Cyclostationary impulsive noise mitigation algorithm for narrowband powerline communications

ARTICLE IN PRESS

JID: FI

[m1+;December 13, 2019;19:54]

Available online at www.sciencedirect.com

Journal of the Franklin Institute xxx (xxxx) xxx www.elsevier.com/locate/jfranklin

Cyclostationary impulsive noise mitigation algorithm for narrowband powerline communications Ying-Ren Chien a, Sendren Sheng-Dong Xu b,c,∗, Shao-Hang Lu b a Department

of Electrical Engineering, National Ilan University, Yilan 26047, Taiwan Institute of Automation and Control, National Taiwan University of Science and Technology, Taipei 10607, Taiwan c Center for Cyber-Physical System Innovation, National Taiwan University of Science and Technology, Taipei 10607, Taiwan

b Graduate

Received 5 February 2019; received in revised form 11 October 2019; accepted 24 October 2019 Available online xxx

Abstract The dominant noise in narrowband powerline communication (NB-PLC) systems is cyclostationary impulsive noise (IN), which severely degrades communication performance. Conventionally, a frequency shift (FRESH) filtering approach exploits the cyclostationary effect to extract the IN component in the received signal. In this paper, we propose a three-stage adaptive FRESH-based cyclostationary noise estimator to make a reliable IN estimation. Inspired by the active noise cancellation method, we firstly employ a FRESH filter to generate a reference IN signal. In addition, considering the cyclostationarity of the IN, we exploit the time-averaged objective function to adjust the coefficients of the FRESH filter. Then, this reference IN signal is fed into an adaptive estimator to further refine the estimation of IN. Lastly, the refined IN estimations are rectified by a hard thresholding device such that significant and reliable IN estimations can be leached out. By subtracting the reliable IN estimations from the received signal, we mitigate the impact of IN on the data transmission for NB-PLC systems. Computer simulation is conducted based on the IEEE 1901.2 specification and we use real measured IN data to verify the effectiveness of our proposed method. The results have shown that the improvement in terms of the normalized mean squared error can be up to 7.57 dB. © 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.

∗ Corresponding author at: Graduate Institute of Automation and Control, National Taiwan University of Science and Technology, Taipei 10607, Taiwan. E-mail address: [email protected] (S.S.-D. Xu).

https://doi.org/10.1016/j.jfranklin.2019.10.026 0016-0032/© 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.

Please cite this article as: Y.-R. Chien, S.S.-D. Xu and S.-H. Lu, Cyclostationary impulsive noise mitigation algorithm for narrowband powerline communications, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.10.026

ARTICLE IN PRESS

JID: FI

2

[m1+;December 13, 2019;19:54]

Y.-R. Chien, S.S.-D. Xu and S.-H. Lu / Journal of the Franklin Institute xxx (xxxx) xxx

1. Introduction Narrowband powerline communication (NB-PLC) technology has been applied to many engineering fields [1], such as smart grid systems [2,3], sensor network deployment [4], intra-vehicular communication [5–7], and smart street lighting systems [8]. However, in the frequency band used by NB-PLC systems, the multipath effect caused by the powerline channel [9] and the dominant additive cyclostationary impulsive interferences significantly degrade the performance of NB-PLC receivers. One of the most common approaches to alleviate the impact of impulsive noise (IN) is the adaptive filtering method, which is widely applied to other varied fields, such as control [10–13], communication [14,15], image processing [16], and estimation [17,18]. In [10], the authors have proposed a time-varying output feedback control method that is able to deal with the unmeasurable or unknown system states. Besides the adaptive filtering method, the fuzzy theory has been exploited for developing adaptive techniques for nonlinear or non-stationary systems as well [19–23]. In [19], a systematic approach to construct C1 and C∞ controllers for stochastic nonlinear systems [24] was presented. Furthermore, [20] has reported the method to construct an adaptive output feedback controller to deal with the stochastic nonholonomic systems. For some challenging noisy environments, such as intra-vehicular applications, some previous reports resorted to the robust and optimal control [22,25,26] or fuzzy control [23]. In [22], the authors have proposed a reliable fuzzy tracking control method with aperiodic measured information. In [23], the authors have proposed an adaptive fuzzy sliding mode controller to adapt the input saturation and to compensate for the matched uncertainty. Moreover, fuzzy techniques have been incorporated into the research of powerline communications [27]. Note that the cyclostationary signals make the conventional least mean square (LMS) and normalized LMS (NLMS) algorithms fail to reach low steady-state errors. The impact of the cyclostationarity on the adaptive filtering algorithms can be alleviated by resorting to the variable step-size method. In [28], a combined step-size approach has been proposed as a variable step-size algorithm to deal with the cyclostationary inputs case. However, this algorithm neglects the cyclostationarity when deriving the weight updating recursions. Recently, a frequency-shift (FRESH) filtering has been applied to orthogonal frequency division multiplexing (OFDM) signal recovery for NB-PLC systems [29]. By exploiting the cyclostationary properties of both the IN and transmitted OFDM signal, two sets of FRESH filters were used to extract the noise and signal, respectively. In this case, the FRESH filters are used to realize a linear periodically timevarying (LPTV) system. In [30], the authors proposed a time-averaged least mean squared (TM-LMS) algorithm to train the coefficients of the FRESH filter and applied it to solving the channel estimation problem in NB-PLC systems. However, the authors did not discuss the IN estimation and mitigation issues. The cyclostationary property has been modeled as a set of switching autoregressive (AR) filters. In [31], the authors have applied a Bayesian nonparametric inference framework to estimating the switching AR models. However, this approach requires a predistortion filter at the transmitter. Such requirements do not comply with some existing specifications, such as IEEE 1901.2. The sparse structure of the time-domain noise was exploited to design the nonparametric mitigation algorithm for the periodic IN [32]. However, this approach requires to solve a compressive sensing problem and the cyclostationarity of the IN is neglected. The multiple cyclic regression method was used to reconstruct periodic IN [33]. Unfortunately, a computationally intensive recursive least-squared (RLS) algorithm is needed to train a two-level recursive filter. Recently, spatial correlation across multiple receive phases was exploited to further improve the performance of the FRESH filters [34]. Note that Please cite this article as: Y.-R. Chien, S.S.-D. Xu and S.-H. Lu, Cyclostationary impulsive noise mitigation algorithm for narrowband powerline communications, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.10.026

