On design criteria and construction of noncoherent space ...mohammad/papers/IT2003.pdf · the information about the statistics of the fading [12]. It can be easily shown that, e.g.,

2332 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 10, OCTOBER 2003

On Design Criteria and Construction of NoncoherentSpace–Time Constellations

Mohammad Jaber Borran, Student Member, IEEE, Ashutosh Sabharwal, Member, IEEE, andBehnaam Aazhang, Fellow, IEEE

Abstract—We consider the problem of digital communicationin a Rayleigh flat-fading environment using a multiple-antennasystem, when the channel state information is available neitherat the transmitter nor at the receiver. It is known that at highsignal-to-noise ratio (SNR), or when the coherence interval ismuch larger than the number of transmit antennas, a constellationof unitary matrices can achieve the capacity of the noncoherentsystem. However, at low SNR, high spectral efficiencies, or forsmall values of coherence interval, the unitary constellations losetheir optimality and fail to provide an acceptable performance.

In this work, inspired by the Stein’s lemma, we propose to usethe Kullback–Leibler (KL) distance between conditional distribu-tions to design space–time constellations for noncoherent commu-nication. In fast fading, i.e., when the coherence interval is equalto one symbol period and the unitary construction provides onlyone signal point, the new design criterion results in pulse ampli-tude moduclation (PAM)-type constellations with unequal spacingbetween constellation points. We also show that in this case, thenew design criterion is equivalent to design criteria based on theexact pairwise error probability and the Chernoff information.

When the coherence interval is larger than the number oftransmit antennas, the resulting constellations overlap withthe unitary constellations at high SNR, but at low SNR theyhave a multilevel structure and show significant performanceimprovement over unitary constellations of the same size. Theperformance improvement becomes especially more significantwhen an appropriately designed outer code or multiple receiveantennas are used. This property, together with the facts that theproposed constellations eliminate the need for training sequencesand are most suitable for low SNR, makes them a good candidatefor uplink communication in wireless systems.

Index Terms—Fading channels, multiple-antenna systems, non-coherent constellations, space–time modulation, wireless commu-nications.

I. INTRODUCTION

EXPLOITING propagation diversity by using multipleantennas at the transmitter and the receiver in wireless

communication systems has been recently proposed and studiedusing different approaches [1]–[8]. In [1], [2], it has been shownthat in a Rayleigh flat-fading environment, the capacity of amultiple-antenna system increases linearly with the smaller of

Manuscript received October 30, 2002; revised May 5, 2003.M. J. Borran was with the Department of Electrical and Computer Engi-

neering, Rice University, Houston, TX 77005 USA. He is now with Nokia Mo-bile Phones, Irving, TX 75039 USA (e-mail: [email protected]).

A. Sabharwal and B. Aazhang are with the Department of Electrical and Com-puter Engineering (MS-366), Rice University, Houston, TX 77005 USA (e-mail:[email protected]; [email protected]).

Communicated by T. L. Marzetta, Guest Editor.Digital Object Identifier 10.1109/TIT.2003.817431

the number of transmit and receive antennas, provided that thefading coefficients are known at the receiver. In a slowly fadingchannel, where the fading coefficients remain approximatelyconstant for many symbol intervals, the transmitter can sendtraining signals that allow the receiver to accurately estimatethe fading coefficients; in this case, the results of [1], [2] areapplicable.

In fast fading scenarios, however, fading coefficients canchange into new, almost independent values before beinglearned by the receiver through training signals. This problembecomes even more acute when large numbers of transmitand receive antennas are being used by the system, whichrequires very long training sequences to estimate the fadingcoefficients. Even if the channel does not change very rapidly,for applications which require transmission of short controlpackets (such as RTS and CTS in IEEE 802.11), long trainingsequences have a large overhead (in terms of the amount oftime and power spent on them), and significantly reduce theefficiency of the system. A noncoherent detection scheme,where receiver detects the transmitted symbols without havingany information about the current realization of the channel, ismore suitable for these fast fading scenarios.

The capacity of the noncoherent systems has been studied in[5], [6], where it has been shown that at high signal-to-noiseratio (SNR), or when the coherence intervalis much greaterthan the number of transmit antennas, capacity can beachieved by using a constellation of unitary matrices (i.e.,matrices with orthonormal columns). Optimal unitary constel-lations are the optimal packings in complex Grassmannianmanifolds [9]. These packings are usually obtained throughexhaustive or random search, and their decoding complexityis exponential in the rate of the constellation and the blocklength (usually assumed to be equal to the coherence intervalof the channel), or linear in the number of the points in theconstellation.

In [7], a systematic method for designing unitary space–timeconstellations has been proposed, however, the resulting con-stellations still have exponential decoding complexity. A groupof low decoding complexity real unitary constellations has beenproposed in [8]. These real constellations are optimal when thecoherence interval is equal to two (symbol periods) and numberof transmit antennas is equal to one. However, the proposedextension to large coherence intervals or multiple transmit an-tennas does not maintain their optimality.

In this paper, we consider constellations of orthogonal (ratherthan orthonormal) matrices, and propose a new design criterion

0018-9448/03$17.00 © 2003 IEEE

BORRAN et al.: DESIGN CRITERIA AND CONSTRUCTION OF NONCOHERENT SPACE–TIME CONSTELLATIONS 2333

for noncoherent space–time constellations based on the Kull-back–Leibler (KL) distance [10] between conditional distribu-tions. By imposing a multilevel structure on the constellation,we decouple the design problem into simpler optimizations anduse the existing unitary designs to construct the optimal multi-level constellation. Our motivations for considering nonunitaryconstellations and proposing the new design criterion are thefollowing.

