Interleaver Design for Small-Coupling-Length Spatially Coupled Protograph LDPC-Coded BICM Systems Over Wireless Fading Channels

As a class of convolutional error-correction codes (ECCs), spatially coupled protograph low-density parity-check (SC-PLDPC) codes with a sufficiently large coupling length can approach the channel capacity under belief propagation (BP) decoding. However, such codes suffer from poor performance in the case of small coupling lengths. To address the above weakness, we investigate the optimization and analysis of SC-PLDPC-coded bit-interleaved coded modulation (BICM) systems over Rayleigh fading channels. Specially, inspired by the unequal-error-protection (UEP) mechanism of <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>-ary phase-shift keying/quadrature amplitude modulation (<inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula>-PSK/QAM), we put forward a novel interleaver design scheme, referred to as <italic>spatial position match mapping (SPMM) scheme</italic>. Based on the mutual information (MI) analysis, the proposed interleaver can be used to significantly boost the performance of the SC-PLDPC codes in BICM systems. Furthermore, we propose a block-scheduling protograph-based extrinsic information transfer (BS-PEXIT) algorithm to estimate the convergence performance (i.e, decoding threshold) of the SPMM-aided SC-PLDPC codes in BICM scenarios. Simulation results are well consistent with the theoretical analyses, and show that the proposed SPMM-aided SC-PLDPC-coded BICM systems are superior to the state-of-the-art counterparts.


I. INTRODUCTION
Bit-interleaved coded modulation (BICM) is of particular effectiveness to achieve both capacity-approaching performance and high transmission rate in various communication scenarios, e.g., wireless communications [1]- [4], vehicular communications [5], and aeronautical communications [6]- [8]. As one of the most critical techniques in BICM systems, error-correction codes (ECCs) must be carefully designed and optimized in order to enhance the reliability and quality of data transmission. In particular, protograph modulations. For instance, an interleaving scheme, called variable-degree matched mapping (VDMM), has been proposed for Gray-mapped BICM systems over additive white Gaussian noise (AWGN) channels [10] and Rayleigh fading channels [11]. Furthermore, an exhaustive-search-based method has been incorporated into VDMM so as to find the best permutation between the variable nodes (VNs) and the labeling bit positions [12]. In parallel with the above works, a variety of VN-degree-distribution-based and weightdistribution-based interleaving schemes have been developed in order to further enhance the performance of BICM systems [13], [14].
Motivated by the superiorities of PLDPC codes, a class of convolutional-like PLDPC codes, called spatially coupled PLDPC (SC-PLDPC) codes, has been widely investigated. Such codes not only benefit from relatively low complexity, but also approach the maximum-a-posteriori (MAP) thresholds under believe propagation (BP) decoding [15]- [17]. Thereby, the SC-PLDPC codes may have great potential to be incorporated into BICM systems [18]. In the case of a sufficiently large coupling length, the SC-PLDPC codes can achieve MAP-approaching performance due to the decoding-wave (i.e., threshold saturation) phenomenon [19]. Unfortunately, such codes have a large codeword length, which cannot satisfy the low-latency requirement of modern wireless communication systems. Besides, the SC-PLDPC codes suffer from relatively weaker decoding thresholds when the coupling length is small or moderate [20]. To deal with the above issue, some research effort has been devoted to the design and analysis of small-coupling-length SC-PLDPC codes [21]- [23] and small-constraint-length SC-PLDPC codes [24], [25] in order to achieve satisfactory decoding performance. In addition, the investigation regarding the performance improvement of finite-coupling-length SC-PLDPC codes in BICM scenarios has been conducted [26]. To be specific, an optimized bit-to-symbol mapping scheme, which combines bit allocation and shortened pattern, has been proposed for finite-coupling-length SC-PLDPC codes so as to improve their performance. Nevertheless, to the best of our knowledge, few works have been reported to investigate the interleaver design for SC-PLDPC-coded BICM systems under fading environments from the perspective of mutual information (MI) analysis.
In this paper, we study the design and analysis of SC-PLDPC-coded BICM systems over ergodic Rayleigh fading channels. 1 To be specific, we first propose a novel interleaver design scheme for SC-PLDPC-coded BICM systems, called as spatial position match mapping (SPMM) scheme, which guarantees that the VNs with different iterative convergence performance are properly mapped to the labeling bit positions with different bit-plane MIs. The proposed SPMM scheme can enhance the decoding-wave phenomenon in the joint demapping-and-decoding process, and thus significantly accelerate the convergence speed of the a-posteriori MIs for the SC-PLDPC codes in BICM systems. For further investigation, we propose a block-scheduling PEXIT (BS-PEXIT) algorithm to predict the convergence performance (i.e., decoding threshold) of the SPMM-aided SC-PLDPC codes in BICM scenarios. Theoretical analyses and simulation results show that our proposed SPMM-adied SC-PLDPC-coded BICM systems achieve better performance than the conventional counterparts.
The remainder of this paper is organized as follows. To begin with, the SC-PLDPC-coded BICM system model over Rayleigh fading channels is introduced in Section II. Then, the SPMM interleaving scheme is put forward in Section III. Section IV gives the BS-PEXIT algorithm for the SC-PLDPC-coded BICM systems. Simulation results are presented in Section V and conclusions are drawn in Section VI.

