1 Introduction

Nowadays, PLC systems are attracting more interests since they can be used for transmitting data over the entire residential or industrial area. They can offer high speed data, image, voice and video services to the customer end. Density, ubiquity and low cost implementation are the main advantages of PLC technology [1]. Unfortunately, the transmission of broadband signals over the PLC channel is facing several challenges. Two major impairments to the PLC systems are due to multipath propagation accompanied and impulsive noise [2, 3]. Multipath propagation is characterized by both slow fading and uncorrelated Rayleigh/Ricean fading scenarios [4]. Impulsive noise is characterized by asynchronous or cyclo-stationnary scenarios. For its performance against multipath and noise effects, OFDM is widely adopted in PLC channels [5–7].

Significant efforts have been conducted during the past decade to model and fit the PLC channel model. A number of channel model proposals can be found in literature with different network locations, topologies, frequency bands, etc. An outstanding top-down model proposed by Zimmermann and Dostert [8] came up with a channel model (0.5–20 MHz) based on physical signal propagation effects in PLC networks with different branches and impedance mismatching. Tlich et al. [9] have proposed a random channel model (1–100 MHz) based on extensive real measurements. In [10], Melit et al. used bond graph theory to model the propagation of PLC signals through a power transformer. This kind of modeling relies on the use of energy and its conservation to model the transmission of low level PLC signals. In a different manner, Galli [11] used time-domain channel analysis to model the PLC channel (1.8–30 MHz) including average channel gain and RMS delay spread. By comparing the electrical lines with twisted-pair and coaxial cable, the authors demonstrated comparable statistical performance. In [12], Tonello introduced a refined representation of Zimmermanns model and present a new model known as bottom-up model. Note that the noise over PLC channels can be classified into three categories including colored background, background noise and impulsive noise that can be cyclostaionary or asynchronous. This last one is the most destructive; it has very high instantaneous power and wide frequency spectrum, leading to high BER. A set of works in the literature has modeled the asynchronous impulsive noise at the PLC receiver by Middleton class A distribution [13]. However, Middleton’s class A model may not be the most suitable candidate to describe impulsive noise in the PLC environment since it is originally designed for man-made interference [14]. Based on measurements in both industrial and residential buildings, different models have been proposed to characterize the impulsive noise such as alpha-stable distribution [15], Bernoulli-Gaussian model [16] and Gaussian mixture distribution [17].

In this paper, we propose a PLC channel estimation and decoding scheme using a turbo coded OFDM and a noise mitigation stage, which we henceforth refer to as turbo coded adaptive impulsive noise compensator (TC-AINC). The proposed configuration is to do all the multi-stage iterations of the impulsive noise compensator first before the output is fed into the turbo decoder which is based on maximum a posteriori (MAP), to immediately recover transmitted symbols. The major contribution of this work can be summarized in the following points; First, a 15-paths PLC channel model and a Poisson–Gaussian distribution are used to model a PLC network. Second, a preliminary estimation process of OFDM signals based on OFDM tones is developed. Third, a new adaptive clipping/blanking function to better detect the impulsive bursts is introduced. Furthermore, the proposed method presented in our system is a non-data-aided scheme in the sense that the estimation of the impulsive noise is essentially based on the estimation of the SINR and the PAPR of the received noisy signal. Fourth, the impulsive bursts are adaptively estimated at the turbo decoder by using a preliminary estimation process and by making use of redundancy introduced by the turbo encoder without using additional information.

The rest of this paper is arranged as follows. A brief description of some related works is introduced in Sect. 2. In Sect. 3, the PLC channel characteristics, OFDM transmission technique, impulsive noise models, and turbo coding are introduced. In Sect. 4, the proposed TC-AINC system is detailed. The computer simulation results and a deep analysis are provided in Sect. 5. Finally, the conclusion is given in Sect. 6.

2 Related works

Several noise mitigation methods have been proposed. Conventional approaches for impulsive noise cancellation in PLC channel are generally based on time-domain treatment [18, 19], in which case the noise impulses can be deleted using nonlinear techniques such as clipping, blanking or clipping/blanking [20]. The performance of these techniques cannot be appreciated since they give poor communication performance. Author of [21] presented an attractive idea to compensate impulsive bursts in wireless channels in frequency-domain after OFDM demodulation and channel equalization. However, the performance of the scheme is highly declined when it is applied to PLC channel, when the impulsive noise magnitude is raised, and when higher order modulation is used. In [22], a time-domain pre-processing mean filter namely composite comparison filter is proposed to detect and reject impulsive bursts over wireless channels. It is based on the estimation of impulsive noise statistics. However, it presents poor performance when it is applied to PLC.

