Double-Layer Low-Density Parity-Check Codes over Multiple-Input Multiple-Output Channels

Abstract

We introduce a double-layer code based on the combination of a low-density parity-check (LDPC) code with the multiple-input multiple-output (MIMO) system, where the decoding can be done in both inner-iteration and outer-iteration manners. The present code, called low-density MIMO code (LDMC), has a double-layer structure, that is, one layer defines subcodes that are embedded in each transmission vector and another glues these subcodes together. It supports inner iterations inside the LDPC decoder and outeriterations between detectors and decoders, simultaneously. It can also achieve the desired design rates due to the full rank of the deployed parity-check matrix. Simulations show that the LDMC performs favorably over the MIMO systems.

1. Introduction

Multiple-input multiple-output (MIMO) transmission has been identified as one of the most practical methods to combat fading and to increase the capacity of wireless channels in the recent years [1–3]. In order to ensure a desired quality of transmission, an outer error correction code can be employed additionally to decrease the occurring errors in imperfect channels. In this case, an appropriate low-density parity-check (LDPC) code [4–7] may be elegantly used for MIMO channels. Actually, the decoding of LDPC codes over MIMO channels can be well dealt with an iterative manner, where probabilistic information should be exchanged iteratively between MIMO detector and LDPC decoder. At the receiver, the detector has to calculate the maximum posteriori probability for each bit. One possible realization of this calculation is called sphere decoder [8]. Much effort has been taken to reduce the complexity of the detection process by introducing constraints in space and/or time. A modified sphere decoder was proposed in [6], which sacrifices performance to reduce the complexity of detection.

The complexity of the MIMO detector depends on the number of bits mapped to a transmission vector (symbols) transmitted simultaneously via multiple antennas. In order to reduce its complexity, the low-density MIMO code (LDMC) was presented with the constraints for each transmitting vector [4, 5, 9]. For an LDMC, the parity-check matrix has to be divided into two layers. One layer defines subcodes which are embedded in each transmitting vector, and another glues these subcodes together. The goal of the double-layer structure is that the first layer reduces the complexity of the decoder, while the second layer has to be handed by an outer decoder. Unfortunately, the double-layer structure of the original LDMC only supports the outer-iterations and it cannot bolster the inner-iterations inside the decoder. This limits the performance and the flexibility of the decoding at the receiver. It seems that a systematic LDMC design that stands by both outer-iterations and inner-iterations for double-layer decoder still remains an open problem.

In this paper, we introduce a novel double-layer decoder for a benefit of the LDMC design criterion over MIMO channels. One advantage of this structure is that the yielded parity-check matrix of an LDMC has full rank and hence can achieve the desired code rates. Another is that the present LDMC can be decoded with low complexity since it supports both inner-iterations and out-iterations at the receiver.

This paper is organized as follows. In Section 2, we introduce the LDPC coded MIMO channels. Some properties of the LDMC are presented for the benefits of the construction. Section 3 describes the conventional LDPC code with multiple parity-check (MPC) encoders. After that an LDMC based on the MPC encoder is designed in Section 4. Simulation results are given in Section 5. Finally, Section 6 concludes this paper.

2. System Model

At the output of the LDPC encoder, the encoded 𝑀_𝑡𝑄 bits, 𝐬=[𝑏₁,…,𝑏_𝑀𝑡𝑄]∈{0,1}^𝑀𝑡𝑄, are mapped into a group of 𝑀_𝑡 constellation symbols 𝐱=[𝑥₁,…,𝑥_𝑀𝑡]∈𝒜^𝑀𝑡, where each symbol 𝑥_𝑗 is taken out of constellations 𝒜 of size 2^𝑄. The sequence of constellation symbols is passed to the transmit filter, and then sent through 𝑀_𝑡 transmit antennas. At the receiver, the detector has to calculate the maximum posteriori probability for each bit, and this probability will be passed to the LDPC decoder for iterative decoding. After a finite number of inneriterations in the LDPC decoder, hard decisions, which are the estimated codewords, are made at the decoder. Soft information from the LDPC decoder is then passed back to the MIMO detector to produce the updated probabilities. The loop between the LDPC decoder and the MIMO detector is called outer-iteration.

