bit-interleaved coded ofdm with signal space diversity: subcarrier

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 3, MARCH 2007 1137

Bit-Interleaved Coded OFDM With SignalSpace Diversity: Subcarrier Grouping and

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

Abstract—This paper investigates the application of bit-interleaved coded modulation and iterative decoding (BICM-ID)in orthogonal-frequency-division-multiplexing (OFDM) sys-tems with signal space diversity (SSD) over frequency-selectiveRayleigh-fading channels. Correlated fading over subcarriers isassumed. At first, a tight bound on the asymptotic error perfor-mance for the general case of precoding over all subcarriersis derived and used to establish the best achievable asymptoticperformance by SSD. It is then shown that precoding over sub-groups of at least subcarriers per group, where is the numberof channel taps, is sufficient to obtain this best asymptotic errorperformance, while significantly reducing the receiver complexity.The optimal joint subcarrier grouping and rotation matrix designis subsequently determined by solving the Vandermonde linearsystem. Illustrative examples show a good agreement betweenvarious analytical and simulation results.

Index Terms—Bit-interleaved coded modulation, frequency-selective fading, iterative decoding, orthogonal frequency-divisionmultiplexing (OFDM), performance bound, signal space diversity(SSD).

I. INTRODUCTION

THE main problem in the design of a communicationssystem over a wireless link is to deal with multipath

fading, which causes a significant degradation in terms of boththe reliability of the link and the data rate. An efficient wayto combat fading is to apply diversity techniques. The basicidea behind various diversity techniques (e.g., time, frequency,and space) is to provide statistically independent copies of thesame transmitted information at the receiver and appropriatelyprocess them to make the detection more reliable.

Signal space diversity (SSD) can provide performance im-provement over fading channels by increasing the diversityorder of a communications system [1], [2]. Basically, in SSD,

Manuscript received January 8, 2006; revised April 21, 2006. The editorcoordinating the review of this paper and approving it for publication is Dr.Mounir Ghogho. This work was supported in part by Discovery Grants fromthe Natural Sciences and Engineering Research Council of Canada (NSERC).N. H. Tran would also like to acknowledge the University of Saskatchewan’sGraduate Scholarship and the Fellowship received from TRlabs-Saskatoon.Part of this work was presented at the IEEE International Symposium onInformation Theory (ISIT), Seattle, WA, July 9–14, 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/TSP.2006.887107

each group of consecutive symbols is first mapped to anelement of an -dimensional ( -dim) constellation, which isgenerally carved from an -dim lattice, and then a rotation ma-trix is applied to the lattice constellation in order to maximizeboth the diversity order and the minimum product distance ofthe -dim lattice [2].

Applications of SSD have been considered in various un-coded orthogonal-frequency-division-multiplexing (OFDM)systems [3]–[8]. For example, the work in [6] studied uncodedOFDM with linear block precoding based on the mean cutoffrate. The investigation of linear constellation precoding appliedto uncoded OFDM in [5] indicates that SSD with a specificallyselected group of subcarriers can achieve the same maximummultipath diversity and coding gains as with all the availablesubcarriers. This technique is then applied to various uncodedOFDM systems [3], [4], [7], [8]. Surprisingly, to date, thesolution for subcarrier grouping has not been solved explicitly.For example, it is not clear if the specific subcarrier groupingsuggested in [5] is the unique and general solution.

Employing channel coding to further improve the per-formance of OFDM systems has also been carried out. Forexample, it is shown in [9] that low-density parity check (LDPC)codes with space–time-coded OFDM can provide a significantperformance improvement as compared to uncoded systems.Recently, bit-interleaved coded modulation (BICM) [10], [11]has also been considered for OFDM systems in [12]–[14]. Inparticular, the work in [13] briefly studied the effect of SSDin coded OFDM systems with a rather unrealistic assumptionof statistically independent fading over subchannels. Since thebandwidth occupied by each subcarrier is selected to be smallerthan the coherence bandwidth of the channel (in order to yieldthe flat fading effect over each subcarrier), or equivalently, thenumber of subcarriers is larger than the number of channel taps,the channel coefficients of subcarriers are generally correlated[15]. A more general OFDM system that employs joint channelcoding and SSD (via linear precoding) has also been presentedin [16].