Another popular approach to protect against impulsive noise is based on error correction code (ECC) including low-density parity-check codes (LDPC) [23], LDPC convolutional codes [24] and codes turbo codes (TC) [25] for the following two reasons. They can provide results closed to Shannon limit capacity with acceptable complexities and the check relationships of block codes can embody the constraints, which help to suppress the additive noises caused by PLC channels. Recent trends in coding PLC systems support and promote the deployment of TC and probabilistic soft-decision iterative decoders [26–28]. An application of adaptive robust turbo equalization for PLC systems has been studied in [26]. The performance of double binary TC has been examined over an impulsive noise environment, as in the case of a PLC channel [27]. The characteristics of high spread random (HSR) and quadratic permutation polynomial based (QPP-based) interleavers for turbo coding have been examined over an impulsive noise PLC channel [28].

Other techniques for impulsive noise mitigation exploit the sparsity of the noise in time-domain. In [29, 30], an impulsive noise is estimated and suppressed based on a convex programming technique used for arbitrary sparse signal reconstruction, which is observed through projections onto a small-dimensional space in background noise. In [31], impulsive bursts estimation and compensation are performed based on a guard band null sub-carriers. Unlike using the l1 minimization as most of the popular compressive sensing based algorithms, this approach exploits both the specific structure of this problem and the available a prior information for sparse signal recovery. A non-parametric impulsive noise mitigation based on sparse Bayesian learning (SBL) is presented in [32]. In [33], a noise reduction scheme based on a factor-graph representation is proposed and the noise compensation is performed using a joint channel, impulse, symbol and bit estimation. Even if the sparsity based algorithms have a good noise reduction capability, they have the inconvenience of being more computationally complex.

3 System model

3.1 PLC channel model

In this paper, we use a top-down PLC channel model. This model considers the PLC channel like a black box and a set of measurements are gathered by exciting the channel with a reference signal in either time-domain or frequency-domain. To construct a model that matches the experimental results, complex fitting approaches are used. The modeling PLC channel has a bandwidth of 500 KHz to 20 MHz, and its frequency response in terms of the total transmission characteristic is expressed as [8]

$$\begin{aligned} H(f)=\sum \limits _{i=1}^{N}g_{i}\cdot \exp \left( -(a_{0}+a_{1}f^{\kappa })d_{i}\right) \cdot \exp \left( -j2\pi fd_{i}/v_{p} \right) ,\nonumber \\ \end{aligned}$$
(1)

where, for N relevant paths, \(g_{i}\) is considered as a weighting coefficient over the ith path with distance \(d_{i};\) \(a_{0}\) and \(a_{1}\) are attenuation factors; \(\kappa \) is the exponent of the attenuation generally selected in the interval [0.2,1]. The final term is considered as the propagation delay, with \( v_{p}=\frac{C_{0}}{2}\) indicating the velocity of propagation (\(C_{0}\) velocity of light). It should be noted that the PLC channel model (1) shows an excellent agreement between theoretical parameters and measured values. For a PLC system with 15 paths, attenuation parameters \(a_{0}=0,\) \(a_{1}=2.5\times 10^{-9},\) \(\kappa =1,\) and weighting factor \(g_{i}\) along the ith path with distance \(d_{i}\) are given in Table 1. The frequency response and impulse response of the PLC channel are shown in Figs. 1 and 2, respectively.

Table 1 Parameters \({g}_{i}\) and \({d}_{i}\) for the PLC channel for each path [8]
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Fig. 1
figure 1

Frequency response of the PLC channel

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Fig. 2
figure 2

Impulse response of the actual PLC channel

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The theoretical and measured results show that: (i) There is a high convergence between the theoretical channel model in (1) and the measured PLC channel response; (ii) The attenuation introduced by the PLC channel is intensely increased with increasing the frequency, since this channel is a frequency selective. Thus, the OFDM modulation, which is very convenient for the selective fading channels, is a good candidate for PLC systems; and (iii) The representative time spread for the PLC channel is about \({\mu }s\)–\(2~{\mu }s\). Consequently, the guard interval (GI) duration requirement is not very strict [34].

The PLC channel model used in this work has a similar or a closed bandwidth as many recent PLC standards, including HomePlug 1.0 [35], HomePlug AV [36], Universal PowerLine Association (UPA) [37] and even HomePlug GP specification developed specifically to support Smart Grid Applications. For example, the UPA protocol uses a frequency bandwidth of 020 MHz and can be configured to work in a bandwidth of 030 MHz as an optional choice [37]. Moreover, the HomePlug V1.0 works in a range of 4.3–25 MHz.