We consider a MIMO communication system with 𝑀_𝑡 transmit antennas and 𝑀_𝑟 receive antennas. The MIMO channel model is described as 𝐲=𝐇𝐱+𝐍, (1) where 𝐲∈ℂ^𝑀𝑟×1 is the complex received signal vector, 𝐇∈ℂ^{𝑀𝑟×𝑀𝑡} is the channel fading matrix with independent entries that are complex Gaussian distributed with zero mean and unit variance, 𝐍∈ℂ^𝑀𝑟×1 is the complex white Gaussian noise with variance 𝜎² per dimension, 𝐱∈ℂ^𝑀𝑡×1 is the complex transmitted signal vector that satisfies the component-wise energy constraint 𝐸(|𝑥_𝑖|²)=𝐸_𝑠/𝑀_𝑡, and the notation 𝐸_𝑠 represents the total transmitted power. The channel information 𝐇 is assumed to be known to the receiver but not the transmitter.

Let 𝐛 denote a set of 𝑀_𝑡𝑄−1 transmitted bits, which exclude the 𝑘th bit 𝑏_𝑘 while 𝐱(𝐛,𝑏_𝑘) is a vector of 𝑀_𝑡 components containing the symbols corresponding to 𝐛 and 𝑏_𝑘. For a given received signal vector 𝐲, we perform the bitwise detection [10] to determine the posteriori probability of bit 𝑏_𝑘, forall𝑘∈{1≤𝑘≤𝑀_𝑡𝑄}, by calculating 𝑃_𝑀→𝐿𝑏_𝑘=_𝐛1√2𝜋𝜎^𝑀𝑟𝑒^{−‖𝐲−𝐇𝐱(𝐛,𝑏𝑘)‖2/(2𝜎2)}×_{𝑏𝑗∈𝐛}𝑃_𝐿→𝑀𝑏_𝑗 (2) or its log-likelihood ratio 𝐿_𝑀→𝐿𝑏_𝑘𝑃=log^𝑀→𝐿𝑏_𝑘=1𝑃_𝑀→𝐿𝑏_𝑘=0, (3) where ‖⋅‖² denotes the norm square of a vector, the subscript 𝑀→𝐿 denotes the message passed from the MIMO detector to the LDPC decoder, and 𝐿→𝑀 denotes the message from the LDPC decoder to the MIMO detector.

The remaining problem of this iterative decoding is the complexity of the MIMO detection. Next, we will show that the complexity of the MIMO detection can be reduced by introducing some constraints in the parity-check matrix of the LDPC codes, which are also called the LDMCs.

The LDMC is a kind of the linear block codes that can be described with a parity-check matrix 𝐇_𝑐 of the size 𝑚×𝑛, subject to the constraint 𝐇_𝑐𝐱=𝟎. The LDMC parity-check matrix can be described by two layers with 𝐇_𝑐=𝐇_𝑔𝐇_𝑒=𝐇_𝑔𝐇^′_𝑒⋯00⋱00⋯𝐇^′_𝑒. (4) The first layer 𝐇_𝑔 is a sparse parity-check matrix, and the second layer 𝐇_𝑒 defines multiple independent subcodes. Each subcode corresponding to 𝐇^′_𝑒 has a block length no more than 𝑀_𝑡𝑄.

As the aforementioned structure, each transmission vector carries 𝑀_𝑇𝑄 bits. For the transmission, it has to be guaranteed that all bits of a subcode corresponding to 𝐇^′_𝑒 should be transmitted within one transmission vector. Thus, each transmission vector 𝐛 carries an embedded code corresponding to 𝐇^′_𝑒. The action of 𝐇_𝑔 is to glue all embedded codes together. Since the matrix 𝐇_𝑔 is sparse and the embedded codes corresponding to the parity-check matrix 𝐇^′_𝑒 define code constraints to each MIMO transmission vector, the overall-coding process is called the LDMC coding scheme or LDMC.