ARTICLE IN PRESS

JID: FI

[m1+;December 13, 2019;19:54]

Y.-R. Chien, S.S.-D. Xu and S.-H. Lu / Journal of the Franklin Institute xxx (xxxx) xxx

3

Elgenedy et al. [34] considered the cyclostationarity of the IN and derived a optimal solution for minimizing the time-averaged MSE. However, this solution necessitates the time-averaged statistics about the input auto-correlation matrix, cross-correlation vector, and the computation of matrix inversion. In this paper, we propose an adaptive FRESH-based cyclostationary noise mitigation algorithm, which consists of three stages. First, we apply a FRESH filter to extracting the denoised cyclostationary IN from the received signal by leveraging the cyclostationary property of the IN. Then, this reference impulsive signal is fed into an adaptive estimator to produce a refined IN estimation. Finally, a hard-thresholding device is used to null the samples in the refined IN whose magnitude are less than a pre-defined threshold. The rectified signals are the estimated cyclostationary IN. The contributions of this paper are summarized as follows. First, instead of using mathematical models to generate the IN samples, we adopt the real measured IN for NB-PLC systems to validate the effectiveness of IN mitigation algorithms. Second, the conventional FRESH filtering method [35] suffers from poor IN estimation. We leverage the idea of active noise cancellation and treat the output of conventional FRESH filter as the reference noise signal such that we can further purify the quality of IN estimation. Simulation results have shown that our proposed method improves the normalized mean squared error (NMSE) of the IN estimation by around 2.72 to 7.57 dB for different values of Eb /N0 . From the bit error rate (BER) performance perspective, if we fixed the BER at 10−2 , the required Eb /N0 for the conventional blanking method [36], FRESH filtering method, and our proposed method are approximately 19.1 dB, 17.2 dB, and 15 dB, respectively. This implies that our method offers the remarkable Eb /N0 gain of 2.2 dB and 4.1 dB over the FRESH approach and blanking method, respectively. Third, we consider the cyclostationarity of the IN and adopt a time-averaged least-mean-square (TA-LMS) algorithm to train the FRESH filter. To the best of the authors’ knowledge, no work has reported the results of IN mitigation ability with respect to the FRESH filter, which is trained by the TA-LMS algorithm. The remainder of this paper is organized as follows. Section 2 describes the system models, including the powerline channel and noise models. Section 3 details the proposed adaptive FRESH-based cyclostationary noise estimator. Section 4 presents our simulation results, in which we refer to the specifications in IEEE 1901.2 [37] for the establishment of NB-PLC systems. Conclusions are drawn in Section 5. 2. System models Fig. 1 illustrates a block diagram of the system model. The information bits feeds the forward error correction (FEC) encoder and then enters a mapper such that the transmitted data symbols S[k] are uniformly selected from differential binary phase shift keying (DBPSK) modulation. It is assumed that the data symbols are independent and identically distributed with zero-mean and unit-variance. After the virtual carrier addition and serial-to-parallel (S/P) conversion, a block length of NFFT symbols are fed through the OFDM modulation block using the inverse fast Fourier transform (IFFT). Output of the IFFT block can be expressed as: s[n] = √

N FFT −1 1 S[k]e j2πkn/NFFT , NFFT k=0

(1)

where n = 0, 1, . . . , (NFFT − 1) and NFFT is the number of subcarriers. Note that the data subcarriers are arranged in Hermitian symmetry such that the outputs of the IFFT are real Please cite this article as: Y.-R. Chien, S.S.-D. Xu and S.-H. Lu, Cyclostationary impulsive noise mitigation algorithm for narrowband powerline communications, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.10.026

ARTICLE IN PRESS

JID: FI

4

[m1+;December 13, 2019;19:54]

Y.-R. Chien, S.S.-D. Xu and S.-H. Lu / Journal of the Franklin Institute xxx (xxxx) xxx FEC Encoder

S / P

Virtu t al carrier addition

Map a per

IFFT

P / S

CP insertion

Powerline Channel Cyscolstationary r Impulsive Noise

AWGN

FEC Decoder

P / S

Demap a per

FEQ

FFT

S / P

CP removal

Fig. 1. Block diagram of the system model.

numbers. After the parallel-to-serial (P/S) block, a cyclic prefix (CP) to mitigate the multipath effect caused by the powerline channel is preappended. The transmitted signals then pass through a powerline channel where the received signal r[n] is corrupted by multipath effects h[n], the background white Gaussian noise v[n] and the dominant cyclostationary IN η[n]. At the receiver side, an impulsive cyclostationary noise estimation algorithm is applied to computing estimations of the dominant noise η[ ˆ n]. The estimated noise is then subtracted from the received signal to obtain a denoised signal. After removing the CP from the denoised signal, a fast Fourier transform (FFT) block is used to transform its input signal into the frequency-domain. Then, a frequency-domain equalization (FEQ) is applied to equalizing the multipath channel. Then, the equalized signals undergo P/S conversion, demapping, and FEC decoding to generate estimations of the stream of transmitted bits.

