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LDMC design for low complexity MIMO detection and efficient decoding
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 187 (2018)
Abstract
Lowdensity multipleinput multipleoutput code (LDMC) can reduce the complexity of treesearch detection in MIMO systems. In this paper, we present a new modified progressive edgegrowth (PEG) algorithm to construct large girth LDMCs, which are referred to as PEGLDMCs. We analyze the complexity of the LDMC constrained sphere decoding (SD) and show that the LDMC constrained SD detection can be used for high reliability of the data transmission in MIMO systems. Furthermore, we propose two new efficient iterative decoding algorithms for LDMCs, which are high speed serial decoding and fast convergence shuffled decoding. Finally, we compare the bit error rate (BER) performance of PEGLDMCs to that of the existing LDMCs. The simulation results show that the PEGLDMCs can achieve better BER performance than that of the existing LDMCs.
Introduction
Multipleinput multipleoutput (MIMO) techniques have been widely studied during the past decades as they can provide significant multiplexing and diversity gains in transmission. Therefore, MIMO techniques have been identified as one of the most practical methods to combat fading and to increase the capacity of wireless channels in the recent years [1–4].
In order to provide high reliability of the data transmission in MIMO systems, special attention has to be paid in the receiver design. Maximumlikelihood (ML) detection is the optimal detection for MIMO systems [5, 6]. However, the complexity of the exhaustive searching in ML detection is too high for practical use. Therefore, suboptimal but low complexity detections were proposed, which include linear detection, such as zeroforcing (ZF) [7, 8], minimum meansquare error (MMSE) [9], and nonlinear detection, such as sphere decoding (SD) [10]. The performance of MMSE detection is better than that of ZF detection, due to the fact that the correlation between the noises in ZF detection is neglected in the projection operation. However, both the ZF detection and the MMSE detection suffer from serious performance degradation with respect to the optimal ML detection. On the other hand, the SD detection draws attention of researchers as it can achieve a nearML detection performance [11]. The main idea of the SD detection is that it only searches the ML candidate symbols which lie inside a specific sphere. For a given sphere size, the detection complexity of SD detection in MIMO systems becomes very high when the number of candidates is large.
Practically, errorcorrection codes are usually applied in MIMO systems. In particular, lowdensity paritycheck (LDPC) codes have drawn attention of many researchers due to their Shannon limit approaching performance [12–14]. As a subclass of LDPC codes, lowdensity MIMO codes (LDMCs) were presented in [15, 16], which reduce the complexity of the ML detection by introducing constraints in each transmission vector. The transmission vector is defined as the set of symbols which are transmitted simultaneously via multiple antennas. In [17], LDMCs with lowencoding complexity are further presented.
Motivated by the nearML detection performance of the SD detection and the complexity reduction property of the LDMCs, in this paper, we study the design of LDMCs and the concatenation strategy of LDMCs and SD detection in MIMO systems.
Furthermore, we propose more efficient iterative decoding algorithms for LDMCs. It is well known that the iterative decoding converges to the optimal solution provided that the paritycheck matrix of the code is of large girth [18]. Therefore, LDMCs of large girth are particularly desirable. Hence, we propose a modified progressive edgegrowth (PEG) algorithm to construct LDMCs of large girth. The contributions of this paper are as follows:

We propose a new modified PEG algorithm to construct large girth LDMCs for good performance.

We analyze the complexity of SD detection in the LDMCcoded MIMO systems, which indicates that the LDMCconstrained SD detection can be used for MIMO systems with large number of transmit antennas.

