Introduction

Polar codes, first proposed by Arikan in 20091, have been shown to be a forward error correction coding scheme that can theoretically achieve the symmetric capacity of the binary-input discrete memoryless channel (BDMC)2,3,4, and are currently adopted as the control channel coding scheme for enhanced mobile broadband (eMBB) scenarios in 5G communication systems. In terms of decoding algorithms, Prof. Arikan proposed the successive cancellation (SC) decoding algorithm along with the publication of polar codes, which has a relatively low complexity but suffers from the defect of decoding error propagation. To reduce this decoding performance defect, A. Vardy proposed the successive cancellation list (SCL) decoding algorithm5. A belief propagation (BP) based decoding algorithm was also proposed in the same period6. Furthermore, the use of other error- correcting codes cascaded with polar codes was shown to be more effective in reducing the error propagation problem in polar codes decoding7,8,9. In7, Reed-Solomon (RS) codes were cascaded with polar codes, and the decoding performance of this scheme was significantly improved over the use of polar codes alone. In8, a selective polar codes approach is proposed to divide the polarized bit channels into three categories, i.e., noiseless channels, full-noise channels, and intermediate state channels, and the intermediate state channels are secondarily polarized to achieve decoding. The channel scenarios assumed by these decoding algorithms above are performed under a flat channel.

Orthogonal frequency division multiplexing (OFDM) is a multi-carrier modulation technique that converts frequency selective fading to flat fading on the unit subcarrier through parallel subcarrier data modulation. In contrast to conventional single-carrier modulation, OFDM is remarkably robust to frequency selective fading channels. As an influential technology for 5G mobile communication systems, OFDM has gained wide attention for its advantages such as high spectrum utilization and resistance to multipath fading10,11,12. The control channel of the 5G standard13,14 uses a control information transmission scheme based on polar codes combined with OFDM.

There is limited research on the decoding of polar-coded multicarrier OFDM system. In 2018, Rahim et al. analyzed the performance of polar codes combined with OFDM techniques in different channel models (Rayleigh, Rician, AWGN)15 They conducted a comparison of the decoding performance between Polar-encoded OFDM and Turbo-encoded OFDM under identical conditions. However, the overall decoding strategy did not take into account the frequency selectivity characteristics induced by OFDM system in multipath wireless channels, resulting in suboptimal soft-decision error correction performance. For OFDM modulation, although the channel on the single subcarrier exhibits a flat fading characteristic, the channel frequency response is different among the multiple subcarriers due to the fact that the individual subcarriers are still affected by frequency selective fading, resulting in a large difference in the signal-to-noise ratio (SNR) of each subcarrier. T. A. Abose et al.16 specifically investigate two polar-coded multicarrier system: offset quadrature amplitude modulation-based filter bank multicarrier (FBMC-OQAM) and OFDM. Their study takes into account channels experiencing Doppler shift, but overlooks the impact of multipath frequency-selective channels. Furthermore, the consecutive arrangement of the output information bits from a polar codes encoder makes the information susceptible to continuous burst errors during transmission due to frequency-selective fading induced by multipath effects. The studies referenced in15,16 overlooked the variations in noise power among subcarriers following equalization and failed to propose strategies to mitigate the risk of consecutive burst errors in the transmission of polar-coded information. To address both of these limitations, the present paper investigates the decoding scheme for polar codes in OFDM modulation system experiencing multipath frequency-selective fading.

The contributions of this paper are mainly in the following three aspects:

  1. (1)

    As the disparity in the SNR across subcarriers after OFDM equalization, the channel state information (CSI) in the frequency domain is used to mitigate the degradation of the decoding performance of polar codes. Specifically, the log-likelihood ratio (LLR) soft information for polar decoding is computed by integrating both the magnitude-frequency response, (alternatively termed as CSI), on each subcarrier of the OFDM system and the demapping output. This integration enhances the reliability of the input data for the decoder.

  2. (2)

    In comparison to low-density parity check (LDPC) codes, the decoding performance of polar codes is highly susceptible to burst errors. Therefore, the interleaving and de-interleaving techniques are employed to significantly enhance the resilience against burst errors that are typically caused by frequency-selective fading. Through interleaving, consecutive transmitted bits are dispersed, and these dispersed bits are then rearranged back to their original order at the receiver using de-interleaving. The occurrence of severe consecutive errors is mitigated during transmission bursts, thereby enhancing the overall transmission performance of the system.