2.1. Powerline channel model We adopted the multipath model proposed by Zimmerman and Dostert [38] as the channel model for a typical powerline channel. The frequency response of the multipath model is expressed as N

H(f ) =

path 

αi · e{−(a0 +a1 f

ρ

)di }

· e{− j2π f τi } ,

(2)

i=1

where Npath denotes the number of paths, a0 and a1 are attenuation parameters, ρ is the exponent of the attenuation factor, α i is the weighting factor of the ith path, di is the length of the ith path, and τ i is the delay associated with the ith path. Please cite this article as: Y.-R. Chien, S.S.-D. Xu and S.-H. Lu, Cyclostationary impulsive noise mitigation algorithm for narrowband powerline communications, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.10.026

ARTICLE IN PRESS

JID: FI

[m1+;December 13, 2019;19:54]

Y.-R. Chien, S.S.-D. Xu and S.-H. Lu / Journal of the Franklin Institute xxx (xxxx) xxx

5

2.2. Dominant cyclostationary IN model Using field measurement on powerline channels, it has been identified that the dominant noise for NB-PLC systems can be modeled as a cyclostationary process. Some existing research results have modeled the noise as additive cyclostationary Gaussian noise (ACGN) with a period that is equal to a half the alternating current (AC) period [3,39–41]. The ACGN η[n] is a cyclostationary process, whose amplitude at the same phase of the AC voltage is Gaussian distributed with zero mean. In addition, the variance of this process ση2 [n] can be expressed as a cyclical function of period TAC /2, in which TAC is the period of the AC source, i.e., η[n] ∼ N (0, ση2 [n]),

(3) ση2 [n]

ση2 [n

where n is the time index, = + kNη ], and k ∈ Z. The period Nη = TAC /2 · fs , in which fs is the sampling rate of the cyclostationary noise. The periodicity leads to a periodic instantaneous autocorrelation function Rη [n, m] such that Rη [n, m] = Rη [n + kNη , m]. Note that the instantaneous autocorrelation function Rη [n, m] of a random process η[n] can be defined as   Rη [n, m] = E η[n + m/2]η∗ [n − m/2] , (4) where E{·} represents the expectation operation and the superscript ∗ denotes conjugate operation. Due to this periodicity, we can express Rη [n, m] in terms of its Fourier series (FS) expansion as follows:  Rη [n, m] = Rη [m; αk ]e j2παk n/ fs , (5) αk

where α k is referred to as the cyclic frequency associated with the non-zero FS coefficients [42]. Thus, the cyclic auto-correlation function Rη [m; α k ] can be expressed as follows: L/2 1  Rη [n, m]e− j2παk n/ fs . L→∞ L n=−L/2

Rη [m; αk ] = lim

(6)

Note that the Fourier transform of Eq. (6) is the so-called cyclic spectrum density (CSD) and can be expressed as follows: Sη ( f ; αk ) =

∞ 1  Rη [m; αk ]e− j2π f m/ fs . fs m=−∞

(7)

By taking the two-dimension Fourier transform of Eq. (4), the spectral correlation density (SCD) function can be expressed as: Sη (α, f ) =

∞ ∞ 1   Rη [n, m]e− j2παn/ fs e− j2π f m/ fs . fs2 n=−∞ m=−∞

(8)

Note that the SCD function represents the power distribution of the signal η[n] over both spectral frequency f and cyclic frequency α. To measure the correlations in spectral components separated by cyclic frequencies α, the cyclic spectral coherence (CSC) function can be defined as follows: E{dη( f + α/2)dη( f − α/2)} γη ( f ; α) =   (9)   , E |dη( f + α/2)|2 E |dη( f − α/2)|2 Please cite this article as: Y.-R. Chien, S.S.-D. Xu and S.-H. Lu, Cyclostationary impulsive noise mitigation algorithm for narrowband powerline communications, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.10.026

ARTICLE IN PRESS

JID: FI

6

[m1+;December 13, 2019;19:54]

Y.-R. Chien, S.S.-D. Xu and S.-H. Lu / Journal of the Franklin Institute xxx (xxxx) xxx

e1[n] r[ n ]

FRESH filter

d1[n]

[ n]

Adaptive Estimator

[ n]

Hard Thresholding

ˆ[n]

x[n]

e2 [n]

Proposed Adaptive FRESH-based Cyclostationary Noise Estimator Fig. 2. Block diagram of the proposed adaptive FRESH-based estimator.

where dη( f + α/2) and dη( f + α/2) denote the spectral increments of signals η[n] at frequencies at ( f + α/2) and ( f − α/2), respectively. Higher value of γ η (f; α) implies the higher strength of cyclostationarity at cyclic frequency α. To evaluate the impact of the ACGN on the BER performance of a NB-PLC system, we further adopt a parameter called Gaussian-to-impulsive-noise ratio (GINR) as follows: := σv2 /ση2 ,

(10)

where σv2 denotes the variance of the zero-mean additive white Gaussian noise (AWGN) v[n]. The value of GINR indicates the strength of the IN. A smaller value of means a higher magnitude of the IN. A typical value of the GINR is around 20 dB for NB-PLC applications [43]. When evaluating the BER performance under different values of energy per bit to noise power spectral density ratio (Eb /N0 ), we keep the value of GINR fixed and normalize the averaged power of η[n] to ση2 = σv2 /GINR. Thus, in the high signal-to-noise (SNR) region, the magnitude of the induced ACGN is relatively lower than that in the low SNR region.

3. Proposed cyclostationary noise mitigation algorithm Once the reference signal is available to the conventional active noise canceler (ANC) [44], the adaptive algorithm is able to exploit the correlation between the primary signal and the reference signal so that the noise components within the primary signal can be extracted. Fig. 2 illustrates the block diagram of the proposed adaptive FRESH-based estimator for cyclostationary noise. Our approach consists of three stages: FRESH filtering, cyclostationary noise leaching, and hard-limiting. First, the received signal r[n] passes through a FRESH filter. The main task of the FRESH filter is to generate a reference cyclostationary noise η[ ˜ n], which is highly correlated with η[n] in r[n]. Next, we supply the reference signal η[ ˜ n] to an adaptive estimator such that the cyclostationary noise η[n] can be extracted from the received signal r[n]. However, the adaptive estimator gives a worse estimation for components with small magnitude in real cyclostationary noise η[n]. A hard thresholding device is utilized to keep the components with large magnitude in η[n], i.e., to null the components whose magnitude is less than a predefined threshold in η[n]. Please cite this article as: Y.-R. Chien, S.S.-D. Xu and S.-H. Lu, Cyclostationary impulsive noise mitigation algorithm for narrowband powerline communications, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.10.026

ARTICLE IN PRESS

JID: FI

[m1+;December 13, 2019;19:54]

Y.-R. Chien, S.S.-D. Xu and S.-H. Lu / Journal of the Franklin Institute xxx (xxxx) xxx

e j2

0n

FIR h1,0 [n] r[ n ]

e j2

e

j2

7

d1[n]

1n

⁞

FIR h1,1[n] ( K 1) n

+ ⁞

− [ n]

FIR h1,K 1[n] e1[n] Fig. 3. Block diagram of the adaptive FRESH filter with (2K + 1) branches.