3.2 Transmission model

Figure. 3 illustrates a framework of the turbo coded OFDM communication system over a multipath PLC channel under additive channel noise. The binary data stream block with length \(N_{c}/2,\) \(\mathbf {b}(n)=[b_{0}(n),b_{1}(n),\ldots ,b_{\frac{N_{c}}{2}-1}(n)]^{T}\) at time n are turbo encoded using a rate of 1 / 2, producing a coded data stream \(\mathbf {C}(n)=[C_{0}(n),~C_{1}(n),\ldots ,~C_{N_{c}-1}(n)]^{T}\) of length \(N_{c}.\) The coded data stream is then permuted by a random interleaver, generating a coded data stream of independent symbols \(\mathbf {X}(n)=[X_{0}(n),~X_{1}(n),\ldots ,~X_{N_{c}-1}(n)]^{T}.\) Each data stream is mapped to a complex data symbol by using either BPSK or QAM mapper. For the sake of reducing the inter-symbol interferences (ISI) and maintaining the orthogonality of the OFDM-PLC signal in the multipath PLC channel [38], a GI or cyclic prefix (CP) is incorporated in the end of each OFDM block in time-domain after OFDM modulation. Finally, the obtained signal is transmitted through the noisy PLC channel with frequency response H(f).

Fig. 3
figure 3

Structure of the turbo coded OFDM base-band transmission model

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3.3 Coded OFDM model

Let us consider an OFDM model with \(N_{C}\) sub-carriers (tones) partitioned into \(N_{d}\) data tones, \(N_{p}\) pilot tones and \(N_{{\varPhi }}\) null tones. Each tone is modulated using a finite alphabet symbol. We use \(\mathbf {S}(n)=[S_{0}(n),\ldots ,S_{N_{c}-1}(n)]^{T}\) to denote the OFDM symbol’s tone vector where the component \(S_{k}(n)\) may be considered as the scalar transmitted symbol on the k pilot of the OFDM symbol during the block instant n.

Recall that, in OFDM modulation, received signal after discrete Fourier transform (DFT) is described by

$$\begin{aligned} R_{k}(n)=H_{k}(n)S_{k}(n)+W_{k}(n), \quad k=0,\ldots ,N_{c}-1, \end{aligned}$$
(2)

where \(\mathbf {R}(n)=[R_{0}(n),\ldots ,R_{N_{c}-1}(n)]^{T}\) is the frequency-domain output vector,\(\ \mathbf {H}(n)=[H_{0}(n),\ldots ,H_{N_{c}-1}(n)]^{T}\) is the frequency-domain channel vector, and \(\mathbf {W}(n)=[W_{0}(n),\ldots ,W_{N_{c}-1}(n)]^{T}\) is the frequency-domain noise vector. We note that \(S_{k}(n)=p\) for all pilot tones, where \(p\in \mathbf {C}\) is a known pilot symbol; \(S_{k}(n)=0\) for all null tones; and \(S_{k}(n)=X_{k}(n)\) for all data tones, where \(X_{k}(n)\) is the interleaved bit. We note also that \(\mathbf {W}(n)=\mathbf {Z}(n)+\mathbf {I}(n)\) where \(\mathbf {Z}(n)=[Z_{0}(n),\ldots ,Z_{N_{c}-1}(n)]^{T}\) is the frequency-domain of AWGN term and \(\mathbf {I}(n)=[I_{0}(n),\ldots ,I_{N_{c}-1}(n)]^{T}\) is the frequency-domain of impulsive noise term, respectively.

3.4 Additive channel noise

Interference generated by the impulsive noise at the PLC receiver are usually modeled by Middleton class A distributions [13] or a Poisson–Gaussian model [39].

  • Middleton’s noise model: is generally described by two factors: the impulse index and the Gaussian-to-impulse noise power ratio (GIR). The noise present at a power-line channel is distinguished between Gaussian background noise and impulsive noise. According to this model, the overall noise is a sequence of independent and identically distributed complex random variables with the PDF given by [13]:

    $$\begin{aligned} P_{A}(x)=\sum \limits _{m=0}^{\infty }e^{-A}\frac{A^{m}}{m!}\frac{1}{\sqrt{2\pi }\sigma _{m}}\exp \left( \frac{-x^{2}}{2\sigma _{m}^{2}}\right) , \end{aligned}$$
    (3)

    where A is the impulse index and \(\sigma _{m}^{2}\) is the variance expressed as

    $$\begin{aligned} \sigma _{m}^{2}=\sigma ^{2}\left( \frac{m/A+{\varGamma }}{1+{\varGamma }}\right) , \end{aligned}$$
    (4)

    with \({\varGamma }=\sigma _{G}^{2}/\sigma _{I}^{2}\) is the GIR. \(\sigma _{G}^{2}\) and \(\sigma _{I}^{2}\) are Gaussian background noise power and impulse noise power, respectively. The sources of impulsive noise are distributed according to a Poisson distributed sequence whose PDF is characterized by the impulse index A.