We propose two efficient iterative decoding algorithms for LDMCs for high speed and fast convergence decoding.
This paper is organized as follows. In Section 2, we review MIMO systems and LDMCs. In Section 3, we introduce a modified PEG algorithm to construct large girth LDMCs. In Section 4, we compare the complexity of LDMCconstrained SD detection with that of the MMSE and ZF detections. In Section 5, we introduce two new shuffled decoding algorithms for LDMCs. Examples of the LDMCconstrained SD detection in MIMO systems and the corresponding simulation results are given in Section 6. Finally, Section 7 concludes the paper.
Notation: bold lower case letters represent vectors, while bold upper case letters denote matrices. (·)^{T}, (·)^{H}, and ∥·∥ denote transpose, Hermitian, and norm operations, respectively. \(\mathbb {C}^{m\times n}\) stands for the complex space of size m×n.
Preliminaries
This section introduces some background concepts that will be used throughout the paper.
MIMO systems
Let M_{t} be the number of transmit antennas and M_{r} be the number of receive antennas. The source binary information bits b=[b_{1},⋯,b_{nR}] are first encoded using an LDPC encoder to generate a codeword \(\mathbf {w}=\left [\mathbf {w}_{1}, \mathbf {w}_{2}, \cdots, \mathbf {w}_{\frac {n}{M_{t}Q}}\right ]\), where R is the code rate and n is the length of the codeword. Then, each group of M_{t}Q coded bits \(\mathbf {w}_{i}=[b_{1},\cdots,b_{M_{t}Q}]\phantom {\dot {i}\!}\) is mapped to a group of M_{t} transmission vector \(\phantom {\dot {i}\!}\mathbf {x}=[x_{1},\cdots,x_{M_{t}}]^{T}\), where x_{j} is taken out of the constellations of size 2^{Q}. x is then passed to the transmit filter and sent through the M_{t} transmit antennas. x is denoted as the transmission vector in the following.
Without loss of generality, we express the system model as
Here, \(\mathbf {y}\in \mathbb {C}^{M_{r}\times 1}\) is the complex received signal vector. \(\mathbf {H}\in \mathbb {C}^{M_{r}\times M_{t}}\) is the Rayleigh fading channel matrix with independent entries that are complex Gaussian distributed with zero mean and unit variance. H is assumed to be known to the receiver, but not to the transmitter, and the channel coefficient randomly varys in time. \(\mathbf {z}\in \mathbb {C}^{M_{r}\times 1}\) is complex white Gaussian noise with variance σ^{2} per dimension.
Referring to [22–24], the received signal is iteratively detected and decoded by mutually exchanging soft information between the detector and LDPC decoders. The detection computes the loglikelihood ratios (LLRs), L_{i}, for each coded bit by using
where i=1,2,…,n. Transforming (2) into a treesearch problem and using the SD detection allows efficient computation of the LLRs [22, 23].
Let (3) denote the distribution of the searching candidates in the searching space,
each of whose elements is the number of constellation symbols to be searched at each antenna. Here, η_{P}+η_{Q}=M_{t}. η_{Q} denotes the number of candidates, which are considered to be fully searched, and η_{P} denotes the number of candidates are not required to be searched. It is clear that if η_{Q}=M_{t}, the SD detection becomes the ML detection.