• The long coherence timerequirement for the optimalityof the unitary constellations makes them less desirable forhigh-mobility scenarios. For example, all of the unitaryconstellations of [6]–[8] have been designed for the caseof . For , they provide only one signal point,which is obviously incapable of transmitting any informa-tion. The capacity of discrete-time fast Rayleigh-fadingchannels has been studied in [11], and has beenshown to be greater than zero. It has also been shownthat the capacity-achieving distribution for is dis-crete, with a finite number of points, and one of themalways located at the origin. In general, long coherenceinterval means a slowly fading channel (in which case atraining-based coherent transmission might be more de-sirable), whereas the noncoherent constellations are usu-ally needed when channel changes rapidly and training isdifficult or expensive (in terms of the amount of time andpower spent on that).

• The high-SNRrequirement for the optimality of the uni-tary constellations implies low power efficiency. This isbecause the capacity is a logarithmic function of the av-erage power, and thus, a linear increase in the power re-sults in only a logarithmic increase in the capacity. Con-sidering the power limitations in the battery-operated de-vices, the high-SNR requirement cannot be easily satisfiedby mobile devices.

• It appears that the unitary designs are not completely usingthe information about the statistics of the fading [12]. Itcan be easily shown that, e.g., in the case of real constel-lations for and , a non-Bayesian approach(i.e., assuming that fading is unknown, with no informa-tion about its distribution), would result in a unitary de-sign. If the statistics of the fading process are known at thetransmitter and the receiver (e.g., if the channel is assumedto be block Rayleigh flat fading, with fading coefficientsfrom the distribution ), then a design criterionwhich takes this knowledge into account more efficiently,would result in better constellations (in terms of error rateperformance).

• A common performance measure for evaluating differentconstellations in communication systems is the averagesymbol error probability. However, the expressions for theaverage error probability are usually very complicated,and do not provide much insight into the design problem.Therefore, we initially consider the pairwise error prob-ability as our performance and design criterion. Unfortu-nately, the exact expression and even the Chernoff upperbound for the pairwise error probability do not seem to be

tractable for a general noncoherent constellation. More-over, except for the special case of unitary constellations,the pairwise error probabilities are not symmetric. The KLdistance is an asymmetric distance, is relatively easy to de-rive even for general multiple-antenna constellations, andis equal to the best achievable error exponent using hy-pothesis testing (Stein’s lemma [10]).

For the above reasons, we propose the use of KL distance be-tween conditional distributions as the design criterion, and con-struct constellations for single- and multiple-antenna systemsin fast and block fading channels which outperform the existingunitary constellations.

The main contributions of this paper are summarized in thefollowing.

1) The relation between the KL distance and the maximum-likelihood (ML) detector performance: Using the fact thatthe KL distance is the expected value of the likelihoodratio, we show that, for a large number of observations(e.g., receive antennas in our case), the performance ofthe ML detector is related to the KL distance between theconditional distributions (Lemma 3). We also show that insome cases , for any number of receive antennas,the KL-based design criterion is equivalent to the designcriteria based on the exact pairwise error probability andthe Chernoff bound (Section III).

2) KL-based design criterion: We derive the KL distancebetween the conditional received distributions corre-sponding to different transmit symbols for the generalcase of multiple-antenna systems (see (31)), and proposea design criterion based on maximizing the minimum ofthis KL distance over the pairs of constellation points.We also derive the simplified distance for orthogonalconstellations (see (34)).

3) Multilevel structure and decoupled optimization: By im-posing a multilevel unitary structure on the constellation,we further simplify the KL distance, and decouple the de-sign problem into simpler optimizations. As a result, wecan use any existing unitary design at the levels of themultilevel constellation, and find the optimum distribu-tion of the points among the levels, and their radiuses. Infast fading, i.e., when and the unitary construc-tion provides only one signal point, the new design crite-rion results in pulse amplitude moduclation (PAM)-typeconstellations with unequal spacing between constella-tion points (Section V-A). When the coherence interval islarger than the number of transmit antennas, the resultingconstellations overlap with the unitary constellations athigh SNR, but at low SNR they have a multilevel struc-ture and show significant performance improvement overunitary constellations of the same size (Sections V-C andV-D).

The rest of this paper is organized as follows. In Section II, weintroduce the model for the system being considered throughoutthis paper. In Section III, we derive the exact expression and theChernoff upper bound for the pairwise error probability in thefast fading scenario, and show that they result in the same de-


sign criterion as the one suggested by the KL distance. In Sec-tion IV, we derive the KL distance between conditional distri-butions of the received signal, and propose the design criterionbased on that. In Section V, we present noncoherent construc-tions for several important cases and compare their performancewith known unitary space–time constellations. We show thatthe new constellations can provide significant performance im-provement compared to the unitary constellations, especially atlow SNR and when multiple receive antennas are used. Finally,in Section VI, we bring some concluding remarks.

II. SYSTEM MODEL

We consider a communication system with transmit andreceive antennas in a block Rayleigh flat-fading channel with

coherence interval of symbol periods (i.e., we assume that thefading coefficients remain constant during blocks ofconsecu-tive symbol intervals, and change into new, independent valuesat the end of each block). We use the following complex base-band notation:

(1)

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

fading coefficients, and is the matrix of the additivereceived noise. Elements of and are assumed to be sta-tistically independent, identically distributed circular complexGaussian random variables from the distribution . Weintentionally avoid using the scaling factor of of [7] to ac-count for the desired SNR (or average power constraint on theconstellation). We will see in Section V that the structure of theoptimal constellation depends on the actual value of the SNR,and constellations of the same size at different SNRs are not nec-essarily scaled versions of each other. Therefore, we capture theSNR factor in the matrix itself, and use the power constraint

where ’s are the elements of the signal matrix.With the above assumptions, each column of the received ma-

trix is a zero-mean circular complex Gaussian random vectorwith covariance matrix . Therefore, the conditionalprobability density function (pdf) of , the th column of thereceived matrix , can be written as

(2)

Since the columns of are statistically independent, we havethe following expression for the conditional pdf of the wholereceived matrix

(3)

Note that the superscript is not an exponent for. It onlyspecifies the column size of and emphasizes the statisticalindependence of its columns. Later, when calculating the pair-wise error probabilities and error exponents, we will find thisnotation convenient.