To enhance the performance of the LDMC coding scheme, we need to reduce the complexity of the MIMO detecting and the LDPC decoding in the combined MIMO transmission system, while keeping the overall data rate and the capacity approaching communication performance. As described, a straightforward approach is to embed constraints for the involved bits within one transmission vector. A simple constraint on several bits is a single parity-check (SPC) constraint. Namely, the matrix 𝐇^′_𝑒 would become the parity-check matrix of an SPC code. Thus if we constraint the involved information bits in one SPC constraint, the overall possibility of detection is 2^{𝑀𝑇𝑄−1}. Furthermore, if we constraint in 𝑞 parity-check constraints embedded within one transmission vector, the overall possibility of detection will downscale to 2^{𝑀𝑇𝑄−𝑞}.

In order to encode efficiently, a quasicyclic structure was proposed for the parity-check matrix of LDMCs [4, 5]. In the quasicyclic structure, the sparse matrix 𝐇_𝑔 consists of cyclic-permutation matrices and zero matrices. Since the subcodes 𝐇^′_𝑒 in transmission vectors are independent and should not be connected to each other, as shown in (4), the second layer 𝐇_𝑒 consists of only identity matrices and zero matrices. Therefore, 𝐇_𝑒 has no cyclic-permutation structure. Consequently, the flexibility of 𝐇_𝑒 may be reduced, which results in many short cycles. Therefore, 𝐇_𝑒 is only employed in the detection, not in the decoding. Since 𝐇_𝑔 alone is too sparse to be applied in the iterative decoding, there is no inner iterations in the decoder in [5]. Therefore, how to address the design of inner iterations in the LDMC is still a retained problem. In the next section, we will propose a simple LDMC to solve this problem.

3. LDPC Code Based on Multiple Parity-Check Codes

Recently, Baldi et al. have proposed an LDPC coding scheme on the basis of multiple serially concatenated multiple parity-check (M-SC-MPC) codes [11]. This code offers flexible code rate and code length with low encoding complexity. Let 𝑛_𝑖, 𝑘_𝑖, and 𝑟_𝑖 denote code length, information length, and parity-check length of the 𝑖th MPC component code, respectively. The encoder of the 𝑖th MPC component code can be implemented as 𝑟_𝑖 SPC encoders. The matrix cells of the 𝑖th encoder are filled in column-wise order. The first 𝑟_𝑖−𝑠_𝑖 cells, with 𝑠_𝑖=𝑘_𝑖 mod 𝑟_𝑖, are unused. The parity bit 𝑝_𝑗 is calculated by XORing the elements of the row, which is stored in the last column, at the same row. We obtain a parity-check matrix 𝐇_𝑖 for the 𝑖th component code, consisting of a row of ⌈𝑛_𝑖/𝑟_𝑖⌉ identity matrices of size 𝑟_𝑖×𝑟_𝑖, and the first 𝑟_𝑖−𝑠_𝑖 columns are shortened. The yielded matrix 𝐇_𝑖 forms a block-row of the parity-check matrix 𝐇_𝑐 of the M-SC-MPC code. The M-SC-MPC code has information length 𝑘=𝑘₁, parity-check length 𝑚=∑^𝑀_𝑖=1𝑟_𝑖 and the code length 𝑛=𝑘+𝑚 can be viewed as LDPC codes.