  3. (3)

    Employing these schemes individually, the decoding performance did indeed improve, but only to a relatively limited extent. In further research, we discovered that when we implemented the two schemes in combination within the system, the decoding performance was significantly and substantially enhanced. Simulation results obtained across a variety of real-world channel models indicate that, when compared to conventional decoding schemes, the two proposed schemes individually exhibit a certain degree of decoding gain improvement (approximately 1–2 dB). Moreover, the combined application of these two schemes leads to a substantial decoding gain enhancement (at least 6 dB), outperforming both the conventional scheme and their standalone performances.

For the notation used in this letter, a character with superscript and subscript, e.g. \({\textbf{u}}_{1}^{N}\) denote a bit sequence or an information vector such as \({\textbf{u}}_{1}^{N}=\left( u_1,u_2,\cdots ,u_i,\cdots ,u_N \right).\) The operator \(\otimes n\) denotes the n-th Kronecker power. The rest of the paper is organized as follows. In System model, we introduced the stages of our polar-coded OFDM system model. In Polar codes basics, we revisited the fundamentals of polar codes, establishing a solid foundation for our subsequent discussions. In Proposed schemes, we proposed two improved schemes and detailed the novel contributions that distinguish our approach from existing methodologies. And finally, Simulation results and Conclusions of this work are presented.

System model

The polar-coded OFDM system’s block diagram is shown in Fig. 1, and a description of each component of the system block diagram is given in the following subsections.

Fig. 1
figure 1

Block diagram of polar-coded OFDM system.

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Transmitter

The transmitter of the system consists of a polar codes encoder, interleaver, binary phase shift keying (BPSK) digital constellation mapper and inverse fast Fourier transform (IFFT). The codeword \({\textbf{x}}_{1}^{N}=\left( x_1,x_2,\cdots ,x_i,\cdots ,x_N \right)\) from polar codes encoder is passed through the packet interleaver to obtain \({{\textbf{x}}^{\prime }}_{1}^{N}=\left( {x^{\prime }}_1,{x^{\prime }}_2,\cdots ,{x^{\prime }}_i,\cdots ,{x^{\prime }}_N \right).\) Then, each bit of the interleaved codeword sequence \({{\textbf{x}}^{\prime }}_{1}^{N}\) is modulated onto each subcarrier of the OFDM signal using a BPSK mapper. Finally, the modulated data on all subcarriers \({\textbf{y}}_{1}^{N}=\left( y_1,y_2,\cdots ,y_i,\cdots ,y_N \right)\) are transformed from frequency domain to time domain by IFFT, and then transmitted to multipath channel.

Equivalent channel model in frequency domain

As the signal is transmitted through the channel to the receiver, it is subject to multipath effects that cause frequency selective fading. The equivalent frequency response of a multipath channel in an OFDM system17 is given by

$$\begin{aligned} H_i=\sum \nolimits _{m=1}^M{a_me^{-j2\pi if_0\tau _m}}, \end{aligned}$$
(1)

where i and \(f_0\) denote the subcarrier sequence number and subcarrier spacing of an OFDM system, respectively. M is the number of paths in multipath channel, \(m\in \left\{ 1,2,\cdots ,M \right\}\). \(a_m\) and \(\tau _m\) denote the complex gain and channel delay of the m-th (\(m\in \left\{ 1,2,\cdots ,M \right\}\)) path, respectively. In a multipath channel, the amplitude-frequency response reflects the fading degree of each OFDM subcarrier in the frequency domain, and those with lower responses are more likely to experience errors.