This paper is also concerned with the SSD of OFDM sys-tems using BICM with iterative decoding (BICM-ID) over fre-quency-selective Rayleigh-fading channels. Different from thework in [16], here the problem of subcarrier grouping and ro-tation matrix design are jointly addressed to optimize the errorperformance while keeping the complexity at minimum. Cor-related fading over subchannels is considered. At first, a gen-eral tight bound on the asymptotic error performance for pre-coding over all subcarriers is derived. Based on this bound,

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1138 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 3, MARCH 2007

Fig. 1. Block diagram of a BICM-ID for OFDM with SSD.

design criterion to achieve the best asymptotic performance withSSD is established for a given constellation and mapping rule.It is then shown that SSD can just be implemented over a setof subcarrier groups of at least elements for the same op-timum performance as in the case of precoding over all sub-carriers, where is the number of channel taps. Subsequently,the joint optimum subcarrier grouping and rotation matrix de-sign for SSD is derived.

The remaining of the paper is organized as follows. Section IIdescribes the general structure of OFDM systems usingBICM-ID with SSD and their corresponding system model.Based on this system model, Section III presents the perfor-mance analysis for the general case when SSD is employedover all subcarriers. Distance criterion for given signalconstellation, signal mapping and rotation matrixis subsequently established. The best achievable asymptoticperformance by using SSD and the optimum joint subcarriergrouping and rotation matrix design are derived in Section IV.Illustrative analytical and simulation results are presented anddiscussed in Section V. Section VI concludes the paper.

Some notations used in the paper are as follows: All the vec-tors are row-wise. The bold lower letters indicate vectors, whilethe bold capital letters are used for matrices. The superscriptsand denote transpose and conjugate transpose, respectively;

is an diagonal matrix with diagonalelements , ; denotes the determinant of asquare matrix; and stands for the all-zero vector.

II. SYSTEM MODEL

Fig. 1 illustrates the application of BICM-ID in an OFDMsystem that also employs SSD, which is similar to the systemmodel studied in [16]. The information sequence is first en-coded into a coded sequence . The coded sequence is theninterleaved by a bitwise interleaver to become the interleavedsequence . Consider OFDM system with subcarriers. Eachgroup of interleaved coded bits are mapped to an originalOFDM symbol with the mapping rule ,where , , with is a conventional two-dimensional(2-D) constellation of size . Clearly, the symbol can be con-sidered as a signal point in a complex -dimensional constel-lation , which includes in total of signal points.

For simplicity, assume the mapping rule is employed inde-pendently in each component , even though the multidimen-sional mapping technique proposed in [17] can be applied. Therotated sequence is obtained as

(1)

where can be any matrix with complex elementswhich satisfies the power constraint as follows:

(2)

In general, the number of carrier can be very large, whichleads to very high decoding complexity. The very attractive so-lution known as subchannel grouping, was proposed in [5] foruncoded OFDM systems to reduce the complexity at the re-ceiver. The basic idea of this approach is to divide all thecarriers into nonintersecting subsets and then apply the rotationmatrix with a much smaller size to each subset. Following thenotations in [5], denote the set of indexes of all the carriersas . Assume that . Then, carrierscan be divided into nonintersecting subsets .Each subset , , includes carrier indexes

. A rotation matrix with size is thenapplied to each group to obtain a sequence of size .sequences form the rotated sequence . Clearly, the sub-carrier grouping approach is a special case of the general modelin (1) with the following equivalent rotation matrix :

(3)

where is a permutation matrix constructed from rows, , of .

By employing a specific subcarrier grouping with equalsthe number of channel taps , it is shown in [5] that both diver-sity and coding gains [but not bit error rate (BER)] of uncodedOFDM can be preserved as in the case of . This result issimply due to the fact that for OFDM, there are totally inde-pendent paths, giving at most a diversity gain of .

The rotated sequence is passed to the IDFT block and theninserted with a cyclic prefix of length . Assume that DFT

TRAN et al.: BIT-INTERLEAVED CODED OFDM WITH SIGNAL SPACE DIVERSITY 1139

processing1 and cyclic prefix removal are properly carried outat the receiver with coherent detection. Since OFDM convertsthe broadband frequency-selective fading channel into flatsubchannels, the received signal can be presented as follows:

(4)

Here, each entry of is a complexwhite Gaussian noise with independent components havingtwo-sided power spectral density of . The matrix

contains the correlated fadingcoefficients in its diagonal, where

(5)

In (5), the channel vector contains the channelgains of all the taps, where each , , is mod-eled as a circularly symmetric complex Gaussian random vari-able. It is assumed that the channel gains remain constant withinone OFDM symbol and change independently from one OFDMsymbol to the next. This is a reasonable assumption when asufficiently long interleaver is used to break the correlation ofthe channel in time. Furthermore, within each OFDM symbol,the general case of correlated channel taps is considered by as-suming that the channel vector has a full-rank correlation ma-trix [3]–[5]. Note that the model of indepen-dent and identically distributed (i.i.d.) channel taps consideredin [12], [15], and [18] with and the model withan exponential power profile of the channel taps in [19] are justspecial cases of the above general model. For convenience, thecoefficient is rewritten as follows:

(6)

where

(7)

for .As shown in Fig. 1, the receiver of the system includes the

soft-input soft-output (SISO) demodulator and the soft-inputsoft-output channel decoder for the convolutional code. TheSISO channel decoder uses the MAP algorithm in [20]. Similarto decoding of Turbo codes, here the demodulator and thechannel decoder exchange the extrinsic information of thecoded bits and through an iterative process.After being interleaved, and become thea priori information and at the input of theSISO decoder and the demodulator, respectively. The total aposteriori probabilities of the information bits can be computedto make the hard decisions at the output of the decoder aftereach iteration.

The optimal soft-output demodulator, called the maximum aposteriori probability (MAP) demodulator [11], [21], can be im-plemented for each group of subcarriers . However, in the gen-eral case of the rotation matrix , the complexity of the MAPdemodulator grows exponentially with the number of coded bits

1When N is a power of 2, IFFT and FFT can be efficiently implemented.

per OFDM symbols , which becomes quickly intractablefor medium to large values of . When subchannel groupingapproach is used, the MAP demodulator can be applied for eachgroup of subcarriers , which significantly reduces the com-plexity, especially when . The suboptimal low-com-plexity yet effective method proposed in [22] using the min-imum mean-square error (MMSE) receiver and the sigma map-ping, or the Gaussian approximations proposed in [23] can beattractive alternatives.

III. PERFORMANCE EVALUATION

In this section, given a constellation and the mapping rule, the asymptotic bit error probability (BEP) of OFDM sys-

tems with BICM-ID and SSD is investigated for the generalrotation matrix . The evaluation of such asymptotic

BEP performance shall be carried out based on the assumptionof error-free feedback from the decoder to the demodulator asnormally done in the analysis of BICM-ID [11], [17] with theperfect interleaver (i.e., an interleaver of infinite length). First,the union bound of the BEP performance for a rate- con-volutional code, a complex -dim constellation and a map-ping rule can be written in a general form as [10]

(8)

In (8), is the total information weight of all error events atHamming distance and is the free Hamming distance ofthe code. The function is the average pairwise errorprobability, which depends on the Hamming distance , the con-stellation , and the mapping rule . In the following, the func-tion is computed from the pairwise error probability(PEP) of two codewords.

Let and denote the input and estimate sequences, respec-tively, with the Hamming distance between them. These bi-nary sequences correspond to the sequences and , whose ele-ments are OFDM symbols in . Without loss of generality, as-sume that and differ in the first consecutive bits. Hence,and can be redefined as sequences of OFDM symbols as

and . Also let ,where , , represents thepath gains that affect the transmitted symbol . More specifi-cally, for each channel realization , one has

(9)

where are tap gains at channel realizationand is given in (7).The two OFDM symbols and correspond to the two

rotated OFDM symbols and , i.e., and. Then, the PEP conditioned on can be computed2 as

follows:

(10)

2The Q-function is defined as Q(x) = (1=p2�) expf�t =2gdt.


where is the squared Euclidean distance betweenthe two received signals corresponding to and conditionedon and in the absence of the additive white Gaussian noise.This distance is given by

(11)

where is the th row of . Clearly,are correlated. Following the same procedure in [12] and [18],substituting (9) into (11), one has

(12)

Letting , it then follows that

(13)

where . Furthermore, let the square rootof the correlation matrix be . Then the channel vectorcan be whitened as follows [3]–[5]:

(14)

where the elements of are complex zero-meanGaussian random variables with unit variance. It then followsfrom (13) that

(15)

where . Since is a Hermitian matrix,it can be decomposed with the eigenvalue decomposition as

(16)

where is unitary and is a diagonal matrix whose diagonalelements are the eigenvalues of . Then, one has

(17)

Clearly, the elements of are also complex Gaussianrandom variables with zero-mean and unit variance. Substi-tuting (17) into (10), the PEP conditioned on is then given by

(18)

Invoking the Gaussian probability integraland averaging (18) over

gives

(19)

Furthermore, using the equality ,where is a Rayleigh random variable with , onehas

(20)

where

(21)

Since it is assumed that the channel realization changesindependently, the variables can be considered as i.i.d.random variables. Furthermore, owning to the success of de-coding step as normally seen in the analysis of BICM-ID sys-tems, the assumption of error-free feedback from the decoderto the demodulator can be made in order to analyze the asymp-totic error performance. This assumption implies that one needsto consider only the pairs of OFDM symbols and whoselabels differ in only 1 bit. Then, the function can becomputed by averaging over the constellation as follows:

(22)where the expectation is over all pairs of OFDM symbols and

in whose labels differ in only 1 bit. The valuesare the eigenvalues of , where

(23)

and

(24)

with

(25)

A straightforward way to compute the expectation operationin (22) is as follows:

(26)


where is the symbol in whose label differs in only one bitat the position compared to the label of .