If an interleaver is inserted before each MPC encoder, an LDPC code can be constructed with larger girths [12]. The parity-check matrix 𝐇_𝑐 can be designed in columnwise or rowwise to avoid short cycles. In the next section, we will introduce a modified progress edge-growth algorithm (PEG) to construct 𝐇_𝑐 see Algorithm 1. Each component matrix 𝐇_𝑖 has row weight ⌊𝑛_𝑗/𝑟_𝑗⌋ in 𝑟_𝑖−𝑠_𝑖 rows, row weight ⌈𝑛_𝑗/𝑟_𝑗⌉ in the remaining 𝑠_𝑖 rows. Also, it has column weigh one in 𝑛_𝑖 columns and column weight zero in the remaining 𝑛−𝑛_𝑖 columns. The interleaver of the 𝑖th MPC encoder is constructed according to the structure of designed parity-check matrix 𝐇_𝑖.

(1) 𝑞=0. (2)𝐲=∅ (3) for 𝑙=0 do (4)  arrange the check node degrees of the   check node group 𝐠𝑙 in nondecreasing order   𝐱𝑖=[𝑥0𝑖,𝑥1𝑖,…,𝑥𝑟𝑖𝑖−1]=[𝜌⌊𝑛𝑖/𝑟𝑖⌋,𝜌⌊𝑛𝑖/𝑟𝑖⌋,…,𝜌⌈𝑛𝑖/𝑟𝑖⌉]. (5)  end for (6)  for 𝑗=0 to 𝑛−1 do (7)  if 𝑗≥𝑛−𝑚 then (8)   𝐬𝑐={𝑐𝑗−(𝑛−𝑚),…,𝑐𝑚−1}⧵𝐲. (9)  else (10)    𝐬𝑐={𝑐0,𝑐1,…,𝑐𝑚−1}⧵𝐲. (11)  end if (12)  for 𝑘=0 to 𝑀−1 do (13)   if 𝑘=0 then (14)    if 𝑗≥𝑛−𝑚 then       𝐞0𝑣𝑗← edge (𝑐𝑖,𝑣𝑗), where 𝐞0𝑣𝑗 is the first edge    incident to 𝑣𝑗. This edge corresponds to the “1” in    the diagonal line of matrix 𝐇𝑝. (15)    else     𝐞0𝑣𝑗← edge (𝑐𝑖,𝑣𝑗), where 𝑖=(𝑗+(𝑟1−𝑠1))    mod𝑟1. (16)    end if (17)   else    expand a subgraph from 𝑣𝑗 up to depth 𝑙 under   the current graph setting 𝐧𝑙𝑣𝑗 such that 𝐧𝑙𝑣𝑗∩𝐬𝑐≠∅   but 𝐧𝑣𝑙+1𝑗∩𝐬𝑐=∅, or the cardinality of 𝐧𝑙𝑣𝑗 stops   increasing, then 𝐞𝑘𝑣𝑗← edge(𝑐𝑖,𝑣𝑗), where 𝐞𝑘𝑣𝑗 is the   𝑘th edge incident to 𝑣𝑗 and 𝑐𝑖 is a check node picked   from the set 𝐧𝑙𝑣𝑗∩𝐬𝑐 having the lowest check-node   degree. (18)   end if (19)   if the degree of 𝑐𝑖 equals 𝑥𝑞𝑘 then      Add 𝑐𝑖 into 𝑌.       𝐱𝑖←𝐱𝑖⧵𝑥𝑞𝑘.       𝑞=𝑞+1. (20)   end if      find out which check node group 𝐠𝑖 includes 𝑐𝑖.    𝐬𝑐←𝐬𝑐⧵𝐠𝑖. (21)   end for (22)  end for

Algorithm 1 

Modified PEG algorithm.

Theorem 1 (full-rank criteria). The parity-check matrices of LDPC codes on the basis of the MPC encoders are of full rank, and hence they can achieve the desired code rates.

Proof. If all the rows of the component parity-check 𝐻_𝑖 are summed, respectively, a vector will be obtained with the first 𝑛_𝑖 elements are ones and the remaining 𝑛−𝑛_𝑖 elements are zeros. Then it is obvious that any row of 𝐇_𝑐 cannot be obtained by a linear combination of the other rows in 𝐇_𝑐. Therefore, we can get the desired code rate 𝑅=𝑘/𝑛.