Receiver

The receiver side consists of a fast Fourier transform (FFT), a channel equalizer , a BPSK demapper, a deinterleaver and a polar codes decoder. The received signal is converted from time domain to frequency domain by separating the orthogonal subcarriers via FFT. Considering the multipath and noise effects on the modulated data as it passes through the channel, the received data \(r_i\) on the i-th subcarrier after FFT transformation can be expressed as18:

$$\begin{aligned} r_i=H_i\cdot y_i+N_i. \end{aligned}$$
(2)

Here, \(H_i\) is the frequency response on the i-th subcarrier. \(N_i\) is the independent and identically distributed additive white Gaussian noise of the channel. Based on the channel frequency response, the channel equalizer can compensate for the information affected by frequency selective fading. After the equalization and the BPSK demapping, the received information is converted to obtain the LLR soft information \({\textbf{L}}_{1}^{N}=\left( L\left( {x^{\prime }}_1 \right) ,L\left( {x^{\prime }}_2 \right) ,\cdots ,L\left( {x^{\prime }}_i \right) ,\cdots ,L\left( {x^{\prime }}_N \right) \right)\) of each interleaved bit \({x^{\prime }}_i\), where \(L\left( {x^{\prime }}_i \right)\) is defined in Eq. (11). Finally, the resultant LLR soft information \(L^{\prime }\left( x_i \right)\), after deinterleaver, is input into the polar codes decoder for error correction.

Polar codes basics

Encoding principle

The parameters \(\left( N,K,{\textbf{a}},u_{{\textbf{a}}^{{\textbf{c}}}} \right)\) are commonly used to define a polar code, where N is the codeword length of the polar code, K is the information bit length, \({\textbf{a}}\) is the index set of the information bit sequence generated by channel reliability estimation. \({\textbf{a}}^{{\textbf{c}}}\) is the complement of \({\textbf{a}}\), and it represents the index set of the redundant bit. \(u_{{\textbf{a}}^{{\textbf{c}}}}\) represents the values of the transmitted redundant bits. The bit rate R is then calculated as \(R=K/N\).

The input bit sequence \({\textbf{u}}_{1}^{N}=\left( u_1,u_2,\cdots ,u_i,\cdots ,u_N \right)\) of a polar encoder consists of an information bit sequence and a redundant bit sequence. Then, the codeword of polar codes can be obtained from the input bit sequence by linear mapping18, which is denoted as

$$\begin{aligned} {\textbf{x}}_{1}^{N}={\textbf{u}}_{1}^{N}{\textbf{G}}_N. \end{aligned}$$
(3)

Here \({\textbf{x}}_{1}^{N}\), \({\textbf{u}}_{1}^{N}\) and \({\textbf{G}}_N\) present the codeword, original bit sequence and the generation matrix, respectively. \(N=2^n\) and the generation matrix is represented as

$$\begin{aligned} {\textbf{G}}_N={\textbf{B}}_N{\textbf{F}}^{\otimes n}, \end{aligned}$$
(4)

where \({\textbf{B}}_N\) denotes the bit-reversal permutation matrix, and the polarization kernel matrix \({\textbf{F}}\) is denoted as

$$\begin{aligned} {\textbf{F}}=\left[ \begin{matrix} 1& 0 \\ 1& 1 \\ \end{matrix} \right] . \end{aligned}$$

Decoding algorithm

The SC decoding algorithm is a commonly used decoding algorithm for polar codes1. The decoder utilizes the LLR soft information generated at the demodulation stage of the receiver as the basis for deciding the input information bits. The estimate of the i-th (\(i\in \{1,2,\cdots ,N\}\)) bit \({\hat{u}}_i\) is determined one by one based on the transition probability \(W_{N}^{(i)}(\hat{{\textbf{y}}}_{1}^{N},\hat{{\textbf{u}}}_{1}^{i-1}|{\hat{u}}_i)\). Where \(\hat{{\textbf{y}}}_{1}^{N}=\left( {\hat{y}}_1,{\hat{y}}_2,\cdots ,{\hat{y}}_i,\cdots ,{\hat{y}}_N \right)\) is the output bit sequence of the equalizer and \(\hat{{\textbf{u}}}_{1}^{N}=\left( {\hat{u}}_1,{\hat{u}}_2,\cdots ,{\hat{u}}_i,\cdots ,{\hat{u}}_N \right)\) is the decoded bit sequence. Specifically, the estimate of the equalized bit \({\hat{u}}_i\) is given by1

$$\begin{aligned} {\hat{u}}_i= {\left\{ \begin{array}{ll} h_i\left( {\textbf{y}}_{1}^{N},\hat{{\textbf{u}}}_{1}^{i-1} \right) \,\,{if}\,\,i\in {\textbf{a}},\\ 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,i\in {\textbf{a}}^{{\textbf{c}}}.\\ \end{array}\right. } \end{aligned}$$
(5)

