A Novel Multi-Dimensional Mapping of 8-PSK for BICM-ID

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1 A Novel Multi-Dimensional Mapping of 8-PSK for BICM-ID Nghi H. Tran and Ha H. Nguyen Department of Electrical Engineering, University of Saskatchewan Saskatoon, SK, Canada S7N 5A9 Abstract Employing multi-dimensional constellation and mapping to improve the error performance of bit-interleaved coded modulation with iterative decoding (BICM-ID) has recently received a lot of attention, both in single-antenna and multipleantenna systems. To date, except for the cases of BPSK and QPSK constellations, good multi-dimensional mappings have only been found by computer searching techniques. This paper introduces an explicit algorithm to construct a good multi-dimensional mapping of 8-PSK for improving the asymptotic performance of BICM-ID systems. By comparing the performance of the proposed mapping with an unachievable lower bound, it is conjectured that the proposed mapping is the globally optimal mapping. The superiority of the proposed mapping over the best conventional (two-dimensional) mapping and the multidimensional mapping found previously by computer search is also demonstrated. I. INTRODUCTION Signal constellation and mapping play an important role in determining the bit error rate (BER) performance of a BICM-ID system. Good one- and two-dimensional mappings have been proposed for single-antenna systems [] [6] as well as multiple-antenna systems [7], [8]. More recently, considerable attention has been paid to the use of multidimensional (multi-d) mapping to further improve the error performance of BICM-ID. In particular, reference [9] studies hypercube mappings of QPSK constellation for BICM-ID in single-antenna systems, where it is shown that a significant coding gain can be achieved without any bandwidth nor power expansion. A parallel research work was also carried out in [0] but it only concentrates on an AWGN channel. The technique of multi-d mapping has also been extended to BICM-ID for multiple-antenna systems in [] [3]. It is observed in [3] that performance of the system with multi-d mapping can achieve near turbo-code performance with only a simple convolutional code. To the best of our knowledge, except [9] and [0], all research work related to the mapping design problem is only based on some searching techniques. For example, the bruteforce computer searching can be carried out for the singleantenna systems that use a low-order signal constellation and the conventional two-dimensional (-D) mapping []. In the case of multi-d mapping with a high-order constellation, exhaustive computer search to find the optimal mapping is impossible due to the huge complexity of the search. As an example, with a very simple single-antenna system employing 8-PSK, there are 64! = possible 4-D mappings. For such a system, the binary switching algorithm (BSA) [4] can then be used. However, it should be emphasized that the BSA only gives locally optimal mappings. Furthermore, for a large constellation (such as high-order multi-d constellations), the BSA quickly becomes intractable and a more complicated searching inside a selected list must be applied [3]. Another searching technique, related to the quadratic assignment problem (QAP), is also presented in [8]. This technique, however, has the same disadvantages with that of the BSA for larger constellations. The contribution of this paper is to introduce a novel design of multi-d mapping for BICM-ID that employs 8-PSK constellation. The proposed mapping exploits the symmetry of the 8-PSK constellation and the simple fact that an 8-PSK constellation can be decomposed into two QPSK constellations. In a -dimensional signal space, it can be easily verified that the proposed mapping is indeed the globally optimal mapping. For a higher dimension, unfortunately, whether the proposed mapping is the globally optimal mapping still remains to be answered. Nevertheless, by comparing the performance of the proposed mapping with an unachievable lower bound, it is shown that there is only a very small gap between the performance of the proposed mapping and the bound. It is therefore conjectured that the proposed mapping is actually the globally optimal mapping. {r} u Demodulator Encoder II. SYSTEM MODEL TRANSMITTER c ~ c {si} Interleaver Modulator Channel P ( c~ ; I) P( c; O) Interleaver Fig.. P( c~ ; O) Deinterleaver RECEIVER P( c; I) Block diagram of a BICM-ID system. SISO Decoder P( u; O) Fig. shows the block diagram of a BICM-ID system. For /06/$0.00 (c) 006 IEEE

