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

Transcription

1 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 3, MARCH Bit-Interleaved Coded OFDM With Signal Space Diversity: Subcarrier Grouping and Rotation Matrix Design Nghi H. Tran, Student Member, IEEE, Ha H. Nguyen, Senior Member, IEEE, and Tho Le-Ngoc, Fellow, IEEE Abstract This paper investigates the application of bitinterleaved coded modulation and iterative decoding (BICM-ID) in orthogonal-frequency-division-multiplexing (OFDM) systems with signal space diversity (SSD) over frequency-selective Rayleigh-fading channels. Correlated fading over subcarriers is assumed. At first, a tight bound on the asymptotic error performance for the general case of precoding over all subcarriers is derived and used to establish the best achievable asymptotic performance by SSD. It is then shown that precoding over subgroups of at least subcarriers per group, is the number of channel taps, is sufficient to obtain this best asymptotic error performance, while significantly reducing the receiver complexity. The optimal joint subcarrier grouping and rotation matrix design is subsequently determined by solving the Vandermonde linear system. Illustrative examples show a good agreement between various analytical and simulation results. Index Terms Bit-interleaved coded modulation, frequencyselective fading, iterative decoding, orthogonal frequency-division multiplexing (OFDM), performance bound, signal space diversity (SSD). I. INTRODUCTION THE main problem in the design of a communications system over a wireless link is to deal with multipath fading, which causes a significant degradation in terms of both the reliability of the link and the data rate. An efficient way to combat fading is to apply diversity techniques. The basic idea behind various diversity techniques (e.g., time, frequency, and space) is to provide statistically independent copies of the same transmitted information at the receiver and appropriately process them to make the detection more reliable. Signal space diversity (SSD) can provide performance improvement over fading channels by increasing the diversity order of a communications system [1], [2]. Basically, in SSD, Manuscript received January 8, 2006; revised April 21, The editor coordinating the review of this paper and approving it for publication is Dr. Mounir Ghogho. This work was supported in part by Discovery Grants from the Natural Sciences and Engineering Research Council of Canada (NSERC). N. H. Tran would also like to acknowledge the University of Saskatchewan s Graduate Scholarship and the Fellowship received from TRlabs-Saskatoon. Part of this work was presented at the IEEE International Symposium on Information Theory (ISIT), Seattle, WA, July 9 14, N. H. Tran and H. H. Nguyen are with the Department of Electrical Engineering, University of Saskatchewan, Saskatoon, SK S7N 5A9, Canada ( nghi.tran@usask.ca; ha.nguyen@usask.ca). T. Le-Ngoc is with the Department of Electrical and Computer Engineering, McGill University, Montreal, QC H3A-2A7, Canada ( tho@ece.mcgill. ca). Digital Object Identifier /TSP each group of consecutive symbols is first mapped to an element of an -dimensional ( -dim) constellation, which is generally carved from an -dim lattice, and then a rotation matrix is applied to the lattice constellation in order to maximize both the diversity order and the minimum product distance of the -dim lattice [2]. Applications of SSD have been considered in various uncoded orthogonal-frequency-division-multiplexing (OFDM) systems [3] [8]. For example, the work in [6] studied uncoded OFDM with linear block precoding based on the mean cutoff rate. The investigation of linear constellation precoding applied to uncoded OFDM in [5] indicates that SSD with a specifically selected group of subcarriers can achieve the same maximum multipath diversity and coding gains as with all the available subcarriers. This technique is then applied to various uncoded OFDM systems [3], [4], [7], [8]. Surprisingly, to date, the solution for subcarrier grouping has not been solved explicitly. For example, it is not clear if the specific subcarrier grouping suggested in [5] is the unique and general solution. Employing channel coding to further improve the performance of OFDM systems has also been carried out. For example, it is shown in [9] that low-density parity check (LDPC) codes with space time-coded OFDM can provide a significant performance improvement as compared to uncoded systems. Recently, bit-interleaved coded modulation (BICM) [10], [11] has also been considered for OFDM systems in [12] [14]. In particular, the work in [13] briefly studied the effect of SSD in coded OFDM systems with a rather unrealistic assumption of statistically independent fading over subchannels. Since the bandwidth occupied by each subcarrier is selected to be smaller than the coherence bandwidth of the channel (in order to yield the flat fading effect over each subcarrier), or equivalently, the number 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 channel coding and SSD (via linear precoding) has also been presented in [16]. This paper is also concerned with the SSD of OFDM systems using BICM with iterative decoding (BICM-ID) over frequency-selective Rayleigh-fading channels. Different from the work in [16], here the problem of subcarrier grouping and rotation matrix design are jointly addressed to optimize the error performance while keeping the complexity at minimum. Correlated fading over subchannels is considered. At first, a general tight bound on the asymptotic error performance for precoding over all subcarriers is derived. Based on this bound, X/$ IEEE

