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Novel polarcoded spacetime transmit diversity scheme over Rician fading MIMO channels
EURASIP Journal on Wireless Communications and Networkingvolume 2018, Article number: 24 (2018)
Abstract
In this paper, a polarcoded spacetime transmit diversity (STTD) scheme is proposed in order to improve the performance of multipleinput multipleoutput (MIMO) system. In Rician fading MIMO channels, the corresponding polarcoded STTD system can be equivalent to a single transmission channel for each polar code bit. Density evolutions for the polarcoded STTD systems are proposed based on the analyses of the single transmission channel. The proposed density evolutions provide preferable guidance to construct polar codes for the polarcoded STTD systems. Simulation results show that the BER performance of polarcoded STTD system is significantly improved as the number of antennas increases and the influence of the variable Rician Kfactor of MIMO channels gradually vanishes. The proposed 2 × 2 polarcoded STTD system can provide better frame error rate (FER) performance than lowdensity paritycheck (LDPC)based system under low and mid code rates, and the average advantage is 0.3 dB approximately when the code rate is 0.25 and 0.2 dB when the code rate is 0.5.
Introduction
Polar code introduced by Arıkan can achieve the capacity of the memoryless symmetric channel with low encoding and decoding complexity [1]. A successivecancellation (SC) decoding specialized for polar codes is proposed at the same time with the complexity O(N logN), which can provide acceptable bit error rate (BER) performance. The belief propagation (BP) decoding for polar codes is discussed in [2], and a modified BP decoder is given in order to improve the performance of polar codes for short lengths. The performance of BP decoder can also be improved by adapting the paritycheck matrix [3]. Polar codes with cyclic redundancy checkaided (CRCaided) successivecancellation list (SCL) decoding [4] can provide the optimal BER performance, and the complexity is reduced in [5]. Polar codes with CRCaided list decoding can outperform stateoftheart lowdensity paritycheck (LDPC) codes at short block lengths [6]. Thus, polar code, the highly efficient and reliable channel code, has many potential applications in wireless communications.
Multipleinput multipleoutput (MIMO) systems have been widely applied in modern communication systems and will be still applied as a key technology for the nextgeneration wireless system with the number of antennas going large [7]. Diversity is an efficient technique to combat multipath fading in MIMO channels. The effect of transmit and receive diversity can be enhanced as the number of antennas increases [8, 9]. Spacetime transmit diversity (STTD) can achieve a full diversity gain with a simple structure, which can be regarded as a process of coding. The basic spacetime coding (STC) model for two transmit antennas is Alamouti’s scheme [10]. To improve the performance, channel coding is adopted in MIMO systems. STTD has been successfully combined with LDPC codes [11, 12] and Turbo codes [13], which enhanced the errorcorrecting performance but also induced huge encoding and decoding complexity. The complexity is mainly caused by the cost of LDPC and Turbo encoding and decoding algorithms. Thus, the lowcomplexity polar code can be a better choice to combine with STTD in MIMO systems.
Density evolution (DE) is an important analytical tool for graph codes with messagepassing decoding algorithms. The DE for polar codes based on the message passing in SC decoding graph is proposed in [14], which provides a preferable guidance to construct polar codes for obtaining better performance. In the process of the code construction, the selection of an information set is a key factor that impacts the performance of polar codes. Imprecise selection of the information set will aggravate the performance. The DE for polar codes provides a way to select the information set precisely for any discrete memoryless channel (DMC). An efficient way to construct polar codes based on DE is proposed in [15], which reduces the calculation cost.
In this paper, we extend the previous works on Rayleigh fading MIMO channels [16]. We analyze polar codes and STTD system in Rician fading MIMO channels of which the transmitter has known channel distribution information (CDI) and the receiver has channel side information (CSI). Some analyses and simulations about Rician fading channels and Kfactors are given in [17]. In this paper, we consider the flat Rician fading channel model which is a suitable model for the fluctuations of the signal envelope in those narrowband multipath fading channels where there is a direct lineofsight (LOS) path between the transmitter and the receiver [18]. The main contributions of this paper are stated as follows. Firstly, the polarcoded STTD system model is proposed, which provides the guidance on how to concatenate the polar coding and STTD. With the analyses, we convert the proposed system to an equivalent single fading transmission channel for each polar code bit. It can help construct suitable polar codes for the proposed system and improve the performance. Secondly, the parameters of the equivalent fading channel are calculated based on the Rician distribution. The DE for the polarcoded STTD system is proposed based on the parameters. We describe the DEs for the systems with different numbers of antennas in this paper. Furthermore, the proposed DE guides the construction of polar codes in the systems. We describe how to construct suitable polar codes for proposed system based on the DEs. At last, simulation results are provided for the BER performance of polarcoded STTD systems in Rician MIMO channels with variable Rician Kfactors. Simulation results of frame error rate (FER) performance are also provided with different code rates. This paper includes a performance comparison of the proposed polarcoded STTD system and the stateoftheart LDPCbased system.