Assuming a signal set of size, , and defining, the ML detector for this system will have

the following form:

where (4)

If , then the probability of error in ML detection of(detecting given that was transmitted) is given by

(5)

where we have used the notation to denote the proba-bility of set with respect to the pdf. Now, if we also assumethat and are transmitted with equal probabilities, then theaverage probability of error in ML detection will be given by

(6)

It is known [10] that this average pairwise error probability de-cays exponentially with the number of independent observa-tions, and the rate of this exponential decay is given by the Cher-noff information [10] between the conditional distributions.

For , even though (5) and (6) are no longer exact, wewill still use them as an approximation for the pairwise errorprobability, which will, in turn, be used to derive the designcriterion for space–time constellations.

For the special case of unitary transmit matrices, i.e., when, the exact expression and Chernoff upper

bound for the pairwise error probability were calculated in [6].However, the corresponding expressions for the general case ofarbitrary matrix constellations do not seem to be easily tractable.In the next section, we will calculate these expressions for thecase of a single transmit antenna in fast fading, and will showthat they result in the same design criterion.

Before proceeding to the next section and deriving the ex-pressions for the pairwise error probabilities, we notice that theconditional pdf in (3) depends on only through the product

. Since the pairwise error probability in (5) (and also ingeneral the average error probability of the ML detector) is de-termined only by the conditional pdfs, it is clear that the perfor-mance of the noncoherent multiple-antenna constellation alsodepends on the constellation matrices only through theproducts . If , using the Cholesky factor-ization where is a lower triangularmatrix, for any constellation of matrices one can finda corresponding constellation of matrices with the samepairwise and average error probabilities. Also, if , usingthe singular value decomposition, we can write ,where and are and unitary (orthonormal)matrices and is an real nonnegative diagonal matrix.Defining , we have . This means thatfor any constellation of arbitrary matrices, one can find aconstellation of orthogonal matrices with the same pair-wise and average error probabilities. Since the average power of


the constellation also depends only on the expected value of theproduct

(7)

the new constellation will also have the same average power asthe original constellation. Therefore, (similarly to the result of[5] for capacity) we have the following result.

Theorem 1: Increasing the number of transmit antennas be-yond does not improve the error probability performance ofthe noncoherent multiple-antenna systems. Furthermore, the op-timum error probability performance can be achieved by a con-stellation of orthogonal matrices.

Therefore, in the remainder of this paper, we will assume that, and that the constellation matrices are orthogonal.

III. PAIRWISE ERRORPROBABILITY FOR FAST FADING

Throughout this section, we will assume that there are onlytwo signal points in the constellation, i.e., . As mentionedin the previous section, there is no gain in using more transmitantennas than . Therefore, we also assume that . Withthis assumption, the transmit matrix is simply a complex scalar.We denote this scalar by. The conditional probability densityof the received signal given the transmitted symbol is given by

(8)

In the following, we derive the exact expression forin this case.

Lemma 1: For a single-antenna communication system (i.e.,) in a fast-fading environment (i.e., ), the

pairwise error probability of noncoherent ML detector is givenby

for

for

(9)

Proof: Using (5) and (8) (with ), and assuming that, we have

where

(10)

and

(11)

Similarly, it can be shown that if , we have

This completes the proof.

Theorem 2: For a single transmit antenna communicationsystem (i.e., ) in a fast-fading environment (i.e., ),the pairwise error probability of noncoherent ML detector isgiven by

for

for(12)

where is as in (11).Proof: Proof is by induction, and is given in Appendix A.

It is interesting to notice that, in this case, the two pairwiseerror probabilities, i.e., and are notequal, even in the limit as or . Thefollowing two identities can be verified easily:

(13)

(14)

For , the above equalities reduce to the following:

This discontinuity can be explained as follows. With the MLdetector, the decision region corresponding to each constella-tion point is the region in which that constellation point has thelargest likelihood among all of the constellation points. The like-lihood functions in this case (binary signaling, single transmitantenna, fast fading, and noncoherent detection) are complexGaussian pdfs with zero mean and variance , for

. The boundary of the two decision regions is the line alongwhich the two likelihood functions are equal. For , thisboundary is a circle in the complex plane centered at the origin,with radius , where is as in (10). Fig. 1 shows a crosssection of the likelihood functions and the boundary of the de-cision regions for and . Fig. 2 shows the radius ofthe boundary versus , when . As we can see in these


Fig. 1. The likelihood functions forjs j = 0, js j = 1, andN = 1.

Fig. 2. The radius of the decision region versusjs j for js j = 1 andN = 1.

figures, the boundary is not defined when , becausein this case the two likelihood functions are equal everywhere.However, both right and left limits of the boundary radius (as

) exist and are, in fact, equal. It can be easily verified

that the limit is equal to the standard deviation of the commondistribution, or

(15)


Fig. 3. Pairwise error probabilities for single antenna in fast fading.

The probability of mistaking for is the volume under theconditional pdf corresponding to in the decision region of .The discontinuity of the pairwise error probability comes fromthe fact that, even though the radius of the boundary of the de-cision regions does not have a jump discontinuity at ,the decision region of itself suddenly changes from outsidethe boundary circle to inside the circle, as changes from avalue smaller than to a value larger than . Since the vol-umes under the Gaussian density function in the two regions ofinside and outside a circle with radius equal to the standard de-viation of the distribution are different, the left and right limitsof the pairwise error probability are also different, resulting ina jump discontinuity. This discontinuity can be seen in Fig. 3which shows the two pairwise error probabilities as a functionof for ( and correspond toand , respectively).

Another interesting point to observe in Figs. 1 and 2 is thatthe signal points may not belong to their respective decision re-gions. As we see in Fig. 1, the radius of the boundary of thedecision region is greater than one (around ), whereasboth constellation points have magnitudes smaller than or equalto one. This is not the case for large values of as shown inFig. 2, e.g., for , where the radius of the boundary is lessthan . Fig. 2 also compares the radius of the boundary of thedecision region with the arithmetic mean of the magnitudes ofthe constellation points (probably the most intuitive, yet incor-rect, value for the radius of the boundary). As we see, for largevalues of , the radius of the boundary is much smaller thanthe arithmetic mean.