II. SYSTEM MODEL A. SC-PLDPC-CODED BICM SYSTEMS
In this paper, we consider a BICM system with the use of SC-PLDPC codes, whose diagram is depicted in Fig. 1. First, the information bits (Inf. bits) are encoded to a coded bit sequence of length n by an SC-PLDPC encoder. Then, such a coded bit sequence is fed to an M -ary modulator after a specific permutation by an interleaver. Moreover, every m = log 2 M coded bits are mapped to a complex modulated symbol under a given constellation set χ, whose output is expressed as x i (i = 1, 2, ...n/m). These modulated symbols will be passed through an uncorrelated flat Rayleigh fading channel and their corresponding received signals can be written as where α i is the complex Gaussian-distributed fading factor; γ i is a complex Gaussian noise with zero mean and variance At the receiver, the received signals are processed by a soft-input soft-output (SISO) demapper and SC-PLDPC decoder. More specifically, the demapper and decoder are implemented via the max-sum approximation of the logdomain maximum a-posteriori probability (Max-log-Map) algorithm and the belief propagation (BP) algorithm [20], respectively. VOLUME 8, 2020

B. SC-PLDPC CODES
A protograph can be regarded as a small Tanner graph [27] G = (V, C, E) consisting of three different sets, i.e., a VN set V, a check node (CN) set C, and an edge set E. The cardinalities of V, C, and E correspond to N v , N c , and E, respectively. In a protograph, each edge e i,j ∈ E connects a VN v j ∈ V to a CN c i ∈ C, and parallel edges are permitted. Furthermore, a protograph with a code rate R = (N v − N c )/N t can be defined by a base matrix B = (b i,j ) of size N c × N v , where b i,j represents the number of edges connecting c i to v j , and N t is the number of transmitted VNs. It should be noted that there is no punctured VNs in the case of N t = N v . An expanded protograph, namely derived graph, can be generated by performing a lifting operation for its corresponding protograph, and thus a protograph code with arbitrary length can be easily obtained. 2 Based on a given protograph, its corresponding SC protograph can be built by applying a specific edge-spreading rule. Specifically, the formulation starts from replicating the protograph L times, and associating each duplicate with a time index t (t = 0, 1, . . . , L − 1). Then, one can connect the edges emanating from the VNs of protograph at times t (t = 0, 1, . . . , L − 1) over the CNs of protograph at times t   Fig. 2(a) shows the formulation process of (3, 6, L) coupled chain corresponding to a (3, 6) SC-PLDPC code with coupling length L, where the black circles and white squares represent the VNs 2 Actually, the lifting procedure can be implemented by exploiting a modified progress-edge-growth (PEG) algorithm [28]. and CNs, respectively. In particular, the base matrix B (3,6) and corresponding submatrices B (3,6) s (s = 0, 1, 2) for the (3,6) protograph are denoted by B (3,6) = [3 3], B