  • Poisson–Gaussian model: In this model, impulsive noise in PLC channel is modeled by using Poisson process. This means that impulsive noise will occur according to a Poisson distribution with a rate \({\uplambda }\) units per second, leading to a probability of an occurrence of k arrivals in unit time that can be defined by [40]:

    $$\begin{aligned} p\left( t\right) =p_{k}\left( T=t\right) =\frac{\left( {\uplambda }t\right) ^{k}}{k!}\exp \left( -{\uplambda }t\right) , \quad k=0,1,2,\ldots \end{aligned}$$
    (5)

    Depending on this model, the whole noise symbol \(w_{k}\) can be explicit as \(w_{k}=z_{k}+i_{k}\), where \(z_{k}\) is the additive white Gaussian noise (AWGN) with per-component power \(\sigma _{z}^{2},\) and \(i_{k}\) is the impulsive noise term specified as:

    $$\begin{aligned} i_{k}=b_{k}g_{k}, \end{aligned}$$
    (6)

    where \(b_{k}\) represents the arrival of impulses according to Poisson process, which characterizes the rate of arrival of the impulsive noise, and \(g_{k}\) is the white Gaussian process with mean zero and variance \(\sigma _{i}^{2}.\) The PDF of \(w_{k}\) can be expressed as

    $$\begin{aligned}&P_{p}(w_{R},w_{I}) =(1-p)\cdot G(w_{R},0,\sigma _{z}^{2})\cdot G(w_{I},0,\sigma _{z}^{2}) \nonumber \\&\quad \cdot p\cdot G(w_{R},0,\sigma _{w}^{2}+\sigma _{i}^{2})\cdot G(w_{I},0,\sigma _{w}^{2}+\sigma _{i}^{2}), \end{aligned}$$
    (7)

    where \(w_{R}\) and \(w_{I}\) are the real and imaginary parts of the total noise, respectively and

    $$\begin{aligned} G(x,m_{x},\sigma _{x}^{2})=\frac{1}{\sqrt{2\pi }\sigma _{z}}\exp (-(x-m_{x})^{2}/2\sigma _{x}^{2})) \end{aligned}$$
    (8)

    is the Guassian density with mean \(m_{x}\) and variance \(\sigma _{x}^{2}.\)

The Poisson–Gaussian model is adopted in this paper as the impulsive noise since it can be an efficient representation of impulsive noise in practical power-line networks.

4 Receiver signal model design

In this section, we design computationally efficient message-passing signal receiver that performs close-to-optimum bit decoding, which, as we shall see, involves suppressing the impulsive noise and estimating the codebits. The proposed turbo receiver method can be considered as an iterative decoder structure that combines both adaptive impulsive noise compensator algorithm and turbo decoding algorithm. At each iteration, extrinsic information from the noise compensator and channel estimator is fed into the turbo decoder, and then their extrinsic information is fed back to the channel estimator. By doing so, the proposed method presents not just the error-correction ability, but the suppression of impulsive noise peak power as well. The structure of the proposed turbo receiver in this paper for PLC systems is illustrated in Fig. 4.

Fig. 4
figure 4

Structure of the proposed turbo receiver

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The received signal is obtained after removing the CP, demodulating the \(N_{c}\) samples of each OFDM block and demapping data tones to the nearest positions in constellation plot. Then, the obtained signal passes through the adaptive impulsive noise compensator block, resulting in an estimated signal \(\hat{\mathbf {R}}(n).\) In the turbo decoder, \(\hat{\mathbf {R}}(n)\) and parity bits and apriori signal produced by the soft input-soft output (SISO) decoder are used in the decoding procedure, resulting in an equalized symbol sequence \(\mathbf {D}(n).\) Then, \(\mathbf {D}(n)\) is passed through a channel deinterleaver, resulting in a deinterleaved equalized symbol sequence \(\mathbf {Z}(n).\) Finally, \(\mathbf {Z}(n)\) is applied to the MAP decoder to calculate the LLR of the posteriori probabilities. Hence, turbo decoding is recursively applied, which makes use of the sub-optimum MAP algorithm. The details of the adaptive impulsive noise compensator algorithm with turbo decoding are shown in the following subsections.