For a large value of , the computation of (26) becomes in-tractable due to the huge number of OFDM symbols in . Forsimplicity, consider the mapping that is implemented inde-pendently and identically for each signal component in the 2-Dconstellation, denoted by . First, by interchanging the summa-tions in (26), one can write as a sum of terms as

(27)

where

(28)is essentially obtained by averaging over all pairs of andwhose labels differ in 1 bit at positions fromto . Note that the inner sum in (28) is taken overpossible symbols , which might still appear computationallyimpossible when and are large. Fortunately, by indepen-dently mapping for each component of OFDM symbols, it iseasy to verify that there is only one distinct component between

and at the th position. Hence, for a given , one has

(29)

where and are the th components of and , respectively.By adding the index , the matrix in (24) is then com-puted as

(30)

It then follows that

(31)where are the eigenvalues of

. It can be seen that for a given ,depends only on the th components of and .

Therefore, can be denoted as .Observe that can be any signal point in the 2-D constella-

tion and is also a signal point in whose label differs inonly 1 bit at precisely position compared tothat of . Therefore, instead of averaging over cases of

as in (31), can be computed more efficiently byaveraging over cases of . Furthermore,can be written without the subscript for and as follows:

(32)

where and are two signal points in whose labels differ inthe position and are the eigenvalues of

with

(33)

for a given . Therefore, the average in (26) can be computedmuch easier as follows:

(34)

Applying (34) in (22), the function can be efficientlycomputed with a high accuracy via a single integral. It can beobserved that is computed by essentially averaging overthe 2-D constellation , instead of . This is due to the as-sumption that the mapping is implemented independently foreach signal component in . This also suggests that the function

can be denoted as .To give an insight on how to design the matrix and

good mappings for a given constellation, use the inequalityto approximate the function

as

(35)

where

(36)

The parameter then can be used to characterize theinfluence of the rotation matrix , the constellation , and themapping rule to the asymptotic BEP performance of BICM-IDin OFDM with SSD. In particular, for a given constellationand the mapping , one would prefer the rotation matrixthat minimizes . It can be observed that the param-eter depends only on the magnitudes of the elementsin at a specific signal-to-noise ratio (SNR) (i.e., given ).This is due to the fact that the error bound derived earlier is forthe asymptotic performance of the systems. At very high SNR,(i.e., ), the design criterion can be made simpler andmore meaningful as follows:

(37)

Furthermore, for a given , one has the following equality:

(38)


where are eigenvalues of . Thus, the aboveequality simplifies the design parameter in (37) to the followingone, which is independent of the correlation matrix

(39)

In the next section, the optimal choice of in terms of boththe error performance and the decoding complexity based on thedesign criterion in (39) is discussed in more details.

IV. OPTIMAL ROTATION MATRIX AND

SUBCARRIER GROUPING

This section addresses the design problem to obtain the op-timal matrix in terms of both error performance and decodingcomplexity for a given constellation and mapping. To do so, theultimate error performance achieved with rotation matrix isinvestigated first.

A. Optimal Rotation Matrix

By interchanging the summations in (39), isrewritten as

(40)

Next, for each pair of signal points and in whose labelsdiffer in only 1 bit at position , , define the param-eter as follows:

(41)

Observe that the matrix is positive semidefinite.Therefore, the eigenvalues of in (33) are all non-negative. Furthermore, given with eigenvalues

, one has the following property:

(42)

Since the diagonal elements of are all one, it thenfollows that

(43)

Summing up the eigenvalues for alland under the power constraint in (2), one has

(44)Using Cauchy inequality, it is straightforward to showthat in (41) achieves the minimum value when

for all and . Therefore, one obtains thelower bound for in (40) as follows:

(45)

Define the class of rotation matrices that achieves the abovelower bound as the optimal set of . This means that the bestasymptotic error performance of BICM-ID in OFDM with SSDis achieved by using any in this class.

Assume that there exists at least one matrix in the op-timal set. Then, one has the following necessary and sufficientcondition.

Condition 1: The necessary and sufficient condition forhaving the optimal matrix is that all the eigenvalues of thematrix in (33) equal to for all .

