ieee transactions on communications, vol. 55, no. 4, …

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 4, APRIL 2007 703

Coded Unitary Space–Time Modulation WithIterative Decoding: Error Performance and Mapping

DesignNghi H. Tran, Student Member, IEEE, Ha H. Nguyen, Senior Member, IEEE, and Tho Le-Ngoc, Fellow, IEEE

Abstract—This paper studies the bit error probability of codedunitary space–time modulation with iterative decoding where nei-ther the transmitter nor the receiver knows the channel fading co-efficients. The tight error bound with respect to the asymptotic per-formance is first analytically derived for any given unitary con-stellation and mapping rule. Design criteria regarding the choiceof unitary constellation and mapping are then established. Fur-thermore, using the unitary constellation obtained from orthog-onal design with quadrature phase-shift keying (QPSK or 4-PSK)and 8-PSK, two different mapping rules are proposed. The firstmapping rule gives the most suitable mapping for systems that donot implement iterative processing, which is similar to a Gray map-ping in coherent channels. The second mapping rule yields the bestmapping for systems with iterative decoding. In particular, analyt-ical and simulation results show that with the proposed mappingsof the unitary constellations obtained from orthogonal designs, theasymptotic error performance of the iterative systems can closelyapproach a lower bound which is applicable to any unitary con-stellation and mapping.

Index Terms—Bit-interleaved coded modulation (BICM), errorbound, error performance, iterative decoding, signal mapping,space–time modulation, unitary constellation.

I. INTRODUCTION

RECENTLY, major research efforts have been made to mul-tiple-input multiple-output (MIMO) systems that employ

multiple antennas at both the transmitter and the receiver. Thisis due to the fact that MIMO systems can provide a significantimprovement for a wireless communications link, with respectto both the reliability of the link and the data rate, especiallyin a Rayleigh fading environment. Based on the assumption ofperfect channel state information (CSI) at the receiver (but notat the transmitter), the theoretical framework in [1] shows thatMIMO systems can achieve a much higher channel capacity

Paper approved by H. Jafarkhani, the Editor for Wireless Communicationsof the IEEE Communications Society. Manuscript received October 15, 2005;revised April 4, 2006 and July 5, 2006. This work was supported in part byDiscovery Grants from the Natural Sciences and Engineering Research Councilof Canada (NSERC). The work of the first author was supported in part by theUniversity of Saskatchewan under the Graduate Scholarship, and in part by TR-Labs Saskatoon under a Fellowship. This paper was presented in part at theIEEE Wireless Communications and Networking Conference, Las Vegas, NV,April 2006.

N. H. Tran and H. H. Nguyen are with the Department of Electrical Engi-neering, University of Saskatchewan, Saskatoon, SK S7N 5A9, Canada (e-mail:[email protected]; [email protected].

T. Le-Ngoc is with the Department of Electrical and Computer Engineering,McGill University, Montreal, QC H3A 2A7, Canada (e-mail: [email protected]).

Digital Object Identifier 10.1109/TCOMM.2007.894092

compared with that of single-antenna counterparts. Assumingperfect information of fading coefficients at the receiver, twospace–time coding methods, namely, space–time trellis codesand space–time block codes, were introduced in [2], [3] to ob-tain the diversity gains in both time and spatial domains. Sincethen, a considerable amount of work has studied both uncodedand coded MIMO systems under this assumption.

Unfortunately, due to the rapidly changing conditions infading environments, perfect knowledge of the fading coef-ficients might not be available at the receiver. In [4], a noveltreatment to compute the capacity of the wireless channelwhere neither the transmitter nor the receiver knows the fadingcoefficients has been performed. It was shown in [4] and [5]that the class of unitary space–time constellations is the mostsuitable constellation with respect to the channel capacity forthis type of channels. Furthermore, by evaluating the asymp-totic union bound (AUB), it was pointed out in [6] that theuse of unitary signal constellations results in the minimizationof the AUB. Various unitary constellation designs have beenpresented [7]–[10]. The design criterion used in these studieswas mainly based on the pairwise error probability proposedin [5] for uncoded systems, which is related to the complexGrasmannian space as opposed to the Euclidean space in thecase of coherent channels.

In order to achieve the channel capacity, it is widely acknowl-edged that the use of channel coding should be incorporatedin the systems. To the best of our knowledge, this combina-tion was first considered in [11] where a turbo code is concate-nated with the unitary signal constellation given in [7]. Sim-ulation results in [11] indicate that the error performance canbe significantly improved as compared with the uncoded sys-tems. More recently, effects of signal mapping on the error per-formance of coded MIMO systems with bit-interleaved codedmodulation and iterative decoding (BICM-ID) were studied in[12]. However, no analytical evaluation was provided, and onlya few signal mappings obtained by computer search were in-vestigated. A similar work considering trellis-coded modulation(TCM) and unitary constellation was also investigated in [10].By a clever construction of the unitary constellation based onorthogonal design, the asymptotic performance is analyticallyquantified in [10]. The design in [10] is limited to a number oftransmit antennas because of the required orthogonality.The number of block intervals of the constellation under consid-eration is .

Motivated from the above observations and discussions, thispaper analytically evaluates the asymptotic bit error probability

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704 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 4, APRIL 2007

Fig. 1. Block diagram of a unitary space–time coded modulation with iterativedecoding.

(BEP) for unitary bit-interleaved space–time coded modulationwith iterative decoding. The evaluation is based on the assump-tion of ideal feedback from the channel decoder to the demodu-lator and can be applied for any given unitary constellation. Sim-ulation results of the bit-error-rate (BER) performance confirmsthat the derived asymptotic BEP is very tight at high signal-to-noise ratio (SNR). Based on the asymptotic performance, designcriteria for the choice of unitary constellation and mapping arealso established. Using the class of unitary constellations ob-tained from orthogonal designs with 4-PSK and 8-PSK signalsets [10], the optimal mappings with respect to the asymptoticperformances of systems with and without iterative processingare then introduced. It is shown that the use of the proposedmappings for iterative systems results in error performance thatis very close to the lower bound of the asymptotic performancefor any unitary constellation and mapping.