4.1 Adaptive impulsive noise compensator algorithm

It can be summarized in the following steps:

  • Step 1. First, we assume ideal channel estimation, i.e., \(\hat{\mathbf {H}}(n)\equiv \mathbf {H}(n)\) where \(\hat{\mathbf {H}}(n)\) is the estimated value of the frequency-domain channel vector of the OFDM symbol \(\mathbf {H}(n).\) Based on (2), estimation of total noise term is done as follows

    $$\begin{aligned} \hat{W}_{k}(n)=\hat{H}_{k}(n)(R_{k}(n) \hat{H}_{k}^{-1}(n)-\hat{S}_{k}(n)), \end{aligned}$$
    (9)

    where \(\hat{\mathbf {H}}^{\mathbf {-1}}(n)=[\hat{H}_{0}^{-1}(n),\ldots ,\hat{H}_{N_{c}-1}^{-1}(n)]^{T}\ \)is the inverse channel frequency response. It should be noted that our idea is to estimate impulse noise based on a preliminary estimation of transmitted baseband symbol \(\hat{S}_{k},k=0,1,\ldots ,N-1\). This last one is derived from the equalizer output as follows:

    1. 1.

      OFDM tones, which are used for data transmission, are demapped to the nearest positions in constellation plot.

    2. 2.

      OFDM tones, which should be silent, are set to zero.

  • Step 2. The variance of the total noise is estimated by

    $$\begin{aligned} \hat{\sigma }^{2}=\frac{1}{N}\sum \limits _{k=0}^{N-1}\left| \hat{w}_{k}(n)\right| ^{2}, \end{aligned}$$
    (10)

    where \(\hat{w}_{k}(n)\) is the time-domain representation of \(\hat{W}_{k}(n)\) determined by means of inverse discrete Fourier transform (IDFT).

  • Step 3. The impulsive noise samples are estimated using a new clipping/blanking function defined as:

    $$\begin{aligned} \hat{\imath }_{k}(n)=\left\{ \begin{array}{l@{\quad }l} 0,&{}\text {if}~~ \left| \hat{w}_{k}(n) \right| <T_{1}, \\ \frac{2\hat{w}_{k}(n)-T_{1})}{(T_{2}-T_{1})^{2}}, &{}\text {if}~~ T_{1}<\left| \hat{w}_{k}(n)\right| <T_{2}, \\ \hat{w}_{k}(n), &{}\text {otherwise,} \end{array} \right. \end{aligned}$$
    (11)

    with \(T_{1}=\hat{\sigma }(\mu -\frac{1}{2})\) and \(T_{2}=\hat{\sigma }(\mu +2)\ \)are the lower and upper thresholds corresponding to small probability of false detection, respectively. Figure 5 shows the function of the proposed non-linear (clipping/blanking) estimator. The parameter \(\mu \) is performed according to the following expression

    $$\begin{aligned} \mu =\eta \exp \left( \frac{\text {SINR}}{\text {PAPR}}\right) , \end{aligned}$$
    (12)

    where \(\eta \) is a positive constant, SINR represents the signal to impulsive noise ratio for received signal \(R_{k}\), it can be calculated using (13), where \(N_{0}\) is the PSD of the impulse noise,

    $$\begin{aligned} SINR_{R_{k}}=\frac{\left| \hat{H}_{k}\right| ^{2}}{N_{0}+\left| I_{k}\right| ^{2}} \end{aligned}$$
    (13)

    and the PAPR is defined as the ratio between the maximum power occurring in OFDM symbol to the average power of the same OFDM symbol, given by

    $$\begin{aligned} PAPR=\frac{\max \left| r_{k}(n)\right| ^{2}}{E\left[ \left| r_{k}(n)\right| ^{2}\right] }. \end{aligned}$$
    (14)
    Fig. 5
    figure 5

    Non-linear impulsive noise estimator defined by (11)

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  • Step 4. Now, the time-domain representation of the estimated received signal \(\hat{r}_{k}(n)\) is derived by subtracting the estimated impulsive noise \(\hat{\imath }_{k}(n)\) from the time-domain representation of the received signal \(r_{k}(n)\) as follows

    $$\begin{aligned} \hat{r}_{k}(n)=r_{k}(n)-\hat{\imath }_{k}(n). \end{aligned}$$
    (15)

  • Step 5. The above steps are then repeated in order to get the best performance in term of BER and MSE.

