g)

where and .(c) and . Then,

and as a result, we attain

(66h)

Comparing (66) with (45) gives (65) as required. Now, using thegeometrical and arithmetical mean inequality and then applyingLemma 4 to (65) and (64) result in

(67)

where

All the above discussions can be concluded as Property 2.Property 2: When , one of the optimal solution to

Problem 2 is that and that , , and aredetermined as follows.(1) If is even, then , ,

and

Furthermore, the optimal coding gain is given by

(2) If is odd, then , and


Actually, Property 2 also suggests us that for , the op-timal UFCP designed by Proposition 5 is still optimal in thesense of maximizing the coding gain.[C]. . In this case, . We examine the

following possibilities:(a) If either or , then we can have either

(68a)

or

(68b)


(b) If either or , then similar to situation(a), we can obtain

(68c)

since according to Lemma 3.(c) If and , then

and .In this case, one of the following two possibilities mustoccur:(a) If either or , then

we have either

(68d)

or

(68e)

(b) If neither nor belong to the samegroup, then by the pigeonhole principle, there existsan such that one of the following four state-ments must be true:i) and . As a result, weattain

(68f)

ii) and . Similarly, wecan arrive at

(68g)

iii) and . Then

(68h)

iv) and . Also, we canhave

(68i)

Now, comparing (68) results in a common upper bound onin Case 2 as follows:

since , , andby Lemma 3. Following the same trick

as the case when , using the geometrical and arithmeticalmean inequality first and then Lemma 4 arrive at the fact that

Therefore, it follows from this that


where we have used Lemma 5 in the last step. All the abovediscussions can be concluded as the following property:Property 3: For given and , we have

Properties 1, 2, and 3 lead us to giving the following theoremas one of the optimal solutions to Problem 2.

Theorem 3: One of the optimal solutions to Problem 2 isgiven as follows:(1) If , then

is the 4-QAM constellation and .Moreover, the optimal coding gain is.

(2) If , thenis the 32-QAM constellation and .

Moreover, the optimal coding gain is.

(3) If is even, then , ,and


(4) If is an odd integer exceeding 4, then ,, and


The proof of Theorem 3 is given in Appendix E. The maximumcoding gains using the optimal UFCPs determined by Theorem3 are listed in Table I for various transmission bit rates, whichis also shown in Fig. 3. Some observations on Theorem 3 aremade as follows.

Fig. 3. Optimal coding gains versus transmission bit rates.

TABLE IMAXIMUM CODING GAINS FOR DIFFERENT TRANSMISSION

BIT RATES USING OPTIMAL DESIGNS

(1) Theorem 3 tells us that the training scheme based onthe Alamouti code using the 4-QAM and 32-QAM con-stellations is optimal when either one bit or 2.5 bits perchannel use is transmitted.

(2) In spite of the fact that from Proposition 5 we knowthat increasing the number of the groups is increasingthe minimum Euclidian distance of the constellation ,the accumulated minimum Euclidian distance along thetwo transmitter antennas between two distinct groups isalways equal to 8. In addition, increasing the numberof groups is also increasing the size of the constella-tions and thus, increasing the energies of the three cornerpoints. As a result, the UFCP code using four groupscannot enable the optimal coding gain.

VI. SIMULATIONS

In this section, we carry out computer simulations and com-pare the error performance of the unitary UFCP code designproposed in this paper with those of other schemes in the liter-ature which can be used in a small noncoherent MISO systemhaving two transmitter antennas and a single receiver antenna,where channel state information is completely unknown at boththe transmitter and the receiver and the coherence time is

. All the schemes that we would like to compare hereare described as follows:


(a) Differential unitary code based on Alamouti codingscheme and PSK constellations. This design with the fastclosed-form ML decoder was proposed in [54], [56] andtwo unitary codeword matrices are and

(69)

where and are randomly, independently and equallylikely chosen from the -ary and -ary PSK constel-lations, respectively, with the two integers and deter-mined as follows:

(70)

For the necessity of performance comparison and de-coding with the GLRT receiver, these two unitary ma-trices are normalized and then stacked into one codewordmatrix, which is denoted by ,

(71)

where the normalization constant assures.

(b) SNR-efficient training Alamouti code. This SNR-efficienttraining scheme using the Alamouti code was presentedin [86]. The codeword matrices are characterized by

(72)

where and are randomly and equally likely chosenfrom either the -ary and -ary PSK constellations orcross QAM constellations, respectively, with the determi-nation of the two integers and being the same as(70). The energy constant is normalized in such a waythat . Here, the optimal average energydistribution over the training phase and communicationphase is attained by maximizing the training efficiency[8], [40], [86].