2 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 with SSD is established for a given constellation and mapping rule. It is then shown that SSD can just be implemented over a set of subcarrier groups of at least elements for the same optimum performance as in the case of precoding over all subcarriers, is the number of channel taps. Subsequently, the joint optimum subcarrier grouping and rotation matrix design for SSD is derived. The remaining of the paper is organized as follows. Section II describes the general structure of OFDM systems using BICM-ID with SSD and their corresponding system model. Based on this system model, Section III presents the performance analysis for the general case when SSD is employed over all subcarriers. Distance criterion for given signal constellation, signal mapping and rotation matrix is subsequently established. The best achievable asymptotic performance by using SSD and the optimum joint subcarrier grouping and rotation matrix design are derived in Section IV. Illustrative analytical and simulation results are presented and discussed in Section V. Section VI concludes the paper. Some notations used in the paper are as follows: All the vectors are row-wise. The bold lower letters indicate vectors, while the bold capital letters are used for matrices. The superscripts and denote transpose and conjugate transpose, respectively; is an diagonal matrix with diagonal elements, ; denotes the determinant of a square matrix; and stands for the all-zero vector. II. SYSTEM MODEL Fig. 1 illustrates the application of BICM-ID in an OFDM system that also employs SSD, which is similar to the system model studied in [16]. The information sequence is first encoded into a coded sequence. The coded sequence is then interleaved by a bitwise interleaver to become the interleaved sequence. Consider OFDM system with subcarriers. Each group of interleaved coded bits are mapped to an original OFDM symbol with the mapping rule,,, with is a conventional two-dimensional (2-D) constellation of size. Clearly, the symbol can be considered as a signal point in a complex -dimensional constellation, which includes in total of signal points. For simplicity, assume the mapping rule is employed independently in each component, even though the multidimensional mapping technique proposed in [17] can be applied. The rotated sequence is obtained as can be any matrix with complex elements which satisfies the power constraint as follows: In general, the number of carrier can be very large, which leads to very high decoding complexity. The very attractive solution known as subchannel grouping, was proposed in [5] for uncoded OFDM systems to reduce the complexity at the receiver. The basic idea of this approach is to divide all the carriers into nonintersecting subsets and then apply the rotation matrix with a much smaller size to each subset. Following the notations in [5], denote the set of indexes of all the carriers as. Assume that. Then, carriers can be divided into nonintersecting subsets. Each subset,, includes carrier indexes. A rotation matrix with size is then applied to each group to obtain a sequence of size. sequences form the rotated sequence. Clearly, the subcarrier grouping approach is a special case of the general model in (1) with the following equivalent rotation matrix : is a permutation matrix constructed from rows,,of. By employing a specific subcarrier grouping with equals the number of channel taps, it is shown in [5] that both diversity and coding gains [but not bit error rate (BER)] of uncoded OFDM can be preserved as in the case of. This result is simply due to the fact that for OFDM, there are totally independent paths, giving at most a diversity gain of. The rotated sequence is passed to the IDFT block and then inserted with a cyclic prefix of length. Assume that DFT (1) (2) (3)

3 TRAN et al.: BIT-INTERLEAVED CODED OFDM WITH SIGNAL SPACE DIVERSITY 1139 processing 1 and cyclic prefix removal are properly carried out at the receiver with coherent detection. Since OFDM converts the broadband frequency-selective fading channel into flat subchannels, the received signal can be presented as follows: Here, each entry of is a complex white Gaussian noise with independent components having two-sided power spectral density of. The matrix contains the correlated fading coefficients in its diagonal, In (5), the channel vector contains the channel gains of all the taps, each,, is modeled as a circularly symmetric complex Gaussian random variable. It is assumed that the channel gains remain constant within one OFDM symbol and change independently from one OFDM symbol to the next. This is a reasonable assumption when a sufficiently long interleaver is used to break the correlation of the channel in time. Furthermore, within each OFDM symbol, the general case of correlated channel taps is considered by assuming that the channel vector has a full-rank correlation matrix [3] [5]. Note that the model of independent and identically distributed (i.i.d.) channel taps considered in [12], [15], and [18] with and the model with an exponential power profile of the channel taps in [19] are just special cases of the above general model. For convenience, the coefficient is rewritten as follows: for. As shown in Fig. 1, the receiver of the system includes the soft-input soft-output (SISO) demodulator and the soft-input soft-output channel decoder for the convolutional code. The SISO channel decoder uses the MAP algorithm in [20]. Similar to decoding of Turbo codes, here the demodulator and the channel decoder exchange the extrinsic information of the coded bits and through an iterative process. After being interleaved, and become the a priori information and at the input of the SISO decoder and the demodulator, respectively. The total a posteriori probabilities of the information bits can be computed to make the hard decisions at the