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Joint channel and impulse noise estimation based on compressed sensing and Kalman filter for OFDM system
EURASIP Journal on Advances in Signal Processing volumeÂ 2023, ArticleÂ number:Â 105 (2023)
Abstract
Impulse noise (IN) widely exists in many communication systems, which seriously affects the performance of OFDM communication systems. A joint channel and IN estimation method based on all subcarriers is designed. This method uses a sparse Bayesian learning (SBL) algorithm incorporating forwardâ€“backward Kalman filter (FBKalman) to tackle the problem of joint channel and IN estimation and data detection for OFDM systems. Firstly, the channel impulse response and IN are regarded as unknown sparse vectors, and a SBL framework using all subcarriers is proposed to estimate the unknown vector. The SBL theory is used based on the prior distribution of variables, and then the forwardâ€“backward joint system is established, which applies the data detection simultaneously. We also propose the FBKalman implementation algorithm by using the expectation maximization updates. Explicit expressions of mean and covariance matrix of the posterior distribution are derived in the Estep. Simulation results show that the proposed algorithm improves the normalized mean square error and bit error rate performance of OFDM system in the presence of IN communication environment.
1 Introduction
IN always exists and is more harmful especially in OFDM system such as digital subscriber lines [1], power line communication (PLC) [2], wireless communication [3], and underwater acoustic communication [4]. IN model such as Gaussian mixture (GM), Middleton Class A [5], and BernoulliGaussian [6] can cover most scenarios where IN occurs in PLC systems, and \(\alpha\)stable distribution model can be used to describe the some shortterm spikes existed in shallow water acoustic channel [7].
Although OFDM is inherently more resistant to IN than single carrier modulation, its influence extends to all subcarriers and if the power of IN exceeds a certain threshold, it will lead to a sharp decline in system performance [8]. It follows that in practical applications, the IN should be properly suppressed in the receiver side before detection.
Numerous algorithms have been proposed. One type of commonly used IN mitigation algorithms is nonlinear filtering in which the received signal samples contaminated by IN is then adjusted by using nonlinear blanking or clipping method. The drawback of such algorithms is that it not only suppresses the interference, but also destroys the original signal. Moreover, such algorithms need to use the prior statistics of IN to derive the optimal threshold, which is very difficult to obtain in practice, and the suboptimal threshold is usually set by using an empirical value. Another kind of suppression algorithms is based on compressed sensing (CS) theory. These algorithms are proposed base on the sparsity of IN and CIR [9,10,11,12,13,14], and their performances are superior to traditional blanking and clipping algorithms [15]. However, these algorithms have high computational complexity. In [16], the generalized approximate message passing (AMP) algorithm is used to solve the problems of joint channel coefficients, IN, and data symbols estimation. However, the algorithms based on prior information about CIR and IN, which is difficult to obtain. In [17], the distribution of CIR and IN are modeled as Gaussian Mixture (GM) model, which is the prior information of AMP algorithm. Nevertheless, this assumption does not apply to other channels. In [18], a SBL algorithm combined with Kalman filter is proposed to solve the joint channel and IN estimation problem in OFDM systems. Compared with SBL algorithm, it improves the estimation performance of the system. However, the estimation performance is limited by the number of pilots, increasing the number of pilots will reduce the spectral efficiency of OFDM system.