Observe that the matrix is diagonalizable. There-fore, it has equal eigenvalues if and only if it is a diagonal matrixwith the diagonal elements equal to . More specifically,letting us define , one has

for all (46)

with is identity matrix of size . Let be the elementat the th row and the th column of an Hermitian matrix

. Then, is given as

(47)

For convenience, let , and thematrix with the element , with ,

, of is

(48)

Straightforwardly, (46) is satisfied if and only if

for all (49)

At this point, it is natural to ask a question that whether theoptimal set of rotation matrices is empty or not? Furthermore,if the set is not empty, what should be the most suitable interms of the decoding complexity?

Consider the class of rotation matrix such that all the entriesequals in magnitude as follows:

for all (50)

Obviously, . Furthermore, for any value of, , one has

(51)

It then follows that . Therefore,there exists at least one optimal rotation matrix in which allthe entries are equal in terms of the magnitudes.


By using the above optimal matrix , the best error per-formance in terms of asymptotic performance of BICM-ID inOFDM with SSD can be achieved. Unfortunately, if such an op-timal matrix is applied, the OFDM symbols need to be codedover all subcarriers. Therefore, the receiver complexity be-comes very high (e.g., increasing exponentially with ). In thefollowing subsection, the subchannel grouping approach is con-sidered. It is then shown that a much simpler method can be usedwithout sacrificing the error performance.

B. Subcarrier Grouping

Consider the subcarrier grouping with groups, . Then, a rotation matrix

with elements , , , is applied to each group.Without loss of generality, the following assumption can bemade:

whenwhen (52)

Any element of the rotation equivalent rotation matrix in (3)can be represented as

if for

(53)

For a given , define to be the matrix whoseelement is

(54)

Also, for a given c, define the matrix. To achieve the best asymptotic

performance, one has the following equivalent condition tothat of (49):

for all (55)

Clearly, the above condition not only depends on the rotationmatrix but also on the way the subgroup is formed (i.e.,how to choose for each ). It has been shown in previoussection that when , there exists at least one optimalsolution for . However, in the view of receiver complexity,one would prefer the solution for in which the value of isas small as possible. The following theorem provides the lowerbound for .

Theorem 1: The optimal solution of the rotation matrixdoes not exist when .

Proof: Assume that there exist an optimal solution when, i.e., . Construct the square matrix

with size including rows from the second row to theth row of . More specifically, the element is

, , . It followsfrom (55) that

(56)

The determinant of is computed as

(57)

where

......

. . ....

(58)

with , . Observe thatis Vandermonde matrix of order . Its determinant can

be computed in a particularly simple form as follows [24]:

(59)

Since , one has . There-fore, the determinant of and cannot be 0. It thenfollows that the only solution for (56) is , whichcontradicts the condition that in (55). Thetheorem is thus proved.

The result stated in Theorem 1 is predicable, since there arein total of independent fading channels. Now, the optimal so-lution of subcarrier grouping and rotation matrix for the caseof will be given explicitly. Clearly, is the mostdesirable value in terms of the decoding complexity. Further-more, another condition is , which can be easilymet by picking up a suitable number of subcarriers .

Now consider the case . It is clear that the matrixin (54) equals in (58). Let . The

necessary and sufficient condition for the existence of optimalsolution in (55) then becomes

for all (60)

The above system is exactly the linear Vandermonde system.Since the determinant of Vandermonde matrix is not 0,there is a unique solution for . Therefore, it is expected that

are identical for all . The solution of (60)is closely related to Lagrange’s polynomial interpolation for-mula, which is generally presented in [25]. First, construct theset of polynomial of degree , , asfollows:

(61)

where is the coefficient corresponding to the orderof . The solution of (60) is just the matrix inverse timesthe right-side , which is [25]

(62)


where and are the th and the th elements of and ,respectively. Since the vector , the solution of(60) is simply

(63)

where can be computed from (61) as

(64)

Therefore, one has

(65)

Each term in the above product can be computed as

(66)

It then follows that

(67)

Since , a necessary condition for in (67) isthat it is a real number. In the following, two cases of will beconsidered separately.

• is odd: It is then straightforward to see that is real ifand only if

(68)

Therefore

(69)

with is an integer. Consider for any and such that. It then follows that

(70)

Thus

(71)

Combining with the first condition in (52), one has the fol-lowing unique form of the elements of for a given :

(72)

Using the second condition in (52), the necessary conditionfor the existence of optimal solution related to subcarriergroups is

(73)

for all and .• is even: It is then straightforward to see that is real

if and only if

(74)

Therefore

(75)

Similar to the case of odd , it is easy to show that theelements of subcarrier groups also satisfy the conditionin (73).