Where \(h_i\) denotes the function used to determine the estimate of \({\hat{u}}_i\), which can be described as

$$\begin{aligned} h_i\left( {\textbf{y}}_{1}^{N},\hat{{\textbf{u}}}_{1}^{i-1} \right) ={\left\{ \begin{array}{ll} 0 \,\,if\,\,L_{N}^{\left( i \right) }\left( {\textbf{y}}_{1}^{N},\hat{{\textbf{u}}}_{1}^{i-1} \right) \geqslant 0,\\ 1 \,\,otherwise.\\ \end{array}\right. } \end{aligned}$$
(6)

\(L_{N}^{\left( i \right) }({\textbf{y}}_{1}^{N},\hat{{\textbf{u}}}_{1}^{i-1})\) is denoted as the LLR of i-th subchannel of polar code and is defined as1

$$\begin{aligned} L_{N}^{\left( i \right) }\left( \hat{{\textbf{y}}}_{1}^{N},\hat{{\textbf{u}}}_{1}^{i-1} \right) \triangleq \ln \frac{W_{N}^{\left( i \right) }\left( \hat{{\textbf{y}}}_{1}^{N},\hat{{\textbf{u}}}_{1}^{i-1}|u_i=0 \right) }{W_{N}^{\left( i \right) }\left( \hat{{\textbf{y}}}_{1}^{N},\hat{{\textbf{u}}}_{1}^{i-1}|u_i=1 \right) }\,\, \end{aligned}$$
(7)

The magnitude of the LLR value reflects the degree of certainty that the decision of \({\hat{u}}_i\) is 0 or 1. According to the recursion principle of channel splitting, the value of LLR can be calculated as1

$$\begin{aligned} \left\{ \begin{aligned}&L_{N}^{(2i-1)}\left( \hat{{\textbf{y}}}_{1}^{N},\hat{{\textbf{u}}}_{1}^{2i-2} \right) =f\left( L_{\small {\small {\frac{N}{2}}}}^{(i)}\left( \hat{{\textbf{y}}}_{1}^{\small {\frac{N}{2}}},\hat{{\textbf{u}}}_{1,o}^{2i-2}\oplus \hat{{\textbf{u}}}_{1,e}^{2i-2} \right) ,L_{\small {\frac{N}{2}}}^{(i)}\left( \hat{{\textbf{y}}}_{\small {\frac{N}{2}}+1}^{N},\hat{{\textbf{u}}}_{1,e}^{2i-2} \right) \right) ,\\&L_{N}^{(2i)}\left( \hat{{\textbf{y}}}_{1}^{N},\hat{{\textbf{u}}}_{1}^{2i-1} \right) =g\left( L_{\small {\frac{N}{2}}}^{(i)}\left( \hat{{\textbf{y}}}_{1}^{\small {\frac{N}{2}}},\hat{{\textbf{u}}}_{1,o}^{2i-2}\oplus \hat{{\textbf{u}}}_{1,e}^{2i-2} \right) ,L_{\small {\frac{N}{2}}}^{(i)}\left( \hat{{\textbf{y}}}_{\small {\frac{N}{2}}+1}^{N},\hat{{\textbf{u}}}_{1,e}^{2i-2} \right) ,\hat{{\textbf{u}}}_{2i-1} \right) .\\ \end{aligned} \right. \end{aligned}$$
(8)

Where \(\hat{{\textbf{u}}}_{1,o}^{2i-2}\) represents the decoded bit sequence of odd bit indexes, and \(\hat{{\textbf{u}}}_{1,e}^{2i-2}\) represents the decoded bit sequence of even bit indexes, respectively.The function \(f\left( \cdot \right)\) and \(g\left( \cdot \right)\) are defined as

$$\begin{aligned} f(a,b)=ln\!\,\left( \frac{1+e^{a+b}}{e^a+e^b} \right) \end{aligned}$$
(9)

and

$$\begin{aligned} g(a,b,u_s)=(-1)^{u_s}a+b. \end{aligned}$$
(10)

Where a and b are the LLRs and \(u_s\) are the estimated bit values used in the operation, \(u_s\in (0,1).\)