The remainder of the paper is organized as follows. Section 2 describes the preliminaries of polar codes and spacetime coding. Section 3 proposes the model of polarcoded STTD system. DEs for polarcoded STTD systems are also proposed. Simulation results of the polarcoded STTD systems are provided in Section 4. Finally, Section 5 concludes the paper.
Preliminaries
Successivecancellation (SC) decoding
The SC decoding specialized for polar codes is proposed in [1] with the complexity O(N logN). The SC decoder outputs bit by bit estimation of the sources. When we need to decode the ith source bit u_{ i }, the prior bits \(u_{1}^{i1}\) which have been decoded are regarded as known constants and substituted into the decoding calculation. In SC decoding calculations, the loglikelihood ratio (LLR) for the ith bit is defined as
where \(u_{1}^{i~~1}~=~(u_{1},...,u_{i~~1})\) denotes an (i − 1)dimensional source vector and \(y_{1}^{N}~=~(y_{1},...,y_{N})\) denotes the output of the channels. The LLR for each bit can be calculated based on the recursive formulas as follows,
where \(\hat {u}_{1,o}^{i}\) and \(\hat {u}_{1,e}^{i}\) denote the subvectors consisting of elements of \(\hat {u}_{1}^{i}\) with odd and even indices and ⊕ denotes modulo2 addition. Then, the ith bit can be determined as 0 or 1 after obtaining the corresponding LLR.
The SCL decoding algorithm is proposed in [4] as an upgrade version of the SC. SCL can be regarded as a joint SC algorithm and maximum likelihood (ML) algorithm. In the process of SCL decoding, source bit is not decoded immediately at each step. Instead, a maximum of l candidate paths are considered. For each source bit u_{ i }, SCL doubles the number of decoding paths by pursuing both u_{ i } = 0 and u_{ i } = 1 options and then preserves the most likely l paths in a list and discards others. When all source bits are traversed, the most reliable path is chosen as the output of decoder. The complexity of SCL decoding is O(lN logN). SCL decoding is often combined with CRC for improving the performance of decoding. The CRCaided SCL decoding provides better FER performance than other decoding methods according to [19]. An LLRbased SCL decoding provided in [20] is a numerically stable implementation of SCL decoding, which can be formulated in the LLR domain.
In [14], each step of SC is regarded as a BP decoding, where a decoding process of ith bit is regarded as a depthn tree. Furthermore, the decoding tree is characterized by the binary expansion b_{ n }⋯b_{1} of (i − 1), such that nodes at deptht of the decoding tree are check nodes and variable nodes if b_{ t }=0 and b_{ t }=1, respectively. As shown in Fig. 1, the decoding tree for 4th bit is shown by thick solid lines when n=3. Known edges are shown by dashed lines. Edges which are not used for decoding of the ith bit are shown by thin solid lines. The binary expansion of (i − 1) is 011. The nodes at depth1 and depth2 correspond to variable nodes, and the nodes at depth3 are check nodes. LLRs are recursively evaluated in the tree based on the formulas (2) and (3) at check and variable nodes, respectively. Thus, density evolution, an analytical tool for graph codes, can be adopted for polar codes based on the LLRs.
Density evolution for polar codes
Polar codes are generated by source bits including variable information bits and fixed frozen bits. The indices of information bits form an information set. The selection of the information set is a key factor that impacts the performance of polar codes. The process of channel polarization provided in [1] shows the basis of the information set selection. The polarized channel W_{ N } combined by N DMCs W is split to parallel subchannels \(\left \{W_{N}^{(i)}\right \}\) whose capacities are different. DE for polar codes guides the selection of the information set by ranking the probability of incorrect messages of subchannels. The indices of the subchannels with lowest probabilities of the incorrect messages are selected as the information set. The probability of incorrect messages can be obtained by calculating the probability distribution functions (PDFs) of LLRs passing in the decoding graph, where the PDFs are regarded as the densities. For example, the LLR of the ith subchannel \(W_{N}^{(i)}\) is regarded as a variable, then the density of the subchannel can be expressed as \(\mathbf {a}_{N}^{(i)}(z)\) that is the PDF of the variable. When all zero bits are transmitted and the channel W is symmetric, the probability of incorrect messages of the ith subchannel can be expressed as \(P_{e}(i)=\int _{\infty }^{0}\mathbf {a}_{N}^{(i)}(z)\mathrm {d}z\). The densities passing in the SC decoding graph can be calculated as follows,
