IEEE TRANSACTIONS ON BROADCASTING, VOL. 64, NO. 1, MARCH EXIT-Chart Aided Design of Row-Permutation Assisted Twin-Interleaver BICM-ID

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1 IEEE TRANSACTIONS ON BROADCASTING, VOL. 64, NO. 1, MARCH EXIT-Chart Aided Design of Row-Permutation Assisted Twin-Interleaver BICM-ID In-Woong Kang, Hyoung-Nam Kim, Member, IEEE, and Lajos Hanzo, Fellow, IEEE Abstract Twin-interleaver bit-interleaved LDPC coded modulation (BICM) has been widely adopted and studied in the context of digital terrestrial transmission (DTT) systems. Extrinsic-information transfer (EXIT) charts have been used as an analysis tool to evaluate the iterative decoding performance and to design a twin-interleaver BICM system. Since BICM using iterative decoding (BICM-ID) exhibits capacity-approaching decoding performance, developing the conventional twininterleaver BICM DTT system to a BICM-ID system has also been attempted. When considering the twin-interleaver BICM-ID systems, there are a couple of aspects that should be taken into account. We will demonstrate that anti-gray mapping should be adopted for improving the iterative decoding performance instead of the classic Gray symbol mapping. However, the previous literature related to designing the associated row-permutation interleaver (RPI) was mainly focused on BICM systems relying on Gray mapping. Hence, we will design the RPI of the twin-interleaver BICM-ID DTT systems using an anti-gray mapping. Explicitly, we use: 1) the EXIT-chart analysis that accurately visualizes the overall iterative decoding and demapping performance of the twin-interleaver BICM-ID DTT systems using anti-gray mapping and 2) a preprocessing stage for eliminating duplicate RPI candidates by using a parameter that roughly predicts the EXIT-chart analysis result before the EXIT-chart analysis is involved. By the elimination of the duplicates, over 99% of the initial candidates of the RPIs can be removed from the original M! candidates, where M stands for the number of modulated bits in a symbol. Given this drastically reduced number of candidates, the proposed design method finds novel RPIs, having superior bit-error ratio performances over the conventional RPIs in the context of the DVB-T2 standard. Index Terms BICM-ID systems, digital terrestrial transmission (DTT) system, EXIT-chart analysis, interleaver, iterative decoding. I. INTRODUCTION BIT-INTERLEAVED coded modulation (BICM) has been in the spotlight as benefit of its near-capacity decoding Manuscript received December 17, 2016; revised March 28, 2017; accepted April 21, Date of publication May 29, 2017; date of current version March 2, This work was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology under Grant 2014R1A1A (Corresponding authors: Lajos Hanzo; Hyoung-Nam Kim.) I.-W. Kang and H.-N. Kim are with the Department of Electronics Engineering, Pusan National University, Busan , South Korea ( hnkim@pusan.ac.kr). L. Hanzo is with the School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, U.K. Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TBC performance and flexibility resulting from the use of various modulation schemes [1], [2]. Accordingly, the digital terrestrial transmission (DTT) standards, such as the digital video broadcasting-terrestrial second generation (DVB-T2) [3], the DVB-next generation handheld (DVB-NGH) [4], [5], and the Advanced Television Systems Committee 3.0 (ATSC3.0) [6], [7], have adopted BICM systems operating at diverse transmission rates and robustness against channel impairments. When using relatively large alphabets, such as 256- quadrature amplitude modulation (QAM), the sub-channel capacities of bits in a symbol differ from each other. In addition, since the irregular LDPC codes of the DTT systems also provide different error correction capabilities, which are represented by the variable node degrees (VND) within a codeword, the unequal constellation sub-channel capacities and the unequal LDPC error protection capabilities must be jointly optimized. For the sake of the feasibility, the DVB- T2 standard has adopted a pair of cascaded interleavers in the BICM aided system [3]. The first one is a block interleaver designed for converting uniformly distributed variable node degree (VND) distribution of the irregular LDPC code used, which influences the error correction capability of the corresponding VND, to a non-uniform distribution so that the codewords result in the required error correction capabilities in conjunction with bits of the symbols [8], [9]. Accordingly, the unequal-protection bits of the modulated symbols can be assigned to the variable nodes represented by the coded bits by permuting the row orders of the VND distribution with the aid of the second row-permutation interleaver (RPI). The unequal error probabilities can be readily equalized with the aid of the popular extrinsic information transfer (EXIT) chart analysis [10] [14], the EXIT-chart analysis visualizes the evolution of the mutual information (MI) iteratively transferred between inner and outer decoders [11], [13] [15]. To elaborate a little further, without a feedback loop, the variable node (VN) and the check node (CN) decoders may be viewed as the inner and the outer decoders, respectively [11], [13]. By contrast, the inner and outer decoders of BICM-ID are the soft demapper and the iterative LDPC decoder, respectively [14]. When designing the RPI of the DTT standards, various symbol mapping schemes with different