From Fig. 3, we observe that in the two disjoint regionsand , the pairwise error probability

is a monotonic function of as defined in (11). Therefore,assuming , minimizing the pairwise error probabilityis equivalent to minimizing .

The next proposition gives the Chernoff bound on the expo-nent for pairwise error probability in this case.

Proposition 1: Consider a single transmit antenna commu-nication system (i.e., ), in a fast-fading environment(i.e., ). The largest achievable exponent for the averageprobability of error (i.e., the Chernoff information [10]) for thissystem is given by the following expression:

(16)

where is as in (11).Proof: See Appendix B.

Notice that since , and and have the samesign, the Chernoff distance in (16) is well defined, and since

for , it is always greater than or equalto zero.

Fig. 4 shows the exact average error probability, and theChernoff bound for the average error probability given by

, for . As we see, the Chernoffbound is also a monotonic function of in the two regionsof and . We will see later in Section V-A,that the KL distance between the two conditional distributionscorresponding to the two different transmitted symbols is givenby

(17)


Fig. 4. Average error probability and the Chernoff bound for single antenna in fast fading.

This expression is also a monotonic function ofin the two re-gions of and . Therefore, the three differentcriteria of a) minimizing the maximum of the exact pairwiseerror probability, b) maximizing the minimum of the Chernoffdistance, and c) maximizing the minimum of the KL distance,are all equivalent to minimizing the maximum of (assuming

), and will result in the same constellation. In SectionV-A, we design new constellations based on this design crite-rion and compare their performance with the performance of theconventional PAM constellations.

IV. DESIGN CRITERION

It is known [5] that the capacity-achieving signal matrix forthe noncoherent systems can be written as , whereis a isotropically distributed unitary matrix, and isan independent real, nonnegative, diagonal matrix. Itis also known [6] that when the SNR is high or when ,the capacity can be achieved by a unitary signal matrix (i.e., asignal matrix with orthonormal columns obtained by settingto a deterministic multiple of the identity matrix).

Based on these results, unitary space–time modulation hasbeen studied in [6], where expressions for the exact pairwiseerror probability as well as the Chernoff upper bound for thepairwise error probability of unitary matrices have been derived.These expressions suggest a design criterion based on mini-mizing the singular values of the product matrices overall pairs of the constellation points. In [8], it has been shown thatthis design criterion is approximately equivalent to maximizingthe minimum so-calledsquare Euclidean distancebetween sub-spaces spanned by columns of the constellation matrices. It can

be shown (see Appendix F) that the square Euclidean distancebetween subspaces is equivalent to thechordal distance, as de-fined in [13]. Therefore, unitary designs can be considered aspackings in the complex Grassmannian manifolds [13].

The problem with the unitary constellations is that they areoptimal only at high SNR or when . These require-ments are rather restrictive, and cannot be met in many situa-tions of practical importance. Operation at high SNR means lowpower efficiency, which is in contradiction with the low powerrequirements of the wireless systems. On the other hand, largecoherence interval means a slowly fading channel, in which casea training-based coherent signaling might be more desirable. Infact, the main motivation for noncoherent communication is todeal with fast-fading scenarios where training is either impos-sible or very expensive (in terms of the fraction of time and en-ergy spent on that). Even if the coherence interval is large, be-cause of the exponential growth of the constellation size and (inmost cases) decoding complexity with, one might decide todesign a constellation for a block length with is much smallerthan . At low SNR, or when the block length is not much largerthan , the unitary designs lose their optimality and fail to pro-vide a desirable performance. For these reasons, in this work wedo not assume a unitary structure on the constellation matrices,and try to design noncoherent signal sets of matrices with or-thogonal (rather than orthonormal) columns.

Unlike the case of unitary constellations, the pairwise errorprobability of the noncoherent ML detector, which is approxi-mately given by (5), does not appear to be tractable in the moregeneral case of orthogonal matrices. Even the Chernoff distance(see Appendix B for the definition and an example), which de-termines the exponential decay rate of the average pairwise error


probability of the ML detector [10], does not seem to admita simple closed-form expression for arbitrary orthogonal mul-tiple-antenna constellations. Therefore, inspired by the Stein’slemma [10], we will use, as our performance criterion, the upperbound on the exponential decay rate of the pairwise error prob-ability, given by the KL distance [10] between conditional dis-tributions. If and are two pdfs on the probability space

, then the KL distance between them is defined as

(18)

where denotes expectation with respect to. We also usethe notation to denote the probability of setwith respect to the pdf.

Stein’s lemma [10] relates the KL distance with the pairwiseerror probabilities of hypothesis testing.

Lemma 2 (Stein’s Lemma):Let bedrawn independent and identically distributed (i.i.d.) accordingto the pdf on . Consider the hypothesis test betweenand , where and are pdfs on , and

. Let be an acceptance region for Hypothesis 1.Denote the probabilities of error by

(19)

(20)

and define

(21)

Then

(22)

In other words, the best achievable error exponent forwith the constraint that is smaller

than a given value, is given by . It turnsout that this error exponent is not achieved with the ML de-tector, but with a detector which is highly biased in favor of thesecond hypothesis. Nevertheless, it serves as an upper bound onthe pairwise error exponent of the ML detector. The followinglemma shows that the performance of the ML detector is, infact, related to the KL distance between the distributions. (Aswe saw in Section III, at least in fast fading, i.e., when ,the KL-based design criterion is equivalent to the designcriterion based on the exact pairwise error probability and alsothe Chernoff bound.)