The search over bit possibilities within x scales exponentially with the number of antennas M_{t} and the constellation size Q. In addition, the complexity of SD detection is very high when the number of bits mapped to the transmission vector and the constellation are large. Then, the complexity of the SD detection is the major problem for applying SD detection to MIMO systems with large number of transmit antennas. In the next subsection, we will describe that the LDMC is a subclass of LDPC codes and can reduce the complexity of the SD detection for MIMO systems.
Lowdensity MIMO codes
LDMCs are linear block codes which can be described by Pw^{T}=0, where P is a paritycheck matrix of n columns and m rows. Therefore, the overall code rate is R=1−m/n. The LDMC paritycheck matrix can be described by two layers as
The first layer P_{g} is a sparse paritycheck matrix, while the second layer P_{e} is defined by multiple, unconnected subcodes Pe′. Each subcode Pe′ corresponding to w_{i} has a code length Ne′≤M_{t}Q. Let N_{g}=n/Ne′ denote the number of subcodes in an LDMC codeword. The size of Pe′ is q×M_{t}Q and the size of P_{g} is (m−N_{g}q)×n.
As mentioned before, each transmission vector x carries M_{t}Q bits. It has to be guaranteed that all bits of a subcode Pe′ are transmitted within one transmission vector x. Thus, each transmission vector x carries an embedded code Pe′, i.e., \(\mathbf {P}_{e}'\mathbf {w}_{i}^{T}=0\).
It has been shown that, the SD detection achieves nearML performance if \(\eta _{Q}\geq \sqrt {M_{t}}1\) [19]. Therefore, it requires at least \(2^{Q(\sqrt {M_{t}}1)}\) times searching to achieve nearML performance. With q constraints in LDMCconstrained SD detection, the number of searching in LDMCconstrained SD will downscale to \( 2^{Q(\sqrt {M_{t}}1)q}\) [15, 16].
LDMC design based on PEG algorithm
It was stated in [18] that large girths facilitate iterative decoding and impose a respectable minimum distance bound that enhances decoding performance. Among the existing approaches for constructing LDPC codes, one of the most successful approaches is PEG algorithm [17]. Recently, several PEG algorithms have been proposed [25–27], which can be applied to construct P_{e}. For a given symbol node s_{j}, define its neighborhood within depth l, \(N^{l}_{s_{j}}\), as the set consisting of all check nodes reached by a subgraph (or a tree) spreading from symbol node s_{j} within depth l. Its complementary set, \(\bar {N}^{l}_{s_{j}}\), is defined as \(V_{c} \backslash N^{l}_{s_{j}}\), or equivalently \(\bar {N}^{l}_{s_{j}}\bigcup N^{l}_{s_{j}}=V_{c}\). Denote the symboldegree sequence by
in which \(d_{s_{j}}\) is the degree of symbol node s_{j}, 0≤j≤n−1 and \(d_{s_{0}}\leq d_{s_{1}}\cdots \leq d_{s_{n1}}\). The PEG algorithm constructs a Tanner graph by operating progressively on symbol nodes to establish edges required by D_{s}. To establish an edge incident to s_{j}, the PEG algorithm first spreads a tree from s_{j} up to maximum depth l and then chooses a check node c_{i} at random from the check nodes of the lowest degree in \(\bar {N}^{l}_{s_{j}}\). In this section, we introduce a modified PEG algorithm to construct large girth LDMCs, which is referred to as PEGLDMC algorithm and the corresponding codes as PEGLDMCs in this paper.
Let D_{g} and D_{e} denote the degree sequence associated with P_{g} and P_{e}, respectively. The paritycheck matrix P of LDMC can be constructed by the PEGLDMC algorithm with the following steps.