3.1. FRESH filtering It is known that cyclostationary IN can be modeled by a linear FRESH filter [35]. The details of an adaptive linear FRESH filter are depicted in Fig. 3. The received signal r[n] passes through a set of frequency shifters e j2παk n with k = 0, 1 . . . , K − 1. Note that α k is the k-th cyclic frequency. We apply the TA-LMS algorithm to adjusting the coefficients of each finite impulse response (FIR) filter in the FRESH filter [30]. Note that the TA-LMS is derived by minimizing the time-averaged measure-square error (TA-MSE) for jointly cyclostationary signals. The object function is different from the conventional MSE. Conventionally, the adaptive algorithms, such as the block LMS (B-LMS) algorithm [45], use the steepest descent algorithm to minimize the MSE and derive their updating recursions. Even though the weight updating equation of the B-LMS is similar to the TA-LMS, they update the weight coefficients with different periods. The B-LMS updates weight coefficients every block but the TA-LMS updates the weight coefficients every sample. Thus, the convergence rate of TA-LMS is superior to the B-LMS. In addition, the cyclostationarity of the IN motivates us to apply the TA-LMS to training the FRESH filters. Assume that we form an M × 1 vector h1 [n] by cascading all the K FIR filters as follows:  T h1 [n] = h1T,0 , h1T,1 , . . . , h1T,K−1 , (11) where T denotes the transpose operation and h1,k is an L1 × 1 vector, which corresponds to the coefficients of the kth FIR filter; M = K L1 is the total number of coefficients of the FRESH filter. Similarly, we stack the input regress vector associated with the kth FIR filter, which is denoted as x1,k [n], into an M × 1 vector x1 [n]. The vth element of the input regress vector associated with the uth FIR filter is denoted as (x1 [n] )(u·L1 +v) = r[n − v]e j2παu (n−v) ,

(12)

where u ∈ {0, 1, . . . , K − 1} and v ∈ {0, 1, . . . , L1 − 1}. Linear estimation of the signal of interest (SOI) d1 [n] with period Nη is calculated by η[ ˜ n] = h1T [n]x[n].

(13)

Please cite this article as: Y.-R. Chien, S.S.-D. Xu and S.-H. Lu, Cyclostationary impulsive noise mitigation algorithm for narrowband powerline communications, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.10.026

ARTICLE IN PRESS

JID: FI

8

[m1+;December 13, 2019;19:54]

Y.-R. Chien, S.S.-D. Xu and S.-H. Lu / Journal of the Franklin Institute xxx (xxxx) xxx

The TA-MSE associated with this linear estimator over Nη samples is defined as    J (h1 ) : = E |d1 [n] − η[ ˜ n]|2 Nη = cd1 − 2cTx1 d1 h1 + h1T Cx1 h1 , (14)    where · Nη represents the time-averaging operation over Nη samples and d1 [n] is the desired signal for training the FRESH filter; d1 [n] is the reference noise data and it comprises the IN and background   noise. This can be obtained by exploiting pilot  symbols [29]; we denote cd1 := E |d1 [n]|2 Nη , cx1 d1 := E{x1 [n]d1 [n]} Nη , and Cx1 := E x1 [n]x1T [n] .   By taking the gradient of J(h1 ) with respect to h1 , we can derive the iterative updating equation for h1 [n] as follows:

h1 [n + 1] = h1 [n] + μ1 cx1 d1 − Cx1 h1 [n] , (15)   where μ1 is a fixed step-size. To obtain a practical number of iterations for Eq. (15), we use the past Nη data to approximate the time-averaged correlations Cx1 and cx1 d1 as follows:   Nη −1  ˆ x1 [n] := 1 Cx1 ≈ C x1 [n − k]x1T [n − k] = X1 [n]X1T [n], N η k=0  

(16)

where X1 [n] := √1 [x1 [n], x1 [n − 1], . . . , x1 [n − N0 + 1]] is an M × Nη matrix and Nη

c x1 d1 

Nη −1 1  ≈ cˆ x1 d1 [n] := x1 [n − k]d1 [n − k] = X1 [n]d1T [n], Nη k=0 

where d1 [n] := √1

Nη

(17)

 T d1 [n], d1 [n − 1], . . . , d1 [n − Nη + 1] is a Nη × 1 vector.

Thus, we can rewrite Eq. (15) as follows: h1 [n + 1] = h1 [n] + μ1 X1 [n]e1 [n],

(18)

where e1 [n] = d1 [n] − X1T [n]h1 [n] is an Nη × 1 error vector. 3.2. Adaptive estimator The adaptive estimator is implemented using an FIR filter with L2 taps. Its coefficients are denoted as h2 [n] = [h2,0 , h2,1 , . . . , h2,L2 −1 ]T [n]. We adopt the NLMS algorithm to adjust its coefficients as follow: h2 [n + 1] = h2 [n] + μ2

x2 [n]e2 [n] , T x2 [n]x2 [n] + δ2

(19)

where μ2 is a fixed step-size, x2 [n] = [η[ ˜ n], η[ ˜ n − 1], . . . , η[ ˜ n − L2 + 1]]T is the input regressor vector with length L2 , e2 [n] = r[n] − η[n] is the error signal, η[n] = h2T [n]x2 [n] is the output of the adaptive estimator, and δ 2 is a small positive regularization parameter. Please cite this article as: Y.-R. Chien, S.S.-D. Xu and S.-H. Lu, Cyclostationary impulsive noise mitigation algorithm for narrowband powerline communications, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.10.026

ARTICLE IN PRESS

JID: FI

[m1+;December 13, 2019;19:54]

Y.-R. Chien, S.S.-D. Xu and S.-H. Lu / Journal of the Franklin Institute xxx (xxxx) xxx