Proposed schemes

Integrating CSI scheme

In general, polar codes soft decision decoders that incorporate soft information exhibit superior error correction capabilities compared to those that do not1. The bit-wise soft information utilized in conventional polar code decoders15 is derived from the demodulated signal outputted by the equalizer. Importantly, this method of soft information calculation does not impart any performance loss during decoding in flat Additive White Gaussian Noise (AWGN) channels. In a polar-coded OFDM system, the data to be transmitted are modulated in the frequency domain. Although the channel on a single subcarrier exhibits a flat fading characteristic, the channel frequency response may vary significantly in both amplitude and phase across different subcarriers. This variation results in frequency-selective fading and differing the SNR levels in the demodulated data of each subcarrier following equalization. In this study, CSI in the frequency domain, which reflects the SNR level of each demodulated data bit, is utilized to mitigate the degradation of decoding performance observed in polar codes. Specifically, the LLR soft information for polar decoding is calculated by incorporating this channel state information along with the demapping output, thereby enhancing the reliability of the decoder’s input soft information.

The bit LLR soft information is frequently used as an interface between demodulator and decoder, and the LLR of the i-th bit output by the demodulator is expressed as

$$\begin{aligned} L\left( {x^{\prime }}_i \right) =L\left( \left. {x^{\prime }}_i \right| {\hat{y}}_i \right) =\ln \frac{p\left( {\hat{y}}_i|{x^{\prime }}_i=0 \right) }{p\left( {\hat{y}}_i|{x^{\prime }}_i=1 \right) }=\frac{2{\hat{y}}_i}{\sigma _{i}^{2}}. \end{aligned}$$
(11)

Here, \(p\left( {\hat{y}}_i|{x^{\prime }}_i \right)\) denotes the probability density of obtaining \({\hat{y}}_i\) at the receiver when the corresponding transmitted bit \({x^{\prime }}_i\) takes the value 0 or 1. \(\sigma _{i}^{2}\) is the random noise power on the i-th bit. The average noise power on each bit in the AWGN channel is equal. Therefore, the conventional receiver algorithm directly decodes \({\hat{y}}_i\) as the LLR soft information of the i-th bit.

At the receiver of an OFDM system, the received signal is transformed by the fast Fourier transform into a frequency domain symbol \(r_i\). Assuming that the channel state information \(H_i\) is known, the equalized data \({\hat{y}}_i\) is obtained from Eq. (2) as15

$$\begin{aligned} {\hat{y}}_i=\frac{r_i}{H_i}=y_i+\frac{N_i}{H_i}. \end{aligned}$$
(12)

The frequency selective fading is compensated after channel equalization, but the average power of the noise after equalization is different due to different frequency responses on different subcarriers. The average power of the noise on subcarrier i after equalization is given by

$$\begin{aligned} \sigma _{i}^{2}=\frac{\sigma _{N}^{2}}{\left| H_i \right| ^2}. \end{aligned}$$
(13)

Let \(\sigma _{N}^{2}\) represent the average power on each subcarrier before equalization, where there will be no significant difference in the average power among them. However, due to the frequency selectivity caused by multipath propagation, the magnitude-frequency response \(\left| H_i \right|\) on each subcarrier of OFDM will vary with the index i of the subcarriers. Consequently, there may be significant differences in the average noise power \(\sigma _{i}^{2}\) of the subcarriers received by the demodulated signal after equalization.

To this end, we propose a Integrating CSI scheme that takes into account channel state information as the input for polar codes decoder. We then substitute the noise power after equalization, as calculated in Eq. (13), into Eq. (11) to derive the LLR that incorporates channel state information as

$$\begin{aligned} L^{\prime }\left( {x^{\prime }}_i \right) =L^{\prime }\left( \left. {x^{\prime }}_i \right| {\hat{y}}_i,H_i \right) =\frac{2{\hat{y}}_i}{\sigma _{i}^{2}}\,\,=\frac{2{\hat{y}}_i}{\sigma _{N}^{2}}\cdot {\left| H_i \right| ^2}. \end{aligned}$$
(14)

The larger the magnitude-frequency response \(\left| H_i \right|\), the smaller the value of \(\sigma _{i}^{2}\) and therefore the larger the value of \(L^{\prime }\left( {x^{\prime }}_i \right)\), which means that the information is more reliable. In turn, the smaller the value of \(\left| H_i \right|\), the less reliable the information. As a result, in this paper, the channel state information of the multipath fading channel in the frequency domain is incorporated to enhance the reliability of the LLR, as shown in Eq. (14). Compared with the traditional receiver which directly uses the demodulated signal output from the equalizer to calculate LLR, CSI characterizes the attenuation factor of the signal along each transmission path. The gain \(\left| H_i \right|\) provided by CSI further reflects the SNR level of each demodulated data bit, thereby enabling a more precise estimation of the probability of each bit being either 0 or 1. This enhanced estimation improves the accuracy of the LLR calculations and ultimately enhances the decoding performance.