constellation sizes have been investigated [16] [19]. In [16], a beneficial method was proposed for designing the RPI of Gray-coded QAM-modulated BICM DTT systems by exploiting the density evolution (DE) based analysis tool. Although the DE analysis is indeed, capable of the evolution of the iteratively c 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 86 IEEE TRANSACTIONS ON BROADCASTING, VOL. 64, NO. 1, MARCH 2018 processed soft information in the LDPC decoder tracking, this imposes a high computational complexity. Hence, Yan et al. [17] advocated an EXIT-chart aided procedure for designing the RPI of Gray-coded QAM- and APSK BICM systems in [17] and [18], respectively. Specifically, it was suggested to amalgamate the soft demapper and the variable node (VN) decoder termed as the evnd into a joint the inner decoder, while viewing the check node (CN) decoder as the outer decoder of the iterative decoding structure. Based on this inner and outer decoder structure, the transfer of extrinsic information between the two decoders is then visualized by the EXIT chart. However, there is a paucity of literature on twin-interleaver BICM using the iterative decoding (BICM-ID), except for [19], where a symbol mapping was considered. Since the BICM-ID scheme generates its feedback information from the LDPC decoder s output for the soft demapper s input, Cheng et al. [19] introduced a new evnd scheme for incorporating the feedback information into the conventional evnd structure and modified the corresponding EXIT chart. Although this technique was directly developed from the previously formulated EXIT-chart of the twin-interleaver BICM systems [19], anti-gray symbol mapping should be considered instead of the Gray symbol mapping, because it provides a better decoding performance [20], [21]. Moreover, since the number of RPI candidates is as high as M!, where M stands for the number of bits per modulated symbol, the RPI design potentially imposes an excessive computational complexity. Against this background, we propose a novel RPI design methodology relying on EXIT-chart analysis for the twininterleaver BICM-ID systems. The main contributions of the proposed method can be summarized as follows: 1) the EXIT-chart analysis of twin-interleaver BICM-ID systems associated with anti-gray symbol mapping is formulated, where the soft demapper and the iterative LDPC decoder are treated as the inner and the outer decoders; and 2) a method of drastically reducing the number of RPI candidates (over 99% from the original M! candidates) before performing the EXIT-chart analysis is proposed. Explicitly, we reduce the number of RPI candidates by introducing a parameter for efficiently predicting the EXIT-chart analysis result. The rest of this paper is organized as follows. Section II describes the twin-interleaver BICM-ID system model. The EXIT-chart analysis regarding the twin-interleaver BICM-ID systems is developed in Section III. In Section IV, we eliminate the duplicate RPI candidates before our EXIT-chart analysis is explained. Finally, our RPI designs and the associated decoding performances are verified by computer simulation in Section V. Our conclusions are offered in Section VI. II. TWIN-INTERLEAVER BIT-INTERLEAVED CODED MODULATION USING ITERATIVE DECODING The block diagram of the twin-interleaver BICM-ID system is shown in Fig. 1. At the transmitter side, the LDPC encoder encodes the information bit stream, u = {u 1, u 2,...,u Lu }, Fig. 1. Fig. 2. Block diagram of the twin-interleaver BICM-ID system. Two concatenated interleavers in BICM-ID systems. into the codeword, c = {c 1, c 2,...,c Lc } where L u and L c denote the lengths of the u and c sequence, respectively. The LDPC code rate is defined as R = L u /L c. As the benefit of their superior error correcting performance, irregular LDPC codes having multiple variable node degrees (VNDs) are used. Fig. 2 shows how the LDPC coded codeword is interleaved by a pair of concatenated bit-level interleavers. The first bit interleaver is a block interleaver (BI) that randomizes the codeword c for the sake of dispersing the burst errors. In addition, the BI scrambles c so that it can have different proportions of VND values for different number of modulation levels. Here, the variation of the VND proportions with the number of modulation levels, m, and the VND values, d v, can be organized in rows and columns of a two-dimensional distribution, respectively. Once the BI generates the non-uniform VND distribution, m, the second interleaver, namely the rowpermutation interleaver (RPI), divides c into M sub-channel block interleaved codewords, c BI m where m = 1, 2,...,M 1, and scrambles the order of the modulation level index calculated as c RPI m = c BI RP (m), in order to obtain the optimum m that best matches the associated sub-channel capacities. Here, RP denotes the row-permutation sequence of the RPI and RP (m) is its mth element. After interleaving, M interleaved bits are grouped together and they are mapped to symbols (s = {s 1, s 2,...,s Ls }) according to the anti-gray mapping rule applied, as represented by the mapping sequence, z AG ={z AG 1, zag 2,...,zAG }. Here, the number of symbols in s 2 is calculated as L M s = L c /M. As an example, the change of VND distributions is given in Table I by the BI for a 64-QAM constellation and a 1/2-rate LDPC code. The VND distribution is determined depending