9

3.3. Hard-limiting The operation of the thresholding device can be expressed as follows:

η[n], if |η[n]| ≥ T η[ ˆ n] = , 0, otherwise

(20)

where T is the threshold. 3.4. Computational complexity The major computational complexity is incurred by the weight-updating Eqs. (15) (TALMS) and (19) (NLMS). Note that the computational cost is high if we directly implemented the TA-LMS by Eq. (15). To further reduce the computational complexity for training the FRESH filter, we can rewrite Eq. (16) into recursive form as follows: Nη −1 1  ˆ Cx1 [n] = x1 [n − k]x1T [n − k] Nη k=0  ⎧ ⎫ Nη −1 ⎬  1 ⎨ = x1 [n]x1T [n] + x1 [n − k]x1T [n − k] ⎭ Nη ⎩ k=1

  ˆ x1 [n − 1] + 1 x1 [n]x1T [n] − x1 [n − Nη ]x1T [n − Nη ] . =C Nη 

(21)

Similarly, Eq. (17) can be implemented as the following recursive form:  1  x1 [n]d1 [n] − x1 [n − Nη ]d1 [n − Nη ] . cˆ x1 d1 [n] = cˆ x1 d1 [n − 1] + Nη  

(22)

Thus, the TA-LMS requires (4M 2 + 4M ) multipliers and (3M 2 + 3M ) adders. As for the adaptive estimator, the NLMS algorithm needs 4L2 multipliers and (4L2 − 2) adders where L2 is the length of the estimator. 4. Simulation results To evaluate the effectiveness of our proposed method, we refer to the IEEE 1901.2 specification and build a simulation system as shown in Fig. 1. The FEC encoder consists of two channel encoders. The outer channel encoder is a Reed–Solomon (255, 239) encoder and the inner channel encoder is a 1/2 rate convolutional encoder with constraint length of 7 and with generator polynomials 171octal and 133octal . The modulation scheme is differential binary phase shift keying (DBPSK). The length of the FFT is 256, in which 64 points are for virtual subcarriers and 192 points are for data carriers. The length of the CP is 64-point. The sampling rate fs is 4 × 105 samples/sec and the IN repeats every 8 ms, i.e., Nη = 3200. The channel model used in the simulation is shown in Fig. 4. In addition, we assume the coefficients of the FEQ are ideal. The noise data used is the real measured noise [46]. The corresponding trace file with the aggregated additive noise at Eb /N0 = 0 dB is illustrated in Fig. 5. The results of the cyclic spectral analysis are shown in Fig. 6. Please cite this article as: Y.-R. Chien, S.S.-D. Xu and S.-H. Lu, Cyclostationary impulsive noise mitigation algorithm for narrowband powerline communications, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.10.026

ARTICLE IN PRESS

JID: FI

10

[m1+;December 13, 2019;19:54]

Y.-R. Chien, S.S.-D. Xu and S.-H. Lu / Journal of the Franklin Institute xxx (xxxx) xxx

a 0.15 0.1 0.05

Amplitude

0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35

1

1.5

2

2.5

3

3.5

4

4.5

5

6

5.5

6.5 10-5

Time (sec)

b 0 -2 -4

Magnitude (dB)

-6 -8 -10 -12 -14 -16 -18 -20

0.5

1

1.5

2

2.5

3

3.5

Frequency (Hz)

4

4.5

5

5.5

6 105

Fig. 4. Powerline channel model. (a) Time domain impulse response and (b) magnitude response. Please cite this article as: Y.-R. Chien, S.S.-D. Xu and S.-H. Lu, Cyclostationary impulsive noise mitigation algorithm for narrowband powerline communications, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.10.026

ARTICLE IN PRESS

JID: FI

[m1+;December 13, 2019;19:54]

Y.-R. Chien, S.S.-D. Xu and S.-H. Lu / Journal of the Franklin Institute xxx (xxxx) xxx -40

-60

150

-80

100 -100

50 -120

0

5

Amplitude

2

10

15

20

25

Power/frequency (dB/Hz)

Frequency (kHz)

200

11

30

Time (ms)

1 0 -1 -2

Fig. 5. The aggregated additive noise at Eb /N0 = 0 dB with = 0.01. (Top: time-frequency diagram; bottom: time domain trace.)

The number of branches of the adaptive FRESH filter is K = 19 and the length of each FIR filter is L1 = 50, which means the total number of taps of the FRESH filter is M = K × L1 = 950. The length of the adaptive estimator is L2 = 50. The performance metric is the NMSE, which is defined as follows:   E |η[n] − η[ ˆ n]|2   NMSE = . E |η[n]|2

(23)

Fig. 7 (a) and (b) shows the evolution of the NMSE for the low Eb /N0 and high Eb /N0 values, respectively. Note that the step-size used for training the FRESH filter (μ1 ) is chosen as 0.06, and the step-size used for training the adaptive estimator (μ2 ) is chosen as 0.06 and 0.01 for the low Eb /N0 and high Eb /N0 cases, respectively. In the low Eb /N0 case, the FRESH filter is able to suppress the IN by 16 dB, while our proposed method can further improve it by an additional 6 dB. On the other hand, in the high Eb /N0 case, the improvement of our method in terms of NMSE over the conventional FRESH filter is reduced by only 2 dB. Due to in the high Eb /N0 case, the strength of the IN is much lower than that in the low Eb /N0 case. Thus, the noise estimation becomes worse in the high Eb /N0 case than that in the low Eb /N0 case. Note that the learning curves of the NMSE are converged in a few iterations for both low and high Eb /N0 cases. Comparison of the resulting NMSE is shown in Fig. 8(a), which confirms that our proposed method outperforms the conventional FRESH filter, especially when the value of Eb /N0 is low. Depending on the value of Eb /N0 , the improvement in terms of the NMSE is at least 2.72 dB and can be up to 7.57 dB. Please cite this article as: Y.-R. Chien, S.S.-D. Xu and S.-H. Lu, Cyclostationary impulsive noise mitigation algorithm for narrowband powerline communications, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.10.026

JID: FI

12

ARTICLE IN PRESS

[m1+;December 13, 2019;19:54]

Y.-R. Chien, S.S.-D. Xu and S.-H. Lu / Journal of the Franklin Institute xxx (xxxx) xxx

Fig. 6. Cyclic spectral analysis of the aggregated additive noise at Eb /N0 = 0 dB with = 0.01. (a) Cyclic spectral density and (b) cyclic spectral coherence.