II. SYSTEM MODEL

The general block diagram of a unitary space–time codedmodulation system with iterative decoding is shown in Fig. 1.The information sequence is first encoded into a coded se-quence using a rate- convolutional code. The coded se-quence is then interleaved by a bit-wise interleaver to becomethe interleaved sequence . Based on the mapping rule , eachblock of coded bits is mapped to one of matrices ,

. Here, each matrix , , is aunitary matrix that satisfies the condition ,

, where denotes conjugate transpose and isthe identity matrix of size . Note that is the numberof transmit antennas, is the number of rows of a space–timecode matrix, and . The set is simplyreferred to as the constellation hereafter. The group ofcoded bits carried by each signal point is called the labelof . Furthermore, the Hamming weight of a label is definedas the number of bits “1” in the label, whereas the label Ham-ming distance between two labels is the number of bit positionsthat the two labels differ. Assume that the receiver is equippedwith antennas. Following the same notations in [4], at thetime index , the matrix of received signals corre-sponding to the transmitted signal can be written as

(1)

where is one of unitary matrices in the constellation .The matrix is an matrix of the fading coefficientswhose entries are .1 The matrix is a ma-trix representing additive white Gaussian noise (AWGN) whoseentries are also . The normalization factorin (1) ensures that the average SNR at each receive antenna is

, independent of . The spectral efficiency of the system istherefore information bits per channel use.

The receiver of the system includes the maximum a posterioriprobability (MAP) soft-output space–time demodulator and thesoft-input soft-output (SISO) channel decoder. The detailed al-gorithm for the demodulator is described in [11], while the SISOchannel decoder uses the MAP algorithm in [13]. Similar to thedecoding of Turbo codes, the space–time demodulator and thechannel decoder exchange the extrinsic information of the codedbits and through an iterative process. Afterbeing interleaved, and become the a priori in-formation and at the inputs of the SISO decoderand the demodulator, respectively, as can be seen in Fig. 1. Thetotal a posteriori probabilities of the information bits can becomputed to make hard decisions at the output of the decoderafter each iteration.

III. PERFORMANCE EVALUATION

The union bound of the BEP for a rate- convolutionallycoded unitary space–time modulation can be written in a generalform as [14]

(2)

In (2), is the total information weight of all of the error eventsat Hamming distance and is the free Hamming distanceof the code. The function is the average pairwiseerror probability, which depends on the Hamming distance ,the unitary constellation , and the mapping rule . In the fol-lowing, the function is computed from the pairwiseerror probability (PEP) of two codewords.

Let and denote the input and decoded sequences, respec-tively, with Hamming distance between them. Without loss ofgenerality, assume that and differ in the first consecutivebits. Furthermore, with the use of a sufficiently long interleaver,it can be assumed that each of these different bits appears ina different block of bits before mapping to the constella-tion symbols. Therefore, the binary sequences and corre-spond to codewords and , defined as sequences of unitarysignal points scaled by , namely and

. Here, and , , belong tothe constellation . Assuming the channel is memoryless andfollowing similar analysis as in [5], the PEP between two code-words and can be computed as follows:

(3)

1Here, CN (0; 1) denotes a circularly symmetric complex Gaussian randomvariable with variance 1/2 per dimension.

TRAN et al.: CODED UNITARY SPACE–TIME MODULATION WITH ITERATIVE DECODING 705

Applying the closed-form analytical expression presented in [5],the above PEP can be computed based on a one-dimensional(1-D) integral as

(4)

where

(5)

In (5), , with , is the th singular

value of the matrix . Since is real-valued, by taking the real-part of (4) gives [5]

(6)

Owing to the success of iterative decoding steps as normallyseen in BICM-ID systems, one is most interested in the asymp-totic performance to which the iterations converge [15], [16].Such asymptotic performance can be obtained by assuming thatthe iterations between the SISO decoder and the demodulatorwork perfectly or, equivalently, one has perfect a priori infor-mation of the coded bits fed back to the demodulator. Considerthe th bit position . With the ideal knowledge of theother coded bits carried by the transmitted symbol , it can beseen that the error happens only when the labels of anddiffer at the th bit position [15], [16]. Furthermore, observe that

are i.i.d. random variables. Then, instead of averaging overall pairs of codewords and , the union bound onfrom [14] can be computed by averaging over the constellation

as follows:

(7)

where the expectation in (7) denotes the average over allsignal points , , as well as all the bit positions,

, i.e.,

(8)

The quantity in (7) and (8) is defined for each pairas follows:

(9)

where is the th singular value of. Here, with a slight abuse of notation,

returns the index of the unitary signal point in whoselabel differs at only the th position compared with the labelof . Note that the expression of is similar to thatof in (5). The major difference, however, is thatis associated with a specific unitary signal point and a

specific label position , rather than the time index. The aboveanalysis thus shows that the function can be easilycalculated in determining the asymptotic performance.

To give an insight on how to design good mappings, the sim-pler Chernoff upper bound in [5] is applied to in (6)to yield

(10)

Similar to the analysis presented earlier, the functioncan then be approximated as

(11)

where

(12)Obviously, the parameter characterizes the effect of theunitary constellation and the mapping rule to the asymptoticperformance of a unitary space–time coded modulation with it-erative decoding. More specifically, the smaller this parameteris, the better the asymptotic performance becomes.

The design criterion can be made simpler and more mean-ingful with the observation that the Chernoff bound dependsdominantly on the large SNR as suggested in [7]. At high SNR(i.e., ), using the approximation when

in (12) and cancelling common factors, the design pa-rameter for a given number of receive antennas, , is then mod-ified to

(13)Observe that the design criterion under consideration is far dif-ferent from that of a coherent system, which is mainly relatedto the Euclidean distances. The parameter does not de-pend on SNR, and it needs to be minimized.