3 KANG et al.: EXIT-CHART AIDED DESIGN OF ROW-PERMUTATION ASSISTED TWIN-INTERLEAVER BICM-ID 87 TABLE I VND DISTRIBUTIONS FOR 6-BIT MODULATION SCHEMES WITH AND WITHOUT THE BLOCK INTERLEAVER important MI terms have to be defined and their transfer from/to the soft demapper and the LDPC decoder also should be formulated. In this respect, this section defines the MI terms based on [14] and introduces more accurate EXIT-chart analysis for the family of twin-interleaver BICM-ID systems. both on the modulation level index, m, and on the VND values, d v. It is shown that the uniformly distributed m associated with the modulation level is reorganized after the first bit interleaver. This non-uniform VND distribution leads to the unequal error protection characteristics of the different modulation levels. This unequal error protection property of the LDPC code has to be properly matched to the unidentical bitwise channel capacities of the high-order modulation schemes applied. This matching can be performed by permuting the order of rows of m by the second interleaver of Fig. 2. At the receiver side, the received symbol can be expressed as r = hs+n, where s is impaired by the channel fading h and by the zero-mean additive white Gaussian noise (AWGN), n, having a variance of σn 2. The soft demapper (the inner decoder) results in the soft information represented in the form of loglikelihood ratio (LLR) values, which are fed into the LDPC decoder (the outer decoder) according to the Log-maximum a-posteriori (MAP) demapping algorithm [22], [23] as follows: L DEM e,m (p) log exp r hx 2 M [1 2b x l ]LDEM a,l (p) x X 0 m σ 2 n log exp r hx 2 x X 1 m σ 2 n l=1,l =m M l=1,l =m [1 2b x l ]LDEM a,l (p), (1) where L DEM e,m (p) and LDEM a,m (p) denote extrinsic and apriori LLR values for the mth bit in the pth symbol, while b x l and Xm b {0,1} denote the value of the lth bit in symbol x and the subset of X ={x k k = 1, 2,...,2 M } whose mth bit is equal to b {0, 1}. The output L DEC p of the LDPC decoder is then used for generating the final hard-decision results. The difference between L DEC p and the bit deinterleaver output, L DEC a,is defined as the extrinsic LLR value L DEC e, which is fed back into the soft demapper after being interleaved. III. EXIT-CHART ANALYSIS TWIN-INTERLEAVER BICM-ID SYSTEMS The EXIT-chart analysis tracks the evolution of the amount of information measured in terms of the average mutual information (MI). In the twin-interleaver BICM-ID systems, four A. Computing Mutual Information Relying on the Ergodicity Theorem As a measure of the amount of information at the decoder (or demapper) output, the mutual information between the transmitted bit stream and the decoded (or demapped) LLR values can be acquired under the assumption of having a known distribution for the LLR values. If the probability density function (PDF) of the LLR values is given and the information bits are equiprobable, the average MI can be expressed as [14] I(x, L) = 1 2 x= 1,+1 + p L (ξ X = x) 2 p L (ξ X = x) log 2 dξ, (2) p L (ξ X = 1) + p L (ξ X =+1) where p L (ξ X = x) denotes the conditional PDF of L given x, while X denotes a random variable representing the information bit x. As shown in (2), the above equation requires the apriori knowledge of p L (ξ X = x). The distribution can be assumed to obey the Gaussian distribution, provided that the interleaver length is sufficiently high. By assuming that L indeed obeys the Gaussian distribution with the mean of σl 2 /2 x and the variance of σl 2,(2) can be rewritten as I σ (σ L ) = I(x; L) = 1 E { log 2 (1 + exp(l)) } ) 2 1 (ξ σ L2 2 = 1 exp 2πσL 2σ 2 L log 2 [1 + exp( ξ)]dξ. (3) It has been shown in [10] and [11] that the mutual information of (3) is a monotonically increasing function, hence it is invertible and since (3) is a function of the noise variance, its compact expression can be formulated as: J(σ ) = I σ (σ L = σ ). (4) Hence, the function J(σ ) can be used for characterizing the mutual information between the transmitted information bit stream and the received LLR values, which can be modeled as L = σ 2 2 x + n L, (5) where n L N(0,σ 2 ). For simplicity, approximate closed-form expressions of the function J(σ ) and their inverse functions may be found in [19] and [24].

4 88 IEEE TRANSACTIONS ON BROADCASTING, VOL. 64, NO. 1, MARCH 2018 and Ie CND [17], [18] by taking into account Ie DEM.TheEXIT function of the evnd block of Fig. 3 can be formulated as M m d vie,m VND (d v) Ie VND d v D v m=1 =, (8) M m d v d v D v m=1 where D v stands for the set of variable node degrees and Ie,m VND is the sub-channel mutual information corresponding to the mth modulation level defined as [17], [24] Ie,m VND (d v) = J ( (d v 1) [ J 1( I CND e )] 2 + σ 2 m ). (9) Fig. 3. Iterative demapping and decoding system model for EXIT-chart analysis. The ensemble average in (3) can be replaced by the sample average relying on the ergodicity theorem according to [12]. Then (3) can be rewritten as I(x, L) 1 1 N N log 2 (1 + exp [ x(n) L(n)]), (6) n=1 where N denotes the total number of bit samples averaged. Given this approximation, the MI can be acquired even when the LLR values do not follow the Gaussian distribution. B. Conventional EXIT-Chart Analysis for Twin-Interleaver BICM-ID Systems In Fig. 3, four MIs defined in our iterative demapping and decoding system context are illustrated, while L DEC a,m, L CND e, and L VND e denote the mth sub-channel s a priori LLR values for the LDPC decoder, the extrinsic LLR values of the VN decoder, and the extrinsic LLR values of the CD decoder, respectively. Furthermore, Ie,m DEM stands