Case 2:
Similarly, the SC-PLDPC codes can be also constructed from irregular protographs. Referring to Fig. 2(b), a repeat-jagged-accumulate (RJA) SC-PLDPC code is constructed by its corresponding RJA protograph [29], whose base matrix B RJA and submatrices B RJA s (s = 0, 1) are given as For a given coupling length L, the code rate R L of an SC-PLDPC code can be given by 33502 VOLUME 8, 2020 where n c and n v are the numbers of CNs and VNs in the corresponding base matrix, respectively; R = 1 − N c /N v is the code rate of the original protograph code. Case 3: Different from the above two cases, the formulation process of (q, a) RA SC-PLDPC codes can be considered as a special case, because additional q − 1 degree-2 VNs are needed to be added, where q = a and they denote the degree of message-bit-related VNs and the number of edges between the CNs and the message-bit-related VNs [30], [31]. In order to make the formulation process more clear, Fig. 2(c) shows a specific example, i.e., a (5, 5) RA SC-PLDPC code. For a given coupling length L, the code rate of a (q, a) RA SC-PLDPC code can be calculated as

III. INTERLEAVER DESIGN FOR SC-PLDPC-CODED BICM SYSTEMS
A. PROPOSED SPMM SCHEME As the coupling length L tends to infinity, the SC-PLDPC codes can attain desirable threshold improvements. But a large L leads to a large codeword length, which makes the SC-PLDPC codes rather difficult to meet the low-complexity requirement in wireless communication scenarios [30], [32].
To address this weakness, some research works related to the performance analysis and structure optimization of finite-coupling-length SC-PLDPC codes have been investigated. For instance, a scaling law has been proposed to predict the finite-coupling-length performance of the SC-PLDPC codes [33]. Moreover, additional encoding structures have been considered incorporating into the boundaries of the finite-coupling-length SC-PLDPC codes so as to enhance their error performance [21], [34]. In addition to the above code-design schemes, the performance of the finitecoupling-length SC-PLDPC codes can also be improved from the perspective of MI convergence. It is well known that the MIs corresponding to the coded bits at both ends of the codeword are capable of realizing fast convergence, while the remaining coded bits suffer from poor convergence performance, which severely degrades the decoding thresholds of the SC-PLDPC codes. With an aim to improve the threshold performance, we propose a novel interleaving scheme, called SPMM scheme, to protect some coded bits based on the UEP property of the bit-to-symbol-mapping process in BICM systems. The proposed SPMM scheme not only can maintain the same implementation complexity as the original systems, but also can enhance the decoding-wave effect so as to accelerate the convergence performance of the SC-PLDPC codes. In the following, we will briefly describe the principle of the proposed SPMM scheme, which can be considered as a two-step design method.

1) MI-BASED BLOCK SCHEDULING
In an M -ary BICM system with the sequential mapping scheme [35], [36], the a-posteriori MIs of all VNs in a given SC protograph can be calculated based on the conventional PEXIT algorithm, and the convergence tendency of each VN can be predicted by analyzing its corresponding MI value, i.e., the larger the MI value of a VN is, the better the convergence performance of the VN can be realized. Then, according to the relationship between the SC-PLDPC code and its corresponding SC protograph, i.e., i) an SC-PLDPC code of length n = n v × p can be obtained by implementing p times lifting operation on its corresponding SC protograph of length n v ; ii) the entire SC-PLDPC codeword can be divided into n v blocks in a sequential order, and each block consisting of p coded bits corresponds to a VN of the SC protograph; the convergence tendency of each block in the SC-PLDPC code can be evaluated. Afterwards, by comparing the MI convergence differences among the n v blocks, one can use a ''high-to-low MI-based scheduling'' rule to adjust the original block sequence {V B,1 , V B,2 , . . . , V B,n v } in a descending order, i.e., the block with the largest MI is placed at the first position, while the other blocks with relatively low MIs are followed. Based on the above operation, one can obtain an optimized block sequence {V B,1 , V B,2 , . . . , V B,n v }, and it will be further processed in the next step.