Theorem 2 (low-complexity criteria). The encoding com-plexity of the LDPC code on the basis of the MPC encoders is upper bounded by ∑^𝑀_𝑖=1(𝑘_𝑖−𝑟_𝑖) and lower bounded by ∑^𝑀_𝑖=1⌈𝑘_𝑖/𝑟_𝑖⌉−1, respectively.

Proof. We evaluate the computation complexity of the encoding processing. There are three cases of encoding processing [13]. If the 𝑖th component encoder generates the 𝑟_𝑖 parity bits in parallel, the 𝑖th component encoder determines 𝑟_𝑖 parity bits with 𝑘_𝑖−𝑟_𝑖 XOR operations. In this case, ∑^𝑀_𝑖=1(𝑘_𝑖−𝑟_𝑖) XOR operations are required to obtain all the parity bits. The encoding processing is accomplished from the 1st MPC component encoder to the 𝑀th MPC component encoder in 𝑀 steps. If the 𝑖th component encoder generates the 𝑟_𝑖 parity bits in serial, the 𝑖th component encoder determines 𝑟_𝑖 parity bits serially with ⌈𝑘_𝑖/𝑟_𝑖⌉−1 XOR operations. In this case, ∑^𝑀_𝑖=1(⌈𝑘_𝑖/𝑟_𝑖⌉−1) XOR operations are required to obtain all the parity bits. Therefore, the encoding processing is completed from the 1st SPC encoder to the 𝑚th SPC encoder in 𝑚 steps. If the encoding processing is a combination of the above-mentioned two cases, the complexity and encoding time is lower bounded by the parallel case and upper bounded by the serial case.

4. LDMCs Based on MPC Codes

To illustrate the double-layer structure of the LDMC, we propose a simple LDMC as an example in this section.

We consider a parity-check matrix 𝐇_𝑐 given by 𝐇_𝑐=𝐇^𝑑𝐇^𝑝=𝐇₁𝐇₂𝐇₃=100110100000011001010000011010111000100101000100001010110110110101001001, (5) where the subscript “T” denotes the transpose of the matrix. This is a parity-check matrix that corresponds to an LDPC code [𝑑₁,𝑑₂,…,𝑑₆,𝑝₁,𝑝₂,…,𝑝₆] of length 12, where 𝑑₁,𝑑₂,…,𝑑₆ are the information bits and 𝑝₁,𝑝₂,…,𝑝₆ are the parity bits. It is by no means a good LDPC code, which is nothing more than a simple example. The parity-check matrix 𝐇_𝑐 in (5) can be divided into three 2×12 submatrices, that is, 𝐇₁, 𝐇₂, 𝐇₃. In each column of 𝐇_𝑖 (𝑖=1,2,3), there is only one “1” element. Due to the special structure of the three submatrices 𝐇₁, 𝐇₂ and 𝐇₃, we define 𝐇_𝑔=(𝐇^T₁𝐇^T₂)^T and 𝐇_𝑒=𝐇₃. Therefore, we first obtain the parity bits [𝑝₅,𝑝₆] from 𝐇₃, then the parity bits [𝑝₃,𝑝₄] from 𝐇₂, and the parity bits [𝑝₁,𝑝₂] from 𝐇₁. The encoding processing of this code includes three steps as follows.

Step 1. Given the information bits [𝑑₁,𝑑₂,…,𝑑₆], compute the parity bits [𝑝₅,𝑝₆] based on the last two columns of the third parity-check matrix 𝐇₃.

Step 2. Given the information bits [𝑑₁,𝑑₂,…,𝑑₆] along with the yielded parity bits [𝑝₅,𝑝₆] in Step 1, compute the parity bits [𝑝₃,𝑝₄] based on the ninth column and the tenth column of the second parity-check matrix 𝐇₂.