The problem of joint channel and IN estimation based on all subcarriers in the presence of IN for OFDM system is discussed. Initially, we regard the CIR and IN as an unknown sparse vector and use a SBL framework that employs all subcarriers to estimate unknown vector. The SBL theory is used based on the prior distribution of variables, and then the forward backward joint system is established, which apply the data detection simultaneously [19]. In addition, we develop an FBKalman implementation algorithm using EM updates to iteratively estimate the unknown parameters. In the Estep, explicit expressions of mean and covariance matrices of the posterior distribution are derived. In the Mstep the unknown parameters are iteratively estimated. Simulation results show that the proposed algorithm improves the channel estimation performance and BER performance of OFDM system in the presence of IN communication environment.
2 System model
Consider a coded frame based OFDM system with N subcarriers, which contains \(N_d\) subcarriers for data transmission, \(N_p\) pilot subcarriers, and \(N_n\) null subcarriers. At the transmitting side, an OFDM frame is composed of \(N_b\) OFDM symbols. At the receiver side, the \(m{\text {th}}\) OFDM symbol \({\varvec{r}}_m \epsilon \mathbb {C}^{N\times 1}\) of the received frame in the frequency domain can be written in a vector form as
where \(\mathbf {D_m}=diag({\varvec{d}}_m)\) is a diagonal matrix with \({\varvec{d}}_m\epsilon \mathbb {C}^{N\times 1}\) as the main diagonal element which represents the \(m{\text {th}}\) transmitted OFDM symbol. \({\varvec{h}}_m\epsilon \mathbb {C}^{N\times 1}\) denotes the frequency domain CIR. \({\varvec{i}}_m\epsilon \mathbb {C}^{N\times 1}\) and \({\varvec{w}}_m\epsilon \mathbb {C}^{N\times 1}\) are the frequency domain IN and background Gaussian noise, respectively. The subscript \(m(m=1,2,...,N_b)\) represents the index of OFDM symbols in a frame.
By introducing \({\varvec{h}}_m=\sqrt{N}\textbf{F}_L{\varvec{h}}_{t,m}\) and \({\varvec{i}}_m=\textbf{F}{\varvec{i}}_{t,m}\), where \(\textbf{F}\epsilon \mathbb {C}^{N\times N}\) represents the normalized discrete Fourier transform matrix, \(\textbf{F}_L\epsilon \mathbb {C}^{N\times L}\) is the submatrix selected by the first L columns of matrix \(\textbf{F}\),where L represents the channel length, \({\varvec{h}}_{t,m}\epsilon \mathbb {C}^{L\times 1}\) and \({\varvec{i}}_{t,m}\epsilon \mathbb {C}^{N\times 1}\) denote the time domain CIR and IN, respectively. EquationÂ (1) can be rewritten as
3 Joint channel and impulsive noise estimation
In this section, based on SBL framework and Kalman filter, we propose the following new algorithm to jointly estimate impulse noise and channel. Let \(\varvec{\Phi _m}=[\sqrt{N}\textbf{D}_m\textbf{F}_L,\textbf{F}] \epsilon \mathbb {C}^{N\times (L+N)}\) and \(\varvec{\theta _m}=({\varvec{h}}_{t,m}^{T},{\varvec{i}}_{t,m}^{T})^T \epsilon \mathbb {C}^{L+N}\), Eq. (2) can be expressed as
Because the CIR \({\varvec{h}}_{t,m}\) and IN \({\varvec{i}}_{t,m}\) are regarded as unknown sparse vectors, \(\varvec{\theta }_m\) formed by \({\varvec{h}}_{t,m}\) and \({\varvec{i}}_{t,m}\), can also be regarded as an sparse vector. Noting that the matrix \(\varvec{\Phi }_m\) is an underdetermined matrix, estimating the unknown vector by Eq. (3) translates into a CS problem.
We assume that \(\varvec{\theta }_m\) is unchanged in one OFDM symbol, but changes between symbol to symbol according to a statespace model. Then, taking Eq. (3) as the observation equation, the state equation can be expressed as
where \(\textbf{A}\triangleq diag(\rho \textbf{1}_L,\varvec{0}_N)\), the \(\textbf{1}_L\) is a Llength vector with all elements as one, and \(\varvec{0}_N\) is a Nlength vector with all elements as zero. \(\varvec{\theta }_m\sim \mathcal{C}\mathcal{N}(0,\textbf{B}\varvec{\Gamma }_m)\), where \(\varvec{\Gamma }_m=diag(\varvec{\Gamma }_{({\varvec{h}})},\varvec{\Gamma }_{({\varvec{i}})_m})\) and \(\textbf{B}\triangleq diag((1{\rho }^2)\textbf{I}_L,\textbf{I}_N)\). \(\varvec{\Gamma }_{({\varvec{h}})}=diag(\gamma _0,\gamma _1,...,\gamma _{L1})\), \(\varvec{\Gamma }_{({\varvec{i}})}=diag(\gamma _L,\gamma _{L+1},...,\gamma _{L+N1})\) are the covariance matrices of the corresponding CIR and IN, respectively. The excitation noise \({\varvec{v}}_m\) and the observation noise \({\varvec{w}}_m\) are independent Gaussian white noise with zero mean i.e.,\({\varvec{v}}_m\sim \mathcal{C}\mathcal{N}(0,\textbf{B}\varvec{\Gamma }_m)\), \({\varvec{w}}_m\sim \mathcal{C}\mathcal{N}(0,\beta _m\textbf{I}_N)\) and \(\textbf{I}_N\) is an \(N\times N\) identity matrix. \(\rho\) is the correlation coefficient of the state transition, and \(\beta _m\) is a scalar corresponding to the background noise.