From the above results, it is obvious that (73) is the necessarycondition for the existence of the optimal solution in terms ofboth subcarrier grouping and applying the rotation matrix .Furthermore, it is shown in Appendix I that when (73) is sat-isfied, . Therefore, one ends up with the followingunique jointly optimal solution of both subcarrier grouping androtation matrix :

for and

(76)

It should be mentioned that the subcarrier grouping in (76) issimilar to the result in [5] for uncoded OFDM, which was onlyspecifically chosen. On the other hand, here, the jointly optimalsolution for both subcarrier grouping and the rotation matrix

in (76) was presented thoroughly and explicitly for codedOFDM systems.

When , it is simple to see that the subcarrier groupingand the rotation matrix in (76) also guarantee the optimal solu-tion. Such a solution, however, might not be unique.

For iterative decoding systems, it is also of interest to studythe performance of the system after the first iteration (i.e., theperformance of BICM). This is because such performance in-fluences the convergence behavior of BICM-ID. To this end, therotated constellation should be taken into account. Letbe the minimum Euclidean distance of . Two relevant param-eters with respect to the performance of BICM are summarizedas follows.

i) The average number of signal points at the minimum Eu-clidean distance , denoted by . This parameteraffects the performance at low SNR and is given by

(77)

where is the number of signal points at Eu-clidean distance whose label differs at only posi-tion compared to that of . The parameter de-


pends on a specific mapping and should be kept as smallas possible.

ii) The distance parameter that affects the per-formance at high SNR

(78)

where is the nearest neighbor of and their labels differat position . The smaller this parameter is, the better theperformance becomes.

Unfortunately, optimizing the above two parameters forBICM becomes rapidly intractable due to the huge number ofvariables when and increases. For this reason, we restrictour attention to the class of unitary3 rotation matrices . It isnot difficult to see that the parameter is the same withthat of system without SSD. This ensures that the performanceafter the first iteration of BICM-ID system with SSD is similarto the performance of a BICM-ID system without SSD at lowSNR. Furthermore, by following the same analysis as in theprevious section, the minimum value of can beachieved when the condition in (76) is satisfied.

At this point, it is natural to ask whether there exists a classof unitary matrices that satisfies the above design criterionfor any value of . Fortunately, thanks to the algebraic numbertheory, this class of is well studied in the literature. For ex-ample, the unitary matrix with size introduced in [2] and[26] falls into the class of optimal . The matrix is constructedas follows [2], [26]:

......

...(79)

where and. More generally, for any value of , the unitary optimal

matrix can be obtained based on the inverse fast Fourier trans-form (IFFT) matrix as follows [26]:

(80)

where is an arbitrary integer and is the -point IFFTmatrix whose th entry is given by

.

V. ILLUSTRATIVE RESULTS

In this section, analytical and simulation results are providedto confirm the analysis carried out in the previous sections. Thequadrature phase-shift keying (QPSK) modulation scheme withanti-Gray mapping rule is employed. It should be mentionedhere that Gray and anti-Gray mappings are the only two avail-able mappings for QPSK constellation. Anti-Gray mapping ischosen simply due to its superiority in providing a better errorperformance in iterative systems [11], [27]. This fact can also be

3The matrix�� is an unitary matrix if it satisfies�� = III .

Fig. 2. Performance of BICM-ID in OFDM systems: optimal subcarriergrouping and rotation is compared with that without rotation.

confirmed for the coded OFDM systems under consideration bycomparing the parameters in (39) for the two map-pings when the optimal rotation matrix and subcarrier groupingare applied.

All the systems are simulated with subcarriers,which is similar to that considered in HiperLan/2 standard [19].Unless specified otherwise, a rate-1/2, four-state convolutionalcode with generator matrix is applied as an outercode, along with a random interleaver of length 9600 coded bits.Furthermore, in all the computations of the bound for in (8),the first 20 Hamming distances of the convolutional code is re-tained to make sure the accuracy of the bound. Each point in theBER curves is simulated with to coded bits. The fol-lowing first two subsections present the results for the case thatthe channel gains are generated independently for each OFDMsymbol and the optimal MAP demodulator is implemented. Thelast subsection studies the error performances of the proposedsystems in a real channel environment by considering ChannelModel A of HiperLan/2 [19]. Due to the large number of channeltaps specified in this channel model, the high complexity of theoptimal MAP demodulator makes it impractical to implement.To overcome this difficulty, a suboptimal SISO demodulatorbased on the vector Gaussian approximation (GA) developedin [23] is employed.

A. i.i.d. Channels

This subsection considers the i.i.d. channel model, in whichthe correlation matrix .