Preprocessing process :
From the optimized degree distribution for the code rate 1−(m−N_{g}q)/n and code rate 1−q/Ne′, generate the symboldegree sequences \(\phantom {\dot {i}\!}\mathbf {D}_{g}=\{d_{s_{0}},d_{s_{1}},\ldots,d_{s_{(N_{e}'1)}}\}\) and \(\phantom {\dot {i}\!}\mathbf {D}_{e}=\{d_{s_{0}},d_{s_{1}},\ldots,d_{s_{(N_{e}'1)}}\}\), respectively.

Constructing process :

Step 1. Construct q×Ne′ submatrix Pe′ of the form \(\mathbf {P}_{e}'=[\bar {\mathbf {P}}_{e}'~\mathbf {I}]\) with the PEG algorithm and symboldegree sequence D_{e}, where I denotes an identity matrix of size q×q and \(\bar {\mathbf {P}}_{e}'\) is of size q×(Ne′−q).

Step 2. Construct the columns of P_{g}, which are in the same columns as P_{e} in P. If the construction of P is not finished, go back to Step 1. Otherwise, stop the construction.
It is worth pointing out that in the PEGLDMC algorithm, the submatrices P_{e} and P_{g} of the paritycheck matrix P are constructed simultaneously. Therefore, the girth of P is optimized. A highlevel description of the PEGLDMC algorithm is given in Algorithm 1 and the algorithms for steps 1 and 2 are given in Algorithm 2 and Algorithm 3, respectively.
Analysis of detection complexity
In this section, we compare the detection complexity of the LDMCconstrained SD detection in the LDMC coded MIMO system with that of ZF and MMSE detections. To estimate the computational complexity, we use the number of multiplications required in the detection. Rough computational complexity estimations are given in Table 1. The number of additions is relatively small, therefore, it is omitted in the analysis.
Generally, SD is a searching algorithm for detection. It performs an exhaustive search based on setting a radius constraint. This has been proved by Hassibi [11] that the expected total number of points visited by the sphere decoding is proportional to the total number of candidates inside spheres of dimension k=1…m:
where d is radius and Γ(n)=(n−1)! represents the Gamma function. Clearly, ω depends on the radius d. Therefore, the channel state information (CSI) estimation does not impact the complexity in SD.
Specifically, if SD is performed in an LDMC coded MIMO system, the norm ∥y−Hx∥^{2} in the LDMCconstrained SD detection in (2) needs 2(M_{t}+2) multiplications. Thus, the total number of multiplications in the LDMCconstrained SD detection is
On the other hand, the ZF detection is represented by
and the MMSE detection is represented by
From the (8) and (9), it is clear that the MMSE and ZF detections consist of a complex inversion of M_{t}×M_{t} matrix, and some matrix multiplications and additions. M_{t}×M_{t} matrix inverse and matrix multiplication operations are both known to need \(M_{t}^{3}\) multiplications. Consequently, the total number of multiplications in MMSE and ZF detections are both
Note that the computational complexity of linear detector mainly depend on M_{t} while the computational complexity of LDMCconstrained SD detection depends on M_{t}, Q, and q. If q is large enough, it is possible that the computational complexity of LDMCconstrained SD detection is lower than that of ZF and MMSE detections. Let \(M_{t}^{3} \geq 2^{Q(\sqrt {M_{t}}1)q}2(M_{t} + 2)\), we have
If q satisfies condition (11), the computational complexity of LDMCconstrained SD detection with nearML performance can be lower than that of ZF and MMSE detections. From Fig. 1, we can see that the complexity of the LDMCconstrained SD detection is lower than that of the MMSE and ZF detections if we choose q properly. Note that a larger q reduces the size of P_{g} of the LDMC matrix, which is constructed by the PEG algorithm. Among the existing approaches for constructing LDPC codes, the most successful one is the PEG algorithm. The LDPC code constructed by the PEG algorithm is one of the best codes of good errorcorrecting performance. Therefore, a larger q not only leads to worse errorcorrecting performance, but also lower SD detection complexity. Hence, there is a tradeoff between performance and detection complexity by choosing the value q.
Efficient decoding algorithm
In this section, we proposed two new decoding algorithms for LDMCs for high speed decoding and fast convergence decoding. We denote the set of symbol nodes that participate in check node c_{i} by N(i)={j:P_{ij}=1}, and the set of check nodes in which s_{j} participates as M(j)={i:P_{ij}=1}.
The standard belief propagation (BP) decoding algorithm consists of initialization, checknode updating, symbolnode updating, stopping criterion test, and output steps [20]. In the proposed decoding algorithms, the initialization, stopping criterion test, and output steps remain the same as that in the standard BP algorithm, and therefore omitted here.
Highspeed serial decoding
In LDMCs, the code length n can be divided into subcodes Pe′ of length Ne′. In the highspeed shuffled decoding algorithm, the updating of subcodes is processed sequential, but the updating of symbol nodes and check nodes corresponding to each subcode Pe′ remain in parallel. Let \(r_{ij}^{l}\) and \(g_{ij}^{l}\) be the LLRs sent from check node c_{i} to symbol node s_{j}, and sent from the symbol node s_{j} to check node c_{i}, respectively, in the lth iteration.
In the standard BP algorithm, all value \(r_{ij}^{l}\) should be updated before updating all value \(g_{ij}^{l}\). We observe that, since the subcodes Pe′ in transmission vectors should not be connected to each other as shown by (4), all value \(r_{ij}^{l}\) for updating \(g_{ij}^{(l+1)}\) of kth subcode have already been updated before updating \(r_{ij}^{l}\) of the (k+1)th subcode, 1≤k≤(N_{g}−1). Therefore, the updating of \(g_{ij}^{(l+1)}\) for kth subcode can be computed in parallel with the updating of \(r_{ij}^{l}\) for (k+1)th subcode. In the highspeed serial decoding (HSSD), the checknode updating and the symbolnode updating of the HSSD are carried out alternately as follows.
Checknode updating corresponding to P_{g}: for 1≤i≤(m−N_{g}q) and each j∈N(i), process
For 1≤k≤N_{g}, process jointly the following two steps.