2) UEP-BASED GROUP MAPPING
For a given M -ary constellation, there exist M different labels and each of which is formulated by m = log 2 M labeling bit positions 3 Inspired by the protection-level difference among the labeling bit positions (i.e., UEP characteristic), some specific coded bits within the optimized block sequence {V B,1 , V B,2 , . . . , V B,n v } can be protected with relatively higher priorities so as to improve the convergence performance of the entire SC-PLDPC codeword. To be specific, the above optimized block sequence can be equally divided into m groups V G,1 , V G,2 , . . . , V G,m in a sequential order, which ensure that the convergence performance of these m groups can satisfy the condition I G,1 ≥ I G,2 ≥ · · · ≥ I G,m , where I G,k (k = 1, 2, . . . , m) denotes the average MI (AMI) of the k-th group. Afterwards, during the bitto-symbol mapping process, every m coded bits, which are separately extracted from the m different groups, can be respectively assigned into the m labeling bit positions with different protection levels based on a ''reverse water-filling'' rule. In particular, the coded bits, which are extracted from the groups with relatively good convergence performance, are assigned into the labeling bit positions with relatively low protection level. According to the above operation, one can guarantee that the coded bits with poor convergence performance in the SC-PLDPC codeword can be protected with relatively higher priorities and the wave-like convergence can be further enhanced.
In general, the principle of the proposed SPMM scheme for the SC-PLDPC-coded BICM systems is illustrated in Fig. 3. 3 In particular, these m labeling bit positions (i.e.,   One can easily find that the proposed SPMM scheme can substantially exploit the convergence feature of SC-PLDPC codes and the UEP characteristic of labeling bit positions so as to balance the convergence speed and enhance the wave-like convergence. To further clearly elaborate on the proposed SPMM scheme, three specific examples are shown as follows. Example: For the BICM system with the Gray-mapped 16-QAM modulation (see Fig. 4) and the (3, 6) SC protograph of length n v = 24, the convergence performance of all the 24 VNs can be predicted based on the conventional PEXIT algorithm, and the detailed MI convergence tendency can be observed from Fig. 5(a). Then, by implementing the p-times lifting operation on such an SC protograph, a (3, 6) SC-PLDPC code of length n = 24 × p can be obtained, and the convergence tendency of the original block sequence 24 } can also be evaluated. Following the  .e., V G,1 , V G,2 , V G,3 , V G,4 ), can be assigned into 4 different labeling bit positions (i.e., x b 4 , x b 3 , x b 2 , x b 1 ), respectively. Based on the above SPMM scheme, the wave-like convergence of the (3, 6) SC-PLDPC code can be effectively enhanced, and its convergence performance can be further improved.
Likewise, for the Gray-mapped 16-QAM BICM system and the RJA SC protograph of length n v = 24, the convergence performance of all the 24 VNs can be predicted, and which can be found in Fig. 5(b). Exploiting the convergence feature of the RJA SC-PLDPC code generated from the above protograph and the UEP characteristic of labeling bit positions, one can also implement the proposed SPMM scheme on the RJA SC-PLDPC code so as to enhance its wave-like convergence.
To further explore the feasibility of the proposed SPMM scheme, the (5, 5) RA SC-PLDPC code generated from the (5, 5) RA SC protograph of length n v = 24 is also considered. Referring to Fig. 5(c), the convergence tendency of all the 24 VNs in its protograph is shown. Based on the MI convergence difference, the proposed SPMM scheme can also be implemented for the (5, 5) RA SC-PLDPC code in order to improve its threshold performance.
Note also that: • Although we only assume 16-QAM modulation in the three specific examples, the proposed SPMM scheme is also feasible to other high-order modulations, such as the 8-PSK and 64-QAM modulations (see Sect. IV-B).
• To make the above three specific examples more compact and clear, we adopt an unified length n v for three different types of SC protographs. Nonetheless, the proposed SPMM scheme is also applicable to other SC protographs and different lengths n v (see Sect. IV-B).
• In the conventional VDMM scheme [10], the permutation pattern between the VNs and the labeling bit positions follows a fixed order, i.e., the VNs with higher degree are assigned into the labeling bit positions with higher protection level, and thus the VDMM scheme is tailored for the conventional irregular protograph codes and no longer effective for SC protograph codes. The convergence principle of the proposed SPMM scheme is quite different from that of the existing techniques. In our proposed SPMM scheme, the UEP characteristic of the bit-to-symbol-mapping process is exploited to balance the convergence speed of all the VNs as much as possible. For this reason, the proposed SPMM scheme is available for both regular and irregular protograph-based codes, especially for the SC protograph codes.
• In general, the SC-PLDPC codes with a small coupling length L suffer from a rate loss and undesirable decoding threshold, while the SC-PLDPC codes with a large coupling length suffer from high decoding complexity and latency. Using puncturing schemes [21] or incorporating some additional encoding structures (e.g., some VNs) into the boundaries of an SC-PLDPC code [34] can effectively alleviate the rate-loss issue.