Fig. 2 compares the BER performance with one, two, and fiveiterations of the system without SSD and to that of the systememploying optimal grouping and rotation matrix whenchannel taps. The rotation matrix is chosen as in (79) with

. It can be seen that performances of both systemsconverge to the analytical BER bounds. A significant codinggain is also achieved by the system employing SSD.

In Fig. 3, the error performances after one, two, and five it-erations for systems employing optimal grouping for the cases


Fig. 3. Performance of BICM-ID in OFDM systems: optimal subcarriergrouping and rotation for F = 2 and F = 4.

Fig. 4. Performance of BICM-ID in OFDM systems with various carriergrouping approaches.

of and are presented. The two rotation matricesare also chosen as in (79). It can be observed that there is al-most no difference between the performances of the two sys-tems at any iteration and they all converge to the same BERanalytical bound. This results confirms our analysis for the op-timal grouping and rotation matrices with different values of .Clearly, is preferred in terms of receiver complexity.

Fig. 4 illustrates the performance of various grouping ap-proaches. Besides the optimal one obtained in Section IV,we consider two more different approaches, namely naturalgrouping with and average grouping with

. The error performances of systemsemploying the same 2 2 rotation matrix as in the previouscase with one and five iterations are shown. Obviously, theerror bounds of all systems are very tight, which makes themvery useful tool to predict the error performance of BICM-ID

Fig. 5. Performance of BICM-ID in OFDM systems: optimal subcarriergrouping with various rotation matrices ��.

in OFDM with SSD. Though not explicitly shown here for thebrevity of presentation, examining the error bounds plotted overa wider range of SNR reveals that, compared with the averagedand natural groupings, the use of the optimal grouping resultsin coding gains as high as 1 and 2.2 dB at the BER level of

, respectively.Fig. 5 demonstrates the effect of the rotation matrix , where

the optimal grouping is applied. Shown in the figure are the BERperformance after one and five iterations of systems using dif-ferent choices of . These choices include i) an optimal matrix

in (79), ii) a randomly generated :

(81)

and iii) the following optimal rotation matrix that maximizesthe minimum product distance for real unitary transformationwith a full SSD [28], given as

where (82)

Also plotted in this figure are the error bounds for all the systemsunder consideration (the broken lines). Observe that, comparedwith the random and , the use of the optimal rotation matrixresults in coding gains as high as 0.25 and 2.6 dB at the BERlevel of , respectively. The results clearly agree with ouranalysis on the jointly optimum subcarrier group and rotationmatrix . Note also that the error bounds are tight for all thesystems.

Finally, Fig. 6 shows the error performances of two systemsemploying optimal carrier grouping and rotation matrices with

and . In the case of , the 4 4 rotation ma-trix is chosen from (79). Clearly, with optimal grouping androtation matrix, the use of BICM-ID in OFDM with SSD suc-cessfully exploit the advantage of multipath diversity in channelenvironment with the richer scattering effect (i.e., correspondingto a larger number of channel taps).


Fig. 6. Error performances for systems withL = 2 andL = 4: optimal carriergrouping and rotation matrices are employed.

Fig. 7. Error performances for systems operating over and correlated channelswith L = 2: optimal carrier grouping and rotation matrices are employed inboth cases.

B. Correlated Channels

Fig. 7 first compares the BER performance with 1, 2 and5 iterations of systems over the i.i.d. channel and a correlatedchannel with number of channel taps . For the correlatedchannel, the channel correlation matrix is assumed to be

(83)

For both systems, the same optimal carrier grouping and rota-tion matrix are employed. It can be seen that the BER perfor-mances over the correlated channel significantly degrade at anyiterations. Observe that the asymptotic error bound is also verytight for correlated channel. The slopes of the asymptotic errorperformances of both systems are, however, the same, whichconfirms that both systems fully exploit the maximum diver-sity order . By examining the error bounds over a wider

Fig. 8. Error performance comparisons with Channel Model A in HiperLan/2.

range of SNR, it is observed that performance degradation overthe correlated channel transfers to about an additional 1 dB re-quired to achieve the BER level of , which is quite notice-able. Though not shown here due to space limit, the effects ofsubcarrier grouping and rotation matrix over correlated chan-nels are very similar to that over the i.i.d. channels.

C. Channel Model A in HiperLan/2

In Channel Model A of HiperLan/2 [19], the carrier frequencyis set at 5.2 GHz and the mobile’s velocity is 3 m/s, which resultsin the Doppler frequency 52 Hz. Each OFDM symbol du-ration is 4 s, corresponding to the normalized Dopplerfrequency . There are eight channel taps(i.e., ), where the variances in the tap order are givenas {0.4505, 0.3467, 0.1283, 0.0522, 0.0102, 0.0077, 0.0029,0.0014} [5], [19]. The gains of the channel taps are modeledas independent circularly symmetric complex Gaussian randomvariables and they are generated according to the Jakes’ model.This means that there is correlation among channel taps overtime. As mentioned earlier, the large number of channel tapsimplies a very high complexity of the optimal MAP demodu-lator in the systems with rotation. To reduce the receiver com-plexity of such systems, a suboptimal SISO demodulator basedon the vector GA in [23] shall be used. In the case of the systemswithout rotation, the MAP demodulator is still employed.