We assume the following forward and backward systems
where the subscripts f denote the variable in the forward systems and b denote the variables in the backward systems, respectively. In the process of forward Kalman filter, the filtering process is initialized as \(\varvec{\theta }_{00}^f=0\), \(\varvec{\Sigma }_{00}^f={{{\textbf {I}}}_{L+N}}\).
In the prediction step, we have
where \(\varvec{\theta }_{mm1}^f\) denotes the \(m{\text {th}}\) state prediction obtained by using the result of the \({m1}{\text {th}}\) optimal state. \(\varvec{\theta }_{m1m1}^f\) is the result of the \({m1}{\text {th}}\) optimal state. \({\varvec{e}}_m^f\) denotes the difference between the \(m{\text {th}}\) measured and predicted value. \(\varvec{\Sigma }_{mm1}^f\) and \(\varvec{\Sigma }_{m1m1}^f\) are the \(m{\text {th}}\) prior and the \({m1}{\text {th}}\) posterior covariance matrices of estimation errors, respectively.
In the update step, we can compute
where \(\varvec{\theta }_{mm}^f\) is the best estimate of the \({m}{\text {th}}\) state, \(\mathbf {K_m^f}\) is the Kalman gain,and \({\varvec{e}}_m^f\) is given by (9).
The process of backward Kalman filtering is similar to the forward Kalman filtering in (7)â€“(13), except that the former is from time \(m=N_b\) to \(m=1\).
For (5) and (6), we can apply the linear minimum meansquared estimator (LMMSE) for each linear system to estimate \(\varvec{\theta }_m\) in each linear system. For the forward system, the estimated value of \(\varvec{\theta }_m\) is
where \(\varvec{\Sigma }_m^f=(\mathbf {R_{\varvec{\theta }_m}^{1}}+(\varvec{\Phi }_m^f)^H\mathbf {R_{{\varvec{w}}_m^f}^{1}}\varvec{\Phi }_m^f)^{1}\) is the \({m}{\text {th}}\) estimation error matrix, \(\mathbf {R_{\varvec{\theta }_m}}\) and \(\mathbf {R_{{\varvec{w}}_m^f}}\) are the covariance matrices of \(\varvec{\theta }_m\) and \({\varvec{w}}_m^f\), respectively. Similarly, the estimated value of \(\varvec{\theta }_m\) from the backward system is
where \(\varvec{\Sigma }_m^b=(\mathbf {R_{\varvec{\theta }_m}^{1}}+(\varvec{\Phi }_m^b)^H\mathbf {R_{{\varvec{w}}_m^b}^{1}}\varvec{\Phi }_m^b)^{1}\) is the \({m}{\text {th}}\) estimation error matrix, \(\mathbf {R_{{\varvec{w}}_m^b}}\) is the covariance matrix of \({\varvec{w}}_m^b\).
Combining (5) and (6), we have
The LMMSE estimation of \(\varvec{\theta }\) from (16) is given by
The estimation error matrix is
According to equation (14) and (15), (17) and (18) can be simplified, respectively, as
As Kalman filtering is an extension of sequential LMMSE estimation, the forward and backward estimations can be combined following (19) and (20), respectively, as
where \(\varvec{\theta }_{mm}^f\) and \(\varvec{\Sigma }_{mm}^f\) are given by (12) and (13), respectively, \(\varvec{\theta }_{mm}^b\) and \(\varvec{\Sigma }_{mm}^b\) obtained in a similar way. Since there is no prior autocorrelation information about \(\varvec{\theta }\), we set \(\mathbf {R_{\varvec{\theta }}^{1}}=0\) in (19) and (20).
In (7)â€“(13), we see that in addition to the unknown parameters \(\varvec{\Gamma }_m\) and \(\beta\), there are some unknown matrix elements of the matrix \(\varvec{\Phi }\) which just based on unknown data symbol \(\textbf{D}\).
Let \(\varvec{\xi }\triangleq \big \{\{\varvec{\Gamma }_m\},\beta ,\mathbf {D_m}\big \}\) as the set of unknown parameters, the steps of EM algorithm are as follows:
In the Estep, the expected value of joint probability density \(p\big (\{{\varvec{r}}_m\},\{\varvec{\theta }_m\},\{\varvec{\Gamma }_m\},\beta \big )\) under posterior probability distribution \(p(\varvec{\theta }{\varvec{r}})\) is obtained from
where k represents the number of iterations.
In the Mstep, first fix one parameter, solve another parameter, and then solve the optimal parameter alternately as
The likelihood function of (2) can be written
Form (24), we obtain the loglikelihood function as
By analyzing the expression (26), the optimal \(\varvec{\Gamma }_m\) can be obtained as
where \(\mathbf {M_{mm}}=\varvec{\Sigma }_{mm}+\hat{\varvec{\theta }}_{mm}\hat{\varvec{\theta }}_{mm}^H\).
Similarly, \(\beta\) can be optimized
By setting the derivative of (28) with respect to \(\beta\) as zero, we obtain
After obtaining the estimated values of \(\varvec{\Gamma }_m^{(k+1)}\) and \(\beta ^{(k+1)}\), respectively, the estimated values of data symbol \(\mathbf {D_m}\) is derived as follows.