IV. BS-PEXIT ALGORITHM AND PERFORMANCE ANALYSIS FOR SC-PLDPC-CODED BICM SYSTEMS A. BS-PEXIT ALGORITHM
With an aim to accurately predict the convergence performance of SC-PLDPC codes in the SPMM-aided BICM systems, a modified PEXIT algorithm, referred to as BS-PEXIT algorithm, has been proposed in this subsection. Specifically, the BS-PEXIT algorithm not only can be utilized to trace the MI convergence behavior in the iterative decoding process, but also can calculate the minimum signal-to-noise ratio (SNR) (i.e., decoding threshold) that allows the system realizing error-free transmission. To facilitate the illustration of such an algorithm, some concepts are defined as follows.
In an M -ary BICM system, the initial channel loglikelihood ratio (LLR) of the coded bit l k i , which corresponds to the k-th labeling bit position within the i-th symbol x b k i , can be derived as Eq. 7, where χ b k =0 and χ b k =1 denote the subset of the constellation set χ with b k = 0 and that with b k = 1, respectively; P r (·) is the probability density function.
For the given LLR l k i ∼ N (σ 2 ch /2, σ 2 ch ), the MI between the coded bit and its associated l k i is represented by J (σ ch ) and given as The derivation of J (·) and its inverse function J −1 (·) can be obtained in [36]. In addition, several different types of MIs for the SC-PLDPC-coded BICM systems are briefly illustrated as follows.
• I ED is denoted as the extrinsic MI of the demapper. • I AV (i, j) and I AC (i, j) are denoted as the a-prior MI transmitted from c i to v j and v j to c i , respectively.
• I EV (i, j) and I EC (i, j) are denoted as the extrinsic MI transmitted from v j to c i and c i to v j , respectively.
• I APP (j) is denoted as the a-posteriori MI of v j . In particular, the maximum number of iterations in such an algorithm is set to T max . Moreover, there is an exchange of MI from VN to CN and vice versa in each iteration, i.e., I AV (i, j) = I EC (i, j) and I AC (i, j) = I EV (i, j). Based on the aforementioned definitions, the BS-PEXIT algorithm can be illustrated as below.
1) Initialization: At the beginning of the algorithm, there exists an initial E b /N 0 , which is served as an input parameter in the demapper. VOLUME 8, 2020 2) Updating the extrinsic MI of demapper: Based on the SPMM-based Monte-Carlo simulation, the extrinsic LLRs corresponding to the coded bits output from the demapper can be obtained, and they will be scheduled by a specific deinterleaver so as to be an original sequential order corresponding to the SC-PLDPC codes before being interleaved. Then, the extrinsic MIs I j ED (j = 1, 2, . . . , n v ), which are regarded as the initial decoding information of the VNs in the SC-PLDPC decoder, can be calculated by [11] 4) Calculating the extrinsic MI from CN to VN : For i = 1, 2, . . . , n c and j = 1, 2, . . . , n v , the extrinsic MI transmitted from the CN c i to the VN v j is measured by 5) Calculating the a-posteriori MI of VNs: For j = 1, 2, . . . , n v , the a-posteriori MI of VN v j can be calculated by exploiting I AV (i, j) and I j CH , as Based on the BS-PEXIT algorithm, one can get the smallest E b /N 0 , referred to as decoding threshold, that guarantees the a-posteriori MIs of all VNs in an SC-PLDPC code converging to the value of unity. For further illustration, the MI update process of an SC-PLDPC code in the BS-PEXIT algorithm is described in Fig. 6.
Note also that: • Due to the characteristic of the proposed SPMM scheme, i.e., the coded bits are interleaved based on the convergence performance analysis, a specific deinterleaving operation must be considered before calculating the extrinsic MI I ED .
• In the conventional PEXIT analysis, only one extrinsic MI I ED can be obtained. However, in the proposed BS-PEXIT analysis, there exist n v extrinsic MI I j ED (j = 1, 2, . . . , n v ) output from the demapper, because the proposed SPMM scheme is incorporated into the Monte-Carlo simulation.
Remark: Because the Gray constellation is considered as an optimal mapping scheme in the BICM systems [1], we mainly take it into account in this paper. Although we only consider three different types of SC-PLDPC codes for simulations, but the proposed SPMM interleaving scheme and BS-PEXIT algorithm are also applicable to other constellation mappers and other SC-PLDPC codes.