output of the decoder after each iteration. The optimal soft-output demodulator, called the maximum a posteriori probability (MAP) demodulator [11], [21], can be implemented for each group of subcarriers. However, in the general case of the rotation matrix, the complexity of the MAP demodulator grows exponentially with the number of coded bits 1 When N is a power of 2, IFFT and FFT can be efficiently implemented. (4) (5) (6) (7) per OFDM symbols, which becomes quickly intractable for medium to large values of. When subchannel grouping approach is used, the MAP demodulator can be applied for each group of subcarriers, which significantly reduces the complexity, especially when. The suboptimal low-complexity yet effective method proposed in [22] using the minimum mean-square error (MMSE) receiver and the sigma mapping, or the Gaussian approximations proposed in [23] can be attractive alternatives. III. PERFORMANCE EVALUATION In this section, given a constellation and the mapping rule, the asymptotic bit error probability (BEP) of OFDM systems with BICM-ID and SSD is investigated for the general rotation matrix. The evaluation of such asymptotic BEP performance shall be carried out based on the assumption of error-free feedback from the decoder to the demodulator as normally done in the analysis of BICM-ID [11], [17] with the perfect interleaver (i.e., an interleaver of infinite length). First, the union bound of the BEP performance for a rate- convolutional code, a complex -dim constellation and a mapping rule can be written in a general form as [10] In (8), is the total information weight of all error events at Hamming distance and is the free Hamming distance of the code. The function is the average pairwise error probability, which depends on the Hamming distance, the constellation, and the mapping rule. In the following, the function is computed from the pairwise error probability (PEP) of two codewords. Let and denote the input and estimate sequences, respectively, with the Hamming distance between them. These binary sequences correspond to the sequences and, whose elements are OFDM symbols in. Without loss of generality, assume that and differ in the first consecutive bits. Hence, and can be redefined as sequences of OFDM symbols as and. Also let,,, represents the path gains that affect the transmitted symbol. More specifically, for each channel realization, one has are tap gains at channel realization and 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 computed 2 as follows: 2 The Q-function is defined as Q(x) =(1= p 2) expf0t =2gdt. (8) (9) (10)

4 1140 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 3, MARCH 2007 is the squared Euclidean distance between the two received signals corresponding to and conditioned on and in the absence of the additive white Gaussian noise. This distance is given by Invoking the Gaussian probability integral and averaging (18) over gives (11) is the th row of. Clearly, are correlated. Following the same procedure in [12] and [18], substituting (9) into (11), one has (19) Furthermore, using the equality, is a Rayleigh random variable with, one has (12) (20) Letting, it then follows that (13). Furthermore, let the square root of the correlation matrix be. Then the channel vector can be whitened as follows [3] [5]: (14) the elements of are complex zero-mean Gaussian random variables with unit variance. It then follows from (13) that (21) Since it is assumed that the channel realization changes independently, the variables can be considered as i.i.d. random variables. Furthermore, owning to the success of decoding step as normally seen in the analysis of BICM-ID systems, the assumption of error-free feedback from the decoder to the demodulator can be made in order to analyze the asymptotic error performance. This assumption implies that one needs to consider only the pairs of OFDM symbols and whose labels differ in only 1 bit. Then, the function can be computed by averaging over the constellation as follows: (15). Since is a Hermitian matrix, it can be decomposed with the eigenvalue decomposition as (16) is unitary and is a diagonal matrix whose diagonal elements are the eigenvalues of. Then, one has the expectation is over all pairs of OFDM symbols in whose labels differ in only 1 bit. The values are the eigenvalues of, and (22) and (23) (17) Clearly, the elements of are also complex Gaussian random variables with zero-mean and unit variance. Substituting (17) into (10), the PEP conditioned on is then given by with (24) (25) A straightforward way to compute the expectation operation in (22) is as follows: (18) (26)

5 TRAN et al.: BIT-INTERLEAVED CODED OFDM WITH SIGNAL SPACE DIVERSITY 1141 is the symbol in whose label differs in only one bit at the position compared to the label of. For a large value of, the computation of (26) becomes intractable due to the huge number of OFDM symbols in.for simplicity, consider the mapping that is implemented independently and identically for each signal component in the 2-D constellation, denoted by. First, by interchanging the summations in (26), one can write as a sum of terms as (27) and are two signal points in whose labels differ in the position and are the eigenvalues of with (33) for a given. Therefore, the average in (26) can be computed much easier as follows: (28) is essentially obtained by averaging over all pairs of and whose labels differ in 1 bit at positions from to. Note that the inner sum in (28) is taken over possible symbols, which might still appear computationally impossible when and are large. Fortunately, by independently mapping for each component of OFDM symbols, it is easy to verify that there is only one distinct component between and at the th position. Hence, for a given, one has (29) and are the th components of