As far as the asymptotic performance is concerned, it followsfrom (13) that, by using the constellation and mapping such thatall of the singular values , the minimum valueof and, consequently, the lowest asymptotic errorperformance is attained. This lowest asymptotic performancetherefore can serve as the lower bound of the asymptotic perfor-mance for any unitary constellation and mapping scheme. As anexample, the minimum value of can be achieved byusing the unitary constellation , where all signal points col-lapse into only two distinct unitary matrices whose all columnsare orthogonal. The mapping rule is implemented such thatone unitary matrix corresponds to signal points whose la-bels have even Hamming weights. The other unitary matrix cor-responds to the remaining signal points whose labels haveodd Hamming weights. Unfortunately, despite the potential ofgiving the best asymptotic error performance, the use of this


constellation and mapping will result in a very poor perfor-mance at any SNR. This is because the ambiguity in this constel-lation and mapping leads to a very poor error performance afterthe first iteration and, as a consequence, the potential asymptoticerror performance can never be achieved regardless of how highthe SNR can be. This issue is elaborated on in detail when theperformance of the system after the first iteration is considerednext. Nevertheless, the above ambiguity does not happen withcommon and well-designed unitary constellations, such as theones obtained from orthogonal design adopted in this paper.

The above evaluation concentrates only on the asymptoticperformance with the perfect a priori information of coded bitsat the input of the demodulator. For systems implementing itera-tive decoding and demodulation, it is also of interest to study theperformance after the first iteration (i.e., without feedback fromthe decoder to the demodulator). Similar to [14], with no a prioriinformation about the coded bits at the input of the demodulator,the computation of the PEP in (3) is now based onthe assumption that and are the two “nearest” neigh-bors, as opposed to the assumption that and are the twosignal points whose labels differ in only 1 bit made earlier. Thisis a reasonable assumption, since, at high SNR, the decodingerror likely happens between the two “nearest” neighbors. Be-cause the study in [14] considers coherent systems, the distancesbetween signal points are simply the Euclidean distances. Inthe case of noncoherent systems, the analogous distance is thechordal distance [5], [7]. More specifically, the chordal distancebetween the two signal points and in is given as[5], [7]

(14)

where are the singular values of .For each signal , let denote the set of all of thenearest signal points, in terms of the chordal distance, to ,whose labels differ at the th position compared with the labelof . Note that the set generally contains more than onesignal point of . This is because the label of a signal point in

might differ in position other than the th position comparedwith the label of . Following the same steps presented be-fore, one can obtain the following parameter that characterizesthe performance of the system without iterative decoding at highSNR:

(15)

where

(16)

In summary, for a given constellation , the sensible de-sign criterion for the systems without iterative processing is tochoose the mapping which results in the smallest value of

. Furthermore, comparing the two parameters in(15) and in (13) gives the coding gain provided by the

iterative processing. For mappings that yield the same value ofthese two parameters, there is no benefit to run the iteration be-tween the decoder and the demodulator. Also, for the ambiguousconstellation and mapping discussed earlier, it can be verifiedthat . This implies a very poor performance afterthe first iteration and the asymptotic performance promised bythe ambiguous constellation/mapping cannot be practically ap-proached.

Given a unitary constellation, brute-force search can be car-ried out to find the optimal mappings based on the establisheddistance criteria. However, when the size of the constellationis large, this exhaustive search becomes infeasible due to thecomplexity. The binary switching algorithm (BSA) proposed in[17] can then be applied. Application of BSA to find good signalmappings was recently investigated in [12] for the specific uni-tary constellation proposed in [7]. However, the BSA only giveslocally optimal mappings in general. Furthermore, it is observedthat almost all good unitary constellations presented so far in theliterature are based on numerical optimization. Due to the lackof the structure of the unitary constellations, this makes the map-ping design problem intractable in general.

From the above discussion, a simple yet effective construc-tion of unitary constellations based on orthogonal designsproposed in [10] shall be adopted to investigate the mappingproblem. More specifically, by investigating properties of thesingular values for the unitary constellations obtained fromorthogonal designs with 4-PSK and 8-PSK signal sets [10],the optimal and good mappings for both systems, with andwithout iterative processing, are introduced. Interestingly, theresults obtained here are inherited from the mapping designsproposed for multidimensional constellations of BICM-ID incoherent channels [16], [18]. Furthermore, it is shown that, forsystems with iterative processing, using the proposed mappingsmakes it possible to closely approach the lower bound onthe asymptotic performance of any unitary constellation andmapping mentioned earlier.

IV. UNITARY CONSTELLATIONS FROM ORTHOGONAL DESIGNS

AND PROPOSED MAPPINGS

Here, the construction of unitary constellations from orthog-onal designs and conventional -PSK signal set proposed in[10] is adopted to investigate the mapping problem. Moreover,the cases of and are of particular interest.

A. Unitary Constellations From Orthogonal Designs

An important class of unitary constellations corresponds to, which has been studied intensively in the literature.

For example, reference [19] shows that, given a coherence time, the maximum number of degrees of freedom is achieved by

using transmit antennas. Furthermore, it is demonstratedin [20] that the condition guarantees full antennadiversity. The differential unitary space–time code proposed in[20] and [21] is also a special case of the unitary space–timestructure with . Other research on unitary space–timecodes with can be found in [22] and references therein.

The orthogonal construction introduced in [10] also concen-trates on this case, with a very simple but effective design of uni-tary constellations. When , given the number of signal


points , where is a positive integer, the constructionprocedure is as follows. For an integer number , ,define and . Then, the 4

2 unitary signal point is [10]

(17)Observe that the elements and

belong to an -PSK constellation. Asshown in [10], for two unitary signal points andthat correspond to two pairs and , the two singularvalues of are equal and given by

(18)Hereafter, references to a signal point using and the pairsof integers , where and are related to as defined ear-lier, are interchangeable. The extensions for are alsopresented in [10] but are only possible for anddue to the limitation of orthogonal design.

With orthogonal design, all , , are equal.Thus, the index can be omitted and the chordal distance be-tween and in (14) is rewritten as

(19)

Using (19), the design parameter in (13) can be rewritten asfollows:

(20)

where is the chordal distance between andwhose labels differ at only position . On the other

hand, the parameter in (15) can be computed similarlyby averaging over , which can now be given in a compactform as

(21)

Let be the minimum chordal distance between any twodifferent signal points in . Similar to the constellations inthe Euclidean space, the mapping rule of a constellationis called a Gray mapping if the labels of two signal points atchordal distance always differ in only 1 b.

