A Novel and Efficient Mapping of 32-QAM Constellation for BICM-ID Systems

Wireless Pers Commun DOI 10.1007/s11277-014-1848-2 A Novel and Efficient Mapping of 32-QAM Constellation for BICM-ID Systems Hassan M. Navazi Ha H. Nguyen Springer Science+Business Media New York 2014 Abstract Bit-interleaved coded modulation with iterative decoding (BICM-ID) is a bandwidth-efficient technique for both additive white Gaussian noise and fading channels. The asymptotic performance of BICM-ID is strongly determined by how the coded bits are mapped to the symbols of the signal constellation. In this paper an explicit mapping method is presented for 32-QAM using two criteria: (i) maximization of the minimum Euclidean distance between the symbols with Hamming distance one, and (ii) minimizing the number of symbols which have jointly the minimum Hamming distance and the minimum Euclidean distance from each other. Our method is much simpler than the previously-known methods. Compared to previously-known best mapping, the mapping obtained by our method performs significantly better in a BICM-ID system implemented with hard-decision feedback, while its asymptotic performance is almost the same in a BICM-ID system using soft-decision feedback. Keywords Bit-interleaved coded modulation with iterative decoding (BICM-ID) Symbol mapping 32-QAM Euclidean distance Hamming distance 1 Introduction Combining channel coding with high-order modulation is an important design task in spectrally-efficient digital communication systems. Trellis-coded modulation (TCM) is the first bandwidth-efficient coding approach, originally proposed by Ungerboeck in [1]. TCM was designed by combining a convolution code with a high-order modulation in order to maximize the minimum Euclidean distance between coded signal sequences. The design approach of TCM works well for an additive white Gaussian noise (AWGN) channel, but H. M. Navazi (B) Department of Electrical and Computer Engineering, Bonab Branch, Islamic Azad University, Bonab, Iran e-mail: hnavazi@gmail.com; hassan.mohammadnavazi@bonabiau.ac.ir H. H. Nguyen Department of Electrical and Computer Engineering, University of Saskatchewan, Saskatoon, Canada

H. M. Navazi, H. H. Nguyen TCM generally does not perform well over fully-interleaved Rayleigh-fading channels due to its low diversity order. A more effective design of coded-modulation schemes for fading channels was proposed by Zehavi in [2] and it is now commonly known as bit-interleaved coded modulation (BICM). The design approach of BICM introduces an interleaver between the channel encoder and the high-order modulator (or symbol mapper). By doing so, BICM benefits from the bitby-bit interleaving and outperforms the symbol-interleaved TCM on fading channels [2,3]. However, bit-interleaving leads to a random modulation [2,4], which causes a reduction in the free Euclidian distance, hence degrading the performance of BICM over AWGN channels [2,3]. The performance of BICM over AWGN channels can be improved by performing iterative demodulation and decoding at the receiver, which results in the so-called bit-interleaved coded modulation with iterative decoding (BICM-ID) [5 7]. It has also been very well-known that the mapping of coded bits (at the output of a convolutional encoder) to the symbols in the modulation constellation directly impacts the performance of a BICM-ID system. In fact, there have been many research studies on this topic [8 17]. In particular, reference [17] applies the binary switching algorithm (BSA) to find the locally-optimum mappings for BICM-ID. From many aspects, the BSA is practically the best known algorithm to find good mappings for arbitrary two-dimensional constellations. In the BSA, a cost function is defined for each symbol of the constellation. The BSA starts with an initial mapping, then picks the constellation symbol with the highest cost and tries to switch the index (i.e., the bit mapping) of this symbol with the index of another symbol so that the total cost reduction due to the switch is as large as possible [17]. Although the BSA is quite