Since the average power of each subcarrier in OFDM is the same before channel equalization, the formula (14) can be simplified to formula (15).

$$\begin{aligned} L^{\prime }\left( {x^{\prime }}_i \right) \triangleq {\hat{y}}_i\cdot \left| H_i \right| ^2. \end{aligned}$$
(15)

Compared with the traditional scheme, the Integrating CSI scheme based on formula (15) is similar to the calibration recalculation of soft information19,20, which can provide more accurate bit soft information for the decoder of Polar codes.

Interleaving module addition scheme

The multipath effect inherently introduces frequency selectivity among OFDM subcarriers, yet the amplitude of the channel frequency response \(H_k\) on adjacent subcarriers maintains a distinct correlation. Consequently, in scenarios where the amplitude of \(H_k\) across a segment of subcarriers is notably small, the data transmitted on those subcarriers is prone to burst errors. In essence, the demodulated data on these subcarriers from channel equalizer may manifest as consecutive transmission errors. We find that the error correcting ability of polar codes for burst errors is far worse than that for random errors. To this end, a block interleaving technique is designed for polar-coded OFDM systems. Block interleaving is a technique that enables the rearrangement of consecutive bits of information to resist continuous burst errors, but without altering their content. The schematic diagram is shown in Fig. 2.

Fig. 2
figure 2

Schematic diagram of packet interleaving.

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The interleaving process divides the bit sequence encoded by the polar codes evenly into Z groups, where each group contains P bits of data. These Z groups can then be arranged in row order to form a \(Z\times P\) interleaving matrix. The elements of this matrix are subsequently output in column order, and the resulting bit sequence is the interleaved bit sequence.

For example, if the length of a codeword encoded by the polar codes encoder \({\textbf{x}}_{1}^{16}=\left( x_1,x_2,\cdots ,x_{16} \right)\) is 16, it is divided into 4 groups by the block interleaver, and the length of each group sequence is 4. The interleaving matrix we obtained is

$$\begin{aligned} {\textbf{T}}_1=\left[ \begin{matrix} x_1& x_2& x_3& x_4\\ x_5& x_6& x_7& x_8\\ x_9& x_{10}& x_{11}& x_{12}\\ x_{13}& x_{14}& x_{15}& x_{16}\\ \end{matrix} \right] . \end{aligned}$$

The interleaved bit sequence we obtain is \({\textbf{x}}_{1}^{{\prime \;16}} = \left( {x_{1} ,x_{5} ,x_{9} ,x_{{13}} ,} \right.\)\(x_{2} ,x_{6} ,x_{{10}} ,x_{{14}} ,\)\(x_{3} ,x_{7} ,x_{{11}} ,x_{{15}} ,\)\(\left. {x_{4} ,x_{8} ,x_{{12}} ,x_{{16}} } \right).\)

If the interleaved bit sequence is affected by two burst errors in the channel, resulting in continuous error bits in the bit sequence, where the bits affected by the first burst error are \(x_2,x_6,x_{10},x_{14}\), denoted by \({\bar{x}}_2,{\bar{x}}_6,{\bar{x}}_{10},{\bar{x}}_{14}.\) And the bits affected by the second burst error are \(x_{11},x_{15},x_4\), denoted by \({\bar{x}}_{11},{\bar{x}}_{15},{\bar{x}}_{4}.\) The information sequence that reaches the receiving stage through the channel is expressed as \({\textbf{c}}^{\prime \;16 } _{1} = \left( {x_{1} ,x_{5} ,x_{9} ,x_{{13}} } \right.,\)\(\bar{x}_{2} ,\bar{x}_{6} ,\bar{x}_{{10}} ,\bar{x}_{{14}} ,\)\(x_{3} ,x_{7} ,\bar{x}_{{11}} ,\bar{x}_{{15}} ,\)\(\left. {\bar{x}_{4} ,x_{8} ,x_{{12}} ,x_{{16}} } \right).\)