for the mth sub-channel s MI, while the total MI considering all M sub-channels is defined as their average, i.e., I DEM e = 1 M M 1 m=0 I DEM e,m. (7) Finally, Ie VND and Ie CND represent the MIs that describe the amount of information at the output of the VN decoder and the CN decoder in the iterative LDPC decoder. Since the output of the VN decoder is used as the final result of the LDPC decoder output, Ie VND can also be denoted as Ip DEC. Given these four MIs associated with the twin-interleaver BICM-ID systems, we have two MI transfers: the transfer of MIs between 1) the soft demapper and the LDPC decoder, as well as between 2) the VN decoder and the CN decoder in the iterative LDPC decoder. At the current state-of-the-art, there is a paucity of contributions related to the EXIT-chart analysis of the twin-interleaver BICM system. Nonetheless, Yan et al. [17] suggested to amalgamate the soft demapper and the VN decoder of the iterative LDPC decoder into the evndshowninfig.3, and investigated the transfer of Ie VND Here, σm 2 denotes the mth modulation level s sub-channel noise variance defined as [ ] 2. σm 2 = J 1 (I DEM (10) e,m When considering the feedback, I DEM e,m I DEM e,m = I [ c RPI m can be defined as [19] ; LDEM m (Ie VND,σn ], 2 ) (11) where Lm DEM (Ie VND,σn 2 ) denotes the LLR values at the output of the soft demapper, when the MI feedback is Ie VND and the channel noise variance is σn 2. Given the feedback IVND e,the feedback LLR values, L FB, can be evaluated by computing its noise variance σfb 2 = [J 1 (Ie VND )] 2 and then substituting σ in (5) byσ FB. The expression of Ie CND is the same as that of the twininterleaver BICM system operating without feedback as shown in [17] and [18], since Ie CND does not rely on the feedback information. Assuming that the irregular LDPC code of the DTT system has only a single CND value, the EXIT function for the CN decoder and its inverse function can be expressed as ( Ie CND (d c ) = 1 J (d c 1) [ ) J 1( 1 Ia CND )] 2, (12) ( J Ia CND 1 ( 1 I CND )) e (d c ) = 1 J. (13) (dc 1) Based on the evnd structure and on the corresponding definitions of the MI terms shown in (8)-(13), Cheng et al. [19] suggested to construct the EXIT chart based on (8) as the inner decoder and the one based on (13) as the outer decoder for the decoding performance evaluation and for the design of the row-permutation aided interleaver. Although this approach can be simply developed based upon the previous EXIT-chart analysis of the twin-interleaver BICM operating without the feedback loop [17], [18], it remains limited to tracking the main transfer relationship of the MIs between the soft demapper (as the inner decoder) and the LDPC decoder (as the outer decoder). In addition to the above-mentioned issue related to the evnd structure, finding an RPI in the entire set of all the M! original RPI candidates by performing EXIT-chart analysis for each RPI will impose a serious complexity problem. Against this background, we conceive a prediction procedure for performing the EXIT-chart analysis of the twininterleaver BICM-ID systems that more accurately describes

5 KANG et al.: EXIT-CHART AIDED DESIGN OF ROW-PERMUTATION ASSISTED TWIN-INTERLEAVER BICM-ID 89 Fig. 4. The generation of the EXIT curve for the LDPC decoder with the two interleavers. the characteristics of the transfer of MI between the soft demapper and the LDPC decoder. This can be achieved by using the soft demapper and the iterative LDPC decoder as the inner as well as the outer decoder, respectively, and then constructing the EXIT curves of the soft demapper and the LDPC decoder with the aid of two interleavers based on the ergodicity theorem of [14]. Since we involve the conventional log-map demapper for the soft demapper, the generation of the EXIT curve for the soft demapper follows the procedure detailed in [14]. To reduce the design complexity, a technique of finding redundant RPI candidates, which are identical to each other in terms of the EXIT-chart analysis formulated in (8) and (9), is also suggested. By spotting the duplicate RPI candidates, numerous redundant candidates can be eliminated before performing the proposed EXIT-chart analysis. Moreover, this paper proposes a parameter that provides a rough prediction of the EXIT-chart analysis result so that the RPI design procedure can directly eliminate some of the deficient RPIs before performing the actual EXIT-chart analysis. Ia DEC C. Proposed EXIT-Chart Analysis for Twin-Interleaver BICM-ID Systems The EXIT curve proposed for the LDPC decoder associated with the two interleavers, namely the DEC EXIT curves, are depicted in Fig. 4. Considering the M sub-channels of 2 M -QAM modulation schemes, the input mutual information has to be transformed into the sub-channel noise variance, σm 2, where σ m 2 can be determined according to IDEM e by using both the symbol and the bitwise sub-channel capacities. Denoting the symbol and the mth sub-channel capacities by C(ρ z AG ) and C(ρ z AG, m) at an SNR value ρ, and using an anti-gray mapping rule [1], z AG, the corresponding SNR can be expressed as ρ DEM = C 1( M I DEM e z AG). (14) Here, the amount of the symbol level mutual information is approximated as (M Ie DEM ) according to (7). Then Ie,m DEM can be expressed by using ρ DEM as ) Ie,m (ρ DEM = C DEM z AG, m. (15) The desired σm 2 can be determined by using the J-function given in (10). Then the output LLR values L DEM e of the soft demapper can be generated according to (5) by substituting x