effective for constellations of small sizes, its effectiveness diminishes severely for larger constellations when the number of possible mappings quickly increases (in genral, the number of possible mappings is 2 m! for an 2 m -ary constellation). In this paper, an efficient method is presented to find good mappings of 32-QAM constellation for BICM-ID. Our algorithm obtains the new mappings by maximizing the minimum Euclidean distance between the pairs of symbols with Hamming distance one and minimizing the number of the symbols which jointly have the minimum Hamming distance and the minimum Euclidean distance. Compared to the BSA, the proposed algorithm does not require computer search and it yields a mapping that performs considerably better over an AWGN channel when the hard decision feedback is implemented at the iterarive receiver, whereas it has a very similar performance under the soft decision feedback. The rest of the paper is organized as follows. Section 2 reviews BICM-ID system models. Section 3 describesthe proposedmapping method. Section4 presents simulation results and discusses performance comparison. Section 5 concludes the paper. 2 System Model Figure 1 shows the block diagram of a BICM-ID system. The transmitter is built from a serial concatenation of a convolutional encoder, a random bit interleaver and asymbol mapper. A block ofn information bits, denoted by u, is first encoded by a convolutional encoder with rate R.TheN-bit encoded sequence c is interleaved by a random bit interleaver. The interleaved bit sequence v is divided into Q consecutive m-bit vectors, v t = (v 1 t,..., vm t ), t {1,..., Q}. Each vector v t is mapped to a complex constellation symbol x t = μ(v t ) chosen from the 2 m -ary constellation χ by some constellation labeling map μ.

A Novel and Efficient Mapping of 32-QAM Constellation u Encoder c Π v S/P v t Mapper x t Channel u Decoder 1 Π Le ( V ) P/S Demapper y t Π La ( V ) Fig. 1 BICM-ID system model The baseband input/output channel model at time index t can be described as y t = ρ t. x t + n t,whereρ t is a zero-mean, unit-variance complex Gaussian random variable representing the channel gain, and n t is a zero-mean complex Gaussian random variable with variance N 0, which represents the additive white Gaussian noise. Note that the magnitude of ρ t is Rayleigh distributed. The transmitted symbol x t belongs to a constellation χ,which has an average symbol energy of E s. It is assumed that the receiver has the perfect channel state information. Note also that for the case of an AWGN channel, ρ t = 1. At the receiver, from the received signal y t and the apriorilog-likelihood ratios (LLRs) of the coded bits, the demapper computes the extrinsic LLR for each of the labeling bits as described in [8]. After being permuted by the random deinterleaver, the extrinsic LLRs are applied to the channel decoder. The decoder then calculates the extrinsic LLRs on the coded bits using the BCJR algorithm [18]. Then, after interleaving, these LLRs are fed back to the demapper and will be used as the apriorillrs in the next iteration. The feedback mechanism described above uses the soft decisions at the output of the channel decoder. To reduce the computational complexity, the Viterbi algorithm can be used instead of the BCJR algorithm. In such a case, only the binary hard decisions are available at the decoder s output but they can still be fed back as used as the aprioriinformation to refine the operation of the demapper in the next iteration [5]. It should be pointed out that, although having much less complexity, the hard-decision feedback mechanism suffers some performance loss when compared to the soft-decision feedback. 