B. Gray Mapping

For simplicity, we restrict our attention to the case of. The results for higher values of are briefly presented in

Appendix I.First, let denote the conventional Gray mapping rule for-PSK in the Euclidean signal space. Since the th signal point

in -PSK is , the mapping essentially re-lates each integer with a label of bits. Consider a uni-tary signal point . From (18) and (19), it can be

seen that there are four and only four nearest neighbors ofat the distance in , which are and

. Thus, one has the following theorem.Theorem 1: A Gray mapping of a unitary constellation can

be constructed by having the first and the last bits of thelabel of each signal point in as the conventional Graylabels of the th and the th signal points of -PSK, respec-tively.

Proof: The proof follows directly from the fact that thelabel of the th signal point in -PSK differsin only 1 b compared with the label of the th signal point.

As an example, Gray mapping for the case andis as follows:

.In the following, it is demonstrated that the above Gray map-

ping is most suitable for systems that do not implement iterativedecoding, but it performs very poor for systems with iterativeprocessing. To this end, the parameters and areevaluated for two separate cases of 4-PSK and 8-PSK.

1) 4-PSK: First, consider the parameter in (15). Forany , there exists four nearest neighbors at distance

. Therefore, each of these four nearest neighbors must bein one of sets . One then obtains the followinglower bound on for any mapping rule:

(22)

Furthermore, for a Gray mapping, it can be easily verifiedthat, for each , contains one and only one nearestsignal point at the distance to . It follows that

for a Gray mapping, which achieves the abovelower bound. Hence, it can be concluded that the use of a Graymapping results in the best performance for systems withoutiterative processing in the sense that it minimizes .

To see the effect of iterative processing with a Gray mapping,it is convenient to evaluate the parameter in (13). Dueto the properties of the nearest neighbors, it can be shown that

for a Gray mapping. This meansthat, with a Gray mapping, even the perfect a priori informa-tion from the decoder to the demodulator is not useful. There-fore, it is unnecessary to run the iterations when a Gray map-ping is employed. In other words, there should be no differencebetween the error performances with and without iterative pro-cessing. This fact is later confirmed by the analysis based on theextrinsic information transfer (EXIT) chart [23]. Furthermore,it can be verified that with a Gray mapping, isachieved when all the chordal distances in (20)equal the minimum chordal distance . This also results inthe maximum value of over all mapping rules for the or-thogonal design based on 4-PSK. This implies that a Gray map-ping provides the worst asymptotic performance for the systemswith iterative decoding.

2) 8-PSK: When orthogonal design with 8-PSK is employed,similar observations can be made due to the properties of thelabels of the nearest signal points with a Gray mapping. First,


since there are four nearest neighbors of any at the min-imum chordal distance and each carries

bits, one has the following bound for and :

(23)

By comparing the value of with a Gray mapping and theabove bound, it is observed that is only slightly greaterthan the lower bound for various values of . For example,with , , whereas the lower bound is124.3182. Therefore, it is expected that a Gray mapping resultsin almost the best performance for systems without iterative de-coding.

To see how good Gray mapping is with iterative processing,the comparison between and is made. It is alsoobserved that the difference is negligible for a given . Forexample, with , one obtains and

, which indicates that there is only a slightimprovement by performing iterative processing. Furthermore,by examining the distance profile of , it can be seen that forany signal point , there are only four distinct signal pointsin at the minimum chordal distance and twoother signal points in at the second minimum chordal distance

to . This fact givesfor any mapping rule when . Clearly, the upper boundon is only slightly bigger than forGray mapping, indicating that a Gray mapping also performsvery poorly in iterative systems in terms of the asymptotic errorperformance.

It should be mentioned that the effects of a Gray mapping onthe error performance of noncoherent systems with and withoutiterative processing as discussed above are very similar to theobservations concerning BICM systems over coherent channelsin [15].

C. Mappings for Systems With Iterative Processing

This subsection introduces novel mappings that result in verysmall values of for systems that implement iterativedemodulation and decoding at the receiver. More specifically,optimal mappings for unitary constellations obtained from or-thogonal design and 4-PSK are proposed. For orthogonal designbased on 8-PSK, a novel mapping with good performance athigh SNR is provided. It is conjectured that this novel mappingis the globally optimal mapping. Also, for simplicity, attentionis paid only to the case . Extension to higher values of

is briefly discussed in Appendix I.1) 4-PSK: When and , by studying the dis-

tance profile of , it is not hard to recognize that the structure ofis very similar to the structure of a four-dimensional (4-D) hy-

percube [16], although the distances considered for the 4-D hy-percube in [16] are the Euclidean distances. More specifically,the following facts about chordal distances can be establishedfor any signal point .

• Fact 1: There is only one signal point whose chordal dis-tance to is , corresponding to the singular value

. It is itself.• Fact 2: There are four signal points whose chordal dis-

tances to are equal to , corresponding

to the singular value . It can be verified thatis the minimum chordal distance of .

• Fact 3: There are six signal points whose chordal distancesto are , corresponding to the singular value

.• Fact 4: There are four signal points whose chordal dis-

tances to equal , corresponding to thesingular value .

• Fact 5: There is only one signal point whose chordal dis-tance to is , corresponding to the singularvalue .

Note that the chordal distances , , are purposelyindexed such that .

Basically, the objective to obtain the mapping with a smallvalue of can be fulfilled by labelling the signal points in

such that two signal points at a larger chordal distance havea smaller label Hamming distance. Therefore, it follows thatand , the two largest chordal distances of , are of interest.With , rewrite the parameter in (20) as follows:

(24)

where the parameter is associated with each signalpoint , computed as

(25)

With the two largest chordal distances and , the followinglower bound on when follows directly fromFacts 4 and 5:

(26)

Therefore, one obtains the following lower bound on :

(27)

Clearly, the lower bound in (27) is independent of the mapping.If it exists, the mapping that yields this lower bound is the bestone in terms of the asymptotic performance. Furthermore, it iseasy to verify that this optimal mapping is any mapping thatsatisfies the following condition.