Checknode updating: for k·q≤i≤(k+1)·q and each j∈N(i), process (12) and (13).

Symbolnode updating: for k·Ne′≤j≤(k+1)·Ne′ and each i∈M(j), process
$$\begin{array}{@{}rcl@{}} &&g_{ij}^{l}=L_{j}+\sum\limits_{i'\in M(j)\backslash i}r_{ij}^{l} \end{array} $$(14)$$\begin{array}{@{}rcl@{}} &&g_{j}^{l}=L_{j}+\sum\limits_{i\in M(j)}r_{ij}^{l}, \end{array} $$(15)where L_{j} is the LLR of jth bit and initially set L_{j}=(4/N_{0})y_{j}.
Note that compared to the serial BP algorithm, our algorithm has higher decoding speed without increasing the complexity.
Fast convergence shuffled decoding
In the HSSD algorithm, the checknode updating corresponding to P_{g} should be done for only once in each iteration. To increase the convergence speed, we modified the HSSD and propose the fast convergence shuffled decoding (FCSD) algorithm in this subsection. In this decoding algorithm, the checknode updating corresponding to P_{g} will be done at each time the checknode updating corresponding to subcode is done. The checknode updating and the symbolnode updating of the FCSD are carried out as follows. For 1≤g≤N_{g}, process jointly the following two steps:

Checknode updating: for 1≤i≤(m−N_{g}q), k·q≤i≤(k+1)·q and each j∈N(i), process (12) and (13).