(c) Optimal unitary UFCP code. The code design is proposedin this paper and the codeword matrix is of the form:

(73)

for , where the optimal energyscale and three constellations , and are deter-mined according to Theorem 3.

It can be seen that the above three transmission schemes havethe same spectrum efficiency, i.e., each transmission rate is

bits per channel use. To make all error performancecomparisons fair, we decode all the codes using the GLRTdetector, i.e.,

where denotes the respective 4 1 received signal vector foreach kind of the aforementioned coding matrix. All the averagecodeword error rates against SNR are shown in Fig. 4. It isobserved that the optimal unitary UFCP code designed in thispaper has the best error performance among all the three codingschemes.Here, it should bementioned that despite the fact that the error

performance of the optimal unitary UFCP code proposed in thispaper is better than those of the training STBC and the differ-ential unitary code based on the Alamouti coding scheme andthe respective QAM and PSK constellations using the GLRT re-ceiver, the GLRT receiver has to be implemented by performingan exhaustive search, and hence, its decoding complexity is ex-ponential. However, the training code designs are originally in-tended for the training ML receiver, whereas the differentialcode designs are originally intended for the differential receiver.These two receivers are much less complicated than the GLRTreceiver. Here, we also would like to mention the first nonco-herent full diversity unitary code proposed by Tarokh and Kim[62]. This code design is attractive because of the fact that itsencoding and ML decoding complexity is linear.In addition, to put our optimal UFCP code design into per-

spective, we also compare its bit error rate performance withthat of the unitary code designed in [46], which is obtained bynumerically minimizing the correlations of the codeword ma-trices in Grassmann manifold. Since this unitary code has noalgebraic structure, for simulation simplicity, we only considerthe case when its transmission bit rate is one bit per channeluse, and thus, all the codeword matrices can be numerically andefficiently computed. The simulation result is shown in Fig. 5.It can be observed from Fig. 5 that when SNR is lower than12 dB, the UFCP code has slightly better error performancethan the Hochwald and Marzetta’s code, whereas when SNRis greater than 12 dB, the Hochwald and Marzetta’s code hasslightly better error performance than the UFCP code. How-ever, the code in [46] has a disadvantage, i.e., because of thefact that this numerically optimal unitary code lacks an algebraicstructure, both the transmitter and the receiver have to store allthe codeword matrices for encoding and decoding, which is notneeded with the UFCP code.

VII. CONCLUSION AND DISCUSSIONS

In this paper, we have considered a wireless communicationsystem having two transmitter antennas and a single receiverantenna, in which the channel coefficients are assumed to beunknown at either the transmitter or the receiver, but remainconstant for the first four time slots, after which they changeto new independent values that are fixed for the next four timeslots, and so on. For such a system, we have developed a novelconcept called the UFCP for the systematic design of full diver-sity unitary space-time block code. By simply normalizing thetwo Alamouti codes and carefully selecting three constellations,a full diversity unitary code design with a symbol rate hasbeen attained. It has been shown that it is the unique factoriza-tion of constellation pairs that guarantees that the unique identi-fication of both the channel coefficients and the transmitted sig-nals in the noise-free case as well as full diversity in the noise


Fig. 4. Performance comparison of unitary UFCP codes with currently avail-able noncoherent codes. (a) bits per channel use. (b) bitsper channel use. (c) bits per channel use. (d) bits perchannel use.

case. In other words, both the unique identification and full di-versity require that the constellation pair must be designed in

Fig. 5. Bit error rate comparison of the UFCP code and Hochwald andMarzetta’s code [46] with .

such a cooperative way that factorization in the product senseis unique-able. It is for this reason that we have named the codeproposed in this paper as the UFCP code. In addition, to furtherenhance error performance, the optimal unitary UFCP code en-abling the maximum coding gain has been designed from a pairof energy-efficient cross QAM constellations subject to a bit rateconstraint. After a careful examination of the fractional codinggain function, in this paper, we have taken two major steps max-imizing the coding gain.1) The energy scale has been carefully designed to compressthe first three largest energy points of the QAM constella-tions in the denominator of the objective.

2) The two UFCPs have been designed so carefully that theone constellation collaborates with the other two constella-tions through the two transmitter antennas maximizing theminimum of the numerator and at the same time, avoidingthe corner points with the largest energy as many as pos-sible achieving the minimum.