3 Proposed Mapping Algorithm of 32-QAM for BICM-ID The minimum Euclidean distance between any pair of symbols whose labels differ in only one bit has a direct impact on the asymptotic error performance of a BICM-ID system, especially in the case of implementing hard-decision feedback at the iterative receiver. The explanation for this is that at high signal-to-noise ratio (SNR), except for the bit under consideration in the demapping process, the information about the other bits is known via the channel decoding process. Such a minimum Euclidean distance parameter shall also be taken into account in this paper to develop a new mapping algorithm for 32-QAM. The 32-QAM constellation can be separated into pairs of symbols with 17 different Euclidean distances, denoted as d k, k {1,..., 17}, Without loss of generality, it is assumed that d i > d j if i > j. Figure 2 shows the 32-QAM constellation and indicates d k for k = 1, 2, 6, 7. The impact of constellation mapping on the asymptotic performance of a BICM-ID system can be succinctly quantified by the bit-wise distance spectrum given full prior information as introduced [17]. This distance spectrum is defined as:

H. M. Navazi, H. H. Nguyen Fig. 2 Symbol arrangements, partitioning and distances in 32-QAM constellation 1 2 Q 30 29 5 d 1 9 13 17 21 25 6 10 14 18 d 6 22 26 7 11 15 19 23 27 I 8 d 7 12 16 20 d 2 24 28 4 3 31 32 Center -symbols Middle-symbols Side-symbols w 1 (k) 1 M M M i=1 j=1 ( ( d s i, s j i ) = d k ) ; k = 1, 2,...,17, where M = 2 m is the size of the constellation (i.e., modulation order), (A) is 1 if the event A is correct, otherwise it is 0, d(, ) is the distance function and s j i is the symbol whose label differs from the label of s i only at the j-th bit. In general, a mapping that yields a smaller value for a lower-index element of vector w 1 will have better performance in subsequent iterations between the decoding and demapping processes. The key to come up with a good mapping scheme for BICM-ID is to make (i) the minimum Euclidean distance between the pair of symbols whose labels differ in only one bit as large as possible and (ii) the number of the symbols which jointly have the minimum Hamming distance and the minimum Euclidean distance from each other as small as possible. It is not hard to show that for 32-QAM, the minimum Euclidean distance between the symbols whose labels differ in one bit can be made to be at least d 7. As a consequence, we introduce the following definitions in developing our mapping algorithm: (1) Forbidden set of a label: This is a set of labels that are different from the reference label only in one bit. (2) Forbidden set of a symbol: This refers to all the symbols whose Euclidean distances from a given symbol are smaller than d 7. Also, we classify symbols in the 32-QAM constellation as follows (see also Fig. 2): Center-symbols: These are the four symbols in the center of the constellation, namely symbols {s 14, s 15, s 18, s 19 }. Side-symbols: These are the 16 outside symbols {s 1 8 } and {s 25 32 }. Middle-symbols: These are the 12 remaining symbols. If the labels of the center-symbols are determined so that the union of the forbidden sets of them has 16 labels, then, by mapping these 16 labels to the 16 side-symbols, it is possible to obtain a mapping whose minimum Euclidean distance between the symbols with Hamming

A Novel and Efficient Mapping of 32-QAM Constellation Table 1 Decimal values of binary vectors [A] [a 4 a 3 a 2 a 1 ] [a 5 a 3 a 2 a 1 ] [a 5 a 4 a 2 a 1 ] [a 5 a 4 a 3 a 1 ] [a 5 a 4 a 3 a 2 ] 0 0 0 0 0 0 1 1 1 1 1 0 2 2 2 2 0 1 3 3 3 3 1 1 4 4 4 0 2 2 5 5 5 1 3 2 6 6 6 2 2 3 7 7 7 3 3 3 8 8 0 4 4 4 9 9 1 5 5 4 10 10 2 6 4 5 11 11 3 7 5 5 12 12 4 4 6 6 13 13 5 5 7 6 14 14 6 6 6 7 15 15 7 7 7 7 16 0 8 8 8 8 17 1 9 9 9 8 18 2 10 10 8 9 19 3 11 11 9 9 20 4 12 8 10 10 21 5 13 9 11 10 22 6 14 10 10 11 23 7 15 11 11 11 24 8 8 12 12 12 25 9 9 13 13 12 26 10 10 14 12 13 27 11 11 15 13 13 28 12 12 12 14 14 29 13 13 13 15 14 30 14 14 14 14 15 31 15 15 15 15 15 distance 1 is larger than d 6. Based on such an observation, the main idea of the proposed method is that the labels of the center-symbols must have common members in their forbidden sets. As a general rule, the labels which are different just in one bit must always be used for those symbols with a Euclidean distance larger than d 6. Note that the forbidden sets for two different labels either have two common members or have no common member. The proposed mapping method is explained with the help of Table 1. Thefirstcolumn of the table, indicated as A, shows the decimal representations of five-bit binary numbers (a 5 a 4 a 3 a 2 a 1 ) 2. The other columns show the decimal representations of the four-bit binary numbers resulting from eliminating one bit in the five bits of a binary number. Using these columns, the numbers that are different just in one bit can be easily recognized. If two rows of Table 1 have the same value in a given column, then the indices of these rows are different