2 conventional BICM-ID systems, every group of m = log M coded bits is mapped to one signal point in a -D M-ary constellation. In contrast to these conventional systems, here, mncoded bits, n >, are simultaneously mapped to n consecutive M-ary signal points. As a result, a bigger constellation Ψ in n-d signal space is created. Each n-d signal point can be represented by s i =[s i,,s i,,...,s i,n,s i,n ]. Equivalently, s i =[q i,,...,q i,n ], where q i,p =[s i,p,s i,p ], p n, represents the pth conventional -D M-ary symbol. Each signal s i is labelled by l = mn bits as follows: s i a i =(a i,,a i,,...,a i,l )=ξ (s i ) () where a i,k, k l, is either 0 or and ξ denotes the mapping from l bits to one n-d signal, i.e., s i = ξ(a i ). Clearly, any -D mapping used in a conventional system with M-ary constellation is just a special case of the above general mapping. Moreover, there is no change in bandwidth efficiency due to the use of this general mapping. Let r =[r,...,r n ] represent the received signal in a n-d signal space. For a frequency non-selective slowly Rayleigh fading channel and coherent detection, it suffices to write the received signal r p as follows []: r p = g p q i,p + w p () where p n and i is the index of the transmitted signal. In (), w p is complex white Gaussian noise with independent components having two-sided power spectral density N 0 /. The scalar g p is a Rayleigh random variable of unit mean squared value that represents the fading amplitude of the pth M-ary symbol. It is assumed that the channel fades slowly and two cases of fading amplitude are considered: (i) The coefficients g p are different for different M-ary symbols in one n-d signal (fast fading); and (ii) The coefficients g p are constant over n consecutive M-ary symbols, i.e., over the duration of the n-d signal. The channel model in the second case is usually referred to as a quasistatic fading channel [5]. Due to the presence of the interleaver/de-interleaver in a BICM system, the fading coefficients can be modeled as independent and identical distributed (i.i.d) Rayleigh random variables [6]. For the case of AWGN channel, g p equals for all p. The receiver in a BICM-ID system includes the maximum a posteriori probability (MAP) soft-output demodulator and the soft-input soft-output (SISO) channel decoder. The detailed algorithm for the demodulator is described in [], while the SISO channel decoder uses the maximum a posteriori probability algorithm in [7]. Similar to the decoding of Turbo codes, here the demodulator and the channel decoder exchange the extrinsic information of the coded bits P ( c; O) and P (c; O) through an iterative process. After being interleaved, P ( c; O) and P (c; O) become the aprioriinformation P (c; I) and P ( c; I) at the inputs of the SISO decoder and the demodulator, respectively, as can be seen in Fig.. 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. III. DISTANCE CRITERIA AND THE LOWER BOUND FOR THE OPTIMAL MAPPING The union bound of the BEP for a BICM-ID system employing a rate-k c /n c convolutional code, a constellation Ψ and a mapping ξ can be written as [9], [6]: P b c d f(d, Ψ,ξ) (3) k c d=d H where c d is the total information weight of all error events at Hamming distance d and d H is the free Hamming distance of the code. The function f(d, Ψ,ξ) is the average pairwise error probability, which depends on the Hamming distance d, the constellation Ψ and the mapping ξ. By evaluating a tight upper bound on f(d, Ψ,ξ), two distance criteria that characterize the influences of the multi-d signal constellation and mapping to the error performance of BICM-ID were obtained in [9] for the two channel models. Let s i = [q i,,...,q i,n ] and s j(i,k) =[q j(i,k),,...,q j(i,k),n ] denote the two signal points whose labels differ at position k. Thus j(i, k) is understood as an integer index of a signal symbol and it depends on i, k and the specific mapping ξ. The distance criterion for the case of fast