Condition 1: For any signal point in , a signal pointwith chordal distance to has label Hamming distance1 to . Furthermore, there are three other signal points withchordal distance and label Hamming distance 1 to .

As mentioned before, there is a strong similarity between thedistance profiles of the unitary constellation and the hyper-cube constellation considered in [16]. In fact, the mapping thatsatisfies Condition 1 can easily be obtained by modifying the


TABLE ICHORDAL DISTANCE PROFILE OF THE PROPOSED OPTIMAL MAPPING FOR THE UNITARY CONSTELLATION CONSTRUCTED FROM THE ORTHOGONAL DESIGN

AND 4-PSK

Note: The binary number under each signal (in the first row) denotes the proposed mapping. As an example, (4) = [0; 3] is labeled with 1010, (10) = [2; 1]

is labeled with 0010. These two labels differ in only the first bit and the chordal distance between these signals isp2.

best mapping for the 4-D hypercube constellation in [16]. Theprocedure to obtain the best mapping is stated as follows.

Step 1) Consider . Create the constellation withfour pointsand the corresponding labels

. Observe thatthis is Gray mapping with .

Step 2) Construct the constellation with all signal points, , . Each signal

point is then labeled with 3 b , whereis the label of in . This step results

in the mapping rule and it is also a Gray mappingwith .

Step 3) Modify the labeling rule to obtain the labelingrule as follows. Each elementcorresponds to element , where

if is even; otherwise, .As the results of labeling rule , each signal pointin carries 4 b.

Step 4) From each element , construct a signalpoint such that and

to obtain the constellation . It canbe verified that the distance between andis . Each label is thenmodified to , where .

Step 5) The constellation and the best mapping are ob-tained by combining and and their labelsand .

The final results are shown in Table I. It can be seen that,for any signal point , the signal point whose label differsin only 1 b at the first position is at the distance to

. The other three signal points whose labels differ in only1 b at the second, third, and fourth positions are at the distance

to .Observe that the value of in (27) for the optimal

mapping is much smaller than that of a Gray mapping for a

given . For example, with , for theoptimal mapping, which is very close to the minimum value of1 and significantly smaller than the value of fora Gray mapping. This suggests that a significant coding gaincan be achieved by using the optimal mapping over a Graymapping in terms of the asymptotic performance. Regardingthe error performance after the first iteration, although beingoptimal in terms of the asymptotic performance with iterativeprocessing, the proposed mapping performs poorer than Graymapping after the first iteration. This is due to the large valueof (e.g., for ) produced by theproposed mapping.

The optimal mapping for transmit antennas is pre-sented in Appendix I.

2) 8-PSK: For the case of the orthogonal design with 8-PSKconstellation, the globally optimal mapping in terms of mini-mizing the asymptotic performance is unknown. Instead, pro-posed in the following is a mapping that can approach veryclosely a lower bound on the asymptotic performance. The con-struction of this mapping is also strongly related to the con-struction of a novel mapping of the multidimensional 8-PSKconstellation recently proposed in [18] for coherent BICM-IDsystems. Similar to [18], the basic idea behind the constructionof this mapping comes from the simple fact that an 8-PSK canbe decomposed into two 4-PSK signal sets. Thus, there is alsoa strong relationship between the orthogonal design based on8-PSK and the one based on 4-PSK.

Upon examining the distance profile of the constellation, itcan be observed that the optimal mapping, if it exists, is anymapping that satisfies the following condition.

Condition 2: For any signal point in , there is onesignal point with chordal distance and label Ham-ming distance 1 to . Furthermore, there are four other signalpoints with chordal distance and label Ham-ming distance 1 to and one signal point with chordal dis-

tance and label Hamming distance 1 to .


TABLE IIPROPOSED MAPPING OF THE UNITARY CONSTELLATION OBTAINED FROM THE ORTHOGONAL DESIGN AND 8-PSK, WITH FOUR SUBCONSTELLATIONS �� ,

�� , �� , AND ��

Here, the chordal distances , and are the three largestdistances of . Unfortunately, it is not difficult to show that theredoes not exist mapping that satisfies Condition 2. Nevertheless,we shall propose a novel mapping that satisfies a weaker con-dition, stated as Condition 3 below. Also, for convenience inevaluating the proposed mapping, the phrase “hypothetical map-ping” is used to refer to a mapping that satisfies Condition 2.

Condition 3: For any signal point in , there is onesignal point with chordal distance and label Ham-ming distance 1 to . Furthermore, there are two other signalpoints with chordal distance and labelHamming distance 1 to and three signal points with chordaldistance and label Hamming distance 1 to .

The construction of the mapping that satisfies Condition 3 isinherited from the one in [18] and based on the optimal mappingfor the orthogonal design with 4-PSK presented earlier. Specif-ically, for any signal point , associate it with 2 b ,where and . The correspondingdecimal number of is denoted as , where .Clearly, for each , one finds 16 signal points in the sub-constellation , where the chordal distance profile of eachsubconstellation is exactly the same with that of the unitary con-stellation obtained from the orthogonal design and 4-PSK. Withthis observation, the detailed steps for the construction of theproposed mapping are given below.

Step 1) For any signal point , label the first 2 b by, where and .

Step 2) Consider . Since the chordal distance profileof is the same with the orthogonal design and4-PSK, label the last 4 b of any signal point inby the optimal mapping of the orthogonal designand 4-PSK.As a result, all signal points in are labeled and theproperties for the last 4 b are related to two distances

and .Step 3) For each signal point whose label is

, associate it with the signalpoint , where

and . Moreover, label thesignal with .This step ensures that, as far as the last 4 b in thelabels of all the signal points in are concerned,the properties of the optimal mapping obtained forthe orthogonal design and 4-PSK also hold here.Furthermore, any two signal points in two differentsubconstellations whose labels differ in only 1 b atthe first or second positions are always at the chordaldistance .