H. M. Navazi, H. H. Nguyen only in the omitted bit in that column. For example, because numbers 0 and 1 in the first column have the same value in the last column, they are different in bit a 1 which is omitted in the last column. In general, Table 1 is used as follows. Whenever label p is mapped to symbol q in the constellation, the row p shall be marked by q and the rows that are numbered by the forbidden set of p are marked by q. The presence of q in the front of a row indicates that the label on the first column of this row, i.e., the row-number, is used and cannot be used for any other symbol in the constellation. The proposed mapping method is completed in three steps. In the first step, the mapping is applied to the four center-symbols; each of them has the largest number of forbidden symbols. In the second step, the labels are decided for the sixteen side-symbols. In the final step, the mapping rule is applied to the middle-symbols. The detailed operations of the proposed algorithm are described next. 3.1 Proposed Mapping Algorithm for 32-QAM Constellation Step 1 (Mapping of the center-symbols): The four center-symbols are partitioned into twomember subsets so that two symbols in each subset have the Euclidean distance d 1 from each other. Then, the mapping is performed under the following conditions: (1) The labels of each subset have common members in their forbidden sets. In other words, these labels have a Hamming distance 2 from each other. (2) One pair of center-symbols with Euclidean distance d 1, where each of them belongs to a different subset, have a Hamming distance 3 from each other, and the remaining two symbols have a Hamming distance 5 from each other. (3) Any pair of the center-symbols with Euclidean distance d 2 have a Hamming distance 3 from each other. To help the explanation of the next steps, suppose that at the end of Step 1 the center-symbols are divided into two subsets {s 14, s 18 } and {s 15, s 19 } where the Hamming distance between s 14 and s 15, which have a Euclidean distance d 1,is3,and,s 18 and s 19 have a Hamming distance 5 from each other. Consequently, at the end of the first step, each of the center-symbols prohibits three labels individually, and there are four labels each of them is prohibited by two center-symbols jointly. In addition, there are 12 labels that are not prohibited by any center-symbol. Step 2 (Mapping of the side-symbols): In this step, the 16 labels, each of them has been prohibited by at least one center-symbol in the first step, are mapped to the 16 side-symbols. More precisely, among these 16 labels, the 12 labels, each of them has been prohibited exactly by one center-symbol in the first step, are mapped to the side-symbols which belong to the quarter in the opposite side of the forbidding center-symbol (see Fig. 3a, b). Moreover, the four remaining labels, each of them is prohibited by two center-symbols in the first step, are mapped to the symbols in the corner of the constellation, i.e., {s 1, s 4, s 29, s 32 } (see Fig. 3c). The mapping is also applied under the condition that the two labels which are different just in one bit must be mapped to the two-symbols whose Euclidean distance is larger than d 6. It should be noted that the opposite quarter of the constellation for the first and the second quarters are the third and fourth quarters, respectively. Also, the opposite quarters for center symbols s 14, s 15, s 18 and s 19 are the fourth, first, third and second quarters, respectively. It is most convenient to start the mapping of the side-symbols from an opposite quarter of one of the center-symbols that has a Hamming distance 5 from the other center-symbol with Euclidean distance d 1 from it. The step then moves to the quarter of the constellation