V. SIMULATION RESULTS
In this section, we provide some simulation results for three different types of SC-PLDPC codes, i.e., (i) (3, 6) SC-PLDPC code [20], (ii) RJA SC-PLDPC code [29], and (iii) (5, 5) RA SC-PLDPC code [37], with different coupling lengths in the Gray-mapped 8-PSK/16-QAM/64-QAM BICM systems over Rayleigh fading channels. In particular, we assume that the transmitted codeword length of all the above mentioned SC-PLDPC codes is set as n = 4800. Moreover, the maximum numbers of BP iterations is set as T BP = 100.

A. BER PERFORMANCE FOR THE 16-QAM BICM SYSTEMS
For the Gray-mapped 16-QAM BICM systems, we study the simulated bit error rate (BER) of the regular SC-PLDPC code, i.e., (3,6) SC-PLDPC code, with different coupling lengths. As observed from Fig. 7, with the help of the proposed SPMM 4 It should be noted that the Gray constellation cannot be formulated in the 32-QAM BICM systems [38], and thus we consider the Gray-mapped 8-PSK/64-QAM BICM systems in this paper so as to illustrate the effectiveness of the proposed SPMM interleaving scheme and BS-PEXIT algorithm.   Likewise, the SPMM interleaving scheme can make the irregular SC-PLDPC code, i.e., RJA SC-PLDPC code, achieving excellent performance. Referring to Fig. 8, the SPMM-aided RJA SC-PLDPC codes with coupling lengths L = 4, 5, 6 can obtain performance gains of about 0.3 ∼ 0.4 dB compared with the corresponding counterparts. VOLUME 8, 2020 To further validate the proposed SPMM interleaving scheme, we perform simulations on the (5, 5) RA SC-PLDPC codes with coupling lengths L = 6, 8, 10. In Fig. 9, the proposed SPMM interleaving scheme not only can greatly enhance the BER performance of the (5, 5) RA SC-PLDPC codes, but also can further improve their error-floor phenomenon. 5 From the above discussions, the three different BER simulations verify the effectiveness of the proposed SPMM interleaving scheme, and which is consistent well with the proposed threshold analyses in Sect. IV.

B. BER COMPARISON AMONG DIFFERENT INTERLEAVING SCHEMES
To verify the superiorities of the proposed SPMM interleaving scheme, two different types of state-of-the-art counterparts, i.e., VDMM [10] and universal matched mapping (UMM) [14], are used as benchmarks for performance comparison. As seen from Fig. 10, the SPMM interleaving scheme can make the RJA SC-PLDPC code with coupling length L = 6 exhibiting the optimal performance in the Gray-mapped 16-QAM BICM systems. It is noteworthy that the above two counterparts, i.e., the VDMM and UMM interleaving schemes, are only applicable to the irregular protograph-based codes, and thus the RJA SC-PLDPC code is exploited as the baseline.
C. BER PERFORMANCE FOR THE BICM SYSTEMS WITH 8-PSK AND 64-QAM MODULATIONS As a final example, we further perform simulations on the (3, 6) SC-PLDPC code with coupling length L = 9 in the Gray-mapped 8-PSK and 64-QAM BICM systems. As observed from Fig. 11, one can easily find that the 5 It should be noted that the minimum distance of an RA code does not grow linearly with the codeword length, and thus the RA code, especially the RA SC-PLDPC code, suffers from error floor in the high-SNR region [37].  (3, 6) SC-PLDPC code with the proposed SPMM interleaving scheme can achieve better performance in contrast with the original counterpart, while the 8-PSK modulated system is considered. Moreover, similar conclusions can be drawn from the 64-QAM modulated system. This implies that the proposed SPMM interleaving scheme is feasible to the BICM systems with various modulated order.

VI. CONCLUSION
In this paper, we have conducted a comprehensive study on the finite-coupling-length SC-PLDPC-coded BICM systems over Rayleigh fading channels. Specially, we have conceived a novel interleaver design scheme, called SPMM scheme, which can be utilized to significantly improve the error performance of SC-PLDPC codes in BICM systems. In addition, we have developed a modified PEXIT algorithm, called BS-PEXIT algorithm, to predict the asymptotic convergence performance (i.e., decoding threshold) of the SPMM-aided SC-PLDPC codes in the BICM systems. Theoretical analyses and simulation results have demonstrated that the SPMM-aided SC-PLDPC codes can achieve better performance than their original counterparts. Thanks to the above superiorities, the proposed SPMM-aided SC-PLDPC-coded BICM scheme stands out as a promising candidate for future wireless communication applications.