The flowchart of the proposed adaptive impulsive noise compensator is shown in Fig. 6. Here, the impulsive bursts detector executes the operations depicted in (10) and (11). To effectively detect and suppress the impulsive bursts, the above steps are iteratively performed on the compensated version of the output signal \(\hat{r}_{k}, k=1,2,\ldots ,N-1\). This establishes adaptive iterative impulsive noise compensation.

Fig. 6
figure 6

Flowchart of the adaptive impulse noise compensator

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4.2 Turbo decoding

The turbo decoder is generally constructed based on SISO decoder via inteleaver and deinterleaver. A set of powerful algorithms is found in literature to implement SISO decoders, which are mainly classified into two classes [41]: the maximum likelihood (ML) algorithm that relies on the sequence estimation, and the maximum a posteriori probability (MAP) algorithm that uses the bits estimation. It has been shown that the MAP algorithm is an optimal decoding algorithm compared to ML algorithm, and, therefore, more suitable for the PLC channel model. In this work, we choose a SISO decoder that makes use of the MAP algorithm, which is based on finding the a posteriori log likelihood ratio (LLR) for the received signal [41, 42]. For each bit of the selected codebit \(C_{k}\), its LLR is calculated according to

$$\begin{aligned} L(\left. C_{k}\right| \mathbf {Z})= & {} \ln \left\{ \frac{P(\left. C_{k}=+1\right| \mathbf {Z})}{P(\left. C_{k}=-1\right| \mathbf {Z})} \right\} \nonumber \\= & {} \ln \left\{ \frac{\sum \limits _{(s^{\prime },s)\Rightarrow C_{k}=+1}P(s^{\prime },s,\mathbf {Z})}{\sum \limits _{(s^{\prime },s)\Rightarrow C_{k}=-1}P(s^{\prime },s,\mathbf {Z})}\right\} , \end{aligned}$$
(16)

where \((s^{\prime },s)\Rightarrow C_{k}=+1\) denotes the set of transition from state \(s^{\prime }\) to s caused by the bit \(C_{k}=+1\) and similarly, \( (s^{\prime },s)\Rightarrow C_{k}=-1\) for the bit \(C_{k}=-1.\) The term \( P(s^{\prime },s,\mathbf {Z})\) represents the joint probability which can be divided into three parts. The past values of received sequence before the time instant k,  i.e., \(Z_{0}^{k-1};\) the present values at the time instant k,  i.e., \(Z_{k};\) and the future values after the time instant k,  i.e., \(Z_{k+1}^{N_{c}-1}.\) Using the Bayes’ rule and the fact that the channel is assumed memoryless, \(P(s^{\prime },s,\mathbf {Z})\) can be expressed as

$$\begin{aligned} P(s^{\prime },s,\mathbf {Z})= & {} P(s^{\prime },Z_{0}^{k-1})\cdot P(\left. (s,Z_{k})\right| s^{\prime })\cdot P(\left. Z_{k+1}^{N_{c}-1}\right| s), \nonumber \\= & {} \alpha _{k-1}(s^{\prime })\cdot \gamma _{k}(s^{\prime },s)\cdot \beta _{k}(s), \end{aligned}$$
(17)

where \(\alpha _{k}(s^{\prime }),\) \(\beta _{k}(s)\) and \(\gamma _{k}(s^{\prime },s)\) are the probability of the state given the received sequence (forward state metric), the probability of the future received state values given the state of the encoder (backward state metric), and the joint probability of the output bit and state given the previous state (branch metric), respectively.

Substituting (17) in (16), we have

$$\begin{aligned} L(\left. C_{k}\right| \mathbf {Z})=\ln \left\{ \frac{\sum \limits _{(s^{ \prime },s)\Rightarrow C_{k}=+1}\alpha _{k-1}(s^{\prime })\cdot \gamma _{k}(s^{\prime },s)\cdot \beta _{k}(s)}{\sum \limits _{(s^{\prime },s)\Rightarrow C_{k}=-1}\alpha _{k-1}(s^{\prime })\cdot \gamma _{k}(s^{\prime },s)\cdot \beta _{k}(s)}\right\} . \end{aligned}$$
(18)