Rayleigh fading channel is written as [9]: δ A (Ψ,ξ)= l l l ( n s i Ψ k= p= + q i,p q j(i,k),p ) (4) where l = mn is the number of coded bits carried by one n- D signal point. For the quasistatic Rayleigh fading channel, the distance criterion is given as follows [9]: δ B (Ψ,ξ)= l l l s i Ψ k= s i s j(i,k) (5) Note that δ A (Ψ,ξ) depends on the channel signal to noise ratio, while δ B (Ψ,ξ) does not. Furthermore, observe that for the set of mappings whose components q i,p q j(i,k),p > 0, one can use the approximation (+x) x when x to simplify δ A (Ψ,ξ) as: where ˆδ A (Ψ,ξ)= l l δ A (Ψ,ξ) ( ) nˆδa (Ψ,ξ) (6) l n s i Ψ k= p= q i,p q j(i,k),p (7) Since ˆδ A (Ψ,ξ) does not depend on N 0, it is more convenient to use for comparing different mappings as long as q i,p q j(i,k),p > 0. As shown in [9], the two parameters δ A (Ψ,ξ) and δ B (Ψ,ξ) should be made as small as possible to lower the asymptotic bit error rate performance of BICM-ID systems. Loosely speaking, this objective can be fulfilled by labelling the signals in the constellation such that the signal pair at a larger Euclidean distance corresponds to a smaller label Hamming distance. To assess the performance of the proposed mapping, lower bounds on the distance parameters δ A (Ψ,ξ) and δ B (Ψ,ξ) 5005

3 shall be first stated. It will also be demonstrated that there does not exits any mappings that achieve these lower bounds. Consider the 8-PSK constellation with four inter-signal Euclidean distances d, d, d 3 and d 4 in decreasing order, i.e., d > d > d 3 > d 4. By studying the Euclidean distance profile of a multi-d constellation constructed from 8-PSK, it can be shown that the best mapping possible in terms of minimizing δ A (Ψ,ξ) and δ B (Ψ,ξ) should have the following property. Property : For any signal point s, the3n signal points whose labels differ in only bit compared to the label of s include the following: (i) One signal point at squared Euclidean distance nd ; (ii) n signal points at squared Euclidean distance (n )d + d ; and (iii) (n ) signal points at squared Euclidean distance (n )d + d + d. Therefore, one has the following lower bounds: δ A (Ψ,ξ) ( ) (n ) [ ( ) + d + d 3n + n + (n ) ( + d ( + d ) ( + d ) ] ) and δ B (Ψ,ξ) [ n 3n nd + (n )d + ] d (n ) + (n )d + d + d Unfortunately, as proved in [8], there does not exist any mappings that satisfy Property. This implies that the lower bounds in (8) and (9) are not achievable. Alternatively, the next section proposes an algorithm to design the mapping with the following property. Property : For any signal point s, the 3n signal points whose labels differ in only bit compared to that of s include the following: (i) One signal point at squared distance nd ; (ii) n signal points at squared distance (n )d + d ; and (iii) (n ) signal points at squared distance (n )d + d 3. IV. THE PROPOSED MAPPING ALGORITHM First, assign an integer index to each of the signal points in the 8-PSK constellation (see Fig. ). Divide the signal points into two subsets as follows: The odd subset, denoted by O, includes the signal points with odd indices, 3, 5 and 7. The even subset, denoted by E, includes the remaining signal points with even indices 0,, 4 and 6. Clearly, each subset is simply a QPSK or a rotated (by π/4) QPSK constellation with two different inter-signal Euclidean distances d and d 3. For convenience, represent the signal point s Ψ as a string of n integer indices as s =[s,...,s n ], where s i takes a value in E or O. The signal point s is labelled with 3n binary bits (8) (9) as s a =(a,...,a 3n ). For each signal point s, associate with it the binary n-tuple x =(x,...,x n ), computed as: x i = s i mod Also denote the corresponding decimal number of x by X. Note that there are 8 n distinct signal points s, while there are only n distinct n-tuple x. According to, associated to a given n-tuple x are 4 n signal points s. These 4 n signal points are the vertices of the twisted hypercube H X. In other words, the multi-d constellation Ψ can be decomposed into n twisted hypercubes, each containing 4 n signal points. As an example, any signal point s whose components {s i } are all in the even subset E will be found in the hypercube H 0.This is because x =(0,...,0) and X =0. The objective of the proposed algorithm