and, respectively. By adding the index, the matrix in (24) is then computed as (34) Applying (34) in (22), the function can be efficiently computed with a high accuracy via a single integral. It can be observed that is computed by essentially averaging over the 2-D constellation, instead of. This is due to the assumption that the mapping is implemented independently for each 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 inequality to approximate the function as (35) It then follows that (30) (31) 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 constellation and is also a signal point in whose label differs in only 1 bit at precisely position compared to that of. Therefore, instead of averaging over cases of as in (31), can be computed more efficiently by averaging over cases of. Furthermore, can be written without the subscript for and as follows: (36) The parameter then can be used to characterize the influence of the rotation matrix, the constellation, and the mapping rule to the asymptotic BEP performance of BICM-ID in OFDM with SSD. In particular, for a given constellation and the mapping, one would prefer the rotation matrix that minimizes. It can be observed that the parameter depends only on the magnitudes of the elements in at a specific signal-to-noise ratio (SNR) (i.e., given ). This is due to the fact that the error bound derived earlier is for the asymptotic performance of the systems. At very high SNR, (i.e., ), the design criterion can be made simpler and more meaningful as follows: Furthermore, for a given, one has the following equality: (37) (32) (38)

6 1142 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 3, MARCH 2007 are eigenvalues of. Thus, the above equality simplifies the design parameter in (37) to the following one, which is independent of the correlation matrix (39) In the next section, the optimal choice of in terms of both the error performance and the decoding complexity based on the design criterion in (39) is discussed in more details. IV. OPTIMAL ROTATION MATRIX AND SUBCARRIER GROUPING This section addresses the design problem to obtain the optimal matrix in terms of both error performance and decoding complexity for a given constellation and mapping. To do so, the ultimate error performance achieved with rotation matrix is investigated first. A. Optimal Rotation Matrix By interchanging the summations in (39), rewritten as is (40) Next, for each pair of signal points and in whose labels differ in only 1 bit at position,,define the parameter as follows: (41) Observe that the matrix is positive semidefinite. Therefore, the eigenvalues of in (33) are all nonnegative. Furthermore, given with eigenvalues, one has the following property: lower bound for for all and. Therefore, one obtains the in (40) as follows: (45) Define the class of rotation matrices that achieves the above lower bound as the optimal set of. This means that the best asymptotic error performance of BICM-ID in OFDM with SSD is achieved by using any in this class. Assume that there exists at least one matrix in the optimal set. Then, one has the following necessary and sufficient condition. Condition 1: The necessary and sufficient condition for having the optimal matrix is that all the eigenvalues of the matrix in (33) equal to for all. Observe that the matrix is diagonalizable. Therefore, it has equal eigenvalues if and only if it is a diagonal matrix with the diagonal elements equal to. More specifically, letting us define, one has for all (46) with is identity matrix of size. Let be the element at the th row and the th column of an Hermitian matrix. Then, is given as For convenience, let, and the matrix with the element, with,,of is Straightforwardly, (46) is satisfied if and only if (47) (48) for all (49) Since the diagonal elements of follows that (42) are all one, it then (43) At this point, it is natural to ask a question that whether the optimal set of rotation matrices is empty or not? Furthermore, if the set is not empty, what should be the most suitable in terms of the decoding complexity? Consider the class of rotation matrix such that all the entries equals in magnitude as follows: for all (50) Summing up the eigenvalues for all and under the power constraint in (2), one has Obviously,,, one has. Furthermore, for any value of (51) (44) Using Cauchy inequality, it is straightforward to show that in (41) achieves the minimum value when It then follows that there exists at least one optimal rotation matrix the entries are equal in terms of the magnitudes.. Therefore, in which all

7 TRAN et al.: BIT-INTERLEAVED CODED OFDM WITH SIGNAL SPACE DIVERSITY 1143 By using the above optimal matrix, the best error performance in terms of asymptotic performance of BICM-ID in OFDM with SSD can be achieved. Unfortunately, if such an optimal matrix is applied, the OFDM symbols need to be coded over all subcarriers. Therefore, the receiver complexity becomes very high (e.g., increasing exponentially with ). In the following subsection, the subchannel grouping approach is considered. It is then shown that a much simpler method can be used without 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 be made: when when (52) Any element of the rotation equivalent rotation matrix in (3) can be represented as if for For a given,define to be the matrix whose element is (53) (54) Also, for a given c, define the matrix. To achieve the best asymptotic performance, one has the following equivalent condition to that of (49): for all (55) Clearly, the above condition not only depends on the rotation matrix but also on the way the subgroup is formed (i.e., how to choose for each ). It has been shown in previous section that when, there exists at least one optimal solution for. However, in the view of receiver complexity, one would