A Novel and Efficient Mapping of 32-QAM Constellation Q 30 2 Q 1 Q 29 25 5 6 18 26 6 14 14 18 I I I 7 15 19 27 15 19 8 3 ( a) ( b) 31 28 4 ( c) 32 Fig. 3 Mapping of the side-symbols in 32-QAM constellation whose opposite center-symbol has a Hamming distance 3 from the opposite center-symbol of the last quarter. As considered in Step 1, since s 18 and s 19 have a Hamming distance 5 from each other, the side-symbol-mapping should start from the second or third quarter, which are the opposite quarters of symbols s 19 and s 18 respectively. Suppose that the mapping starts from the third quarter. Consequently, the mapping will be consecutively applied to the side-symbols in the first, fourth, and second quarters. The details are as follows: (2-A) Mapping of the side-symbols in the third quarter: There are three labels that have been prohibited in the first step by only s 18. Two of these labels, each has a Hamming distance 2 from s 15, are mapped to s 7 and s 8. Then, the third label which has a Hamming distance 4 from s 15 is mapped to s 3. One of the labels has been prohibited in the first step by both s 14 and s 18 and it should be mapped to s 4 considering the condition that the label of s 4 must have a Hamming distance 4 from s 15. (2-B) Mapping of the side-symbols in the first quarter: Among the three labels that are prohibited by only s 15 in the first step, the label prohibited by both s 7 and s 8 is mapped to s 26, the label prohibited by s 7 is mapped to s 25, and the one prohibited by s 8 is mapped to s 30. Furthermore, one of the labels prohibited by both s 15 and s 19 in the first step is mapped to s 29 considering the condition that the label of s 29 must have a Hamming distance 2 from s 14. There is no any other mapping for this sub-step that can lead to a better mapping for 32-QAM. (2-C) Mapping of the side-symbols in the fourth quarter: Among the three labels prohibited in the first step by only s 14, the label prohibited by both s 29 and s 30 is mapped to s 28, the one prohibited by both s 25 and s 29 is mapped to s 31, and the last label, which has a Hamming distance larger than 1 from the side-symbols of the first quarter, is mapped to s 27. Finally, the label prohibited by both s 14 and s 18 in the first step and having a Hamming distance 2 from the symbol s 15 is mapped to s 32. (2-D) Mapping of the side-symbols in the second quarter: Among the three labels that are prohibited in the first step by only s 19, the label with Hamming distance 4 form s 14 is mapped to s 2, the one prohibited by both s 27 and s 31 must be mapped to s 5, and the remaining one which is prohibited by both s 27 and s 28 is mapped to s 6. Lastly, the label prohibited by both s 15 and s 19 in the first step and having a Hamming distance 4 from the symbol s 14 is mapped to s 1. Step 3 (Mapping of the middle-symbols): There are twelve labels remain for this step and none of them is prohibited by any center-symbol. In other words, they have a Hamming distance larger than 1 from the center-symbols. Using the results of the previous steps and applying the general rule, these labels are assigned to the middle-symbols. To facilitate the

H. M. Navazi, H. H. Nguyen Fig. 4 Subsets of middle-symbols in 32-QAM constellation for the proposed algorithm in Step 3 A 1 B 1 C 1 B 2 A 2 C 2 D 1 E 1 D 2 E 2 mapping process in this step, the middle-symbols are divided into subsets as shown in Fig. 4. Each of the remaining 12 labels belongs to only one of these subsets. Table 2 describes the implementation of the proposed algorithm, particularly how the subsets are used in the last step of the algorithm. The presence of a subset, for example A 1, in front of any row in the last step of Table 2 means that the label of this row can only be mapped to a symbol which belongs to the marked subset. Figure 5 shows the resulting mapping of the Table 2. 