and L of (5)byσm 2 and by the LDPC codeword, c, respectively, where L DEM e is deinterleaved and decoded, hence resulting in the LDPC decoded LLR values, L DEC p. The extrinsic LLR values are given by L DEC e = L DEC p L DEC a. Once L DEC e has been determined, the mutual information between the information bit stream u and L DEC e can be calculated according to (6). As an example, the EXIT curves of the LDPC decoder (DEC EXIT curves) associated with two RPIs (left) and the capacity curves of the anti-gray mapping (right) applied are plotted in Fig. 5. In the left side of Fig. 5, the soft demapper s EXIT curve associated with an arbitrary anti-gray 16-QAM mapping, z AG =[12, 4, 1, 3, 14, 2, 5, 13, 8, 10, 7, 15, 6, 16, 9, 11], is illustrated and the LDPC decoder s EXIT curves associated with RP =[3, 2, 1, 4] (dashed line) and [2, 4, 3, 1] (dotted line) are plotted. Given the input I DEMe = 0.62, the DEC EXIT curve can be generated by computing I DEMe,m and σ m according to (14) and (15). The apriorimi Ie DEM = 0.62 is multiplied by M = 4, resulting in 2.48 for the input of (14). By using the symbol capacity curve at the right side (solid line) of Fig. 5, its corresponding SNR value is 9 db, as expressed in (14). Then the SNR value is used for determining Ie,m DEM by using the sub-channel capacities (circles for m = 1, squares for m = 2, crosses for m = 3, and asterisks for m = 4) as (15). Then the Ie,m DEMs are transformed into σ m 2 by (10). It is noted that due to using the anti-gray mapping rule, the subchannel capacities are different from each other. Once the σm 2 values have been prepared, the Ie DEC values are numerically computed for the pair of RP s according to the DEC EXIT curve generation illustrated in Fig. 4. Regarding the EXIT curve of the soft demapper (DEM EXIT curve), the EXIT function associated with an anti-gray mapping provides the EXIT curve (solid line), which increases as Ia DEM increases, corresponding to the improvement of the soft demapping performance, as the iterative decoding proceeds. Thanks to the demapper s beneficial EXIT curve shape, there may be an open area between two EXIT curves and the extrinsic mutual information can be transferred from/to the soft demapper and the LDPC decoder. Since the DEM EXIT curve is fixed for a certain noise variance, the attainable decoding performance now purely relies on the given RPI. Contrasting a pair of EXIT curves of the LDPC decoders associated with different RPIs, the EXIT curve of RP =[3, 2, 1, 4] (dashed line) yields an intersection with the DEM EXIT curve, while the DEC EXIT curve with RP =[2, 4, 3, 1] (dotted line) does not. It is shown in Fig. 5 that the Monte-Carlo decoding trajectories marked with thin lines between the DEM and the DEC LDPC curve only exist for RP =[2, 4, 3, 1]. In the sense of the EXIT-chart analysis, the combination of the given soft demapper and the RPI of [2, 4, 3, 1] will provide a better decoding performance than the soft demapper combined with the RPI of[3,2,1,4]. IV. DESIGN OF A ROW-PERMUTATION INTERLEAVER BASED ON EXIT-CHART ANALYSIS A. EXIT-Chart Trajectory Test Given the DEM and DEC EXIT curves of Fig. 5, the decoding performance of the iterative demapping and decoding

6 90 IEEE TRANSACTIONS ON BROADCASTING, VOL. 64, NO. 1, MARCH 2018 Fig. 5. (Left) The EXIT curves for the soft demapper and the LDPC decoders with an anti-gray mapping z AG =[12, 4, 1, 3, 14, 2, 5, 13, 8, 10, 7, 15, 6, 16, 9, 11] and two RP examples. (Right) The symbol and the sub-channel capacities with z AG. system can be estimated by tracking the transfer of MI between two EXIT curves. If a pair of EXIT curves have an open area between them, then the iterative decoding system is capable of achieving a near-error-free decoding performance, and vice versa. Since the DEM EXIT curve moves up and down according to the channel s noise variance, the lowest noise variance (or the SNR value) that creates an open EXIT-tunnel in conjunction with the given DEC EXIT curve is returned as the result, which we refer to as the trajectory test. It is noted that since the DEC EXIT curves do not reach the Ie DEM = 1 value, there must be an intersection between the two EXIT curves. In such a case, the trajectory test is performed before the DEM EXIT curve reaches a value relatively close to Ia DEM = 1[14]. B. Definition of the EXIT-Chart Prediction Parameter P According to the definition of the VN decoder s EXIT functions in (8)-(9), they are determined by the VND distribution, m, and the sub-channel noise variance, σm 2. Among these two parameters, the row-permutation interleaver affects m by adjusting the order of the rows in the distribution. The number of possible permutations is M!, implying that M! cases have to be investigated by the EXIT-chart analysis to find the best RP in terms of the EXIT-chart analysis result. Although a previous study disseminated in [19] tried to eliminate duplicate RP candidates by defining the RPIs associated with the same EXIT curves as the duplicates, there may still remain hundreds of RPI candidates. In this respect, this section proposes a parameter, P, that represents the combination of m and σ m, which can be adjusted by the row-permutation interleaver. Moreover, by the definition of the P values, we can provide a rough prediction of the attainable decoding performance. The RPIs associated with poor performance prediction can be deleted before performing the EXIT-chart analysis. The choice of RP affects the iterative decoding performance by adjusting