prefer the solution for in which the value of is as small as possible. The following theorem provides the lower bound for. Theorem 1: The optimal solution of the rotation matrix does 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 the th row of. More specifically, the element is,,. It follows from (55) that (56) The determinant of is computed as (57) (58) with,. Observe that is Vandermonde matrix of order. Its determinant can be computed in a particularly simple form as follows [24]: (59) Since, one has. Therefore, the determinant of and cannot be 0. It then follows that the only solution for (56) is, which contradicts the condition that in (55). The theorem is thus proved. The result stated in Theorem 1 is predicable, since there are in total of independent fading channels. Now, the optimal solution of subcarrier grouping and rotation matrix for the case of will be given explicitly. Clearly, is the most desirable value in terms of the decoding complexity. Furthermore, another condition is, which can be easily met by picking up a suitable number of subcarriers. Now consider the case. It is clear that the matrix in (54) equals in (58). Let. The necessary and sufficient condition for the existence of optimal solution 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 formula, which is generally presented in [25]. First, construct the set of polynomial of degree,,as follows: (61) is the coefficient corresponding to the order of. The solution of (60) is just the matrix inverse times the right-side, which is [25] (62)

8 1144 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 3, MARCH 2007 and are the th and the th elements of and, respectively. Since the vector, the solution of (60) is simply Using the second condition in (52), the necessary condition for the existence of optimal solution related to subcarrier groups is can be computed from (61) as Therefore, one has Each term in the above product can be computed as It then follows that (63) (64) (65) (66) (73) for all and. is even: It is then straightforward to see that is real if and only if Therefore (74) (75) Similar to the case of odd, it is easy to show that the elements of subcarrier groups also satisfy the condition in (73). From the above results, it is obvious that (73) is the necessary condition for the existence of the optimal solution in terms of both subcarrier grouping and applying the rotation matrix. Furthermore, it is shown in Appendix I that when (73) is satisfied,. Therefore, one ends up with the following unique jointly optimal solution of both subcarrier grouping and rotation matrix : for and (67) Since, a necessary condition for in (67) is that it is a real number. In the following, two cases of will be considered separately. is odd: It is then straightforward to see that is real if and only if Therefore (68) (69) with is an integer. Consider for any and such that. It then follows that Thus (70) (71) Combining with the first condition in (52), one has the following unique form of the elements of for a given : (72) (76) It should be mentioned that the subcarrier grouping in (76) is similar to the result in [5] for uncoded OFDM, which was only specifically chosen. On the other hand, here, the jointly optimal solution for both subcarrier grouping and the rotation matrix in (76) was presented thoroughly and explicitly for coded OFDM systems. When, it is simple to see that the subcarrier grouping and the rotation matrix in (76) also guarantee the optimal solution. Such a solution, however, might not be unique. For iterative decoding systems, it is also of interest to study the performance of the system after the first iteration (i.e., the performance of BICM). This is because such performance influences the convergence behavior of BICM-ID. To this end, the rotated constellation should be taken into account. Let be the minimum Euclidean distance of. Two relevant parameters with respect to the performance of BICM are summarized as follows. i) The average number of signal points at the minimum Euclidean distance, denoted by. This parameter affects the performance at low SNR and is given by (77) is the number of signal points at Euclidean distance whose label differs at only position compared to that of. The parameter de-

9 TRAN et al.: BIT-INTERLEAVED CODED OFDM WITH SIGNAL SPACE DIVERSITY 1145 pends on a specific mapping and should be kept as small as possible. ii) The distance parameter that affects the performance at high SNR (78) is the nearest neighbor of and their labels differ at position. The smaller this parameter is, the better the performance becomes. Unfortunately, optimizing the above two parameters for BICM becomes rapidly intractable due to the huge number of variables when and increases. For this reason, we restrict our attention to the class of unitary 3 rotation matrices.itis not difficult to see that the parameter is the same with that of system without SSD. This ensures that the performance after the first iteration of BICM-ID system with SSD is similar to the performance of a BICM-ID system without SSD at low SNR. Furthermore, by following the same analysis as in the previous section, the minimum value of can be achieved when the condition in (76) is satisfied. At this point, it is natural to ask whether there exists a class of unitary matrices that satisfies the above design criterion for any value of. Fortunately, thanks to the algebraic number theory, this class of is well studied in the literature. For example, the unitary matrix with size introduced in [2] and [26] falls into the class of optimal. The matrix is constructed as follows [2], [26]:... (79) and. More generally, for any value of, the unitary optimal matrix can be obtained based on the inverse fast Fourier transform (IFFT) matrix as follows [26]: (80) is an arbitrary