4 Numerical Results First, Table 3 compares several important parameters concerning the error performance in BICM-ID systems for two 32-QAM mappings: our proposed mapping and the M32 a mapping obtained in [17] by the BSA. As Table 3 shows, our proposed mapping considerably outperforms the M32 a mapping in terms of the bit-wise distance spectrum given full prior information, i.e., w 1. As discussed before, this is a very critical parameter in determining the asymptotic performance of a BICM-ID system with a hard-decision feedback. Figure 6 compares the BER performance of the proposed 32-QAM mapping and M32 a mapping in a BICM-ID system implemented with hard-decision feedback and seven iterations. The channel coding used is a 4-state, rate-1/2 convolutional code with generator sequence (5, 7) 8, whereas the interleaver length is 10,000 bits. It can be seen from Fig. 6 that the proposed mapping considerably outperforms the M32 a mapping over an AWGN channel in the high SNR region. Specifically, the error floor of the proposed mapping is significantly lower than the error floor of the M32 a mapping. Note that the SNR is measured as E b /N 0, where E b is the energy per information bit. In terms of parameter d h 2, which determines the asymptotic performance of a BICM-ID system using soft-decision feedback, Table 3 shows that the proposed mapping performs very close to the M32 a mapping. This theoretical prediction is confirmed by the simulation results in Fig. 7, from which one can observe that the proposed mapping achieves the same BER with just about 0.05 db higher SNR. Note that the convolutional code used for this figure has generator sequences (5, 7) 8 and the interleaver also has a length of 10,000 bits.

A Novel and Efficient Mapping of 32-QAM Constellation Table 2 Implementation of the proposed algorithm [A] [a4a3a2a1] [a5a3a2a1] [a5a4a2a1] [a5a4a3a1] [a5a4a3a2] First step Second step Last step 0 0 0 0 0 0 14 1 1 1 1 1 0 14,18 25,30 32 2 2 2 2 0 1 14,18 4 3 3 3 3 1 1 18 4 4 4 0 2 2 14 25,29 31 5 5 5 1 3 2 15 7 25 6 6 6 2 2 3 7,4 31 B1 11, 22 7 7 7 3 3 3 18 7 8 8 0 4 4 4 14 30,29 28 9 9 1 5 5 4 15 8 30 10 10 2 6 4 5 8,4 28 D1 11, 21 11 11 3 7 5 5 18 8 12 12 4 4 6 6 15,19 29 13 13 5 5 7 6 15 14 14 6 6 6 7 26,29 2 B2 11 15 15 7 7 7 7 15 7,8 26 16 0 8 8 8 8 14 27 17 1 9 9 9 8 3 27,32 A1 10 18 2 10 10 8 9 3,4 27 C1 20,12 13 19 3 11 11 9 9 18 3 20 4 12 8 10 10 19 31,27 5 21 5 13 9 11 10 25 5,1 D2 10, 24 22 6 14 10 10 11 2, 5 E2 22, 17 12 23 7 15 11 11 11 3,7 B1 24, 17

H. M. Navazi, H. H. Nguyen Table 2 continued [A] [a4a3a2a1] [a5a3a2a1] [a5a4a2a1] [a5a4a3a1] [a5a4a3a2] First step Second step Last step 24 8 8 12 12 12 19 28,27 6 25 9 9 13 13 12 30 6,1 A2 10, 23 26 10 10 14 12 13 2,6 A2 21, 20 27 11 11 15 13 13 3,8 E1 23,20 9 28 12 12 12 14 14 19 29 13 13 13 15 14 15,19 1 30 14 14 14 14 15 19 2 31 15 15 15 15 15 26 2,1 C2 17,9 16

A Novel and Efficient Mapping of 32-QAM Constellation Fig. 5 The proposed mapping for 32-QAM Q 11101 11110 01001 01100 10100 11011 10010 10111 01010 00101 11000 10001 00000 00011 00110 01111 00111 01110 01101 11100 11001 10000 I 01011 10110 11111 11010 10101 01000 00010 10011 00100 00001 Table 3 Comparison of the bit-wise distance spectrum given full prior information, w 1, harmonic mean without feedback dh 2 [9], and harmonic mean with ideal feedback d h 2 [9] for 32-QAM mappings 32-QAM Mapping d 2 h d 2 h w 1 Proposed mapping 0.2112 2.7260 [0, 0, 0, 0, 0, 0, 1.75, 1.125, ] M32 a mapping 0.2092 2.7849 [0, 0, 0, 0, 0.25, 0.125, 1, 1.125, ] 10-2 Proposed Mapping M32 a 10-3 10-4 BER 10-5 10-6 10-7 5.9 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 E b /N 0 (db) Fig. 6 BER performance with seven iterations of a BICM-ID system implemented with hard-decision feedback over an AWGN channel: (5, 7) 8 convolutional code and 10,000-bits interleaver Finally, it should be pointed out that our proposed mapping method can be easily adapted to work with the 16-QAM constellation and generalized for higher-order QAM constellations, such as 64-QAM. In fact we were able to generate the optimum (M16 a ) and suboptimum (MSEW) mappings for 16-QAM as reported in [17].