where a_{ W } is the PDF of the initial channel W’s LLR when 0 is transmitted and where ⋆ and \(\boxdot \) are the convolution operations for variable nodes and check nodes, respectively [21]. Based on the densities of all subchannels, the corresponding probabilities of incorrect messages can be calculated. With ranking of the probabilities of incorrect messages, the information set is determined.
Spacetime coding
Spacetime coding is a simple coding technique to achieve transmit diversity. The basic STC model, Alamouti’s scheme proposed for 2 × 1 multipleinput and singleoutput (MISO) channel and 2 × 2 MIMO channel, has simple encoding and decoding operations. The encoding matrix of Alamouti’s scheme is
where the source bits x_{1} and x_{2} in the first column are separately transmitted by the two antennas in the first transmission period and the complex conjugations of them in the second column are separately transmitted in the second transmission period. We assume that each pointtopoint channel in MIMO follows a Rician fading model, where the channel gains are constant across two consecutive transmission periods. The signals received by the jth antenna in two periods can be expressed as
where h_{ ij } is the channel gain between the ith transmit antenna and the jth receive antenna following a Rician distribution and where n_{ jt } is an additive Gaussian noise with zero mean and variance \(\sigma _{n}^{2}\) at the jth receive antenna in the tth period. Decoding of STC is a twostep process, combining and maximum likelihood decoding. We assume that there is only one receive antenna. Combiner combines the received signals based on the estimation of CSI as follows
where \(\tilde {x}_{1}\) and \(\tilde {x}_{2}\) are the outputs of the combiner, and the distributions of them are same obviously, and where source bits x_{1} and x_{2} are isolated from the received signals. Then, the combined signals are sent to maximum likelihood decoder which can give out the source bits x_{1} and x_{2}. The generalized spacetime code model proposed for more antennas can be designed based on orthogonal theories.
Density evolution method for polarcoded STTD system
In this section, we will propose a polarcoded STTD system model in which the estimation of channel information is considered. We assume that the CDI of Rician MIMO channel is known at the transmitter and perfect estimation of CSI is known at the receiver. DEs will be also proposed for polarcoded STTD systems with different numbers of antennas.
Model of polarcoded STTD system
In Rician fading MIMO channel, the model of each pointtopoint channel is comprised of a multiplicative gain coefficient and an additive noise. The gain coefficient follows the Rician distribution \(h_{ij}~\sim ~\frac {h_{ij}}{\sigma _{h}^{2}}\exp \big (\frac {h_{ij}^{2}~+~\beta ^{2}}{2\sigma _{h}^{2}}\big)I_{0}\big (\frac {h_{ij}\beta }{\sigma _{h}^{2}}\big)\), where I_{0}(·) is the zeroorder modified Bessel function of the first kind. β is the amplitude of the specular signal component which can be calculated from the Rician Kfactor as \(\beta ~=~\sqrt {2\sigma _{h}^{2}K}\), in which K is a pure number. The Rician Kfactor is often expressed in decibels (dB) in practice, which will be applied in the simulations in this paper. The additive noise of a pointtopoint channel model follows the Gaussian distribution \(\mathcal {N} \left (0,\sigma _{n}^{2}\right)\).
As shown in Fig. 2, a polarcoded STTD system model is proposed. It can be regarded as a kind of concatenated coding, where the polar codes are the outer codes and the STC codes are inner codes. The outer polar codes are generated based on transmitter’s CDI, then sent to the STC encoder. The output concatenated codes after passing through the Rician fading MIMO channels are obtained by the receive antennas. The received signals are combined by the STC combiner, then decoded by the polar code decoder. The SC, SCL, and other decoders for polar codes can be adopted in the system.
Density evolution for polarcoded STTD system with two transmit antennas and one receive antenna
Now, we focus on the DE for polarcoded STTD system to guide the construction of polar codes in the system. We assume that there are two transmit antennas and one receive antenna in the proposed polarcoded STTD system. Alamouti’s scheme is adopted to construct STC. In the system, the output of the STC combiner can be expressed as,
where x_{ i } is any modulated polar code bit. Each combined signal y_{ i } can be regarded as an output of a fading channel where a polar code bit x_{ i } traverses. Thus, the polarcoded STTD system can be equivalent to a single fading transmission channel for each polar code bit. The gain coefficient of the equivalent channel can be expressed as \(\mathop {\sum }\limits _{q}h_{q}^{2}\). The i.i.d variable h_{ q }^{2} can be regarded as a square sum of two nonzero mean Gaussian variables, so the variable \(\mathop {\sum }\limits _{q}h_{q}^{2}\) follows a noncentral chisquare distribution