integer and is the -point IFFT matrix whose th entry is given by. V. ILLUSTRATIVE RESULTS In this section, analytical and simulation results are provided to confirm the analysis carried out in the previous sections. The quadrature phase-shift keying (QPSK) modulation scheme with anti-gray mapping rule is employed. It should be mentioned here that Gray and anti-gray mappings are the only two available mappings for QPSK constellation. Anti-Gray mapping is chosen simply due to its superiority in providing a better error performance in iterative systems [11], [27]. This fact can also be 3 The matrix 2 is an unitary matrix if it satisfies 22 = I. Fig. 2. Performance of BICM-ID in OFDM systems: optimal subcarrier grouping and rotation is compared with that without rotation. confirmed for the coded OFDM systems under consideration by comparing the parameters in (39) for the two mappings when the optimal rotation matrix and subcarrier grouping are 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 convolutional code with generator matrix is applied as an outer code, 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 retained to make sure the accuracy of the bound. Each point in the BER curves is simulated with to coded bits. The following first two subsections present the results for the case that the channel gains are generated independently for each OFDM symbol and the optimal MAP demodulator is implemented. The last subsection studies the error performances of the proposed systems in a real channel environment by considering Channel Model A of HiperLan/2 [19]. Due to the large number of channel taps specified in this channel model, the high complexity of the optimal MAP demodulator makes it impractical to implement. To overcome this difficulty, a suboptimal SISO demodulator based on the vector Gaussian approximation (GA) developed in [23] is employed. A. i.i.d. Channels This subsection considers the i.i.d. channel model, in which the correlation matrix. Fig. 2 compares the BER performance with one, two, and five iterations of the system without SSD and to that of the system employing optimal grouping and rotation matrix when channel taps. The rotation matrix is chosen as in (79) with. It can be seen that performances of both systems converge to the analytical BER bounds. A significant coding gain is also achieved by the system employing SSD. In Fig. 3, the error performances after one, two, and five iterations for systems employing optimal grouping for the cases

10 1146 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 3, MARCH 2007 Fig. 3. Performance of BICM-ID in OFDM systems: optimal subcarrier grouping and rotation for F =2and F =4. Fig. 5. Performance of BICM-ID in OFDM systems: optimal subcarrier grouping with various rotation matrices 2. in OFDM with SSD. Though not explicitly shown here for the brevity of presentation, examining the error bounds plotted over a wider range of SNR reveals that, compared with the averaged and natural groupings, the use of the optimal grouping results in 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, the optimal grouping is applied. Shown in the figure are the BER performance after one and five iterations of systems using different choices of. These choices include i) an optimal matrix in (79), ii) a randomly generated : (81) Fig. 4. Performance of BICM-ID in OFDM systems with various carrier grouping approaches. of and are presented. The two rotation matrices are also chosen as in (79). It can be observed that there is almost no difference between the performances of the two systems at any iteration and they all converge to the same BER analytical bound. This results confirms our analysis for the optimal grouping and rotation matrices with different values of. Clearly, is preferred in terms of receiver complexity. Fig. 4 illustrates the performance of various grouping approaches. Besides the optimal one obtained in Section IV, we consider two more different approaches, namely natural grouping with and average grouping with. The error performances of systems employing the same 2 2 rotation matrix as in the previous case with one and five iterations are shown. Obviously, the error bounds of all systems are very tight, which makes them very useful tool to predict the error performance of BICM-ID and iii) the following optimal rotation matrix that maximizes the minimum product distance for real unitary transformation with a full SSD [28], given as (82) Also plotted in this figure are the error bounds for all the systems under consideration (the broken lines). Observe that, compared with the random and, the use of the optimal rotation matrix results in coding gains as high as 0.25 and 2.6 db at the BER level of, respectively. The results clearly agree with our analysis on the jointly optimum subcarrier group and rotation matrix. Note also that the error bounds are tight for all the systems. Finally, Fig. 6 shows the error performances of two systems employing optimal carrier grouping and rotation matrices with and. In the case of, the 4 4 rotation matrix is chosen from (79). Clearly, with optimal grouping and rotation matrix, the use of BICM-ID in OFDM with SSD successfully exploit the advantage of multipath diversity in channel environment with the richer scattering effect (i.e., corresponding to a larger number of channel taps).