Fig. 8 (b) shows a comparison in terms of BER performance with different IN mitigation methods. For conventional blanking technique [36], the optimal threshold is determined by an exhaustive searched for the different values of Eb /N0 . However, as the value of Eb /N0 is greater than or equal to 9 dB, the blanking method nulls not only the noise components but also parts of the signal. Therefore, the resulting performance is worse than the case that no noise mitigation method is applied. As expected, because of the proposed method achieves a Please cite this article as: Y.-R. Chien, S.S.-D. Xu and S.-H. Lu, Cyclostationary impulsive noise mitigation algorithm for narrowband powerline communications, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.10.026

ARTICLE IN PRESS

JID: FI

[m1+;December 13, 2019;19:54]

Y.-R. Chien, S.S.-D. Xu and S.-H. Lu / Journal of the Franklin Institute xxx (xxxx) xxx

13

a -8 FRESH Proposed

-10

NMSE (dB)

-12 -14 -16 -18 -20 -22 -24

10

20

30

40

50

60

70

80

90

100

Iteration

b 0

Proposed FRESH

-1

NMSE (dB)

-2 -3 -4 -5 -6 -7 -8

10

20

30

40

50

60

70

80

90

100

Iteration Fig. 7. Evolution of NMSE under two different Eb /N0 values. (a) Eb /N0 = 0 dB and (b) Eb /N0 = 21 dB. Please cite this article as: Y.-R. Chien, S.S.-D. Xu and S.-H. Lu, Cyclostationary impulsive noise mitigation algorithm for narrowband powerline communications, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.10.026

ARTICLE IN PRESS

JID: FI

14

[m1+;December 13, 2019;19:54]

Y.-R. Chien, S.S.-D. Xu and S.-H. Lu / Journal of the Franklin Institute xxx (xxxx) xxx

-2 -4 -6

NMSE (dB)

-8 -10 -12 -14 -16 -18

Proposed FRESH

-20 -22

0

3

6

9

12

15

18

21

Eb/N0 (dB)

(a) 10

0

No Mitigation Blanking FRESH Proposed

10-1

BER

10-2

10

-3

10-4

0

3

6

9

12

15

18

21

Eb/N0 (dB)

(b) Fig. 8. Performance comparison under different Eb /N0 values. (a) NMSE and (b) BER. Please cite this article as: Y.-R. Chien, S.S.-D. Xu and S.-H. Lu, Cyclostationary impulsive noise mitigation algorithm for narrowband powerline communications, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.10.026

ARTICLE IN PRESS

JID: FI

[m1+;December 13, 2019;19:54]

Y.-R. Chien, S.S.-D. Xu and S.-H. Lu / Journal of the Franklin Institute xxx (xxxx) xxx

15

lower values of NMSE than that could be obtained for the FRESH filter, our approach has a better BER performance. For example, with BER set at 10−2 , our method offers an Eb /N0 gain of approximately 2 dB over the FRESH approach. 5. Conclusions In this paper, we presented a FRESH-based adaptive estimator to combat the dominant IN in NB-PLC systems. The proposed FRESH-based adaptive estimator algorithm leverages the cyclostationary property of the IN to extract a reference signal from the received signal. The simulation results have shown that the adaptive estimator with a hard thresholding device can further improve the accuracy of the IN estimation. Comparing our proposed method with the conventional FRESH filter approach, the NMSE improvement of our method can be up to 7.57 dB. In this study, we have assumed that the cyclic frequency α k , which depends on the period of AC source from the power company, is a constant. However, the period of the AC source may not be a constant. For example, the regulation for the power company in Taiwan requires that the variation of the AC period should be within ± 4%. Unfortunately, the performance of FRESH filters can seriously degrade if there exist cyclic frequency errors. Our future works are to develop a variable step-size TA-LMS algorithm to compensate for the cyclic frequency errors and to study the feasibility of our method that is applied to a more complex noise disturbed environments such as Lévy noise [47] and semi-Markov switched stochastic process [48]. Acknowledgments This research was supported by the Ministry of Science and Technology (MOST), Taiwan, under the Grants MOST 108-2221-E-197-010, MOST 107-2221-E-011-145, and MOST 1082221-E-011-154, and in part by the “Center for Cyber-Physical System Innovation” from The Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan. References [1] C. Cano, A. Pittolo, D. Malone, L. Lampe, A.M. Tonello, A.G. Dabak, State of the art in power line communications: from the applications to the medium, IEEE J. Sel. Areas Commun. 34 (7) (2016) 1935–1952, doi:10.1109/JSAC.2016.2566018. [2] A. ElSamadouny, A.E. Shafie, M. Abdallah, N. Al-Dhahir, Secure sum-rate-optimal MIMO multicasting over medium-voltage NB-PLC networks, IEEE Trans. Smart Grid 9 (4) (2018) 2954–2963. [3] M. Nassar, J. Lin, Y. Mortazavi, A. Dabak, I.H. Kim, B. Evans, Local utility power line communications in the 3-500 kHz band: channel impairments, noise, and standards, IEEE Signal Process. Mag. 29 (5) (2012) 116–127, doi:10.1109/MSP.2012.2187038. [4] T. Kim, I.H. Kim, Y. Sun, Z. Jin, Physical layer and medium access control design in energy efficient sensor networks: an overview, IEEE Trans. Ind. Inf. 11 (1) (2015) 2–15, doi:10.1109/TII.2014.2379511. [5] M. Antoniali, M. Girotto, A.M. Tonello, In-car power line communications: Advanced transmission techniques, Int. J. Autom. Technol. 14 (4) (2013) 625–632, doi:10.1007/s12239- 013- 0067- 2. [6] S. Liu, F. Yang, W. Ding, J. Song, Double kill: Compressive sensing based narrowband interference and impulsive noise mitigation for vehicular communications, IEEE Trans. Veh. Technol. 65 (7) (2016) 5099–5109, doi:10.1109/TVT.2015.2459060. [7] N. Lallbeeharry, R. Mazari, V. Dégardin, C. Trebosc, PLC applied to fault detection on in-vehicle power line, in: Proceedings of the IEEE International Symposium on Power Line Communications and its Applications (ISPLC), 2018, pp. 1–5, doi:10.1109/ISPLC.2018.8360233. Please cite this article as: Y.-R. Chien, S.S.-D. Xu and S.-H. Lu, Cyclostationary impulsive noise mitigation algorithm for narrowband powerline communications, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.10.026