The final mapping is shown in Table II, which can be verifiedto satisfy Condition 3. By the symmetry properties of thisproposed mapping, it is also easy to verify that this mappingis immune to the BSA.2 This means that, if the BSA startswith the proposed mapping, it can never find a better one. Fur-thermore, simple computation shows that, when , thevalues of for the proposed and hypothetical mappingsare 1.4436 and 1.1716, respectively, which are quite close.Compared with of the Gray mapping,the parameter of the proposed mapping is obviouslymuch smaller, which suggests that a significant coding gainin terms of the asymptotic performance can be achieved withthe proposed mapping. Analytical and simulation results in thenext section also show that the performance of the proposedmapping can closely approach a lower bound on the asymptoticperformance of any constellation and mapping that correspondsto and greatly outperforms the one with a Graymapping. However, with the large value of (e.g.,

when ), this mapping provides avery poor performance after the first iteration.

D. Discussion on the Convergence Behavior of the ProposedMappings

Until now, it has been demonstrated analytically that the useof the proposed mappings for iterative systems will result ina significant coding gain compared with a Gray mapping, as

2The proof is omitted here for the brevity of presentation.


far as the asymptotic performance is concerned. To achievea perfect convergence (i.e., the BER performances after anumber of iterations can converge to the asymptotic one), it ispointed out in [24] that the BER measured at the input of theSISO decoder during the first iteration must be under a giventhreshold. This threshold varies for different error-correctingcodes. In general, the powerful codes such as turbo-like codesrequire lower thresholds than the standard convolutional codes[24]. Due to their poor performances after the first iteration,the proposed mappings for the iterative systems need a higherSNR to achieve the lower BER thresholds of the more powerfulcodes to make iteration convergence happen. At a very highSNR, the asymptotic coding gains of the proposed mappingsover a Gray mapping can still be observed. However, thishappens at very low BER levels, which is not of practicalinterest. On the other hand, using a Gray mapping can achievethe practical BER level of interest (say to ) at a lowerSNR because of its superior performance after the first iteration.The above discussion suggests that a Gray mapping is the mostsuitable mapping when powerful codes (e.g., Turbo-like codes)are employed. This fact has been observed for systems overcoherent channels using Turbo codes in [24] and low-densityparity-check (LDPC) codes in [25].

V. ANALYTICAL AND SIMULATION RESULTS

Here, we provide analytical and simulation results to confirmthe analysis carried out in the previous sections. For given con-stellation and mapping rule , the error bound computed from(2) and (7) with the first 20 Hamming distances of convolutionalcodes is provided to show the tightness of the bound. The lowerbound applicable for any unitary constellation and mapping isobtained by simply setting and all the singularvalues in (13) to zero. A brief comparison between systemsemploying the orthogonal designs and the proposed mappingsand systems using the systematic design is also made. Finally,convergence analysis of the proposed mappings is provided bymeans of the extrinsic information transfer (EXIT) charts.

A. Orthogonal Design With Proposed Mappings

First, start with the orthogonal design based on 4-PSK andtransmit antennas. A rate-1/2, four-state convolutional

code with generator matrix is used, which resultsin the spectral efficiency of 0.5 b per channel use. A randominterleaver of length 12 000 coded bits is implemented. Fig. 2presents the BER performances of unitary space–time codedsystems with the two proposed mappings. There is only one re-ceive antenna, i.e., . For convenience, the SNR is com-puted as (dB). More specifically, shown in Fig. 2 are the BERof the iterative system with the proposed optimal mapping afterone, four, and ten iterations. In the case of a Gray mapping, theBER is only provided with one iteration since it was observedthat there is no improvement from the second iteration. This isexpected from the analysis made in the previous section and itconfirms that iterative processing is useless for a Gray mapping.The error bounds are also plotted in Fig. 2 to illustrate the tight-ness of the bounds. It can be seen that, for the proposed optimal

Fig. 2. Performance of the unitary coded systems employing the orthogonaldesign based on 4-PSK with different mappings, N = 1.

mapping, the BER converges to the error bound at a practicallevel of . Thus, the bound is useful to predict the error per-formance at high SNR. In the case of a Gray mapping, the boundis seen to underestimate the simulation result. This is only dueto the poor performance of the Gray mapping over the range ofSNR shown in Fig. 2. Though not shown here, it was observedthat the bound is tight at higher SNR where the BER level ofaround is reached.

It can also be seen that, with the use of a very simple convo-lutional code, the system employing the proposed optimal map-ping and iterative processing achieves a significant coding gaincompared to that using Gray mapping, even though it offers avery poor performance after the first iteration. To gauge the per-formance improvement by the proposed optimal mapping, thelower bound on the asymptotic performance of any unitary con-stellation and mapping is also provided in Fig. 2. Observe thatthere is only a slight difference between the two systems. In par-ticular, the performance gap is only about 0.8 dB over the BERrange of to . Though not explicitly shown here, ex-amining the error bounds plotted over a wider range of SNRreveals that compared to the Gray mapping, the use of the pro-posed optimal mapping results in the coding gain of 4.8 dB atthe BER level of . However, it should be pointed out that,when a more powerful code (e.g., a Turbo-like code) is selected,Gray mapping is expected to be the most suitable mapping asdiscussed earlier.

Similar results can be obtained for the case of receiveantennas, which are presented in Fig. 3. Observe that, when

, the BER also converges to the bound but at a lowerlevel of as compared with the case of . Clearly,the bound is still very useful to predict the BER for the systems.The advantage of using the proposed optimal mapping is alsoobvious, where there is only about a 0.8-dB gap between theperformance of the proposed optimal mapping and the generallower bound on the asymptotic performance at the BER level of

. Based on the error bounds plotted over a wider range ofSNR, it is also observed that a coding gain of 4.5 dB at a BER



level of can be obtained with the optimal mapping com-pared with a Gray mapping.