where c is a constant independent of \({\varvec{D}}\), and Tr() denotes the matrix trace. We have
It is reasonable to assume that the CIR vector and IN vector are independent of each other, so that the covariance submatrices \(\varvec{\Sigma }_{2,1}\) and \(\varvec{\Sigma }_{1,2}\) in Eq. (31) can be set to zero matrix.
Denotes \({\varvec{I}}_d\) as the indices of subcarriers with data symbols. By substituting (31) into (30), the iterative update formula of each data symbol \(d_m[i]\), i.e., the \({i}{\text {th}}\) diagonal element of \(d_m\), \(i\in {\varvec{I}}_d\), can be obtained by solving the following problem
where
and \(\textbf{F}[i,:]\) is the \({i}{\text {th}}\) row of \(\textbf{F}\), \(\varvec{\theta }_{mm}\) and \(\varvec{\Sigma }_{mm}\) can be obtained, respectively, by (21) and (22).
The entire algorithm of FBKalman is summarized in Algorithm 1.
4 Complexity analysis
The FBKalman algorithm includes three major steps: forward filtering, backward filtering, and EM parameters estimation. In the forward filtering step, most of the calculations are used to calculate the Kalman gain \(\textbf{K}_m\), which has a complexity of \({O}({N^{3}})\) for each OFDM symbol. The backward filtering step has a computational complexity of \({O}({N^{3}})\) for each OFDM symbol. Moreover, the complexity of the EM step is \({O}(I_1{N^{3}})\), where \(I_1\) is the average number of iterations of the EM process till convergence. Thus, for each OFDM symbol, the computational complexity order of the FBKalman algorithm is \({O}((I_1+2){N^{3}})\).
The JCIKalman algorithm uses pilot subcarriers, the complexity of the filtering step is \({O}({N_p^{3}})\) for each OFDM symbol, and the complexity of the EM step is \({O}(I_2{N_p^{3}})\) for each OFDM symbol, where \(I_2\) is the average number of iterations of the EM process till convergence. So, the computational complexity of this algorithm is \({O}((I_2+1){N_p^{3}})\) per OFDM symbol.
The SBLLS algorithm uses SBL method to remove IN by using null subcarriers, and the CIR is estimated by least square method. The EM step has a computational complexity of \({O}(I_3{N_n^{3}})\) for each OFDM symbol, where \(I_3\) is the average number of iterations of the EM process till convergence. And the complexity of the LS step is \({O}({N_p^{3}})\) for each OFDM symbol. Thus the SBLLS algorithm has a computational complexity of \({O}(I_3{N_n^{3}}+{N_p^{3}})\).
From the above analysis, we see that the FBKalman algorithm using all subcarriers has higher complexity than the JCIKalman algorithm using only pilot subcarriers and the SBLLS algorithm using pilot subcarriers and null subcarriers. However, the simulation results show that the proposed FBKalman algorithm provides best performance in NMSE and BER. Similarly, compared with the original SBL based algorithm [10], the proposed FBKalman algorithm has higher computational complexity and better performance by using filtering and smoothing operations to track timevarying channels and IN. These tradeoffs of performance and complexity are meaningful for practical OFDM systems.
5 Results and discussion
In this section, we demonstrate the performance of the proposed joint channel and IN estimation algorithms. An OFDM system with \(N=256\), \(N_p=44\), \(N_d=162\), and \(N_n=50\) is simulated. The Rayleighfading uncorrelatedscattering model with sparse impulse response [20] is adopted. Each OFDM frame is composed of \(N_b=7\) OFDM symbols.
The noise including IN and background noise is realized by the publicly available software [21], which uses a Gaussian mixture model to simulate the IN distribution, where the probability of each noise component is 0.9, 0.07, 0.03, and the corresponding power of each noise component is 1, 100, 1000. In the noise generated in the simulation, \(7\%\) of the IN components exceed the background noise power by 20 dB, and about \(3\%\) of the IN components exceed the background noise power by 30 dB.
In addition, we also consider IN environment with \(\alpha\)stable distribution. To verify the performance of channel and IN estimation, \(\alpha\)stable IN model with characteristic exponent \(\alpha =1\), skewness parameter \(\beta =0\), scale parameter \(\gamma =0.05\), and location parameter \(\delta =0\) is used in the simulation.
The performance of the FBKalman algorithm is compared with the other two algorithms in this section:

SBLLS: assume that the IN is mitigated by using the SBL method with null subcarriers [13], the tapaware LS method is used to estimate the CIR.

JCIKalman: assume that the IN and CIR are jointly estimated by using the JCIKalman method with pilot subcarriers [18].
FigureÂ 1 compares the channel estimation performance of all the above mitigation algorithms for convolutional coded systems with 1/2 rate using 4QAM. The figure shows that the NMSE of the three algorithms gradually decreases with the increase in signaltonoise ratio (SNR). The proposed algorithm performs better than SBLLS and JCIKalman. Compared with SBLLS, the SNR improvement of the proposed algorithm is more than 10 dB, compared with JCIKalman algorithm, the SNR improvement is about 7 dB. Compared with SBLLS, JCIKalman has a gain of about 4 dB in SNR. It can be seen that the joint channel and IN estimation algorithm has better performance than the single estimation. The reason why the proposed algorithm has better performance is that the forward message and the backward message can provide more useful information.
FigureÂ 2 provides the BER performance for convolutional coded systems using 4QAM. As shown in the figure, the BER of the three algorithms gradually decreases with the increase in SNR. The proposed algorithm is always better than the other two algorithms, and until the SNR approaches 10 dB, the BER of JCIKalman and FBKalman gradually approaches and then to zero. When BER \(={10^{3}}\), the proposed algorithm achieves about 2 dB gain in SNR over JCIKalman and about 7 dB gain in SNR over SBLLS.
FigureÂ 3 compares the channel estimation performance of three mitigation algorithms for uncoded systems using 4QAM. Similar to the coded system, the NMSE of three algorithms decreases with the increase in SNR. The proposed algorithm performs better than the other two algorithms. Compared with SBLLS, the SNR improvement of the proposed algorithm is more than 8 dB, compared with JCIKalman, the SNR improvement is more than 5 dB. Compared with SBLLS algorithm, JCIKalman algorithm has a gain of about 3 dB in SNR. Compared with the coded system, the uncoded system has higher NMSE. It shows that channel coding can effectively improve the signal channel estimation performance.
FigureÂ 4 provides the BER performance for uncoded systems using 4QAM. Similar to the coded system, the figure also shows that the BER of three algorithms decreases with the increase in SNR. It can be seen that the proposed algorithm performs better than the other two algorithms. The uncoded system exhibits higher BER compared to the coded system. It shows that channel coding can effectively improve system performance.
In the IN environment with \(\alpha\)stable distribution, the results of NMSE and BER performance for coded system are shown in Figs.Â 5 and 6, respectively, while the results for uncoded system are shown in Figs.Â 7 and 8, respectively. For coded system, Fig.Â 5 compares the channel estimation performance of three algorithms. It is obviously seen that the proposed algorithm performs better than SBLLS and JCIKalman. Compared with SBLLS algorithm, the SNR improvement of the proposed algorithm is more than 10 dB, compared with JCIKalman algorithm, the SNR improvement is more than 8 dB. FigureÂ 6 provides the BER performance of three algorithms. The figure shows that the proposed algorithm outperforms the other two algorithms. When BER \(={10^{  3}}\), the proposed algorithm achieves about 5 dB gain in SNR over JCIKalman and about 6 dB gain in SNR over SBLLS. For uncoded system, Fig.