The forward state metric \(\alpha _{k}(s)\) and the backward state metric \( \beta _{k-1}(s^{\prime })\) are calculated recursively from \(\gamma _{k}(s^{\prime },s)\) as follow

$$\begin{aligned}&\alpha _{k}(s) =\sum \limits _{s^{\prime }}\gamma _{k}(s^{\prime },s)\cdot \alpha _{k-1}(s^{\prime }), \end{aligned}$$
(19)
$$\begin{aligned}&\beta _{k-1}(s^{\prime }) =\sum \limits _{s}\beta _{k}(s)\cdot \gamma _{k}(s^{\prime },s), \end{aligned}$$
(20)

and the branch metric \(\gamma _{k}(s^{\prime },s)\) can be computed from

$$\begin{aligned}&\gamma _{k}(s^{\prime },s)\nonumber \\&\quad =\frac{1}{\sqrt{2\pi }\sigma }\exp \left[ \frac{ -(Z_{k}-(\hat{H}_{k}^{*}\hat{H}_{k})C_{k}(s^{\prime },s))^{2}}{2\sigma ^{2}}\right] \cdot P(\left. s\right| s^{\prime }), \nonumber \\ \end{aligned}$$
(21)

where \(\hat{H}_{k}^{*}\) is the estimated value of the frequency-domain channel vector and \(*\) denotes the complex transpose. \(\sigma ^{2}\) is the Gaussian noise variance.

Using formulas (19)–(21), and assuming some a priori probabilities for inputs \(-1\) and 1 at all stages, the first decoder calculates LLR for all stages and decodes the received codebits. Then, it passes the LLR values excluding the intrinsic information to the second decoder, which uses these LLR values to calculate a priori probabilities at all stages for both inputs. The second decoder repeats the same process incorporated by the first decoder. We note that \(\alpha _{k}(s),\) \(\beta _{k-1}(s^{\prime })\) and \(\gamma _{k}(s^{\prime },s)\) are updated until the decoded bits become the same.

5 Simulation results

In this section, computer simulations are investigated to check out the performance of the proposed noise reduction method for PLC systems. Performance evaluations are carried out for the adaptive impulsive noise compensation (AINC) algorithm without turbo coding, and for the TC-AINC algorithm. The scenario for both AINC and TC-AINC simulation studies is as follows. A BPSK and a 16-QAM OFDM-based modulation formats are employed with \(N_{c}=2048\) tones and 171 pilot tones equally spaced in a frequency-domain, i.e., each 12th tone is used as pilot while the null tones were placed randomly. The PLC system has a 19.5 MHz bandwidth with a total period \(T_{s}=136~ {\mu }s\) of which \(8~{\mu }s\) constitute the CP \((L=32).\) Constant \(\eta \) of the adaptive impulsive noise cancellation algorithm is fixed to \(\eta =1.35.\) The modulated signal is affected by independent exponentially decaying 15-paths Rayleigh fading and impulsive noise generated according to Middleton class A model and Poisson–Gaussian model, with the occurrence probability of impulsive noise \(p=1~\%,\) \(p=5~\%,\) \(p=10~\%\) and \(p=20~\%\) of transmitting packets. The parameters used in the simulation of the OFDM system are listed in Table 2.

Table 2 Parameters of the simulated OFDM system
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5.1 Performance of the adaptive impulsive noise compensator

In the uncoded system, the performances of the proposed AINC algorithm are checked, in term of BER and MSE. Furthermore, in order to measure the capability of the proposed AINC algorithm for PLC channel infected by impulsive noise, we compare its performance with those obtained from other similar algorithms, including the impulsive noise compensation (INC) algorithm proposed by Zhidkov [21], and the composite comparison value (CCV) based filter proposed by Jia and Meng [22]. The BER performance of the algorithms with BPSK and 16-QAM modulated OFDM through the PLC channel in different impulsive noise scenarios are plotted in Figs.  7 and 8. The simulation results show that there is an inverse fitting between the BER and Eb/No, BER decreases when the Eb/No increases for all the algorithms, with the BPSK modulation having the fattest decrease. Furthermore, the BER curves show that the proposed method significantly outperforms the other algorithms. For example, the proposed compensator achieves 10–17 dB SNR gain over conventional OFDM receivers for Eb/No=0 dB. It outperforms INC and CCV-based filter algorithms by approximately 5–11.5 dB and 6–15 dB respectively. Moreover, these improvements become more important for moderate and high SNR regimes. It should be further noted that the improvements are less important for low impulsive bursts probabilities \(p=1~\%\) and become more important for increased values of p.