Step 3. Given the information bits [𝑑₁,𝑑₂,…,𝑑₆] along with the previous parity bits [𝑝₃,𝑝₄,𝑝₅,𝑝₆], compute the parity bits [𝑝₁,𝑝₂] based on the seventh column and the eighth column of the first parity-check matrix 𝐇₁.

Now we transmit this code through a MIMO system with two transmit antennas and 8PSK modulation scheme. Each transmission vector carries 6 bits which results in 2^𝑀𝑡𝑄=64 possibilities reflecting all bit possibilities. However, because of the parity-check constraints in 𝐇₃, that is, 𝑑₃⊕𝑑₅⊕𝑝₁⊕𝑝₂⊕𝑝₄⊕𝑝₅𝑑=0,₁⊕𝑑₂⊕𝑑₄⊕𝑑₆⊕𝑝₃⊕𝑝₆=0, (6) where ⊕ denotes the XOR operation, we can divide the code word into two transmission vectors, that is, [𝑑₃,𝑑₅,𝑝₁,𝑝₂,𝑝₄,𝑝₅] and [𝑑₁,𝑑₂,𝑑₄,𝑑₆,𝑝₃,𝑝₆]. Each transmission vector contains one parity-check constraint. This results in 2^{𝑀𝑇𝑄−1}=32 possibilities. Therefore, the present LDPC code is an LDMC in essence. In this code, the first layer that glues the subcodes is 𝐇_𝑔 and the second layer that defines the subcode is 𝐇_𝑒=𝐇₃. Note that we can transform 𝐇_𝑒 into the diagonal form by column permutations, which results in the parity-check matrix 𝐇_𝑐=011000101000100100010100111100010010000010101100=111111000000000000111111𝐇^T_𝑔𝐇^T_𝑒^T=𝐇^T₁𝐇^T₂𝐇^T₃^T. (7) It is obvious that the first transformed layer 𝐇_𝑔 is a sparse parity-check matrix and the second layer 𝐇_𝑒 can generate two unconnected subcodes corresponding to 𝐇^′_𝑒=[1,1,1,1,1,1].

We note that if we let 𝐇_𝑔=(𝐇^T₂𝐇^T₃)^T and 𝐇_𝑒=𝐇₁, in a similar way we calculate the parity bits [𝑝₁,𝑝₂] from 𝐇₁, then the parity bits [𝑝₃,𝑝₄] from 𝐇₂, and the parity bits [𝑝₅,𝑝₆] from 𝐇₃. Therefore, the equivalent constructions of this code can be described as the following.

Step E1
given the information bits [𝑑₁,𝑑₂,…,𝑑₆], compute the parity bits [𝑝₁,𝑝₂] based on 𝐇₁.

Step E2
given the information bits [𝑑₁,𝑑₂,…,𝑑₆] along with the yielded parity bits [𝑝₁,𝑝₂], compute the parity bits [𝑝₃,𝑝₄] based on 𝐇₂.

Step E3
given the information bits [𝑑₁,𝑑₂,…,𝑑₆] along with the previous parity bits [𝑝₁,𝑝₂,𝑝₃,𝑝₄], compute the parity bits [𝑝₅,𝑝₆] based on 𝐇₃.

According to the aforementioned LDMC construction based on three matrices {𝐇_𝑖∶𝑖=1,2,3}, the proposed design approach can be generalized with 𝑀 block matrices {𝐇_𝑖∶𝑖=1,…,𝑀}.

Theorem 3. The LDPC code on the basis of the MPC component encoder in (4) is an LDMC, where 𝐇_𝑐=𝐇^𝑑𝐇^𝑝=(𝐇^T_𝑔𝐇^T_𝑒)^T (8) with 𝐇_𝑔=(𝐇^T₁𝐇^T₂⋯𝐇^T_𝑀−1)^T and 𝐇_𝑒=𝐇_𝑀. Generally, the first layer 𝐇_𝑔 can be selected as 𝐇_𝑔=(𝐇^T₁⋯𝐇^T_𝑖−1𝐇^T_𝑖+1⋯𝐇^T_𝑀)^T and the second layer 𝐇_𝑒 can be selected as 𝐇_𝑒=𝐇_𝑖, forall𝑖∈{1,…,𝑀}.