11 TRAN et al.: BIT-INTERLEAVED CODED OFDM WITH SIGNAL SPACE DIVERSITY 1147 Fig. 6. grouping and rotation matrices are employed. Error performances for systems with L =2and L =4: optimal carrier Fig. 8. Error performance comparisons with Channel Model A in HiperLan/2. range of SNR, it is observed that performance degradation over the correlated channel transfers to about an additional 1 db required to achieve the BER level of, which is quite noticeable. Though not shown here due to space limit, the effects of subcarrier grouping and rotation matrix over correlated channels are very similar to that over the i.i.d. channels. Fig. 7. Error performances for systems operating over and correlated channels with L =2: optimal carrier grouping and rotation matrices are employed in both cases. B. Correlated Channels Fig. 7 first compares the BER performance with 1, 2 and 5 iterations of systems over the i.i.d. channel and a correlated channel with number of channel taps. For the correlated channel, the channel correlation matrix is assumed to be (83) For both systems, the same optimal carrier grouping and rotation matrix are employed. It can be seen that the BER performances over the correlated channel significantly degrade at any iterations. Observe that the asymptotic error bound is also very tight for correlated channel. The slopes of the asymptotic error performances of both systems are, however, the same, which confirms that both systems fully exploit the maximum diversity order. By examining the error bounds over a wider C. Channel Model A in HiperLan/2 In Channel Model A of HiperLan/2 [19], the carrier frequency is set at 5.2 GHz and the mobile s velocity is 3 m/s, which results in the Doppler frequency 52 Hz. Each OFDM symbol duration is 4 s, corresponding to the normalized Doppler frequency. There are eight channel taps (i.e., ), the variances in the tap order are given as {0.4505, , , , , , , } [5], [19]. The gains of the channel taps are modeled as independent circularly symmetric complex Gaussian random variables and they are generated according to the Jakes model. This means that there is correlation among channel taps over time. As mentioned earlier, the large number of channel taps implies a very high complexity of the optimal MAP demodulator in the systems with rotation. To reduce the receiver complexity of such systems, a suboptimal SISO demodulator based on the vector GA in [23] shall be used. In the case of the systems without rotation, the MAP demodulator is still employed. Fig. 8 compares the error performances of the proposed system implementing the optimal rotation and subcarrier grouping after one and ten iterations and that of the system without rotation after ten iterations. Due to the high correlation of the channel taps across multiple OFDM symbols, the random interleaver of length coded bits is applied, which corresponds to a frame of 1600 OFDM symbols. The same outer code as used in the previous subsections is applied. To serve as the reference, the error performance after five iterations and the analytical error bound of the proposed system when the channel taps are generated independently across OFDM symbols are provided. As can be seen from Fig. 8, implementing the optimal rotation and grouping significantly improves the

12 1148 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 3, MARCH 2007 error performance of the conventional system that does not have constellation rotation after ten iterations. For example, an SNR gain of about 2 db is realized by the proposed system at the BER level of. Compared with the case of independent fading across OFDM symbols, the error performance is significantly poorer because of the high correlation of the channel taps over time present in Channel Model A of HiperLan/2. VI. CONCLUSION This paper presented a performance analysis of BICM-ID for OFDM systems with SSD over frequency-selective Rayleighfading channels. The design of subcarrier grouping and the rotation matrix in order to reduce the receiver complexity while preserving the error performance is subsequently derived. The analytical expression of the tight performance bound when SSD is employed over all subcarriers allows us to predict the best achievable asymptotic error performance using SSD. From the result of this derivation, a joint optimum subcarrier grouping and rotation matrix was developed based on the linear Vandermonde system. It was demonstrated through both analytical and simulation results that using the proposed optimal grouping and rotation matrix, the best error performance of BICM-ID for OFDM with SSD can be achieved with low-complexity receiver. APPENDIX I THE OPTIMAL VALUES OF When (73) is satisfied, one has Furthermore Hence, in (67) is rewritten as follows: (84) (85) (86) After some manipulations, the product inside the square brackets is computed as (87) the last equality follows from the finite product of sin functions. It then follows that Furthermore, for any Therefore, it is easy to see that (88) (89) (90) ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their helpful comments that improve the presentation of the paper. REFERENCES [1] K. Boulle and J. C. Belfiore, Modulation schemes designed for the Rayleigh channel, in Proc. Conf. Information Sciences Systems, Princeton, NJ, Mar. 1992, pp [2] J. Boutros and E. Viterbo, Signal space diversity: A power and bandwidth efficient diversity technique for the rayleigh fading channel, IEEE Trans. Inf. Theory, vol. 44, pp , Jul [3] Z. Liu, Y. Xin, and G. B. Giannakis, Space-time-frequency coded OFDM over frequency-selective fading channels, IEEE Trans. Signal Process., vol. 50, no. 10, pp , Oct [4] Z. Wang and G. B. Giannakis, Complex-field coding for OFDM over fading wireless channels, IEEE Trans. Inf. Theory, vol. 49, no. 3, pp , Mar [5] Z. Liu, Y. Xin, and G. B. Giannakis, Linear constellation precoding for OFDM with maximum multipath diversity and coding gains, IEEE Trans. Commun., vol. 51, pp , Mar [6] Y. Rong, S. A. Vorobyov, and A. B. Gershman, Linear block precoding for OFDM systems based on maximization of mean cutoff rate, IEEE Trans. Signal Process., vol. 53, no. 12, pp , Dec [7] U. Sripati, B. S. Rajan, and V. Shashidhar, Full-diversity group-spacetime-frequency (GSTF) codes from cyclic codes, in Proc. IEEE Int. Symp. Information Theory, Sep. 2005, pp [8] S. Gowrisankar and B. S. Rajan, A rate-one full-diversity low-complexity space-time-frequency block code (STFBC) for 4-tx MIMO- OFDM, in Proc. IEEE Int. Symp. Information Theory, Sep. 2005, pp [9] B. Lu, X. Wang, and K. R. Narayanan, LDPC-based space-time coded OFDM systems over correlated fading channels: Performance analysis and receiver design, IEEE Trans. Commun., vol. 5, no. 1, pp , Jan [10] G. Caire, G. Taricco, and E. Biglieri, Bit-interleaved coded modulation, IEEE Trans. Inf. Theory, vol. 44, no. 3, pp , May [11] X. Li, A. Chindapol, and J. A. Ritcey, Bit-interleaved coded modulation with iterative decoding and 8-PSK signaling, IEEE Trans. Commun., vol. 50, no. 8, pp , Aug [12] I. Lee, A. M. Chan, and C.