is to obtain the label a for each signal point s that satisfies Property mentioned earlier. The proposed algorithm is given and discussed in the following three main steps. Step : For each signal point s with the associated vector x, the first n bits of its label vector a are simply assigned as: a i = x i = s i mod ; i n () As a direct consequence of Step, all the signal points in the same twisted hypercube H X have the same first n bits in their labels. Step : This step considers the labelling strategy for the special hypercube H 0. More specifically, the remaining n binary labelling bits (a n+,...,a 3n ) for any s in H 0 are determined by applying the the optimal mapping algorithm of a hypercube proposed in [9]. As a result, the signal points in H 0 and their labels possess the following properties [9]: Distance Properties of H 0 : For any signal point s H 0,the n signal points whose labels differ in only bit at positions from (n +) to 3n compared to the label of s arealsoinh 0. Furthermore, among these n signal points, there are signal point at squared distance nd and (n ) signal points at squared distance (n )d + d 3. After Step, 4 n signal points in H 0 have been labelled. Step 3: This final step constructs the labels of all the remaining signal points in Ψ based on the labels of H 0.In particular, for each decimal number X, X n, with the corresponding n-tuple x = (x,...,x n ), first associate each signal point s =[s,...,s n ] H 0 with the signal point z =[z,...,z n ] H X as follows: z j =[s j +4(x x n ) x j ] mod 8, j n () Then label the signal point z H X by: z (x,...,x n,a n+,...,a 3n ) (3) With the above mapping construction, one has the following theorems. Theorem : For any X n, the mapping defined in () and (3) preserves the distance properties of H 0. H X is called a twisted hypercube if each of its faces is a QPSK or a rotated QPSK. 5006

4 Proof : Since s H 0, all components s i are even and (z j x j ) mod =0 (4) This ensures that the signal point z is in the hypercube H X. Clearly, with a given s, the operation in () with ( n ) n- tuple x (x 0)gives( n ) signal points z in the hypercube H X, X n. Furthermore, there is a one-to-one correspondence between s and z (i.e., given x, there is only one pair of s and z that satisfy ()). Consider two signal points s and t in H 0 and the corresponding signal points z and y in H X. It follows from () that s j t j = z j y j, j n, and hence s t = z y (5) From the labelling rule in (3), it is clear that the labels of the two signal points s and z have the same last n bits. It then can be concluded that the distance properties of H 0 stated before are preserved for any hypercube H X. This also implies that such properties are preserved for any signal points in Ψ. Theorem : By the association in () and the labelling rule in (3), the squared Euclidean distance between any two signal points in Ψ whose labels differ in only bit at the position j, j n, is(n )d + d. Proof : Consider any two signal points z and w whose labels differ in only bit at position j. Without loss of generality, assume that: z =[z,...,z n ] (x,...,x j,,x j+,...,x n,x n+,...,x 3n ) (6) w =[w,...,w n ] (x,...,x j, 0,x j+,...,x n,x n+,...,x 3n ) (7) Furthermore, consider the following signal point s H 0 : s =[s,...,s n ] (0,...,0, 0, 0,...,0,x }{{} n+,...,x 3n ) n 0 s (8) It then follows from () that: z k =[s k +4(x +...+x j ++x j x n ) x k ] mod 8 (9) and w k =[s k +4(x +...+x j +0+x j x n ) x k ] mod 8 Therefore the absolute difference between the two integers z k and w k is { 4 mod 8=4, k j z k w k = () 3 mod 8=3, k = j Hence the squared Euclidean distance between two 8-PSK signal points corresponding to the integer indices z k and w k can be determined as { z k w k d =, k j d (), k = j It then can be concluded that: z w =(n )d + d (3) From Theorems and, it is simple to see that the proposed mapping satisfies Property. As aforementioned, the common technique to find good multi-d mappings for BICM-ID relies on the BSA. It is shown in [8] that the proposed mapping is immune (i.e., unaffected) by the BSA. This means that if the BSA starts with the proposed mapping, it can never find a better one. This result also implies that the proposed mapping is at least one of the locally optimum mappings produced by the BSA. Example : As an example, consider the proposed mapping for the simplest case of n =, i.e., the case of 8-PSK constellation in a -D signal space. There are only two hypercubes H 0 and H. Step : Label the first bit for all signal points as follows: 0 for signal points in H 0 and for signal points in H.The result of this step is shown in Fig. -(a). 