In other words, the optimal coding gain has been obtained byconstellations collaboration and energy compression. It is forthis reason that we have also called the optimal UFCP code de-signed in this paper as the energy-efficient collaborative UFCPcode. Computer simulations have demonstrated that error per-formance of the optimal unitary UFCP code presented in thispaper outperforms those of the differential code and the SNR-ef-ficient training code, which, to the best knowledge of the au-thors, is the best code in current literature for the system.As we have seen, the concept of the UFCP plays an impor-

tant role in the systematic design of energy-efficient full diver-sity unitary space-time block codes for the small MIMO systemhaving the two transmitter antennas and a single receiver an-tenna. However, the constructions and properties on the UFCPand the related transmission scheme which have been reportedin this paper are just initiative. Some significant issues still re-main unsolved.1) The construction of the optimal UFCPs for the design ofthe unitary space-time block code has been derived fromthe cross QAM constellations. How about the hexagonalconstellations? since the hexagonal constellations carved


from the Eisenstein integer ring are supposed to be moreenergy efficient than the QAM constellations carved fromthe Gaussian integer ring [83]. Generally, which constel-lation is optimal to generate a unitary UFCP space-timeblock code with the optimal coding gain?

2) Instead of a pair of coprime PSK constellations, whether isthe UFCP constructed in this paper used to systematicallydesign full diversity noncoherent space-time block codesfor a general MIMO system by following the way similarto [71]?

3) The coding scheme which has been adopted in this paper isthe Alamouti scheme. In spite of the fact that the Alamouticode is optimal in many senses for such coherent system, itis not optimal anymore for such noncoherent system, sinceit was proved that unitary codes are optimal for generalnoncoherent MIMO communications, whereas the Alam-outi code resulting from the QAM constellation is not uni-tary in general. Only when the constellation is the PSK,the resulting Alamouti code is unitary. However, the PSKconstellation is not as energy efficient as the QAM con-stellation. In addition, although this paper has proposed asimple method for the design of the unitary code just bynormalizing the two Alamouti codes, a deep insight intothe fractional coding gain function exposures the drawbackof the Alamouti scheme in the nocoherent case, i.e., Toolarge energies are contributed to the denominator. Hence,a question is: Is it possible to find another coding schemethat has the same minimum of the numerator but a smallermaximum of the denominator as the Alamouti scheme?

This paper has just casted a brick so that the jade may beattracted.

APPENDIX

A. Proof of Proposition 5

We consider the following two cases.Case 1: , i.e., there are only two elements in . Let

, where . By Proposi-tion 4, we have with . Infact, and , and thus,

. Let be one of the corner points inwith the largest energy. Without loss of generality, we can al-ways assume . The following three possibilities needto be considered separately.1) . In this case, let is the first nearest neighborof , i.e., , and is the second nearestneighbor of , i.e, . If , then

; If , then; Otherwise,

both and must lie in , and as a result,. At any rate, the minimum

distance of is upper-bounded by

(74)

On the other hand, it can be verified directly by calculationthat the constellation given by (20) satisfies three condi-tions: (a) ; (b) ; (c) and

indeed form a UFCP. Therefore, is optimal.

2) . In this case, has the two nearest neighbors,which are denoted by and , i.e.,

. If one of and , say, , belongs to theGroup- , , then. Otherwise, both and must lie in and as aresult, . Hence,the minimum distance of is always upper-bounded by

(75)

Now, following the argument similar to the Possibility 1of , we can say that the constellation given by(20) is indeed optimal.

3) is even. Similar to the possibility of , wecan prove . In addition, notice that theconstellation determined by (20) has two properties: (a)

; (b) for any two distinct

points and in , where is some complex integer.As a consequence, . Let us now checkwhether or not such a pair of and constitutesa UFCP. Suppose that there exist andsuch that , and then, . If ,then , since . If , then .Combining this with the fact that ,we have , which is impossible. Thus, and

. In other words, the pair of the constellations

and constitutes a UFCP. In conclusion, in this case,

is optimal.4) is odd. Following almost the same discussion as inthe case of even exceeding 2, we can arrive at the factthat the constellation given by (20) is also optimal.

Case 2: . In this case, possibilities for and5 can be verified directly by calculation. In addition, since thepossibility for even greater than 2 is similar to that for oddexceeding 5, here we only provide a proof for the situation

when is an odd number greater than 5. By Proposition 4,with for .