Fig. 8 compares the error performances of the proposedsystem implementing the optimal rotation and subcarriergrouping after one and ten iterations and that of the systemwithout rotation after ten iterations. Due to the high correlationof the channel taps across multiple OFDM symbols, the randominterleaver of length 204 800 coded bits is applied, which cor-responds to a frame of 1600 OFDM symbols. The same outercode as used in the previous subsections is applied. To serveas the reference, the error performance after five iterations andthe analytical error bound of the proposed system when thechannel taps are generated independently across OFDM sym-bols are provided. As can be seen from Fig. 8, implementingthe optimal rotation and grouping significantly improves the


error performance of the conventional system that does nothave constellation rotation after ten iterations. For example, anSNR gain of about 2 dB is realized by the proposed system atthe BER level of . Compared with the case of independentfading across OFDM symbols, the error performance is signif-icantly poorer because of the high correlation of the channeltaps over time present in Channel Model A of HiperLan/2.

VI. CONCLUSION

This paper presented a performance analysis of BICM-ID forOFDM systems with SSD over frequency-selective Rayleigh-fading channels. The design of subcarrier grouping and the ro-tation matrix in order to reduce the receiver complexity whilepreserving the error performance is subsequently derived. Theanalytical expression of the tight performance bound when SSDis employed over all subcarriers allows us to predict the bestachievable asymptotic error performance using SSD. From theresult of this derivation, a joint optimum subcarrier groupingand rotation matrix was developed based on the linear Vander-monde system. It was demonstrated through both analytical andsimulation results that using the proposed optimal grouping androtation matrix , the best error performance of BICM-ID forOFDM with SSD can be achieved with low-complexity receiver.

APPENDIX ITHE OPTIMAL VALUES OF

When (73) is satisfied, one has

(84)

Furthermore

(85)

Hence, in (67) is rewritten as follows:

(86)

After some manipulations, the product inside the squarebrackets is computed as

(87)

where the last equality follows from the finite product of sinfunctions. It then follows that

(88)Furthermore, for any , it is easy to see that

(89)

Therefore

(90)

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewersfor their helpful comments that improve the presentation of thepaper.

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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 (withGraduate Thesis Award) from the University ofSaskatchewan, Saskatoon, SK, Canada, in 2004,all in electrical engineering. Since January 2005,he has been working towards the Ph.D. degree inthe Department of Electrical Engineering at theUniversity of Saskatchewan.

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

Mr. Tran received the Graduate Thesis Award for this M.Sc. degree from theUniversity of Saskatchewan in 2004.

Ha H. Nguyen (M’01–SM’05) received the B.Eng.degree from Hanoi University of Technology, Hanoi,Vietnam, in 1995, the M.Eng. degree from the AsianInstitute of Technology, Bangkok, Thailand, in 1997,and the Ph.D. degree from the University of Mani-toba, 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 Professor in2005. His research interests include digital commu-

nications, spread-spectrum systems, and error control coding.Dr. Nguyen is a registered member of the Association of Professional Engi-

neers and Geoscientists of Saskatchewan (APEGS).

Tho Le-Ngoc (F’97) received the B.Eng. degree(with Distinction) 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, ON, Canada, in 1983.

From 1977 to 1982, he was with Spar AerospaceLimited, Montreal, QC, Canada, where he wasinvolved in the development and design of satellitecommunications systems. From 1982 to 1985,

he was an Engineering Manager of the Radio Group in the Department ofDevelopment Engineering of SRTelecom, Inc., Montreal, QC, Canada, wherehe developed the new point-to-multipoint DA-TDMA/TDM subscriber radiosystem SR500. From 1985 to 2000, he was a Professor at the Departmentof Electrical and Computer Engineering of Concordia University, Montreal,QC, Canada. Since 2000, he has been with the Department of Electrical andComputer Engineering of McGill University. His research interest is in the areaof broadband digital communications with a special emphasis on modulation,coding, and multiple-access techniques.

Dr. 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 is the recipient of the 2004 CanadianAward in Telecommunications Research and recipient of the IEEE Canada Fes-senden Award 2005.

bit-interleaved coded ofdm with signal space diversity: subcarrier

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