4 3 Fig () () 6 () () (a) Step () () 6 () () (b) Step ()00 8 () ] m od a re0 =[ x + 3) 8 5 =( t bits ] las =[ m od a re x + 3) =(6 t bits las ()0 (c) Step 3. ()0 Three steps to construct proposed mapping for 8-PSK constellation. Step : For signal points H 0, label the last two bits with the optimal mapping algorithm in [9], which is simply the anti- Gray mapping of QPSK constellation. The results after this step are shown in Fig. -(b). Step 3: Using the association in () and the labelling rule in (3), label the last two binary bits of the signal points in H. For example, since =(6+3)mod 8, the last binary two bits of the signal point with index are the same with that of the signal point with index 6 and they are (, ). The complete mapping is shown in Fig. -(c). This mapping is exactly one version of the semi-set partitioning (SSP) mapping, which is the best mapping of 8-PSK constellation found by a bruteforce search [], [6]. Example : For the case of n =, the proposed mapping is provided in Table I. Observe that any signal points in the same hypercube H X have the same first labelling bits, while the labelling of the last 4 bits follow from the optimal mapping rule in [9]. Table II compares the parameters ˆδ A (Ψ,ξ) and δ B (Ψ,ξ) of the proposed mapping against the unachievable lower bound when n =, 3, 4 and 8. Note that the parameter ˆδ A (Ψ,ξ) can be used since all components q i,p q j(i,k),p of the proposed and non-exiting mappings are strictly positive. Observe from Table II that the differences in the two distance parameters of the two mappings are very small, especially when n increases. This suggests that there is almost no

5 TABLE I THE PROPOSED 4-D MAPPING. IT INCLUDES 4 TWISTED HYPERCUBE H 0, H, H AND H 3.IN EACH HYPERCUBE, THE SIGNAL POINT s IS REPRESENTED BY A STRING OF TWO INTEGERS. THE CORRESPONDING LABEL a IS A BINARY 6-TUPLE. H 0 H H H 3 s a s a s a s a [0, 0] (0, 0, 0, 0, 0, 0) [0, ] (0,, 0, 0,, 0) [, 0] (, 0,,,, ) [, ] (,,,, 0, ) [0, ] (0, 0,, 0,, 0) [0, 3] (0,,, 0,, ) [, ] (, 0, 0,, 0, ) [, 3] (,, 0,, 0, 0) [0, 4] (0, 0, 0, 0,, ) [0, 5] (0,, 0, 0, 0, ) [, 4] (, 0,,, 0, 0) [, 5] (,,,,, 0) [0, 6] (0, 0,, 0, 0, ) [0, 7] (0,,, 0, 0, 0) [, 6] (, 0, 0,,, 0) [, 7] (,, 0,,, ) [, 0] (0, 0, 0,,, ) [, ] (0,, 0,, 0, ) [3, 0] (, 0, 0, 0,, ) [3, ] (,, 0, 0, 0, ) [, ] (0, 0,,, 0, ) [, 3] (0,,,, 0, 0) [3, ] (, 0,, 0, 0, ) [3, 3] (,,, 0, 0, 0) [, 4] (0, 0, 0,, 0, 0) [, 5] (0,, 0,,, 0) [3, 4] (, 0, 0, 0, 0, 0) [3, 5] (,, 0, 0,, 0) [, 6] (0, 0,,,, 0) [, 7] (0,,,,, ) [3, 6] (, 0,, 0,, 0) [3, 7] (,,, 0,, ) [4, 0] (0, 0,, 0,, ) [4, ] (0,,, 0, 0, ) [5, 0] (, 0, 0,, 0, 0) [5, ] (,, 0,,, 0) [4, ] (0, 0, 0, 0, 0, ) [4, 3] (0,, 0, 0, 0, 0) [5, ] (, 0,,,, 0) [5, 3] (,,,,, ) [4, 4] (0, 0,, 0, 0, 0) [4, 5] (0,,, 0,, 0) [5, 4] (, 0, 0,,, ) [5, 5] (,, 0,, 0, ) [4, 6] (0, 0, 0, 0,, 0) [4, 7] (0,, 0, 0,, ) [5, 6] (, 0,,, 0, ) [5, 7] (,,,, 0, 0) [6, 0] (0, 0,,, 0, 0) [6, ] (0,,,,, 0) [7, 0] (, 0,, 0, 0, 0) [7, ] (,,, 0,, 0) [6, ] (0, 0, 0,,, 0) [6, 3] (0,, 0,,, ) [7, ] (, 0, 0, 0,, 0) [7, 3] (,, 0, 0,, ) [6, 4] (0, 0,,,, ) [6, 5] (0,,,, 0, ) [7, 4] (, 0,, 0,, ) [7, 5] (,,, 0, 0, ) [6, 6] (0, 0, 0,, 0, ) [6, 7] (0,, 0,, 0, 0) [7, 6] (, 0, 0, 0, 0, ) [7, 7] (,, 0, 0, 0, 0) difference between the two systems in terms of the asymptotic performance. V. ANALYTICAL AND SIMULATION RESULTS In all simulations, a simple rate-/3, 4-state convolutional code with generator sequences g = (6,, 6) and g = (, 4, 4) [9] is used. This code together with an 8-PSK constellation yields a spectral efficiency of bits/s/hz. The bit-wise interleaver with a length of,000 coded bits is designed according to the rules outlined in []. Each point in the BER curves is simulated with 0 7 to coded bits. The error floors calculated according to (3) with the first 0 Hamming distances are also drawn to show how the iterations converge to the error floor. The lower-bound corresponding to the non-existing mapping