where I_{n − 1}(·) is the n − 1order modified Bessel function of the first kind. For 2 × 1 polarcoded STTD system, the gain coefficient of the equivalent channel is expressed as h = h_{11}^{2} + h_{21}^{2}. Now, each output y_{ i } can be regarded as a Gaussian random variable with mean h and variance \(h\sigma _{n}^{2}\). The conditional PDF of y_{ i } can be expressed as \(p(y_{i}x_{i},h)~=~\frac {1}{\sqrt {2\pi h\sigma _{n}^{2}}}\exp {\left (\frac {(y_{i}~~hx)^{2}}{2h\sigma _{n}^{2}}\right)}\), and the initial LLR is obtained by
The initial LLR L_{ W } is a Gaussian variable with mean \(2h/\sigma _{n}^{2}\) and variance \(4h/\sigma _{n}^{2}\). Now, we can derive the PDF of the initial LLR by using the distribution of h. We assume that each h_{ ij } is a normalized Rician fading factor. Substituting n = 2 to function (9), we can obtain the distribution of h. Finally, the PDF of the initial LLR can be expressed as
Thus, densities of subchannels can be calculated by (4). Corresponding probabilities of incorrect messages can also be calculated. Then, the information set can be selected.
Now, we focus on the symmetry of the proposed DE, which is a key property when using DE in symmetric channels. The symmetry of DE is defined in [22]. If the density a_{ W }(z) satisfies
the density a_{ W }(z) is symmetric. When substituting (11) to (12), we find that the equation is true and the initial density for 2 × 1 polarcoded STTD system is symmetric.
Density evolution for polarcoded STTD system with two transmit antennas and two receive antennas
We now analyze the polarcoded STTD system corresponding to Rician fading 2 × 2 MIMO channel. The output of the STC combiner can be expressed as
Now, the gain coefficient of the equivalent channel is expressed as h = h_{11}^{2} + h_{21}^{2} + h_{12}^{2} + h_{22}^{2}. Each output y_{ i } can still be regarded as a Gaussian random variable with mean h and variance \(h\sigma _{n}^{2}\). Substitute n = 4 to function (9) to get the distribution of h. The PDF of the initial LLR is as follows
When substituting (14) to (12), we find that the initial density for 2 × 2 polarcoded STTD system is symmetric.
Density evolution for polarcoded STTD system with four transmit antennas and two receive antennas
The generate matrix of STC for four transmit antennas is given as follows,
We adopt this STC scheme in the 4 × 2 polarcoded STTD system. Now, the output of the combiner can be expressed as
The distributions of gain coefficient and additive noise for each output signal y_{ i } are same respectively. The gain coefficient of the equivalent channel is expressed as h = h_{11}^{2} + h_{21}^{2} + h_{31}^{2} + h_{41}^{2} + h_{12}^{2} + h_{22}^{2} + h_{32}^{2} + h_{42}^{2}. We can substitute n = 8 to function (9). Now, the PDF of the initial LLR can be expressed as
This initial density for 4 × 2 polarcoded STTD system is also symmetric obviously.
Results and simulations
We have implemented the DEs for polarcoded STTD system. In this section, we will focus on the BER and FER performance of the proposed polarcoded STTD systems.
BER performance of polarcoded STTD systems in Rician MIMO channels with different Kfactors
Simulations for the BER performance of the polarcoded STTD systems respectively corresponding to 2 × 1, 2 × 2, and 4 × 2 Rician fading MISO/MIMO channels are provided where the polar codes are constructed based on the proposed DEs. The length of the polar codes is 1024, and the rate is 0.5. SC decoding is applied in the BER simulations. Rician fading channels with different Rician Kfactors are adopted in the simulations. We apply BPSK to modulate the signals.
Figure 3 shows the BER performance of polarcoded STTD system with two transmit antennas and one receive antenna corresponding to the 2 × 1 MISO. When the Rician Kfactor in dB approaches positive infinity, the pointtopoint Rician channel in MIMO can be regarded as a Gaussian channel. When it approaches negative infinity, the pointtopoint Rician channel can be regarded as a Rayleigh channel. The BER performance of the 2 × 1 system can reach 10^{−4} when K = 0 dB and E_{ b }/N_{0} is 3.9 dB approximately. Here, we consider that the BER 10^{−4} and FER 10^{−3} are sufficient to meet the actual demand for common scenarios. The similar performance indicators have been also adopted in many relevant works, such as [5, 23, 24] and [25]. As Rician Kfactor increases, the BER performance of the system is significantly improved. For example, there is an approximate 0.2dB improvement on BER performance of the 2 × 1 system as Kfactor increases from 0 to 3 dB.
As the number of transmit and receive antennas increases, the BER performance is significantly improved. As shown in Fig. 4, the BER performance of the 2 × 2 system can reach 10^{−4} when K = 0 dB and E_{ b }/N_{0} is 0.6 dB approximately, which obtains an approximate 3.3dB improvement over the 2 × 1 system. This performance is further improved by 3.2 dB by the 4 × 2 polarcoded STTD system as shown in Fig. 5. On the other hand, as the number of antennas increases, STC becomes a more dominant influence factor on BER performance than variable Rician Kfactor. Figure 4 shows that there is only an approximate 0.1dB improvement on BER performance of the 2 × 2 system as Kfactor increases from 0 to 3 dB. Moreover, this improvement is further reduced in the 4 × 2 polarcoded STTD system as shown in Fig. 5. The Rician Kfactor has less and less influence on BER performance as antenna structure in the system enlarges. The performance difference of the 2 × 1 system is about two orders of magnitude between Rayleigh fading and AWGN channel models. However, for 4 × 2 polarcoded STTD system, the differences of performance with variable Kfactors and fixed E_{ b }/N_{0} are only in an order of magnitude approximately.