Lemma 3: Let be drawn i.i.d. ac-cording to the pdf on . Consider two hypothesis tests, onebetween and , and the other between and

, where and are pdfs on , and

Let

and

denote the likelihood ratios for the two tests, so that the proba-bilities of mistaking for and using the ML detector aregiven by

and

Also, denote by the difference between the two KL dis-tances, i.e., . With theseassumptions, we will have

as (23)

Proof: We have

(24)

Since ’s are assumed to be drawn i.i.d. according to the pdf, by the weak law of large numbers, we have

in probability w.r.t. (25)

Multiplying the numerator and denominator of the argument ofthe function by , we have

(26)

From equations (24)–(26) we have

in probability w.r.t. (27)

which means that for any

as (28)

Let . From (28) we will have

or as (29)

or

as (30)

The above lemma states that, for sufficiently large, withhigh probability the likelihood ratio of the first test is greaterthan the likelihood ratio of the second test, and the ratio of thetwo likelihood ratios grows exponentially with . Recallingthat the error probability of each test is the probability that itscorresponding likelihood ratio is smaller than one, this implies


that for large , the first test will have a lower probability oferror than the second test.

In Lemma 3, is the number of independent observations.In our case, independent observations can be obtained by usingan outer code which operates over several independent fadingintervals, or simply by using multiple receive antennas.

In Appendix C, we show that the KL distance betweenand(obtained by substituting and for in (3)), is given by

(31)

Adopting the KL distance as performance criterion, the signalset design criterion in general will be maximization of theminimum KL distance between conditional distributionscorresponding to the signal points, i.e., assuming equiprobablesignal points

(32)

where

is the Frobenious norm of , or the total power used to trans-mit .

If we denote by the eigenvalues of, the KL distance in (31) can be

written as

(33)

This expression, in spite of its notational simplicity and also itsresemblance to the well-knownrank anddeterminantcriteriaof coherent space–time codes [4], does not provide much in-sight into the design problem. Moreover, the power constraintdoes not appear to be easily expressible in terms of the aboveeigenvalues. Therefore, in the next section, we will try to ap-proach the design problem by imposing some extra constraintson the signal set and directly simplifying the original expressionin (31). Since the actual value of does not affect the maxi-mization in (32), in designing the signal constellations we willalways assume that .

V. SIGNAL SET CONSTRUCTION

In Theorem 1, we showed that any error probability per-formance achievable by a constellation of arbitrary matrices,can also be achieved by a constellation of orthogonal matrices.Therefore, in this work we will only consider matrix constella-tions with orthogonal columns. In Appendix D, we show that

with this assumption, the KL distance expression in (31) canbe written as

(34)

where we have used the notation to denote the th columnof . The expression for KL distance in (34) is not very illu-minating as it is. Therefore, we study the signal set constructionproblem through a series of special cases. These special caseswill provide an understanding of the nature of the KL distancein (31) by breaking it down into simpler components. In mostcases, this results in a systematic technique for constellation de-sign.

Notice that Sections V-B and V-D correspond to single-an-tenna and multiple-antenna unitary constellations, and are spe-cial subcases of Sections V-C and V-E, respectively. It is shownin these sections that, assuming orthonormal signal matrices, theKL-based design criterion reduces to the previously proposeddesign criterion for the unitary constellations [6], [8]. There-fore, unitary designs can be considered as special cases of themore general constellations designed using the KL-based crite-rion. Simulation results corresponding to Section V-B and Sec-tion V-D are presented in Section V-C and Section V-E, respec-tively, where the unitary designs are compared with their multi-level versions designed using the KL-based criterion.

A. Fast Fading

As stated in Theorem 1, there is no gain in using more thantransmit antennas. Therefore, in this case we consider only

single transmit antenna systems, where signal matrices are com-plex scalars. The KL distance of (34) reduces to

(35)

It can be easily verified that, similarly to the pairwise error prob-ability (12) and the Chernoff information (16), the KL distanceis also a monotonic function of in the two regionsof and . Therefore, for a single transmitantenna system in fast fading, maximizing the minimum of theKL distance is equivalent to minimizing the maximum of theexact pairwise error probability as well as the Chernoff bound.The following theorem characterizes the solution to the max-imin problem in this case.

Theorem 3: The solution to the maximin problem (32) forthe case of and , is given by ,where is the largest real root of the polynomial

(36)

Proof: See Appendix E.


Fig. 5. Magnitudes of the optimal signal points for a four-point constellation withM = 1, T = 1, andN = 1.

Notice that, since , , and, this polynomial always has a real root

larger than one.The constellations obtained from Theorem 3 are PAM-type

constellations but with unequal spacing between the signalpoints, and the first point is always at the origin. The interestingfact is that even the relative locations of the constellationpoints depend on the SNR, and two constellations of the samesize designed for different SNR values are not necessarilyscaled versions of each other. Fig. 5 shows the locations ofthe signal points for a four-point constellation versus averagetransmit power. As can be seen, the spacings between pairs ofconsecutive points are not equal. However, in terms of the KLdistance, these points are, by construction, equally spaced. Athigh SNR, the outer points have to be placed farther apart thanthe inner points, to maintain a constant KL distance. Therefore,for a PAM constellation with equally spaced points, the outerpoints have a smaller KL distance than the inner points. In fact,it can be easily shown that in (35), if the ratio of the magnitudesof two constellation points is constant, by increasing SNR theKL distance between those two points converges to a finiteconstant. This results in an error floor for a PAM constellationas shown in Fig. 6. However, as we see in this figure, theoptimal constellation does not see any error floor.

B. , , and for

This is the case of single transmit antenna systems in block-fading environment, with constellation points which are columnvectors and all lie on a sphere in . Since all of the points havethe same magnitude, these constellations can be considered as

single-antenna unitary constellation. The KL distance of (34)reduces to

(37)

where is the inner product operation, and denotes theangle between signal vectors and . This distance dependsonly on the angle between the signal points.

The optimum constellation in this case is obviously the onethat is designed to maximize the minimum angle between sub-spaces spanned by the signal points (or, equivalently, minimizesthe maximum absolute inner product or correlation betweensignal points). This is the same design criterion proposed forunitary constellations [7], [8] if only one transmit antenna isconsidered. Examples of such designs can also be found in [7]and [8].