Proof. The present LDMC is a straightforward extending of the LDMC given in (5). Therefore, the proof of this theorem can be derived in a similar way, which is omitted for simplicity.
For simplicity, we consider the LDMC with 𝐇_𝑔=(𝐇^T₁𝐇^T₂⋯𝐇^T_𝑀−1)^T and 𝐇_𝑒=𝐇_𝑀. We note that there are 𝑟_𝑀 SPC constraints in the parity-check matrix 𝐇_𝑒 that has row weight ⌊𝑛_𝑀/𝑟_𝑀⌋ in the first 𝑟_𝑀−𝑠_𝑀 rows and row weight ⌈𝑛_𝑀/𝑟_𝑀⌉ in the remaining 𝑠_𝑀 rows. Each subcode 𝐇^′_𝑒 involves 𝑞 SPC constraints. We divide 𝑟_𝑚 SPC constraints into 𝑟_𝑚/𝑞 separate subgroups. At the transmitter, each transmission vector involves one of the 𝑟_𝑚/𝑞 subgroups. Then the overall possibilities for the MIMO detection is downscaled from 2^𝑀𝑡𝑄 to 2^{𝑀𝑡𝑄−𝑞}. The larger 𝑞 means the smaller complexity of the MIMO detection. Namely, let 𝑟_𝑚/𝑞=𝑛/𝑛^′_𝑒, and then it is obvious that the larger 𝑛^′_𝑒 implies the smaller complexity of the MIMO detection. Since 𝑛^′_𝑒≤𝑀_𝑡𝑄, the optimal block length of the subcode is 𝑛^′_𝑒=𝑀_𝑡𝑄. (9)
In addition, the parity-check matrix 𝐇_𝑐 of the present LDMCs should satisfy the following two constraints: (1)the 𝐇^𝑝 part of 𝐇_𝑐 contains the lower triangular pattern; (2)in each column of 𝐇_𝑖, forall𝑖∈{1,…,𝑚}, of 𝐇_𝑐, there is only one “1” element.
The first constraint ensures that the present LDMC is linear encodable [14]. The second constraint guarantees that the 𝑟_𝑖 parity bits can be generated by the 𝑟_𝑖 parity-check constraints, simultaneously, in the 𝑖th component encoder corresponding to the matrix 𝐇_𝑖 [11–13]. Observing two constraints, the parity-check matrix 𝐇_𝑐 of the LDMC can be efficiently designed with the modified progress edge-growth (PEG) algorithm.

Remark 4. In the present algorithm, lines 1–5 and lines 20-21 are used to ensure that the parity-check matrix 𝐇_𝑖 has row weight ⌊𝑛_𝑖/𝑟_𝑖⌋ in 𝑟_𝑖−𝑠_𝑖 rows and row weight ⌈𝑛_𝑖/𝑟_𝑖⌉ in the remaining 𝑠_𝑖 rows, which are required for the design of parity-check matrix of an LDMC with a double-layer structure. Line 14 and line 17 are used to guarantee that 𝐇^𝑝 is a lower triangular matrix, which is for the desired code rate. Since there is no interleaver in front of the first component encoder, line 15 is used to ensure that 𝐇₁ consists of identity matrices with the first 𝑟₁−𝑠_𝑖 columns eliminated, as shown in (5). Due to this special structure, the LDPC decoding can be well performed with low complexity.