Actually, , since . Thus, wehave . Let for be ninepoints in the first quadrant around the corner of shown inFig. 6. Using the pigeonhole principle, there exists one Group,say, , including at least three of these nine points, say,

, and . Among these three points, if there exist twoof them lying either in the same row or in the same column,then . Otherwise, these three pointsare located in different rows and different columns, and thus,

. Therefore, in any case, wecan always have . On the other hand, notice thatthe constellation determined by (21) possesses two features: 1)

; and 2) for any two distinct points

and in , where is some complex integer. Hence, wehave . Let us now examine whether such a pair ofthe constellations and forms a UFCP. Suppose thatthere exist and such that ; then, wehave

(76)


Fig. 6. Nine corner points in the cross QAM constellation.

If , then , since . If , then. No matter whatever situation occurs, once we

have substituted into (76), we can always obtainfor some complex integer , where we have used the

fact that . This implies that is an evennumber, which is impossible. Thus, and and

and indeed constitute a UFCP. Therefore, in this case,is optimal. This completes the proof of Proposition 5.

B. Proof of Lemma 3

Since the proof of (28b) is much similar to that of (28a), weonly provide a proof for (28a), which can be fulfilled by consid-ering the following possibilities:1) . In this case, Lemma 2 tells us that and

. Hence, inequality (28a) is reduced to

(77)

If , , , so , inequality(77) is hold. If is an odd number exceeding 3, then byLemma 2, we haveand . Thus, we obtain

,since . Hence, in this case, inequality (77) is also

true. If is an even integer, then by Lemma 2 again, wehave andso that ,since . Hence, inequality (77) is still true.

2) is an odd number exceeding 3. In this case, Lemma 2gives us that and

so that. On the

other hand, if is an odd number not less than , then,similarly, ,since the exponential function is increasing and .Therefore, in this case, we have ,which is equivalent to the fact that .If is an even number not less than , then using Lemma2 again yields

. Hence, inthis case, inequality (28a) holds.

3) is an even number greater than or equal to 2. Similarly, byLemma 2, we can have and (78) atthe bottom of the page. Notice that when , must be2, since . Then, in this case,. Therefore, we can always obtain .

This completes the proof of Lemma 3.

C. Proof of Lemma 4

Here, we only give the proof of (29b), since the proofs of theother inequalities are much similar. To do that, let us considerthe following situations.1) is an even number not less than 4. In this case, since

, both and are either even or odd. If bothand are even, then Lemma 2 provides us with

and . Then,we have

. Since ,. Now, consider the following function in

terms of variable :

Then, the first-order derivative of with respect to isgiven by

Since the exponential function is increasing and, and thus, ,

showing that is an increasing function. Therefore, wehave , i.e., . Con-sequently, inequality (29b) holds in this case. Similarly, if

(78)


both and are odd, we can also prove that (29b) stillholds.

2) is an odd number exceeding 4. Since , eitheris even and is odd or is odd and is even. If is

even and is odd, by Lemma 2, we obtain, and hence,

,where . This leads us to considering afunction:

The first-order derivative of is

for . Since is increasing and, we arrive at the fact that

. In addition, notice that

because of the fact that . As a result,and is increasing. Hence, , i.e.,

. This can also proved to be trueif is odd and is even. Analogously, we can prove that

. Therefore, inequality (29b) holds.This completes the proof of Lemma 4.

D. Proof of Lemma 5

We prove Lemma 5 by considering the following four dif-ferent cases for .1) , where is a positive integer not less than 1. Inthis case, by the definition of and in (27), we have

, and thus, , ,which implies that .

2) . Then, (27) tells us thatis odd, whereas is even. Particularly,when , and . In this special case,

and, resulting in . If , then,

by Lemma 2, we have and

.Since , in this case, we obtain

.3) . From the definition of and given in (27),we know that . This means thatboth and are odd. Then, Lemma 2 gives us

and

.

Since , it follows that.

4) . In this case, is evenand is odd. Especially for ,we have , and thus,and . Hence, we have

in this particular case. If , then by Lemma

2 again, we can attain that

and . Because of

the fact that ,we have .

Now, summing up all the above results ends the proof of Lemma5.

E. Proof of Theorem 3

We prove Theorem 3 by considering the following situations:1) . In this case, we know from Properties 1, 2, and 3that

Therefore, the optimal coding gain is , ,, and and are the 4-QAM constellation.