Â 7 compares the channel estimation performance of three algorithms. Similar to the coded system, the proposed algorithm performs better than the other algorithms. Compared with SBLLS algorithm, the SNR improvement of the proposed algorithm is more than 12 dB, compared with JCIKalman algorithm, the SNR improvement is about 10 dB. FigureÂ 8 provides the BER performance of three algorithms. Similar to the coded system, the proposed algorithm outperforms the other two algorithms.Â When BER \(={10^{  3}}\), the proposed algorithm achieves about 5 dB gain in SNR over JCIKalman and about 9 dB gain in SNR over SBLLS.
FiguresÂ 9 and 10 show the NMSE of three algorithms versus the number of iterations in uncoded system with SNRs of 5 dB and 20 dB, respectively. The figures show that both the FBKalman and JCIKalman algorithms exhibit a rapid convergence rate during the first 15 iterations, with a significant decrease in NMSE, and then the convergence rate gradually slows down and NMSE approaches a stable value. After 40 iterations, both algorithms have reached an almost stable state. While the SBLLS algorithm converges fast, but the NMSE is the largest among three algorithms.
The FBKalman achieves better and more reliable performance than the JCIKalman and SBLLS because it can jointly estimate the channel, IN, and symbols using all subcarriers in the received OFDM symbol. Moreover, the FBKalman filter algorithm captures the time correlation of sparse timevarying channels, so it has better BER performance.
6 Conclusion
In this paper, we discuss the problems of joint sparse channel estimation, IN mitigation, and data detection in the presence of IN for OFDM systems. A joint channel estimation and IN suppression algorithm by using all subcarriers based on SBL and FBKalman is proposed. An efficient implementation algorithm based on EM is utilized to estimate the unknown parameters. Simulation results verify the efficiency of the proposed algorithm.
Although the proposed algorithm improves the estimation performance of OFDM systems in IN background, it comes at the cost of sacrificing algorithm complexity. Our future work includes fast implementation algorithms with robust performance.
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Abbreviations
 AMP:

Approximate message passing
 BER:

Bit error rate
 CS:

Compressed sensing
 CIR:

Channel impulse response
 EM:

Expectation maximization
 FBKalman:

Forwardâ€“backward Kalman filter
 GM:

Gaussian mixture
 IN:

Impulse noise
 LMMSE:

Linear meansquared estimator
 NMSE:

Normalized mean square error
 OFDM:

Orthogonal frequency division multiplexing
 PLC:

Power line communication
 SBL:

Sparse Bayesian learning
 SNR:

Signaltonoise Ratio
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Acknowledgements
This work was supported in part by the International Cooperation Project of the Ministry of Science and Technology under Grant 2018YFE0206500; in part by the National Natural Science Foundation of China under Grant 61571250; and in part by the Zhejiang Natural Science Foundation under Grant LY22F010018.
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Zhao, Y., Li, Y., Shi, S. et al. Joint channel and impulse noise estimation based on compressed sensing and Kalman filter for OFDM system. EURASIP J. Adv. Signal Process. 2023, 105 (2023). https://doi.org/10.1186/s13634023010645
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DOI: https://doi.org/10.1186/s13634023010645