H. M. Navazi, H. H. Nguyen 10 0 10-1 Proposed Mapping M32 a 10-2 BER 10-3 10-4 10-5 10-6 10-7 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 E b /N 0 (db) Fig. 7 BER performance with seven iterations of a BICM-ID system implemented with soft-decision feedback over an AWGN channel: (5, 7) 8 convolutional code and 10,000-bits interleaver 5Conclusion In this paper, an explicit method has been presented to obtain a new and efficient mapping of 32-QAM for a BICM-ID system operating over an AWGN channel. The proposed method maximizes the minimum Euclidean distance between any pair of symbols with Hamming distance one, and simultaneously minimize the number of pairs of symbols which have jointly the minimum Euclidean and Hamming distances. Our method is much simpler than the previously-known methods. It has been shown that the obtained mapping performs remarkably better than the best existing mapping of 32-QAM in a BICM-ID system implemented with hard-decision feedback, while its performance in a BICM-ID system implemented with soft-decision feedback is very close to the performance of the previously-known best mapping, i.e., the M32 a mapping. The proposed method also can also be adapted to work with 16-QAM constellation and generalized for higher-order QAM constellations, such as 64- QAM. Acknowledgments The authors would like to thank Dr. B. Mozaffary from Tabriz University for his contribution in this paper. References 1. Ungerboeck, G. (1982). Channel coding with multilevel/phase signals. IEEE Transactions on Information Theory, 28, 56 67. 2. Zehavi, E. (1992). 8-PSK trellis codes for a Rayleigh fading channel. IEEE Transactions on Communications, 40, 873 883. 3. Caire, G., Taricco, G., & Biglieri, E. (1998). Bit-interleaved coded modulation. IEEE Transactions on Information Theory, 44, 927 946. 4. Sundberg, C.-E. W., & Seshadri, N. (1993). Coded modulation for fading channels: An overview. European Transactions on Telecommunications, 4, 325 334.