For , if we confine ourselves to real constellations, theabove criterion results in the signal set

which is the same as the signal set proposed in [8]. As also men-tioned in [8], these so-called phase-shift keying (PSK) constel-lations have the advantage of low-complexity decoding basedon a single phase calculation and quantization. Therefore, wewill use these constellations for a more general design explainedin the next subsection. Notice that the angle between adjacentpoints is , not . This is because this angle is actuallythe angle between subspaces containing the constellation points,and thus has to be considered modulo.


Fig. 6. Symbol error rate comparison with regular PAM for a four-point constellation withM = 1, T = 1, andN = 1.

C. and

This is the general case for single-antenna constellations. TheKL distance in (34) reduces to

(38)

As can be seen, the KL distance between any two points con-sists of two parts: due to having different magnitudes

(lying on different spheres in ), and dueto the angle between the points (lying on different one-dimen-sional subspaces of ). If two points lie on the same sphere

, and if they lie on the same complex plane (one-dimensional subspace) . In general, the overalldistance is greater than or equal to either of these parts. Thisproperty of the KL distance in (38) suggests partitioning thesignal space into subsets of concentric spheres ofradius containing points, respectively,and defining the intrasubset and intersubset distances as

(39)

and

(40)

Without loss of generality, we can assume thatWith this assumption, and using the fact that if

then , we can reformulate thedesign problem as the following suboptimal maximin problemover , , and :

(41)

where

(42)

The suboptimality of this approach comes from the fact thatin the above formulation of , it is assumed that forany two subsets and , there are two constellation points

and , such that

This assumption is not necessarily true for the optimal constel-lation. However, since , it is guaranteedthat the actual minimum KL distance of the resulting constel-lation from the above optimization will not be smaller than theminimum in (41). Moreover, with this assumption, for a fixed

and fixed such that , we can de-couple the original optimization problem into the following sim-pler problems.

1) For each subset (each ), find the bestconfiguration of points on the surface of theth sphere

, i.e., maximize the minimum intrasubset distance in-side the th subset. Notice that appears inonly through a multiplicative factor, and does not affectthis optimization. Also notice that this step is equivalentto designing a single-antenna unitary constellation of


points (see Section V-B), and any existing unitary design(e.g., [7] or [8]) can be used at this step.

2) Solve the following continuous optimization to find theradiuses of the subsets:

(43)

where is as in (42). This optimization problem can besolved numerically, e.g., using the function ofthe Matlab program (which uses a Sequential QuadraticProgramming method to solve the nonlinear constrainedoptimization problems).

The solution to the problem in (41) can then be obtained bysearching over all possible values for and suchthat . The following proposition can be used tofurther restrict the domain of search.

Proposition 2: The solution of (41) satisfies the followinginequalities:

(44)

Proof: Let be the solutionof (41), with . Suppose, for the sake of contradiction, that

for some . Now, by removingone point from subset and adding it to subset (andrearranging the points in these two subsets to maximize the min-imum intrasubset distances), and specifying the parameters ofthe new constellation by a “” sign (e.g., and

), we will have the following.

1) is an increasing function of the subset radius, anda nonincreasing function of the subset size (number ofpoints in the subset). Therefore, since and

, we have

(45)

and since and we have

(46)

Since all other intrasubset distances and also the in-tersubset distances are not affected by this change, theoverall minimum KL distance of the new constellationwill be greater than or equal to the minimum KL distanceof the original constellation.

2)

Therefore, the average power of the new constellation issmaller than the average power of the original constella-tion.

Now, by appropriately scaling the new constellation, we canmake the average powers of the two constellations equal, andobtain a new constellation which has a larger minimum KL dis-

tance. This is a contradiction with our initial assumption. There-fore, the solution of (41) should satisfy (44).

For a fixed , the collection of all of the possible -tuples, satisfying and (44), can be found

recursively. The details are omitted here for brevity. The opti-mization in (41) can then be solved through the following steps.

1) Set .2) Find the collection of all of the possible -tuples

, satisfying and (44).3) For each member of the above collection, perform the

following steps, and find the best achievable minimumdistance.

a) For each subset (each ), find thebest configuration of points on the surface of the

th sphere , i.e., maximize the minimum intra-subset distance inside theth subset (unitary de-sign).

b) Solve the continuous optimization in (43) to find theradiuses of the subsets.

4) Store the parameters of the constellation with the largestminimum KL distance from the previous step as the bestcandidate with levels.

5) Increase by one. If go to the second step.6) Among all of the above candidates, choose the constel-

lation with the largest minimum KL distance, as the solu-tion of (41).

Assuming that the best unitary constellations (of the typementioned in Section V-B, and) of arbitrary size are known,this approach significantly simplifies the design problem by re-ducing the number of design parameters from (real andimaginary parts of the elements of the constellation vectors), to

(number of the subsets, and radius and number of thepoints in each subset).

Notice that, since unlike the square Euclidean distance, theKL distance does not scale with the average power of the con-stellation, the structure of the optimal constellation based on theabove criterion depends on the actual value of the SNR, andconstellations of the same size at different SNR values are notscaled versions of each other. It is also worthwhile to notice thatat high SNR (for large values of or , or both), we havethe following approximations:

if

if is kept fixed.(47)

This means that increases almost linearly with SNR,whereas either approaches a constant value orincreases at most logarithmically with SNR. As a result, at highSNR, having multiple levels is not desirable, and one shouldonly consider constellations of constant magnitude. In this case,the KL-based design criterion reduces to the design criterionfor the single-antenna unitary constellations (see Section V-B),confirming the high-SNR optimality of the unitary constella-tions.


The decoding of the multilevel unitary constellations pro-posed in this section can be done in a similar way to that oftrellis-coded modulation schemes, i.e., in two steps of “point insubset decoding” and “subset decoding.” If a unitary code withlow decoding complexity, such as the schemes described in [8],is used inside each subset, then the point in subset decoding stepcan be done at a very low cost, and considering the fact thatthe number of subsets is usually much smaller than the size ofthe whole constellation, the overall decoding complexity of thecode will be much lower than the regular ML decoder.