Next, consider the case of the orthogonal design based on8-PSK and . Simulations were carried out for systemswith a rate-2/3, four-state convolutional code whose generatormatrices are and . This gives aspectral efficiency of 1 b per channel use. A random interleaverof length 18 000 coded bits was selected. Fig. 4 presents the BERperformance of the systems employing the Gray mapping and themapping proposed in Table II. The number of receive antennasis . The error bounds are also plotted for comparison.Similar to the case of 4-PSK, only the error performance withone iteration is provided for the Gray mapping, since there areonly very slight improvements from the next iterations. Thisagain confirms that iterative demodulation and decoding isuseless for systems with a Gray mapping. Also, in this figure, theperformance of the proposed mapping is compared against theasymptotic performances of the hypothetical mapping (whichsatisfies Condition 2) and the general lower bound on the asymp-totic performance of any constellation and mapping. It can beseen that the performance gaps at the BER level of are only0.4 and 0.8 dB, respectively. It can also be seen that the BERperformances of the proposed mapping greatly outperform thatof the Gray mapping. Based on the error bounds, it is calculatedthat the coding gain is about 8.5 dB at the BER level of .

B. Comparison of Orthogonal and Systematic Designs

As mentioned before, the mapping problem of the unitaryconstellation obtained from the systematic design has beenstudied in [12] using the BSA. More specifically, the study in[12] only concentrates on the specific unitary constellation with

and introduced in [7]. It is observed in [12]that performance difference due to the use of different mappingrules is not much.

The basic idea of the systematic design is to be able to con-struct the whole unitary constellation by starting from thefirst signal in the constellation, namely . The other signalsare then obtained systematically by successively rotating


in a high-dimensional complex space. This construction can beviewed algebraically, in which the constellation is created bymapping codewords of a linear block code into complex signalmatrices [7]. The construction procedure can be summarized inthe following.

Let be the ring of integers modulo- . Let be a linearblock code over whose dimension is , which has a totalof codewords. The code has a generatormatrix restricted in a systematic form , wherethe elements of the matrix are also in .Using the mapping with ,the set of diagonal matrices with entries

, , is then created. The matricesand the first signal point are then combined to

form the remaining signal points in . For example, withthe th signal point in is determined as ,where the starting signal can be chosen as the columns ofa DFT matrix whose all elements have the same magnitude of

[7].Clearly, the structure of depends largely on the generator

matrix . The sensible design criterion is to maximize the min-imum chordal distance of [7]. As an illustrative ex-ample, consider the case and . Using com-puter search, the unitary constellation with ,and is found. Then, by applying the BSAwith different trials, the following mapping is obtained:

. This BSA mappingyields . Note that, our proposed optimal map-ping for the orthogonal design based on 4-PSK yields

. Although BSA generally gives only a locally optimalmapping, it is expected that the mapping obtained with trialsof BSA is very close to the globally optimal mapping.

Fig. 5 compares the error performance of the two systemsthat employ the orthogonal design/optimal mapping and sys-tematic design/BSA mapping when . The channel code


Fig. 5. Performance comparison between the orthogonal design/optimal map-ping and the systematic design/BSA mapping, N = 1.

is the same rate-1/2 convolutional code mentioned earlier. Morespecifically, plotted in Fig. 5 are the BER performances withone, four, and ten iterations for each system. The error boundsare also provided to show the tightness of the bounds. It can beclearly seen that the BER converges to the error bounds for bothsystems. As expected, the use of the orthogonal design/optimalmapping offers significant coding gains over the systematic de-sign/BSA mapping. Furthermore, it is observed that the systemusing the orthogonal design/optimal mapping is also preferredin terms of the convergence behavior. This observation is furtherconfirmed by the EXIT chart analysis in the next subsection.

C. Convergence Analysis of the Proposed Mappings WithEXIT Charts

The different mappings proposed for coded unitaryspace–time systems can also be analyzed by the histogrammethod, commonly known as the extrinsic information transfer(EXIT) chart [23]. Following the same notations in [23], let

and denote the mutual information that quantify thea priori knowledge and the extrinsic information of the codedbits at the input and output of the demodulator, respectively.Similarly, let and be the mutual information repre-senting the a priori knowledge and the extrinsic informationof the coded bits at the input and output of the SISO decoder,respectively. After being deinterleaved, the extrinsic output ofthe demodulator becomes the a priori input to the decoder, i.e.,

. Furthermore, after being interleaved, the extrinsicinformation of the decoder becomes the a priori informationfor the demodulator, i.e., .

Fig. 6 plots the EXIT charts for the system employing the or-thogonal design/optimal mappings with 4-PSK, , andat dB. The EXIT chart for the systematic design/BSAmapping is also provided for comparison. As can be seen, the it-erative demodulation and decoding is useless for the Gray map-ping as the bit-wise mutual information for the Gray mappingremains almost constant for the whole range of the a priori in-formation . This again confirms the analytical and simula-

Fig. 6. EXIT charts of the systems employing different mappings at� = 6:5 dB, N = 1 and a rate-1/2, 4-state convolutional code.

Fig. 7. EXIT charts of the systems employing the orthogonal design based on8-PSK with different mappings at � = 12 dB,N = 1, and a rate-2/3, four-stateconvolutional code.

tion results shown in the previous sections. On the other hand,there is a significant difference in terms of the bit-wise mutualinformation for the optimal mapping with no a priori informa-tion and with the perfect a priori information. Based on the de-coding trajectory, it takes about six iterations for the iterativedecoding to converge, which matches well with the BER per-formance presented earlier. Furthermore, it is interesting to ob-serve that the orthogonal design/optimal mapping outperformsthe systematic design/BSA mapping regardless of the values ofthe a priori information. Thus, with respect to both the asymp-totic performance and convergence behavior, the orthogonal de-sign/optimal mapping is clearly preferred over the systematicdesign/BSA mapping.

Similar EXIT charts are shown in Fig. 7 (at dB) forboth the Gray and proposed mappings of the orthogonal designbased on 8-PSK. Again, observe that the bit-wise mutual infor-mation for the Gray mapping is almost constant. For the pro-


TABLE IIIPROPOSED OPTIMAL MAPPING FOR THE ORTHOGONAL DESIGN WITH 4-PSK AND N = 4

Note: Each signal point is represented by a string of three integers [k; p; q], and the corresponding label � is a binary 6-tuple.

posed mapping, the bit-wise mutual information approaches 1very closely when there is a perfect feedback from the decoder.It needs about ten iterations to achieve to the asymptotic perfor-mance, which also closely matches with the simulation resultspresented before.