Deinterleaving is the inverse process of interleaving, where the received information sequence is arranged into a deinterleaving matrix of size \(Z\times P\) in column order

$$\begin{aligned} {\textbf{T}}_2=\left[ \begin{matrix} x_1& {\bar{x}}_2& x_3& {\bar{x}}_4\\ x_5& {\bar{x}}_6& x_7& x_8\\ x_9& {\bar{x}}_{10}& {\bar{x}}_{11}& x_{12}\\ x_{13}& {\bar{x}}_{14}& {\bar{x}}_{15}& x_{16}\\ \end{matrix} \right] , \end{aligned}$$

and output in row order to rearrange the original bit sequence \({\mathbf{c}}_{1}^{{16}} = \left( {x_{1} ,\bar{x}_{2} ,x_{3} ,\bar{x}_{4} ,} \right.\)\(x_{5} ,\bar{x}_{6} ,x_{7} ,x_{8} ,\)\(x_{9} ,\bar{x}_{{10}} ,\bar{x}_{{11}} ,x_{{12}} ,\)\(\left. {x_{{13}} ,\bar{x}_{{14}} ,\bar{x}_{{15}} ,x_{{16}} } \right).\) The rearranged LLR information will be decoded by a polar codes decoder.

Thus, the burst errors that result from successive and small value of \(H_{k}\) are randomly dispersed to different bits after the deinterleaving process. This random dispersion significantly improves the decoding performance of the polar codes decoder.

Simulation results

Simulation settings

To evaluate the performance effect, the bit error rate (BER) in communication systems is used as a reference standard to measure the transmission performance of the system in this paper.The proposed two anti-burst error schemes for polar-coded OFDM system and their bit error rate (BER) and frame error rate (FER) performance are simulated on a multipath fading channel. In addition, the Extended Pedestrian A model (EPA), Extended Vehicular A model (EVA) and Extended Typical Urban model (ETU) multipath fading channel model parameters of Long-Term Evolution (LTE) wireless communication standard are set to simulate the real channel environment respectively. The rest of the simulation parameters are shown in Table 1.

Table 1 Polar-coded OFDM system’s parameters.
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Performance comparison

Figure 3 shows a comparison of the BER and FER performance between the conventional scheme and the Integrating CSI scheme in polar-coded OFDM system over a multipath fading channel. Both schemes use the SC decoding algorithm for decoding. With the same simulation parameters, the experimental results show that at the BER of \(6\times 10^{-3}\), the improved scheme utilizing channel state information shows a nearly \(3\,\,\textrm{dB}\) improvement over the conventional decoding scheme. At the FER of \(6\times 10^{-3}\), our proposed scheme shows an approximately \(2\,\,\textrm{dB}\) improvement over the conventional scheme. For the polar-coded OFDM system, the improved scheme achieves significantly better BER and FER performance than the conventional scheme.

Fig. 3
figure 3

Comparison of BER and FER performance before and after adding channel state information.

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Figure 4 illustrates the BER performance of the conventional decoding scheme, the proposed two schemes used individually, as well as the combination of the two schemes in polar-coded OFDM system over a multipath fading channel. Experimental results show that the performance of the proposed Integrating CSI scheme is significantly improved compared to the conventional scheme15 at Eb/No between \(20\,\,\textrm{dB}\) and \(30\,\,\textrm{dB}\). When the Interleaving module addition scheme is added separately and the interleaving matrix is set to \(128^*8\), the transmission performance is also improved compared with the conventional scheme15. Moreover, at Eb/No between 2 and 30 dB, the decoding performance of the Interleaving module addition scheme starts to outperform that of the Integrating CSI scheme by adding channel state information to the decoding side of the receiver. By combining the two proposed schemes, the interleaved sequences in the polar-coded OFDM system add CSI to the soft information computation at the decoder side of the receiver, increasing the certainty of the decoded bit information corresponding to each subcarrier while the information sequences are dispersed to resist successive errors. As a result, the decoding performance of the polar-coded OFDM system’s scheme combined with the two optimization methods has the lowest BER and resulting in an additional coding gain of about \(21\,\,\textrm{dB}\) when the BER is \(2\times 10^{-4}\) under the same Eb/No compared to the proposed scheme alone. The performance of the system is further optimized.