For the special case of real constellations with , theangle between adjacent points in theth subset is simply(see Section V-B), and the maximin problem in (41) is relativelyeasy to solve. The resulting 2-, 4-, 8-, and 16-point constella-tions with average powers of and are shown in Figs. 7and 8. Each axis in these figures actually represents a com-plex plane corresponding to one transmit symbol interval. Thesymbol error rate performance of the 8- and 16-point constella-tions at (SNR 7 dB) are simulated for different valuesof and compared with the corresponding constellations pro-posed in [8]. The results are shown in Figs. 9 and 10. As ex-pected, due to the larger minimum KL distance of the new con-stellations, the exponential decay of the symbol error rate versus

has a much higher rate for the new constellations. The min-imum KL distances of the new constellations are and

for 8- and 16-point constellations, respectively, whereasthe corresponding PSK constellations of [8] have minimum KLdistances of and , respectively.

D. , , and for

This is the case of unitary constellations. Since all of thecolumns of and have the same square magnitude, theKL distance in (34) reduces to

(48)

where and denote the subspaces of spannedby columns of and , respectively, isthe Euclidean distance of vector from subspace ,and is the Euclidean distanceof subspaces

and , as defined in [8]. As can be seen, for unitaryconstellations, the KL-based design criterion reduces to theEuclidean-based design criterion, and, therefore, the newnoncoherent space–time constellations include the existingunitary constellations as a special case.

In Appendix F, we show that theEuclidean distancedefinedin [8] and thechordal distancedefined in [13] are equivalent.

Fig. 7. KL-optimal constellations of size 2, 4, 8, and 16 forM = 1, T = 2,andP = 0:5.

Fig. 8. KL-optimal constellations of size 2, 4, 8, and 16 forM = 1, T = 2,andP = 5.

Therefore, the unitary constellations are, in fact, packings incomplex Grassmannian manifolds. In [9], it has been shownthat, at high SNR, the calculation of capacity of the noncoherentmultiple-antenna channel can also be viewed as sphere packingin the product space of Grassmannian manifolds.


Fig. 9. Symbol error rate for an 8-point constellation forM = 1, T = 2, and SNR= 7 dB.

E. , , and for

The assumption in this case is that each signal matrix is ascalar multiple of a unitary matrix. With this assumption, theKL distance in (31) reduces to

(49)

where is the squareEuclidean distance[8] orchordal distance[13] between the two subspaces andspanned by columns of and , defined as

(50)

denotes the distance between two constellation points whichrepresent the same -dimensional subspace of the-dimen-sional space, and denotes the distance between two constel-lation points with the same power which represent two different

-dimensional subspaces. In general, the overall distance isgreater than or equal to either of these parts. Recalling that theunitary constellations are designed to maximize theEuclideandistancebetween subspaces, the above partitioning of the KLdistance suggests partitioning the signal space into subsets ofunitary constellations, , with columns of square

norm , containing points, respectively.Similar to the approach of Section V-C, we define the intra-subset and intersubset distances as

(51)

and

(52)

Without loss of generality, we can assume that, and solve the simplified maximin problem

(53)

where

(54)

to find the -point multilevel unitary constellation of ma-trices with average power. At each level, we can use any ex-isting unitary construction and substitute, for in (51),the best achievable KL distance with that construction and withsize .

As explained in Section V-C for the case of single transmitantenna, in (53), and are discrete variables, while

are continuous variables. For any fixed value ofand satisfying the specified constraints, (53) reducesto a continuous optimization over , which can besolved numerically. Moreover, as shown in Proposition 2, thesolution of (53) also satisfies the extra constraint

, which can be used to further restrict the domain ofsearch.


Fig. 10. Symbol error rate for a 16-point constellations forM = 1, T = 2, and SNR= 7 dB.

Fig. 11. Performance comparison of one and two transmit antenna systematic constellations of [7] and their multilevel versions versus number of receive antennas.

Similar to the case of single transmit antenna (Section V-C),from (49) we also observe that at high SNR,becomes a con-stant, or its minimum grows at most logarithmically with SNR(when is kept fixed), whereas grows linearly with SNRfor nonzero constellation points from different subspaces. As aresult, at high SNR, becomes the dominant term, and the

KL-based design criterion reduces to theEuclidean-based de-sign criterion of the unitary constellations, confirming the highSNR optimality of the unitary constellations.

Figs. 11 and 12 show the error rate performance comparisonof the proposed constellations with their unitary counterparts.In these examples, we have used the systematic unitary designs


Fig. 12. Performance comparison of one and two transmit antenna systematic constellations of [7] and their multilevel versions versus SNR.

of [7] as the constituent subsets of the multilevel constellations.Fig. 11 shows the block error rate performance of the 16- and32-point two-antenna constellations with and (re-sulting in spectral efficiencies of 1.33 and 1.25 b/s/Hz), respec-tively. The horizontal axis is the number of receive antennas,with SNR kept fixed at 0 dB. As can be seen, the multilevelconstellations can save up to four receive antennas at SNRs aslow as 0 dB.

Fig. 12 compares the performances of 16-point, one and two-antenna constellations for and (resulting in spec-tral efficiencies of 2 and 1.33 b/s/Hz), respectively. The hori-zontal axis is SNR, and the receiver is assumed to have 10 re-ceive antennas. Even if multiple receive antennas are not avail-able, similar gains can be obtained by encoding across severalfading blocks using an outer code. For each point in the curvescorresponding to the multilevel constellations, a separate opti-mization problem with appropriate power constraint has beensolved and the resulting constellation has been used to evaluatethe performance. We observe that the multilevel unitary con-stellation can provide up to 3-dB gain over its correspondingone-level unitary constellation at low SNR. We also notice thatas SNR increases, the two curves become closer, which is ex-pected, recalling the optimality of the unitary constellations athigh SNR.