Fig. 7
figure 7

BER performance comparison of different methods in the case of BPSK modulation for the impulse burst probabilities: a \(p=1~\%\), b \(p=5~\%\), c \(p=10~\%\) and d \(p=20~\%\)

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Fig. 8
figure 8

BER performance comparison of different methods in the case of 16-QAM modulation for the impulse burst probabilities: a \(p=1~\%\), b \(p=5~\%\), c \(p=10~\%\) and d \(p=20~\%\)

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To further quantify the performance of our proposed algorithm, we simulate the noise reduction performance of the proposed method by measuring the average MSE between the received and transmitted signals. A comparison with the two other algorithms described previously is then conducted. The results are shown in Fig.  9 and 10 for BPSK and 16-QAM, respectively. Clearly, the MSE curves for AINC algorithm are generally much lower than the MSE curves for the other two methods under all the impulsive noise scenarios used in the simulation. This implies that the AINC-based denoising signal has much less error compared to the denoised signals with both INC and CCV-based algorithms.

Fig. 9
figure 9

MSE performance comparison of different methods in the case of BPSK modulation for the impulse burst probabilities: a \(p=1~\%\), b \(p=5~\%\), c \(p=10~\%\) and d \(p=20~\%\)

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Fig. 10
figure 10

MSE performance comparison of different methods in the case of 16-QAM modulation for the impulse burst probabilities: a \(p=1~\%\), b \(p=5~\%\), c \(p=10~\%\) and d \(p=20~\%\)

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5.2 Performance of the TC-AINC receiver

In the coded system, we investigate the BER performance of the proposed turbo coded impulsive noise compensator receiver in the PLC system. The turbo code has codebite length \({\approx }\)60,000 and rate 1/2, with coder/decoder implementations as described in Sects. 3.2 and 4.2. We performed SISO decoding as the final step with at most five turbo iterations. Figures 11 and 12 plot the BER for the conventional OFDM, TC, AINC and TC-AINC receivers in the case of BPSK and 16-QAM, respectively. It is clearly seen that the proposed TC-AINC receiver results in further gains. For example, by introducing TC and under Eb/No-QAM= 0, we obtain more than 16 dB over conventional OFDM receivers for all the impulsive burst probabilities. Furthermore, an extra 4–6 dB gain can be achieved over the proposed AINC-OFDM receiver. The improvement becomes more important in moderate and high \(E_{b}/N_{0}\) regimes.

Fig. 11
figure 11

BER performance comparison of different receivers in the case of BPSK for the impulse burst probabilities: a \(p=1~\%\), b \(p=5~\%\), c \(p=10~\%\) and d \(p=20~\%\)

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Fig. 12
figure 12

BER performance comparison of different receivers in the case of 16-QAM for the impulse burst probabilities: a \(p=1~\%\), b \(p=5~\%\), c \(p=10~\%\) and d \(p=20~\%\)

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From another side, we have compared our algorithms, AINC and TC-AINC to other state of the art approaches which use sparsity of the noise in time-domain. Two recent works are selected, the first one is a non-parametric scheme based on SBL [32], the second scheme is a factor-graph-based approach [33]. A comparison of the SNR improvement introduced by each method against conventional OFDM receivers is conducted and results are reported in Table 3. We can see that the proposed AINC approach gives comparable performance in term of SNR improvement, and becomes more significant when TC are introduced, which represents the case of TC-AINC.

Table 3 SNR improvement comparison of proposed algorithm and state of the art approaches
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Table 4 illustrates the complexity performance of different approaches when a data stream of 480,000 bits is transmitted over the PLC channel. The complexity is measured by checking the execution time, the number of iterations, and the number of multiplications and divisions. The different results are obtained using Matlab 8.1 on a 3.3 Ghz intel Core i3 computer with 12 GB of RAM memory. It is shown from Table 4 that AINC can give less time complexity in comparison to the other algorithms since it contains less complex operations. Furthermore, the computational complexity of TC-AINC is acceptable when it is compared to CCV or non-parametric approach since it has a better BER and SNR performance.

Table 4 Computation complexity of AINC and TC-AINC against other algorithms
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6 Conclusion

In this paper, we proposed a TC-AINC receiver for an OFDM broadband PLC system. The proposed adaptive noise mitigation algorithm performs an iterative estimation and suppression of the impulsive noise generated over the PLC channel. Different environmental scenarios were simulated consisting of a variation of impulsive noise level by using different impulse bursts probabilities, different SNR values and two modulation schemes BPSK and QAM. Extensive numerical simulations show that the proposed turbo receiver provides drastic performance gains over recent and well-know existed impulsive noise suppression methods. Through the simulation results, it can be proved that by applying the proposed method, the BER and the average MSE performances of the OFDM-based PLC channel are highly improved.