Fig. 4
figure 4

BER Performance comparison of different decoding schemes for polar-coded OFDM system in EPA channel.

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Figure 5 presents a comparison of the BER performance of various strategies under the parameters of the EVA multipath fading channel model. The experimental results demonstrate that, within the Eb/No range of 8 to 22 dB, the proposed Integrating CSI scheme and Interleaving module addition scheme exhibit significantly improved performance compared to the conventional scheme15, in which the coding gain of the system under BER \(10^{-6}\), the Integrating CSI scheme has been improved by 1.5 dB, and the Interleaving module addition scheme has been improved by 2.5 dB. Furthermore, when the Eb/No exceeds 14 dB, the Interleaving module addition scheme outperforms the Integrating CSI scheme in terms of decoding performance. When both strategies are employed concurrently, the system achieves a significant coding gain of approximately 10 dB at a BER of \(10^{-6}\), relative to the traditional scheme.

Fig. 5
figure 5

BER Performance comparison of different decoding schemes for polar-coded OFDM system in EVA channel.

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Figure 6 presents a comparison of the BER performance of various strategies under the parameters of the ETU multipath fading channel model. The experimental results demonstrate that, within the Eb/No range of 8 to 22 dB, the proposed Integrating CSI scheme and Interleaving module addition scheme exhibit improved performance compared to the conventional scheme15 and the Integrating CSI scheme has always been superior to the Interleaving module addition scheme in BER performance, in which the system coding gain Integrating CSI scheme under BER \(10^{-4}\) has been improved by 4.5 dB, and the Interleaving module addition scheme has been improved by only 1 dB. When both strategies are employed concurrently, the system achieves a significant coding gain of approximately 11 dB at a BER of \(10^{-6}\), relative to the traditional scheme.

Fig. 6
figure 6

BER Performance comparison of different decoding schemes for polar-coded OFDM system in ETU channel.

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Based on the simulation results depicted in Figs. 4, 5 and 6, it is observed that both decoding methodologies exhibit a decoding performance improvement of approximately 1–2 dB compared to the traditional scheme. The two proposed schemes may demonstrate distinct error correction capabilities depending on the specific channel conditions. In particular, under high Eb/No conditions, the scheme involving the addition of interleaving modules demonstrates superior decoding performance for both EPA and EVA channels. Conversely, in the ETU channel, the Integrating CSI scheme consistently surpasses the Interleaving module addition scheme in terms of decoding performance. Moreover, their combined application yields notably superior results with a substantial decoding gain enhancement at least 10 dB over conventional ones.

Implementation analysis

The two proposed solutions do not significantly increase computational complexity. Compared to traditional methods, the Integrating CSI scheme requires an additional multiplication operation to calculate the soft information of each bit. Specific implementation may require wider bit quantization of soft information, potentially increasing the decoder’s complexity. In addition, the implementation of the scheme involving the addition of interleaving modules requires additional RAM units. These factors may collectively contribute to an elevated decoder complexity, , which we will investigate in the future.

Conclusions

In this paper, we introduce two schemes aimed at enhancing the decoding performance of polar-coded OFDM system. The channel state information, which reflects the transmission quality of the OFDM subcarriers, is incorporated to the soft information calculation at the decoder to improve the reliability of the LLR. The packet interleaving module, which rearranges the transmitted bits, is also added to reduce burst errors. Simulation outcomes demonstrate that both schemes individually contribute to improved decoding performance (approximately 1–2 dB) over conventional decoding scheme. Moreover, their combined application yields notably superior results with a substantial decoding gain enhancement at least 10 dB over conventional ones.

In future work, we aim to expand and refine the proposed schemes. Firstly, we intend to explore the adoption of optimized decoding algorithms, specifically SCL and BP, as alternatives to the currently employed SC algorithm. These optimization algorithms are anticipated to further augment the decoding performance of polar-coded OFDM system by enhancing the flexibility and precision of the decoding process. Secondly, we will delve into the potential of employing concatenated coding schemes. Despite the potential increase in computational complexity, the enhanced error correction capability of such schemes is crucial for improving the reliability of the system in real-world channel environments.