the matches between m and σm 2. In order to characterize the match between m and σm 2 with TABLE II m L c FOR M = 4WITH THE 2/3-RATE LDPC CODE AND THE CORRESPONDING BI the aid of a simpler parameter, we introduce a new parameter, P, based on the first moment, P row, of the rows of the distribution, which is formulated as m=m m=1 σ m P row P = m=m m=1 P, (16) row wherewehavep row = d v D v m d v. The generation of the σ m values in conjunction with a given Ie DEM is characterized in (10), (14), and (15). By defining the parameter, P, with the aid of the combination of m and σm 2, a single P value is capable of fully representing the corresponding EXIT chart. In other words, if the P values are different for certain RPIs, then their EXIT-chart analysis results are also different. If the P values are the same, then the RPIs are the same in terms of the EXIT-chart analysis. In Table II, m for M = 4 and a 2/3-rate LDPC code and its associated BI is given as an example. For this example, a symmetric anti-gray mapping rule of z AG =[1, 3, 2, 4, 9, 11, 10, 12, 5, 7, 6, 8, 13, 15, 14, 16] is applied. Thus, the subchannel noise variances are symmetric, i.e., we have σ 1 = σ 3 and σ 2 = σ 4. Due to the symmetry of σ m, there are a number

7 KANG et al.: EXIT-CHART AIDED DESIGN OF ROW-PERMUTATION ASSISTED TWIN-INTERLEAVER BICM-ID 91 TABLE III REPRESENTATIVE RP S AND THEIR P AND ρ R VALUES WITH AN ANTI-GRAY 16-QAM MAPPING TABLE V P AND ρ R VALUES WITH SYMMETRIC ANTI-GRAY 64-QAM CONSTELLATIONS AND 2/3-RATE LDPC CODE TABLE IV COMPARISON BETWEEN N RP,eff AND M! WITH VARIOUS M AND R of duplicate RPI candidates, such as, RP = [1, 2, 3, 4] and RP = [1, 4, 3, 2]. In order to check the redundant nature of the original RP candidates, Table III provides all the M! = 4! = 24 RP candidates and their P values. According to the P values computed, 24 candidates are classified into 6 representative (Rep.) RP candidate groups which have 3 redundant (Red.) RP s. It is noted that the P values of the third and the fourth RP groups have very little difference because 2 and 3 are very similar. The P values associated with an asterisk are larger than the remaining ones. Without eliminating any redundant option of RP, the total number of RP is M! = M i=1 i. Assuming a general anti- Gray mapping, whose σ m values are different from each other, the number of redundant RP s can be found by investigating the duplicate rows of m. In general, the rows of m can be grouped into J groups that share identical a j=1,...j rows. The remaining (M J j=1 a j ) rows are different from any other ones. Given these duplicate rows of m, the number of representative RP s, denoting no-duplicate RP s, can be computed as J ( j 1 N RP,rep = M i=1 a ) J i M!, (17) j=1 a j wherewehave j 1 i=1 a i M. Considering the various code rates (R) of the LDPC code and the number of bits (M) inthe modulated symbols, N RP,eff can be computed and its values are shown in Table IV. Comparing N RP,eff against the original number of M!, a redundant entry reduction of 50.0% (M = 4, all R) to99.3% (M = 8, R = 1/2) has been achieved. When the symmetry of the specific anti-gray mapping is exploited and the small difference between two rows of κ d c m is assumed to be zero, N RP,eff becomes even smaller as seen from (17). j=1 a j Assuming the symmetry of the anti-gray mapping, the number of representative RPIs for the 64-QAM and the 256-QAM constellations are both found to be 24. In the next subsection, we proceed by defining an EXIT-chart prediction parameter P for further reducing the number of representative RPIs. C. Eliminating More RPI Candidates Based on the P Values Once the representative RP s and their P values are prepared, the trajectory test can be performed in order to find the best RP in terms of the EXIT-chart analysis result. Unfortunately, still hundreds of candidates may have to be examined by the trajectory test under various combinations of the modulation constellation size and the code rates even after reducing the number of redundant RPIs compared to the numbers in Table IV. In order to further reduce the number of candidates, the pre-defined parameter P can be invoked. By the definition given in (16), the parameter P can be interpreted as a representative value of all σ m values computed with the aid of the first moment regarding and d v. Recalling that the mutual information of (4) monotonically increases as the input noise variance increases, a larger P value is likely to yield an improved decoding performance. Hence, before performing the EXIT-chart analysis to find the best RPI, the P values can be invoked for eliminating some of the worst RPIs in terms of the P values. For example, Table III presents the P values calculated for the six representative RP s and the resultant SNR values ρ R in db obtained from their trajectory test for a symmetric anti- Gray 16-QAM constellation using a 2/3-rate LDPC code. The number in parenthesizes next to the ρ R values denote the associated ranking. For example, the representative RPI of [3, 1, 4, 2] has the highest ρ R value as well as the highest P value. It is shown that the descending order of the P values does not exactly tally with the increasing order of ρ R values. Nevertheless, the correlated nature of the P and ρ R values