2) . Similarly, Properties 1, 2, and 3 tell us that

Hence, the optimal coding gain is

, and and are the cross 32-QAM constellation.3) is an even integer exceeding 4 and not equal to 10. In thiscase, we consider the following two possibilities:a) If , then Properties 1, 2, and 3 give us

Thus, in this case, we have , , and .b) If , then we have from Properties 1, 2, and 3 that


Recall the following facts.i) , , is the largestenergy among all the points in the -arycross QAM constellation, and is the en-ergy of the second neighbor of the point withthe largest energy in the constellation.

ii) is the largest energy among all the pointsin the -ary cross QAM constellation,while and are the respective energiesof the first and second neighbors of the pointwith the largest energy in the -ary crossQAM constellation.

iii) , is the largest energy amongall the points in the -ary cross QAM con-stellation, and is the energy of the firstneighbor of the point with the largest energy inthe constellation.Hence, we obtain ,and as a result,

. In addition, since

where we have used the facts thatand that if

, we attain. Therefore, in this case,

the optimal coding gain is is.

4) is an odd integer exceeding 4. There are three cases whichneed to be considered: a) , b) , and c) .However, since the discussions on Cases a and b are muchsimilar to the previous cases and , we onlyconsider Case c. In this situation, Properties 1, 2, and 3provide us with

Notice the following facts.a) is the largest energy among all the points in the

-ary cross QAM constellation, and andare the energies of the first and second neigh-

bors of the points with the largest energies in the-ary cross QAM constellations.

b) , is the largest en-ergy among all the points in the -ary cross

QAM constellation, and is the energy of the firstneighbor of the point with the largest energy in theconstellation.

c) is the largest energy among all the points inthe -ary cross QAM constellation, is thelargest energy among all the points in the -arycross QAM constellation, and is the energy ofthe first neighbor of the point with the largest energyin the -ary cross QAM constellation.Now, following the way similar to the previous casewhere is even but not equal to 10, we can provethat is still the optimal codinggain.

This completes the proof of Theorem 3.

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Dong Xia received the B.S. degree in Automation from University of Scienceand Technology of China, Hefei, China, in 2009, the M.S. degree in Electricaland Computer Engineering from McMaster University, Hamilton, ON, Canada,in 2011.He is currently a software engineer at Amazon, Seattle, WA, US. His research

interests include MIMO communications and signal processing.

Jian-Kang Zhang (SM’09) received the B.S. degree in Information Science(Math.) from Shaanxi Normal University, Xi’an, China, M.S. degree in Infor-mation and Computational Science (Math.) from Northwest University, Xi’an,China, and the Ph.D. degree in Electrical Engineering from Xidian University,Xi’an, China, in 1983, 1988, 1999, respectively.He is now Assistant Professor in the Department of Electrical and Computer

Engineering at McMaster University, Hamilton, Ont., Canada. He has hold re-search positions in McMaster University and Harvard University. He is theco-author of the paper which received the “IEEE Signal Processing Society BestYoung Author Award” in 2008. He has served as an Associate Editor for theIEEE SIGNAL PROCESSING LETTERS. He is currently serving as an AssociateEditor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING and the Journal ofElectrical and Computer Engineering.His research interests are in the general area of signal processing, digital

communication, signal detection and estimation, wavelet and time-frequencyanalysis, mainly emphasizing mathematics-based new technology innovationand exploration for variety of signal processing and practical applications, andspecifically, number theory, matrix theory and linear algebra-based variouskinds of signal processing. His current research focuses on transceiver designsfor multi-user communication systems, coherent and noncoherent space-timesignal and receiver designs for MIMO and cooperative relay communications.

Sorina Dumitrescu (M’05) received the B.A.Sc. and Ph.D. degrees in mathe-matics from the University of Bucharest, Romania, in 1990 and 1997, respec-tively.From 2000 to 2002 she was a Postdoctoral Fellow in the Department of Com-

puter Science at the University of Western Ontario, London, Canada. Since2002 she has been with the Department of Electrical and Computer Engineeringat McMaster University, Hamilton, Canada, where she held Postdoctoral, Re-search Associate, and Assistant Professor positions, and where she is currentlyan Associate Professor.Her current research interests include multimedia coding and communi-

cations, network-aware data compression, multiple description codes, jointsource-channel coding, signal quantization. Her earlier research interests werein formal languages and automata theory. Dr. Dumitrescu has held an NSERCUniversity Faculty Award during 2007–2012.

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