JID: FI

16

ARTICLE IN PRESS

[m1+;December 13, 2019;19:54]

Y.-R. Chien, S.S.-D. Xu and S.-H. Lu / Journal of the Franklin Institute xxx (xxxx) xxx

[8] M. Mahoor, F.R. Salmasi, T.A. Najafabadi, A hierarchical smart street lighting system with brute-force energy optimization, IEEE Sens. J. 17 (9) (2017) 2871–2879, doi:10.1109/JSEN.2017.2684240. [9] L. Guerrieri, G. Masera, I.S. Stievano, P. Bisaglia, W.R.G. Valverde, M. Concolato, Automotive power-line communication channels: mathematical characterization and hardware emulator, IEEE Trans. Ind. Electron. 63 (5) (2016) 3081–3090, doi:10.1109/TIE.2016.2516508. [10] Q. Zhu, H. Wang, Output feedback stabilization of stochastic feedforward systems with unknown control coefficients and unknown output function, Automatica 87 (2018) 166–175, doi:10.1016/j.automatica.2017.10.004. [11] H.R. Karimi, Optimal vibration control of vehicle engine-body system using Haar functions, Int. J. Control Autom. Syst. 4 (6) (2006) 714–724. [12] R. Wang, H. Jing, F. Yan, H.R. Karimi, N. Chen, Optimization and finite-frequency H∞ control of active suspensions in in-wheel motor driven electric ground vehicles, J. Frankl. Inst. 352 (2) (2018) 468–484, doi:10. 1016/j.jfranklin.2014.05.005. [13] X. Dong, Y. Zhao, H.R. Karimi, P. Shi, Adaptive variable structure fuzzy neural identification and control for a class of MIMO nonlinear system, J. Frankl. Inst. 350 (5) (2013) 1221–1247, doi:10.1016/j.jfranklin.2013.02.016. [14] X. Chen, L. Wu, Z. Zhang, J. Dang, J. Wang, Adaptive modulation and filter configuration in universal filtered multi-carrier systems, IEEE Trans. Wirel. Commun. 17 (3) (2018) 1869–1881, doi:10.1109/TWC.2017.2786231. [15] H. Ren, G. Zong, H.R. Karimi, Asynchronous finite-time filtering of networked switched systems and its application: an event-driven method, IEEE Trans. Circuits Syst. I Regul. Pap. 66 (1) (2019) 391–402, doi:10.1109/TCSI.2018.2857771. [16] S. Suresh, S. Lal, Two-dimensional CS adaptive FIR Wiener filtering algorithm for the denoising of satellite images, IEEE J. Sel. Topics Appl. Earth Observ. Remote Sens. 10 (12) (2017) 5245–5257, doi:10.1109/JSTARS. 2017.2755068. [17] Y. Li, Q. Shi, H.R. Karimi, Robust L1 fixed-order filtering for switched LPV systems with parameter-dependent delays, J. Frankl. Inst. 352 (3) (2015) 761–775, doi:10.1016/j.jfranklin.2014.11.007. [18] K. Jiang, H. Zhang, H.R. Karimi, J. Lin, L. Song, Simultaneous input and state estimation for integrated motortransmission systems in a controller area network environment via an adaptive unscented Kalman filter, IEEE Trans. Syst., Man, Cybern. Syst. (2018) 1–10, doi:10.1109/TSMC.2018.2795340. [19] H. Wang, Q. Zhu, Global stabilization of stochastic nonlinear systems via C1 and C∞ controllers, IEEE Trans. Autom. Control 62 (11) (2017) 5880–5887, doi:10.1109/TAC.2016.2644379. [20] H. Wang, Q. Zhu, Adaptive output feedback control of stochastic nonholonomic systems with nonlinear parameterization, Automatica 98 (2018) 247–255, doi:10.1016/j.automatica.2018.09.026. [21] Y. Wang, Y. Xia, C.K. Ahn, Y. Zhu, Exponential stabilization of Takagi–Sugeno fuzzy systems with aperiodic sampling: an aperiodic adaptive event-triggered method, IEEE Trans. Syst. Man Cybern. Syst. 49 (2) (2019) 444–454, doi:10.1109/TSMC.2018.2834967. [22] Y. Wang, X. Yang, H. Yan, Reliable fuzzy tracking control of near-space hypersonic vehicle using aperiodic measurement information, IEEE Trans. Ind. Electron. 66 (12) (2019) 9439–9447, doi:10.1109/TIE.2019.2892696. [23] Y. Wang, H.R. Karimi, H. Shen, Z. Fang, M. Liu, Fuzzy-model-based sliding mode control of nonlinear descriptor systems, IEEE Trans. Cybern. 49 (9) (2019) 3409–3419, doi:10.1109/TCYB.2018.2842920. [24] Q. Zhu, Stabilization of stochastic nonlinear delay systems with exogenous disturbances and the event-triggered feedback control, IEEE Trans. Autom. Control 64 (9) (2019) 3764–3771, doi:10.1109/TAC.2018.2882067. [25] H. Zhang, X. Huang, J. Wang, H.R. Karimi, Robust energy-to-peak sideslip angle estimation with applications to ground vehicles, Mechatronics 30 (2015) 338–347, doi:10.1016/j.mechatronics.2014.08.003. [26] S.K. Kommuri, M. Defoort, H.R. Karimi, K.C. Veluvolu, A robust observer-based sensor fault-tolerant control for PMSM in electric vehicles, IEEE Trans. Ind. Electron. 63 (12) (2016) 7671–7681, doi:10.1109/TIE.2016. 2590993. [27] W.K. Wong, H.S. Lim, A robust and effective fuzzy adaptive equalizer for powerline communication channels, Neurocomputing 71 (1) (2007) 311–322, doi:10.1016/j.neucom.2006.12.018. [28] S. Zhang, W.X. Zheng, J. Zhang, A new combined-step-size normalized least mean square algorithm for cyclostationary inputs, Signal Process. 141 (2017) 261–272, doi:10.1016/j.sigpro.2017.06.007. [29] N. Shlezinger, R. Dabora, Frequency-shift filtering for OFDM signal recovery in narrowband power line communications, IEEE Trans. Commun. 62 (4) (2014) 1283–1295, doi:10.1109/TCOMM.2014.020514.130421. [30] N. Shlezinger, K. Todros, R. Dabora, Adaptive filtering based on time-averaged MSE for cyclostationary signals, IEEE Trans. Commun. 65 (4) (2017) 1746–1761, doi:10.1109/TCOMM.2017.2655526. [31] J. Lin, B. Evans, Cyclostationary noise mitigation in narrowband powerline communications, in: Proceedings of the Asia Pacific Signal and Information Processing Association Annual Summit and Conference, 2012, pp. 1– 4. Please cite this article as: Y.-R. Chien, S.S.-D. Xu and S.-H. Lu, Cyclostationary impulsive noise mitigation algorithm for narrowband powerline communications, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.10.026