Remark 5. Since 𝐇_𝑐 is obtained from the modified PEG algorithm, the girth of 𝐇_𝑐 and 𝐇_𝑒 can be enlarged, simultaneously [12]. Therefore, the matrix 𝐇_𝑒 can be used in both detection and inner iteration. Consequently, we can perform multiple out iterations between MIMO detector and LDPC decoder at the receiver. In each out iteration, the MIMO detection is performed once followed by a number of inner iterations of LDPC decoding. The decoding process is halted if the decoder converges to a valid code or a maximum number of iterations is reached. It is obvious that the proposed double-layer LDMC is different from the quasicyclic LDMC [4], where the PEG algorithm is based on the permuted identity matrices.

5. Simulations

So far, we have constructed an LDMC with the modified PEG algorithm in the previous section. In order to simulate the proposed LDMC denoted 𝑐_𝑀 with simplicity, it is assumed that the block length of the LDMC is 𝑛=1920 and the code rate is 0.5. The numbers of the component encoder 𝑀 are 5 and 6, respectively, that is, 𝑀∈{5,6}. Some parameters are given as follows: (1)𝑀=5, 𝑟₁=360, 𝑟₂=360, 𝑟₃=360, 𝑟₄=360, 𝑟₅=480, and 𝐿=⌈𝑛/𝑟₅⌉=4;(2)𝑀=6, 𝑟₁=288, 𝑟₂=288, 𝑟₃=288, 𝑟₄=288, 𝑟₅=288, 𝑟₆=480, and 𝐿=⌈𝑛/𝑟₆⌉=4.

The simulation results are in terms of the frame error rate (FER) with the iterative decoding. We make use of the QPSK modulation and the 16-QAM modulation transmissions over a 4×4 MIMO channels, as shown in Figures 1 and 2. Therefore, the block length of transmission vector is 8 with ⌈𝑛^′_𝑒/𝐿⌉=2. We simulate the proposed codes in two ways. For one hand, we perform 3 outer iterations and 50 inner iterations, shown in Figure 1, for both 𝐜₅ and 𝐜₆. For another, we perform 50 outer iterations and no inner iteration. For a fair comparison, we have also simulated the original LDMCs with a minor modification in [5]. The 𝐇^′_𝑒 part 1111000011110000001100110011001101010101010101010000111111110000, (10) which involves a parity-check matrix [5], is replaced with the matrix 1111000000000000000011110000000000000000111100000000000000001111, (11) which involves 4 parity-check constraints of length 4. Each transmission vector of length 8 involves 2 SPC constraints. This modified code is denoted by 𝐜₀. The performance curve of 𝐜₀ is illustrated with 50 outer iterations and no inner iteration.

Figure 1 

FER performances of the proposed LDMCs 𝐜5 and 𝐜6 with QPSK modulation.

Figure 2 

FER performances of the proposed LDMCs 𝐜5 and 𝐜6 with 16QAM modulation.

It is shown in Figure 2 that three LDMCs, 𝐜₀, 𝐜₅, 𝐜₆, can downscale overall possibilities of the MIMO detection from 2⁸ to 2⁶. The proposed two codes 𝐜₅ and 𝐜₆ perform the similar performances as that of the original LDMC 𝐜₀ when no inner iteration is applied in the LDPC decoder. However, since the present LDMCs support inner iterations in the LDPC decoder, we can improve the performance of these codes by increasing several inner iterations in the LDPC decoder.

6. Conclusion

A double-layer structure LDMC is introduced with the LDPC decoder and the MIMO detector. It provides both inner iterations and outer iterations at the receiver, which gives more flexibility for detecting and decoding than that of the original LDMC. The advantage of the double-layer LDMC is that the parity-check matrix has the full rank to achieve the desired code rate, which can be efficiently employed in a modified PEG algorithm. The present LDMC also reduces the complexity of the MIMO detection due to the low-density of the parity-check matrix. Simulations show that the performance of the LDMC is enhanced with the increasing inner iterations in the LDPC decoder.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (60902044), the Ph.D. Program Foundation of Ministry of Education of China (20090162120070), and the World Class University R32-2010-000-20014-0 NRF, Korea.