8 92 IEEE TRANSACTIONS ON BROADCASTING, VOL. 64, NO. 1, MARCH 2018 TABLE VI P AND ρ R VALUES WITH SYMMETRIC ANTI-GRAY 256-QAM CONSTELLATIONS AND 2/3-RATE LDPC CODE Fig. 7. EXIT charts associated with a symmetric anti-gray 64-QAM constellation given in Table VII. Fig. 6. EXIT charts associated with a symmetric anti-gray 16-QAM constellation given in Table VII. can be exploited for dropping some of the RPIs before performing the EXIT trajectory test. For example, if we create sub-sequences of the order of the three worst RPI groups in terms of two parameters, then these two sets have the same elements for this example, namely the P values of [4, 5, 6] and the ρ R values of [5, 4, 6], although the order of elements is not the same. This suggests that the P value can roughly predict a group of deficient representative RPIs, which do not necessarily have to be examined by the trajectory test. In order to further confirm the validity of the P values, the symmetric anti-gray 64- and 256-QAM mappings RPIs are also tested. In Tables V and VI, the column marked by # represents the representative RPIs and the P values of 24 representative RPIs are sorted in descending order, implying that the best RPI in terms of the P value is indexed as 1, while the worst is indexed as 24. The order of ρ R values is parenthesized next to each ρ R value. For example, observe in Table V that ρ R of db associated with the 9th RPI has the index of 1. This implies that this RPI has the best (i.e., smallest) ρ R value, which is different from the best RPI in terms of the P parameter. Again, similarly to 16-QAM, although the best RPIs in terms of the P and ρ R values are not the same, the groups of deficient RPIs associated with both parameters exhibit useful similarity. In detail, the orders of the worst 12 RPIs in terms of the P values, namely the P-bad RPI set denoted by O P ={13, 14,...,24} exhibit similarity to the set of the order of O ρe ={10, 11, 18, 15, 22, 20, 17, 21, 19, 16, 23, 24}, denoted as ρ R -bad set. More explicitly, only two elements of two sets are different, while the other 10 elements are identical in the pair of sets. More importantly, the different ones are not the higher ranked elements such as that of index 1 representing the best RPI. The same trend is also observed for the 256-QAM mapping. Explicitly, in Table VI, the ρ R -bad set is given by O ρe ={14, 18, 17, 12, 16, 18, 19, 23, 20, 21, 24, 22}. Only the index 12 of the set O ρe, which is a low-ranked element, is different from the set O P. In short, the sets O P and O ρe are similar and the elements of O ρe are ranked low. Accordingly, the RPI design procedure can ignore the representative RPIs associated with O P during the EXIT-trajectory test. V. RESULTS AND COMPUTER SIMULATIONS Once the representative RP candidates are prepared, the best RP in terms of the EXIT-chart analysis can be acquired by contrasting ρ R obtained by the trajectory test. In Figs. 6 8, the best and the worst RPIs DEC EXIT curves and the corresponding DEM EXIT curves along with their ρ R are plotted for symmetric anti-gray 16-, 64-, and 256-QAM mappings. Specifically, the DEC EXIT curves of the best and the worst RPIs are marked with circles as well as crosses and their corresponding DEM EXIT curves are solid and dashed

9 KANG et al.: EXIT-CHART AIDED DESIGN OF ROW-PERMUTATION ASSISTED TWIN-INTERLEAVER BICM-ID 93 TABLE VII THE BEST AND THE WORST RPIS WITH 16-, 64-, AND 256-QAM CONSTELLATIONS AND THEIR CORRESPONDING TRAJECTORY TEST RESULTS Fig. 8. EXIT charts associated with a symmetric anti-gray 256-QAM constellation given in Table VII. Fig. 9. Coded BER performance with the best and the worst RPI under the AWGN channel. The constellation mapping rules and the RPI sequences are given in Table VII. Other simulation parameters are given in Table VIII. lines, respectively. Specifically, the best and the worst RPIs associated with the 16-, 64-, and 256-QAM constellations are given in Table VII. The best RPIs were found to be [3, 1, 4, 2], [1, 2, 4, 3, 5, 6], and [2, 4, 6, 3, 5, 1, 7, 8] for the 16-QAM, 64-QAM, and 256-QAM mappings, respectively. The worst RPIs were found to be [1, 3, 2, 4], [4, 6, 1, 5, 3, 2], and [2, 6, 3, 4, 5, 7, 8, 1] for the 16-QAM, 64-QAM, and 256-QAM mappings, respectively. The BER performance obtained by iterative decoding using the best and the worst RPIs in an AWGN environment is portrayed in Fig. 9. The number of outer iterations between the soft demapper and the LDPC decoder is I outer = 3 and that in the LDPC decoder is I inner = 100. The BICM-ID systems associated with the best RPIs and with 16-, 64-, and 256- QAM exhibit a 0.28, 0.59, and 0.35 db gain, respectively, in terms of the SNR required at BER=10 5 over the worst RPIs. The twin-interleaver BICM systems with the best and the worst RPIs are also examined by simulation under the TU- 6 frequency-selective fading channel model at zero Doppler frequency shift [25] in Fig. 10. The TU-6 channel profile is given in Table VIII. These coded BER curves also demonstrate that, as expected, the best RPIs provide superior decoding performance over the worst ones. Specifically, at BER= 10 5 the SNR gain of the best RPIs over the worst ones recorded for 16-, 64-, and 256-QAM constellations are 0.18, 0.40, and 0.29 db, respectively. In addition, Figs. 11 and 12 are displaying the BER curves with the energy per bit to noise power ratio (E b /N 0 ) in order