JID: FI

ARTICLE IN PRESS

[m1+;December 13, 2019;19:54]

Y.-R. Chien, S.S.-D. Xu and S.-H. Lu / Journal of the Franklin Institute xxx (xxxx) xxx

17

[32] J. Lin, B.L. Evans, Non-parametric mitigation of periodic impulsive noise in narrowband powerline communications, in: Proceedings of the IEEE Global Communications Conference (GLOBECOM), 2013, pp. 2981–2986, doi:10.1109/GLOCOM.2013.6831528. [33] B. Han, C. Kaiser, K. Dostert, A novel approach of canceling cyclostationary noise in low-voltage power line communications, in: Proceedings of the IEEE International Conference on Communications (ICC), 2015, pp. 734–739, doi:10.1109/ICC.2015.7248409. [34] M. Elgenedy, M. Sayed, N. Al-Dhahir, R.C. Chabaan, Cyclostationary noise mitigation for SIMO powerline communications, IEEE Access 6 (2018) 5460–5484, doi:10.1109/ACCESS.2017.2789185. [35] M. Elgenedy, M. Sayed, A.E. Shafie, I.H. Kim, N. Al-Dhahir, Cyclostationary noise modeling based on frequency-shift filtering in NB-PLC, in: Proceedings of the IEEE Global Communications Conference (GLOBECOM), 2016, pp. 1–6, doi:10.1109/GLOCOM.2016.7841712. [36] M. Korki, N. Hosseinzadeh, T. Moazzeni, Performance evaluation of a narrowband power line communication for smart grid with noise reduction technique, IEEE Trans. Consum. Electron. 57 (4) (2011) 1598–1606, doi:10. 1109/TCE.2011.6131131. [37] IEEE standard for low-frequency (less than 500 kHz) narrowband power line communications for smart grid applications, IEEE Std 1901.2-2013 (2013) 1–269. [38] M. Zimmermann, K. Dostert, A multipath model for the power line channel, IEEE Trans. Commun. 50 (4) (2002) 553–559, doi:10.1109/26.996069. [39] M. Katayama, T. Yamazato, H. Okada, A mathematical model of noise in narrowband power line communication systems, IEEE J. Sel. Areas Commun. 24 (7) (2006) 1267–1276, doi:10.1109/JSAC.2006.874408. [40] K. Nieman, J. Lin, M. Nassar, K. Waheed, B. Evans, Cyclic spectral analysis of power line noise in the 3200 kHz band, in: Proceedings of the IEEE International Symposium on Power Line Communications and its Applications (ISPLC), 2013, pp. 315–320, doi:10.1109/ISPLC.2013.6525870. [41] M. Elgenedy, M. Sayed, A.E. Shafie, I.H. Kim, N. Al-Dhahir, Cyclostationary noise modeling based on frequency-shift filtering in NB-PLC, in: Proceedings of the IEEE Global Communications Conference (GLOBECOM), 2016, pp. 1–6, doi:10.1109/GLOCOM.2016.7841712. [42] W.A. Gardner, Cyclostationarity in Communications and Signal Processing, IEEE Press, 1994. [43] M. Sayed, T.A. Tsiftsis, N. Al-Dhahir, On the diversity of hybrid narrowband-PLC/wireless communications for smart grids, IEEE Trans. Wireless Commun. 16 (7) (2017) 4344–4360, doi:10.1109/TWC.2017.2697384. [44] B. Widrow, J.R. Glover, J.M. McCool, J. Kaunitz, C.S. Williams, R.H. Hearn, J.R. Zeidler, J. Eugene Dong, R.C. Goodlin, Adaptive noise cancelling: Principles and applications, Proc. IEEE 63 (12) (1975) 1692–1716, doi:10.1109/PROC.1975.10036. [45] B.K. Mohanty, P.K. Meher, A high-performance energy-efficient architecture for FIR adaptive filter based on new distributed arithmetic formulation of block LMS algorithm, IEEE Trans. Signal Process. 61 (4) (2013) 921–932, doi:10.1109/TSP.2012.2226453. [46] K. Nieman, J. Lin, M. Nassar, K. Waheed, B. Evans, Measured power line noise in the 3-200 kHz band, 2013, (http:// users.ece.utexas.edu/ ∼bevans/ papers/ 2013/ PLCcyclic/ isplc2013_cyclic_noise_analysis.zip), Online; accessed June 5, 2019. [47] Q. Zhu, Stability analysis of stochastic delay differential equations with Lévy noise, Syst. Control Lett. 118 (2018) 62–68, doi:10.1016/j.sysconle.2018.05.015. [48] B. Wang, Q. Zhu, Stability analysis of semi-Markov switched stochastic systems, Automatica 94 (2018) 72–80, doi:10.1016/j.automatica.2018.04.016.

Please cite this article as: Y.-R. Chien, S.S.-D. Xu and S.-H. Lu, Cyclostationary impulsive noise mitigation algorithm for narrowband powerline communications, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j. jfranklin.2019.10.026