-E. W. Sundberg, Space-time bit-interleaved coded modulation for OFDM systems, IEEE Trans. Signal Process., vol. 52, no. 3, pp , Mar [13] D. Rende and T. F. Wong, Bit-interleaved space-frequency coded modulation for OFDM systems, IEEE Trans. Wireless Commun., vol. 4, no. 5, pp , Sep [14] Z. Hong and B. L. Hughes, Robust space-time codes for broadband OFDM systems, in Proc. IEEE Wireless Communications Networking Conf., Mar. 2002, pp [15] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge, U.K.: Cambridge, Univ. Press, [16] Z. Wang, S. Zhou, and G. B. Giannakis, Joint coding-precoding with low-complexity turbo-decoding, IEEE Trans. Wireless Commun., vol. 3, no. 3, pp , May [17] N. H. Tran and H. H. Nguyen, Design and performance of BICM-ID systems with hypercube constellations, IEEE Trans. Wireless Commun., vol. 5, no. 5, pp , May [18] B. Lu and X. Wang, Space-time code design in OFDM systems, in Proc. IEEE Global Telecommun. Conf. (IEEE GLOBECOM), Nov. 2000, pp [19] ETSI Normalization Commitee, Channel models for HIPERLAN/2 in different indoor scenarios, Eur. Telecommun. Standards Inst., Sophia- Antipolis, France, Doc. 3ER10085B. [20] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, A soft-input soft-output APP module for iterative decoding of concatenated codes, IEEE Commun. Lett., vol. 1, no. 1, pp , Jan [21] B. M. Hochwald and S. ten Brink, Achieving near-capacity on a multiple-antenna channel, IEEE Trans. Commun., vol. 51, no. 3, pp , Mar

13 TRAN et al.: BIT-INTERLEAVED CODED OFDM WITH SIGNAL SPACE DIVERSITY 1149 [22] N. H. Tran, H. H. Nguyen, and T. Le-Ngoc, Performance of BICM-ID with signal space diversity, IEEE Trans. Wireless Commun, to appear, accepted for publication. [23] Y. Li, X.-G. Xia, and G. Wang, Simple iterative methods to exploit the signal space diversity, IEEE Trans. Commun., vol. 53, no. 1, pp , Jan [24] D. Sharpe, Rings and Factorization. Cambridge, U.K.: Cambridge Univ. Press, [25] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C. Cambridge, U.K.: Cambridge Univ. Press, 1992 [Online]. Available: cbookcpdf.html [26] Y. Xin, Z. Wang, and G. B. Giannakis, Space-time diversity systems based on linear constellation precoding, IEEE Trans. Wireless Commun., vol. 2, pp , Mar [27] S. ten Brink, J. Speidel, and R. H. Yan, Iterative demapping for QPSK modulation, Electron. Lett., vol. 34, no. 15, pp , Jul [28] P. Dayal and M. Varanasi, An optimal two transmit antenna spacetime code and its stacked extensions, IEEE Trans. Inf. Theory, vol. 51, pp , Dec Nghi H. Tran (S 06) received the B.Eng. degree from Hanoi University of Technology, Hanoi, Vietnam, in 2002 and the M.Sc. degree (with Graduate Thesis Award) from the University of Saskatchewan, Saskatoon, SK, Canada, in 2004, all in electrical engineering. Since January 2005, he has been working towards the Ph.D. degree in the Department of Electrical Engineering at the University of Saskatchewan. His research interests span the areas of coded modulation and iterative decoding. Mr. Tran received the Graduate Thesis Award for this M.Sc. degree from the University of Saskatchewan in 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 Asian Institute of Technology, Bangkok, Thailand, in 1997, and the Ph.D. degree from the University of Manitoba, Winnipeg, MB, Canada, in He joined the Department of Electrical Engineering, University of Saskatchewan, Saskatoon, SK, Canada, in 2001 as an Assistant Professor and was promoted to the rank of Associate Professor in His research interests include digital communications, spread-spectrum systems, and error control coding. Dr. Nguyen is a registered member of the Association of Professional Engineers and Geoscientists of Saskatchewan (APEGS). Tho Le-Ngoc (F 97) received the B.Eng. degree (with Distinction) in electrical engineering and the M.Eng. degree in microprocessor applications from McGill University, Montreal, QC, Canada, in 1976 and 1978, respectively, and the Ph.D. degree in digital communications from the University of Ottawa, ON, Canada, in From 1977 to 1982, he was with Spar Aerospace Limited, Montreal, QC, Canada, he was involved in the development and design of satellite communications systems. From 1982 to 1985, he was an Engineering Manager of the Radio Group in the Department of Development Engineering of SRTelecom, Inc., Montreal, QC, Canada, he developed the new point-to-multipoint DA-TDMA/TDM subscriber radio system SR500. From 1985 to 2000, he was a Professor at the Department of Electrical and Computer Engineering of Concordia University, Montreal, QC, Canada. Since 2000, he has been with the Department of Electrical and Computer Engineering of McGill University. His research interest is in the area of 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, a Fellow of the Engineering Institute of Canada (EIC), and a Fellow of the Canadian Academy of Engineering (CAE). He is the recipient of the 2004 Canadian Award in Telecommunications Research and recipient of the IEEE Canada Fessenden Award 2005.