A Novel and Efficient Mapping of 32-QAM Constellation 5. Li, X., & Ritcey, J. A. (1997). Bit-interleaved coded modulation with iterative decoding. IEEE Communications Letters, 1, 169 171. 6. Li, X., & Ritcey, J. (1998). Bit-interleaved coded modulation with iterative decoding using soft feedback. Electronics Letters, 34, 942 943. 7. Speidel, J., ten Brink, S., & Yan, R. (1998). Iterative demapping and decoding for multilevel modulation. In Proceedings IEEE global conference telecommunications, Sydney, Australia (Vol. 1, pp. 579 584). 8. Chindapol, A., & Ritcey, J. A. (2001). Design, analysis and performance evaluation for BICM-ID withsquare QAM constellations in Rayleighfading channels. IEEE Journal on Selected Areas in Communications, 19, 944 957. 9. Li, X., Chindapol, A., & Ritcey, J. A. (2002). Bit-interleaved coded modulation with iterative decoding and 8PSK signaling. IEEE Transactions on Communications, 50, 1250 1257. 10. Tran, N. H., & Nguyen, H. H. (2007). A novel multi-dimensional mapping of 8-PSK for BICM-ID. IEEE Transactions on Wireless communications, 6, 1133 1142. 11. Simoens, F., Wymeersch, H., Bruneel, H., & Moeneclaey, M. (2005). Multidimensional mapping for bitinterleaved coded modulation with BPSK/QPSK signaling. IEEE Communications Letters, 9, 453 455. 12. Tran, N. H., & Nguyen, H. H. (2006). Design and performance of BICM-ID systems with hypercube constellations. IEEE Transactions on Wireless Communications, 5, 1169 1179. 13. Changyou, G. (2010). A new constellation mapping scheme of MPSK in BICM-ID. In International IEEE 2nd international conference on networks security, wireless communications and trusted computing (Vol. 1, pp. 386 389). 14. Tran, N. H., & Nguyen, H. H. (2006). Signal mappings of 8-ary constellation for bit-interleaved coded modulation with iterative decoding. IEEE Transactions on Broadcasting, 52, 92 99. 15. Tan, J., & Stüber, G. L. (2005). Analysis and design of symbol mappers for iteratively decoded BICM. IEEE Transactions on Wireless Communications, 4, 662 672. 16. Brännström, F., & Rasmussen, L. K. (2009). Classification of unique mappings for 8PSK based on bit-wise distance spectra. IEEE Transactions on Information Theory, 55, 1131 1145. 17. Schreckenbach, F., Gortz, N., Hagenauer, J., & Bauch, G. (2003). Optimization of symbol mappings for bit-interleaved coded modulation with iterative decoding. IEEE Communications Letters, 7, 593 595. 18. Bahl, L. R., Cocke, J., Jelinek, F., & Raviv, J. (1974). Optimal decoding of linear codes for minimising symbol error rate. IEEE Transactions on Information Theory, 20, 284 287. 19. Wesel, R. D., Liu, X., Cioffi, J. M., & Komninakis, C. (2001). Constellation labeling for linear encoders. IEEE Transactions on Information Theory, 47(6), 2417 2431. Hassan M. Navazi was born in Leilan, Eastern Azarbayjan, Iran, in 1984. He received the B.S. degree from Birjand University, Birjand, Iran, in 2009, the M.Sc. degree from Tabriz University, Tabriz, Iran, in 2011, all in electrical engineering. Since October 2011 he has been working as an instructor in the Department of Electrical and Computer Engineering at the Islamic Azad University, Bonab branch, Bonab, Iran. Mr. Navazi s research interests span the areas of wireless communications and information theory.

H. M. Navazi, H. H. Nguyen Ha H. Nguyen (M 01, SM 05) received the B.Eng. degree from the Hanoi University of Technology (HUT), Hanoi, Vietnam, in 1995, the M.Eng. degree from the Asian Institute of Technology (AIT), Bangkok, Thailand, in 1997, and the Ph.D. degree from the University of Manitoba, Winnipeg, MB, Canada, in 2001, all in electrical engineering. He joined the Department of Electrical and Computer Engineering, University of Saskatchewan, Saskatoon, SK, Canada, in 2001, and became a full Professor in 2007. He holds adjunct appointments at the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB, Canada, and TRLabs, Saskatoon, SK, Canada, and was a Senior Visiting Fellow in the School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, Australia during October 2007-June 2008. His research interests include spread spectrum systems, error-control coding and diversity techniques in wireless communications. Dr. Nguyen was an Associate Editor for the IEEE Transactions on Wireless Communications during 2007-2011. He currently serves as an Associate Editor for the IEEE Transactions on Vehicular Technology and the IEEE Wireless Communications Letters.