10 94 IEEE TRANSACTIONS ON BROADCASTING, VOL. 64, NO. 1, MARCH 2018 TABLE VIII SIMULATION PARAMETERS Fig. 12. Coded BER performance under the TU6 fading channel with E b /N 0 scale. Refer Tables VII and VIII for the simulation parameters and the channel profile. Fig. 10. Coded BER performance with the best and the worst RPI under the TU-6 fading channel. The constellation mapping rules and the RPI sequences are given in Table VII. Other simulation parameters are given in Table VIII. VI. CONCLUSION We propose an EXIT-chart aided design procedure for the row-permutation interleavers of twin-interleaver BICM- ID systems. In particular, anti-gray QAM mapping schemes which are known to provide better iterative decoding performance than the classic Gray mapping have been considered. In order to obtain the best RPI in terms of the EXIT-chart analysis, the generation of EXIT curves for the soft demapper and for the iterative LDPC decoder has been proposed. Moreover, this paper also proposed a novel parameter that provides a rough prediction of the EXIT-chart analysis result so that the design procedure can ignore some of bad RPI candidates without high-complexity EXIT-chart analysis. By using the prediction parameter P, the design procedure can eliminate up to 99% of the number of RPI candidates. The optimum RPIs can be found by using the proposed EXIT-chart aided design methodology at a low complexity. Especially, the proposed EXIT-chart prediction parameter can be utilized to reduce the complexity of designing the RPIs of the twininterleaver BICM systems as well as the systems with the iterative decoding. REFERENCES Fig. 11. Coded BER performance under the AWGN channel with E b /N 0 scale. Refer Tables VII and VIII for the simulation parameters and the channel profile. to provide the reception performance with same throughput. With E b /N 0 scale, the same trend of the BER performance is also observed for all the modulations used. [1] L. L. Hanzo, T. Liew, B. Yeap, R. Tee, and S. X. Ng, Turbo Coding, Turbo Equalisation and Space-Time Coding: EXIT-Chart-Aided Near- Capacity Designs for Wireless Channels. Hoboken, NJ, USA: Wiley, [2] G. Caire, G. Taricco, and E. Biglieri, Bit-interleaved coded modulation, IEEE Trans. Inf. Theory, vol. 44, no. 3, pp , May [3] Frame Structure Channel Coding and Modulation for a Second Generation Digital Terrestrial Television Broadcasting System (DVB- T2), Eur. Telecommun. Standards Inst. EN Standard , Apr [4] Digital Video Broadcasting (DVB); Next Generation Broadcasting System to Handheld, Physical Layer Specification (DVB-NGH), Digital Video Broadcasting Final Draft DVB Document A160, Geneva, Switzerland, [5] D. Gómez-Barquero, C. Douillard, P. Moss, and V. Mignone, DVB- NGH: The next generation of digital broadcast services to handheld devices, IEEE Trans. Broadcast., vol. 60,no.2,pp ,Jun [6] ATSC Proposed Standard: Physical Layer Protocol, Adv. Television Syst. Committee Standard S32-230r56, Jun

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Hyoung-Nam Kim (M 00) received the B.S., M.S., and Ph.D. degrees in electronic and electrical engineering from the Pohang University of Science and Technology, Pohang, South Korea, in 1993, 1995, and 2000, respectively. From 2000 to 2003, he was with Electronics and Telecommunications Research Institute, Daejeon, South Korea, developing advanced transmission and reception technology for terrestrial digital television. In 2003, he joined the Faculty of the Department of Electronics Engineering, Pusan National University, Busan, South Korea, where he is currently a Full Professor. From 2009 to 2010, he was with the Department of Biomedical Engineering, Johns Hopkins University School of Medicine, as a Visiting Scholar. From 2015 to 2016, he was a Visiting Professor with the School of Electronics and Computer Engineering, University of Southampton, U.K. His research interests are in the area of digital signal processing, radar/sonar signal processing, adaptive filtering, and bio-medical signal processing, in particular, signal processing for digital communications, electronic warfare support systems, and brain-computer interface. He is a member of IEEK and KICS. Lajos Hanzo (M 89 SM 91 F 04) received the D.Sc. degree in electronics in 1976, the Doctorate degree in 1983, and the Honorary Doctorate degree from the Technical University of Budapest and the University of Edinburgh in 2009 and 2015, respectively. In 2016, he was admitted to the Hungarian Academy of Science. During his 40-year career in telecommunications he has held various research and academic posts in Hungary, Germany, and the U.K. Since 1986, he has been with the School of Electronics and Computer Science, University of Southampton, U.K., where he holds the Chair in telecommunications. He has successfully supervised over 100 Ph.D. students, co-authored 18 Wiley/IEEE Press books on mobile radio communications totalling in excess of pages, published 1663 research contributions at IEEE Xplore, acted both as the TPC and the General Chair of IEEE conferences and presented keynote lectures. He was a recipient of number of distinctions awards. He is currently directing a 60-strong academic research team, working on a range of research projects in the field of wireless multimedia communications sponsored by industry, the Engineering and Physical Sciences Research Council, U.K., the European Research Council s Advanced Fellow Grant and the Royal Society s Wolfson Research Merit Award. He is an Enthusiastic Supporter of industrial and academic liaison and he offers a range of industrial courses. He is also a Governor of the IEEE VTS. From 2008 to 2012, he was the Editor-in-Chief of the IEEE Press and a Chaired Professor with Tsinghua University, Beijing. He has over citations and an H-index of 68. He is a fellow of REng, IET, and EURASIP.