VI. CONCLUSION

The error performance and mapping design for noncoherentcoded unitary space–time modulation have been studied. Atight bound on the asymptotic performance was first derived.Concentrating on the unitary constellations obtained from theorthogonal designs with 4-PSK and 8-PSK signal sets, optimaland good mappings were provided for both systems, with andwithout iterative processing. The analytical derivations andevaluations were substantiated by extensive simulation re-sults. Significant performance gains offered by the orthogonaldesign/proposed mappings over the systematic design/BSAmapping for the iterative systems were clearly illustrated.

APPENDIX I

OPTIMAL MAPPING OF THE ORTHOGONAL DESIGN WITH

4-PSK AND

With , there are in total signal points of theorthogonal design based on -PSK, where . First,represent any integer , , by a set of three integers

as

(28)

where . Each signal point is then givenby [10]

(29)

where . It is observed in [10] that for any twounitary signal points and , the four singular values of

are all equal to

(30)It can be verified easily that the minimum chordal distanceof is achieved when , , and .Therefore, a Gray mapping rule can be implemented easily byassigning the conventional Gray labels of -PSK signal set tothe integers , and , as done in Section IV-B.

To come up with the optimal mapping for the iterative system,the chordal distance profile of needs to be examined. It is ob-served that there is a strong similarity of and the six-dimen-sional hypercube constellation considered in [16]. More specif-ically, the following facts about chordal distances can be estab-lished for any .

• Fact 1: There is only one signal pointwith the maximum chordal

distance to , corresponding to the singularvalue 0.

• Fact 2: There are six signal points with the secondmaximum chordal distance to , corre-sponding to the singular value .

Hence, it follows that the optimal mapping is any mapping thatsatisfies the following condition.

Condition 4: For any signal point in , a signal pointwith chordal distance to has label Hamming distance1 to . Furthermore, there are five other signal points withchordal distance and label Hamming distance 1 to .

An algorithm to construct the mapping that satisfies Condi-tion 4 is provided next. This algorithm is modified from theconstruction of the optimal mapping for presented inSection IV-C and the algorithm to obtain the optimal mappingof the six-dimensional hypercube employed in BICM-ID overthe coherent channels in [16]. The algorithm includes the fol-lowing three simple steps.


Step 1) Consider the subconstellation ,where and . Label each

by 5 b where andare the labels of the th and the th signal

points of 4-PSK with a conventional Gray mapping.It can be verified that this labeling rule of is aGray mapping.

Step 2) Modify the Gray labeling rule as follows.Each label is changed to

, where ifis even; otherwise, . The

resulting labeling rule is denoted by , where eachsignal point in carries 6 b.

Step 3) For every , construct the signalpoint such that ,

and .This construction gives the constellation .The label of each is then as-signed with , where

is the label of the cor-responding signal point . The finaloptimal mapping of is obtained by combining

and and their labels.The final results obtained by the above algorithm are shown

in Table III.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewersfor their helpful comments and suggestions that improved thepresentation of the paper.

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Nghi H. Tran (S’06) received the B.Eng. degreefrom Hanoi University of Technology, Hanoi,Vietnam, in 2002, and the M.Sc. degree fromthe University of Saskatchewan, Saskatoon, SK,Canada, in 2004, both in electrical engineering. Heis currently working toward the Ph.D. degree in theDepartment of Electrical Engineering, University ofSaskatchewan.

His research interests span the areas of coded mod-ulation and iterative decoding.

Mr. Tran was the recipient of the Graduate ThesisAward from the University of Saskatchewan.

Ha H. Nguyen (M’01–SM’05) received the B.Engdegree from the Hanoi University of Technology,Hanoi, Vietnam, in 1995, the M.Eng degree from theAsian Institute of Technology, Bangkok, Thailand,in 1997, and the Ph.D. degree from the University ofManitoba, Winnipeg, MB, Canada, in 2001.

He joined the Department of Electrical Engi-neering, University of Saskatchewan, Saskatoon,SK, Canada, in 2001 as an Assistant Professor andwas promoted to the rank of Associate Professorin 2005. He holds adjunct appointments at the

Department of Electrical and Computer Engineering, University of Manitoba,Winnipeg, MB, Canada, and TRLabs, Saskatoon, SK, Canada. His researchinterests include digital communications, spread spectrum systems anderror-control coding.

Dr. Nguyen currently serves as an Associate Editor for the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS and the IEEE TRANSACTIONS

ON VEHICULAR TECHNOLOGY. He is a Registered Member of the Associationof Professional Engineers and Geoscientists of Saskatchewan (APEGS).


Tho Le-Ngoc (F’97) received the B.Eng. (withDistinction) degree in electrical engineering andthe M.Eng. degree in microprocessor applicationsfrom McGill University, Montreal, QC, Canada, in1976 and 1978, respectively, and the Ph.D. degreein digital communications from the University ofOttawa, Ottawa, ON, Canada, in 1983.

From 1977 to 1982, he was with Spar AerospaceLimited, where he was involved with the devel-opment and design of satellite communicationssystems. From 1982 to 1985, he was an Engineering

Manager with the Radio Group, Department of Development Engineering, SRT-elecom Inc., where he developed the new point-to-multipoint DA-TDMA/TDM

Subscriber Radio System SR500. From 1985 to 2000, he was a Professorwith the Department of Electrical and Computer Engineering, ConcordiaUniversity. Since 2000, he has been with the Department of Electrical andComputer Engineering, McGill University. His research interest is in the areaof broadband digital communications with a special emphasis on modulation,coding, and multiple-access techniques.

Prof. Le-Ngoc is a Senior Member of the Ordre des Ingénieur du Quebec, aFellow of the Engineering Institute of Canada (EIC), and a Fellow of the Cana-dian Academy of Engineering (CAE). He was the recipient of the 2004 Cana-dian Award in Telecommunications Research and the IEEE Canada FessendenAward in 2005.

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