mentioned in previous sections is also plotted. First, for the quasistatic Rayleigh fading channel, Fig. 3 presents the performance of a BICM-ID system employing the proposed 4-D mapping after, 4, 8 and iterations. The performance after iterations of the conventional system using SSP 8-PSK is also shown. It can be seen from Fig. 3 that, the simulation results converge to the error floor bounds for both systems. It is also obvious from Fig. 3 that a significant coding gain is achieved by the proposed mapping over the SSP 8-PSK mapping. Moreover, it can be clearly observed that there is almost no difference between the asymptotic performance of the proposed mapping and that of the non-existing mapping. Since the asymptotic performances and the real BER are very tight at medium and high SNR, the asymptotic performances can be used to accurately predict the BER. Though not shown here, examining the error bounds plotted over wider range of SNR reveals that coding gain offered by the proposed mapping at the BER level of 0 6 is about 3.8 db. Such a coding gain is achieved with a reasonable increase in the receiver complexity which is required to address the soft-output demodulation of BER The proposed 4 D mapping:, 4, 8 and iterations SSP 8 PSK: iterations BER floor Lower bound (non existing mapping) E b /N 0 (db) Fig. 3. BER performance of BICM-ID systems over quasistatic fading channel: 4-state convolutional code and different mappings. a higher-order multi-d constellation. For the case of fast Rayleigh fading channel, Fig. 4 shows the performance of the BICM-ID system using the proposed 4- D mapping with, 4, 8 and iterations and the conventional system with SSP 8-PSK mapping after iterations. Comparing Figs. 3 and 4 indicates that that there is no difference in BER performance between the two models of Rayleigh fading channels if the -D SSP mapping is used. It can be seen from Fig. 4 that the actual BER performance of two systems also converges to the BER floors. Though the convergence happens at a quite low BER for the proposed mapping with, 000 bit interleaver, the analytical asymptotic performance is still very useful to predict the actual BER. A faster convergence can be achieved by using a longer interleaving length. Similar to the 5008

6 TABLE II THE DISTANCE PARAMETERS ˆδ A (Ψ,ξ) AND δ B (Ψ,ξ) FOR THE NON-EXISTING AND THE PROPOSED MAPPINGS. ˆδ A (Ψ,ξ) δ B (Ψ,ξ) n The non-existing mapping e 4 5.8e 5.8e The proposed mapping e 4 7.9e 5 3.9e BER The proposed 4 D mapping:, 4, 8 and iterations SSP 8 PSK: iterations BER floor Lower bound (non existing mapping) E b /N 0 (db) Fig. 4. BER performance of BICM-ID systems over fast fading channel: 4-state convolutional code and different mappings. previous channel model, it can be seen that the gap in terms of the asymptotic BER perforamnce between the proposed mapping and the non-existing mapping is very slight. It can also be observed from the error floor bounds that the coding gain of the proposed mapping is about 9.6dB at the BER level of 0 6 [8]. When compared to the case of quasistatic fading channel, the coding gain is different in about 5.8 db. This observation is similar to the performance of a BICM-ID system with signal space diversity studied in [4]. Finally, it is shown in [8] that the proposed mapping is superior to the mappings found by the time-consuming computer search based on the BSA [0] for both fading channels. VI. CONCLUSIONS An explicit multi-d mapping of 8-PSK has been presented for BICM-ID systems. It was demonstrated that, by employing the proposed mapping in a 4-D signal space, significant coding gains can be achieved over the conventional BICM-ID systems that use the well-known SSP mapping over both considered fading channel models. Such coding gains are achieved with a reasonable increase in the receiver complexity to address the soft-output demodulation of the multi-d constellation. By comparing the asymptotic performance of the proposed mapping with unachievable lower bounds, the proposed mapping is conjectured to be the globally optimal mapping. REFERENCES [] X. Li, A. Chindapol, and J. A. Ritcey, Bit-interleaved coded modulation with iterative decoding and 8PSK signaling, IEEE Trans. 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