VI. CONCLUSION

We considered the problem of noncoherent communicationin a Rayleigh flat-fading environment using a multiple-antennasystem. We derived the design criterion for space–time con-stellations in this scenario based on the KL distance betweenthe distributions of the received signal conditioned on differenttransmitted values. We showed that close-to-optimal constella-

tions according to the proposed criterion can be obtained by par-titioning the signal space into appropriate subsets and using uni-tary designs inside each subset. We designed new noncoherentconstellations based on the proposed criterion, and through sim-ulations, showed that they can provide a substantial improve-ment in the performance over known unitary space–time con-stellations, especially at low SNR and when multiple receiveantennas are used. We showed that unitary designs can be con-sidered as special cases of the proposed constellations when theSNR is high.

APPENDIX AEXACT PAIRWISE ERRORPROBABILITY FOR FAST FADING

In this appendix, we prove that the expression for the exactpairwise error probability of the single transmit antenna systemin fast fading is given by (12). For convenience, we use the fol-lowing notation for the received vector:

(55)

Using (5) and (8), and assuming that , we have

(56)

where is as in (10).


Similarly, for we have

(57)

(since does not have any mass accumulation point).Equation (12) then follows by applying the following lemma

with and using (10) and (11).

Lemma 4: For any , we have

(58)

Proof: The proof is by induction, as follows.For , we have

(59)

which is true, and proven in Proposition 1.Now assume that (58) is true for . We prove that it

will also be true for . Using (8) and the notationdefined in (55), we can write

(60)

Defining the regions , , and as

and

and

we have

and

(61)

The first term in (61) can be calculated as

(62)

where the last equality follows from the fact that we have as-sumed (58) is true for .

The second term in (61) can be calculated as

(63)

where the third equality follows from (59) and (8), and the lastequality follows from the formula of the volume of a -di-mensional sphere with radius,

(64)

Substituting (62) and (63) in (61) shows that (58) is true for. This completes the proof.

APPENDIX BCHERNOFFBOUND FOR THESINGLE-ANTENNA CASE

It can be easily shown that

(65)

Therefore, in the following we only derive the expression for. By definition, the Chernoff information (distance)

between two probability densities and is given by

(66)

Using (8) for the conditional probability densities, we will have

(67)

where


Using (8) again with for , we have

(68)

Substituting (68) in (67) and using (11), we have

(69)

Now, since is a strictly convex functionof for , we can find the minimum by taking a derivativewith respect to and setting it to zero, which results in

for (70)

Substituting this value of as the minimizer in (69) togetherwith (65) results in (16).

APPENDIX CTHE KL DISTANCE

In this appendix, we derive the expression for the KL distancebetween two distributions of form (3). By definition

Using (3) and defining for ,we have

(71)

since ’s are i.i.d.Substituting (2) for and , we will have

(72)

where

Again, using (2) for , we have

(73)

Substituting (73) in (72), we will have

(74)

Equations (74) and (71) result in (31).

APPENDIX DTHE SIMPLIFIED KL DISTANCE FORORTHOGONAL MATRICES

In this appendix, we derive a simplified version of the KL dis-tance in (31) for a constellation of orthogonal matrices,with for . Here is a diagonal ma-trix with its th diagonal element , equal to the magnitudesquare of the th column of , . Using the matrix in-version lemma [14]

(75)

we can write

(76)

Therefore, we will have

(77)

We need to calculate the trace and determinant of this matrix,and substitute for them in (31). To find the trace of (77), wecalculate the trace of each term separately. We have

(78)

and

(79)


To find the trace of the last term in (77), we use the followingidentity which can be easily verified:

(80)

where is an arbitrary square matrix, is a diagonal matrixof the same size as, and denotes a diagonal matrixconstructed from the diagonal elements ofin the same order.Defining , we have

(81)

where is the th diagonal element of , and is the innerproduct operation. Using (80) and (81), we will have

(82)

Equations (77)–(79) and (82) result in

(83)

Now we calculate the determinant of

For this, we use the identity [14]

(84)

to write

(85)

Using (85), we will have

(86)

Substituting (83) and (86) in (31), results in (34).

APPENDIX ETHE OPTIMAL SINGLE ANTENNA CONSTELLATION FOR

FAST FADING

Without loss of generality, assume that. Since is monotonically

decreasing for and monotonically increasing for, with for , it is clear that

the minimum KL distance will occur between a pair of consecu-tive symbols from the above order, in the same order. Moreover,in order to solve (32), it is sufficient to solve the following min-imax problem:

(87)

Defining

we will have

(88)

for , or

which implies

(using the average power constraint). Now, since is amonotonically increasing function of, it is clear that the max-imum of is obtained if and only if , and

. This requires that all of the inequalities in (88) holdwith equality. Therefore, the optimum signal set can be obtainedby setting

where is the largest real number satisfying ,or


APPENDIX FEQUIVALENCE OF THE EUCLIDEAN AND CHORDAL DISTANCES

BETWEEN SUBSPACES

The chordal distancebetween two -dimensional sub-spaces, and , of is defined [13] as

(89)

where and are theprincipal vectorscor-responding to and , respectively, and are recursively de-fined as

(90)

for . We will use te following lemma to provethe equivalence of theEuclideanandchordaldistances.

Lemma 5: If and are two -dimensional subspacesof , and and are theprincipal vectorscorresponding to and , respectively, then

a) and form orthonormal bases forand , respectively, and

b) for .

Proof:

a) By definition, each principal vector has unit norm, and wehave for . By exchanging the roleof and , we also have for .Therefore, we have for .

b) For any given , define

By definition

(91)

where is the projection of on .Therefore, we have

(92)

for , which implies

(93)

for . Similarly

(94)

where . Therefore, we have

(95)

for , which implies

for . Therefore, we have for .

Now, using and as bases for and, by definition of theEuclidean distancebetween subspaces,

we have

(97)

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On design criteria and construction of noncoherent space ...mohammad/papers/IT2003.pdf · the information about the statistics of the fading [12]. It can be easily shown that, e.g.,