FER performance of polarcoded STTD system with different code rates
According to the simulations above, we acknowledge that the performance of 2 × 2 polarcoded STTD system is sufficient for actual needs, and the complexity of it is low relatively. Thus, we focus on the FER performance of the proposed 2 × 2 polarcoded STTD system now. Firstly, we test the FER performance of the polarcoded STTD system and compare it with the FER performance of the LDPCbased STTD system [11]. In the simulations, the block length of the outer polar codes or LDPC codes is 1024, and the code rates are 0.25, 0.5, and 0.75. The Rician Kfactor of the channel is assumed as 0 dB. The SC decoding for the polar codes is still adopted in the simulation. In addition, the CRCaided SCL decoding is adopted in order to show the optimal performance of the system. Twentyfour CRC parity bits are attached to every input frame. The list size of CRCaided SCL is l = 32. At this point, the complexity of the SCL decoding is as O(lN logN) = O(327680). In the simulations, the regular LDPC codes [26] are adopted in the LDPCbased STTD system with the average of variable degree distribution \(\bar {d}_{v}~=~4\). We set the maximum BP iteration as I_{max} = 40 in order to balance the complexity of the BP decoding for LDPC codes with the one of the SCL decoding for polar codes. At this point, the complexity of the BP decoding is as \(O(I_{\text {max}}(2N\bar {d}_{v}))~=~O(327680)\).
As shown in Fig. 6, CRCaided SCL decoding improves the FER performance of polarcoded STTD system significantly. It obtains a 0.3dB improvement than SC decoding approximately for each simulated code rate. When the code rate is R = 0.25, polarcoded STTD system with the help of CRCaided SCL decoding provides an average 0.3 dB advantage over LDPCbased STTD system in FER. When the code rate increases to R = 0.5, LDPCbased STTD system provides better FER performance than polarcoded STTD system with SC decoding under high E_{ b }/N_{0}. However, it cannot exceed polarcoded STTD system with CRCaided SCL decoding, and the FER performance gap is 0.2 dB approximately. When the code rate reaches R = 0.75, LDPCbased STTD system shows advantages under high code rate. Its FER performance is slightly better than that of polarcoded STTD system with CRCaided SCL decoding. Above all, CRCaided SCL decoding can help polarcoded STTD system provide optimal FER performance under low and mid code rates.
Moreover, we make an FER performance comparison of the proposed polarcoded STTD system and the stateoftheart LDPCbased system, in which the stateoftheart LDPC codes provided in Verizon’s document [27] are applied. In the simulations, the block length of the outer polar codes is switched to 2048. According to the document, the length of the LDPC codes in the simulations is 1944 and the code rates 0.5, 0.67, and 0.75 are applied in the simulations. The corresponding averages of variable degree distributions are as \(\bar {d}_{v}~=~3.583\), 3.667, and 3.542, respectively. The Rician Kfactor of the channel is still assumed as 0 dB. The CRCaided SCL decoding is applied with the CRC parity bits 24 and list size l = 32. At this point, the complexity of the SCL decoding is as O(lN logN) = O(720896). Then, we set the maximum BP iteration as I_{max} = 50 in order to balance the complexity of the BP decoding for LDPC codes with the one of the SCL decoding for polar codes approximatively. As shown in Fig. 7, the proposed polarcoded STTD system with the CRCaided SCL decoding performs better than the stateoftheart LDPCbased system when the code rate is R = 0.5 and E_{ b }/N_{0} is not exceeding − 0.1 dB. The stateoftheart LDPCbased system provides better FER performance in waterfall region when the code rate increases to 0.67 and 0.75. The polar code block becomes less reliable than LDPC block with rising the code rate, mainly because that more incompletely polarized subchannels are adopted to contain the information bits, which causes performance loss.
Conclusions
In this paper, a novel polarcoded STTD system is proposed to approach the capacity of MIMO. Density evolutions for our proposed scheme is analyzed in the Rician fading MIMO channels, which provide preferable guidance to construct polar codes. Simulation results show that as the number of antennas increases, the proposed systems will have better BER performance and less influence from the variable Rician Kfactor of MIMO channels, simultaneously. Furthermore, we acknowledge that the 2 × 2 polarcoded STTD systems have a good tradeoff of cost and BER performance in practical applications. With the help of CRCaided SCL decoding algorithm, the proposed polarcoded STTD system with 2 × 2 antennas can provide better FER performance than conventional LDPCbased system under low and mid code rates. The average advantage is 0.3 dB approximately when code rate is 0.25 and 0.2 dB when code rate is 0.5.
Abbreviations
 BER:

Bit error rate
 BP:

Belief propagation
 CDI:

Channel distribution information
 CRC:

Cyclic redundancy check
 CSI:

Channel side information
 DE:

Density evolution
 DMC:

Discrete memoryless channel
 FER:

Frame error rate
 LDPC:

Lowdensity paritycheck
 LLR:

Loglikelihood ratio
 LOS:

Direct lineofsight
 MIMO:

Multipleinput multipleoutput
 PDF:

Probability distribution function
 STTD:

Spacetime transmit diversity
 SC:

Successivecancellation
 SCL:

Successivecancellation list
 STC:

Spacetime coding
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Funding
This work has been supported by the National Natural Sciences Foundation of China (NSFC) under grant 61771158, 61701136, 61525103, and 61371102; the National High Technology Research and Development Program no. 2014AA01A704; the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology under grant HIT. NSRIF. 2017051; and the Shenzhen Basic Research Program under grant JCYJ20160328163327348 and JCYJ20150930150304185.
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BF and JJ conceived and designed the study. SG carried out most of the analyses. BF and SW performed the experiments. QZ reviewed and edited the manuscript. All authors read and approved the manuscript.
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The authors are with the Communication Engineering Research Center, Harbin Institute of Technology (Shenzhen), Guangdong 518055, China.
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Keywords
 Polar code
 Spacetime transmit diversity
 Density evolution
 Rician fading