Symbolnode updating: for k·Ne′≤j≤(k+1)·Ne′ and each i∈M(j), process (14) and (15).
Note that compared to the highspeed serial decoding, this algorithm has faster convergence speed [21].
Simulation results
In this section, we compare the bit error rate (BER) performance of the PEGLDMCs, LDMCs proposed in [17] and original LDMCs [16]. The PEGLDMCs are constructed with the proposed PEGLDMCs algorithm. The code rate is R=0.5. The channel matrix H is assumed to remain constant during the transmission of each codeword. We perform three outeriterations between MIMO detector and LDPC decoder. In each outeriteration, MIMO detection is performed followed by 40 inneriterations inside LDPC decoder. The decoding process is halted if the decoder converges to a valid code or a maximum number of outer and inner iterations are reached. Note that in [16, 17], only the physical layer is considered. Therefore, for the sake of fairness, we have only considered the LDMCs of the same length as that of [16, 17], which are transmitted over the physical layer.
Figure 2 shows the BER performance of proposed PEGLDMCs, LDMCs proposed in [17] and original LDMCs [16]. The transmission is over a 4×4 MIMO channel with 16QAM and 64QAM modulations. The first layer P_{g} is of size 480×1920 and 480×2880 for each modulation, respectively. Therefore, the paritycheck matrix P of the corresponding LDMCs is of size 960×1920 and 1440×2880 for 16QAM and 64QAM. Thus, the code length is n=1920 for 16QAM and n=2880 for 64QAM, respectively. For the proposed PEGLDMCs, the HSSD and FCSD algorithms are both applied for decoding. It can be seen that our proposed PEGLDMCs with HSSD and FCSD algorithms have about 0.3 and 0.5 dB performance gain compared to original LDMCs with 16QAM modulation, and they have about 0.1 and 0.3 dB performance gain compared to LDMCs in [17] with 16QAM modulation. The proposed PEGLDMCs with HSSD and FCSD algorithms also have about 0.4dB and 0.6dB performance gain compared to original LDMCs with 64QAM modulation, and they have about 0.15 and 0.35 dB performance gain compared to LDMCs in [17] with 64QAM modulation. This is due to the reason that the girth of the PEGLDMC is optimized globally by the PEGLDMC algorithm and the FCSD has faster convergence speed. Therefore, the performance of PEGLDMCs with FCSD algorithm is better than that of PEGLDMCs with HSSD algorithm and original LDMCs in [16].
Figure 3 shows the BER performance of proposed PEGLDMCs, LDMCs proposed in [17] and original LDMCs. The simulations are under 4×4 MIMO and 16×16 MIMO with 16QAM modulations. For the proposed PEGLDMCs, the proposed HSSD and FCSD algorithms are also applied for decoding. It can be seen from Fig. 3 that our proposed PEGLDMCs with HSSD and FCSD algorithms have about 0.5 and 0.6 dB performance gain compared to original LDMCs over a 16×16 MIMO channel, and they have about 0.1 and 0.3 dB performance gain compared to LDMCs in [17] over a 16×16 MIMO channel. This is due to the reason that the performance of LDMCs depends on the girth, and our proposed new modified PEG algorithm can construct PEGLDMCs with a girth of 8, which is larger than that of LDMCs proposed in [17].
Conclusions
In this paper, we have introduced the PEGLDMC algorithm to construct LDMCs of large girth. In order to reduce the complexity of detection, SD detection has been adopted into LDMC coded MIMO systems. The complexity of LDMCconstrained SD detection has been analyzed and compared to that of MMSE and ZF detections. Two new decoding algorithms have also been proposed for highspeed decoding and fast convergence decoding. The simulation results have shown that the proposed PEGLDMCs with new decoding algorithms achieved better BER performance than that of LDMCs proposed in [17] and original LDMCs in [16].
Methods/Experimental
The purpose of this work is to reduce the detection complexity in the MIMO systems and to achieve a good BER performance. For this purpose, we use the method which is referred to as LDMCconstrained SD detection. Since LDMCs with large girth have good performance, we propose a new modified PEG algorithm to construct large girth LDMCs. In this paper, we analyze the complexity of SD detection in the LDMC coded MIMO systems, and compare it to ZF detection and MMSE detection. Furthermore, we propose highspeed serial decoding and fast convergence shuffled decoding in order to improve the BER performance.
In this paper, the comparison of BER performance among the PEGLDMCs, original LDMCs [16], and LDMCs proposed in [17] is presented. The PEGLDMCs are constructed with the proposed PEGLDMCs algorithm. The function to generate the LDMCs and proposed PEGLDMCs algorithm for MIMO systems can be made by MATLAB code. Experimental results in this paper had performed by using MATLAB R2016a on Intel Core i74790 @ 3.60GHz platform.
Abbreviations
 BER:

Bit error rate
 BP:

Belief propagation
 CSI:

Channel state information
 FCSD:

Fast convergence shuffled decoding
 HSSS:

Highspeed serial decoding
 LDMC:

Lowdensity MIMO code
 LDPC:

Lowdensity paritycheck
 LLR:

Loglikelihood ration
 MIMO:

Multipleinput multipleoutput
 MMSE:

Minimum meansquare error
 PEG:

Progressive edgegrowth
 SD:

Sphere decoding
 ZF:

Zeroforcing
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Acknowledgements
We gratefully acknowledge the MDMC Lab, Chonbuk National University, Korea and Donghua University, China, which provided the simulation platform.
Funding
This work was supported by National Natural Science Foundation of China (61671143,61631004,61571315), and the Initial Research Funds for Young Teachers of Donghua University.
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HH and XQJ contributed to the main idea, PEG based algorithms and wrote the manuscript. PS and SC designed and carried out the analysis and simulations. MHL and HJ reviewed and edited the manuscript. All authors read and approved the final manuscript.
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Hai, H., Jiang, XQ., Selvaprabhu, P. et al. LDMC design for low complexity MIMO detection and efficient decoding. J Wireless Com Network 2018, 187 (2018). https://doi.org/10.1186/s1363801812026
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DOI: https://doi.org/10.1186/s1363801812026
Keywords
 Lowdensity paritycheck (LDPC) codes
 Multipleinput multipleoutput (MIMO)
 Sphere decoding (SD)
 Lowdensity MIMO codes (LDMCs)