A New Noise Mitigation Scheme in Power Line Communication Systems Based on the Principal Components

International Journal of Information and Computer Science (IJICS) Volume 4, 2015 www.iji‐cs.org doi: 10.14355/ijics.2015.04.003

A New Noise Mitigation Scheme in Power Line Communication Systems Based on the Principal Components Analyses Jiang‐Qian Li China Electric Power Research Institute, Beijing, P. R. of China 100192 hammerstrom@126.com

Abstract Power Line Communication (PLC) systems suffer from too severe noise contamination in the channels. In the paper, a new scheme was proposed based on the principal components analyses (PCA) to mitigate noises in PLC channels. The scheme consists of four steps. In the first step, the PLC signal with one dimension is generalized to a data set with multiple dimensions (degrees). In order to de‐correlate the new data set, generalized signals are whitened with Gaussian noises in the second step.

For the uncorrelated data set, the PCA technology is applied to cancel noises in the transformed domain. As the last step, the PLC signals of interesting are recovered by applying the inverse transform matrix. The results from the simulated seismic signals and the measurements data verify the validity of the proposed scheme.

Keyword Smart Grids; PLC; PCA; Noise Mitigation

Introduction IN China, conventional power systems produce a significant amount of pollution gas emissions mainly due to their use of fossil fuels for power generation and consumption, esp. in Beijing, the capital of China [7].

A communication infrastructure, providing such two‐way communications, is an essential part of the success of the emerging smart grid [6]. Power line communication (PLC) has recently appeared as one of the promising technologies that could potentially be used in such a communication infrastructure [8].

Unlike other communication technologies(for example, a local‐area network (LAN), a digital subscriber line (DSL), and cable), power lines are not meant to be used for communication purposes, but for transmitting one‐directional electric power in the frequency range of 50Hz or 60Hz , Thus it presents a harsh and noisy communications channel, which isn’t favorable for the high‐frequency communications signal transmission with frequency ranging from 0 to 30MHz .

Due to its ubiquity of already existing electrical power delivery networks, PLC is susceptible to numerous appliances and power networks. Several studies have shown that the PLC channel is impaired by different types of noise, which are mostly produced by the electrical appliances in the networks. The isolator switching or breaker in the grids can cause impulse noise [4]; The broadcast transmitters operating at frequencies higher than a few hundred kilohertz, can ingress the narrow band noise, mostly amplitude modulated sinusoidal signals; The switching actions of silicon‐controlled rectifier diodes found in many electrical appliances, and industrial sources directly connected to the supply network, can cause periodic impulsive noise synchronous to the mains frequency [3, 2].

Therefore, noise and interferences mitigation is an important issue in PLC systems [9]. There are lots of papers trying to deal with such challenge by exploiting various statistical properties of the noise and interference [5]. A noise whitening filter is designed to improve the reliability and throughput of PLC systems with a specific noise model [1].

As one of the most valuable results from applied linear algebra, principal component analysis (PCA) has been used abundantly in all forms of data analysis. In general, PCA provides a roadmap for how to reduce a complex data set

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to a lower dimension to reveal the sometimes hidden, simplified dynamics that often underlie it. Therefore, it is often used in data set analyses, not in signal processing. Until now, there is no paper to discuss the application of PCA to noise reduction in seismic data processing.

In the paper, we propose a new scheme based on PCA to mitigate random noise. In the scheme, a PCA‐based signal de‐noising algorithm is utilized to decompose received seismic signals into a linear combination of signal components in the domain, spanned by the received signal’s covariance matrix’s eigenvectors. In such domain, signals of interesting are with greater powers than random noises, and powers of random noises, therefore, can be forced to be zero. By doing so, random noises can be mitigated greatly in the time domain. At the same time, we evaluate the performance of the noise mitigation scheme based on the PCA technology in seismic signal processing applications.

Fundamentals of the Principal Components Analyses Principal component analysis (PCA) has been called one of the most valuable results from applied linear algebra.

PCA is used abundantly in all forms of analysis ‐ from neuroscience to computer graphics ‐ because it is a simple, non‐parametric method of extracting relevant information from confusing data sets. With minimal additional effort, PCA provides a roadmap for how to reduce a complex data set to a lower dimension to reveal the sometimes hidden, simplified dynamics that often underlie it. In general terms, PCA uses a vector space transform to reduce the dimensionality of large data sets. Using mathematical projection, the original data set, which may have involved many variables, can often be interpreted in just a few variables (the principal components). It is therefore often the case that an examination of the reduced dimension data set will allow the user to spot trends, patterns and outliers in the data, far more easily than would have been possible without performing the principal component analysis.

Assume that we start with a data set that is represented in terms of an m  n matrix, X where the n columns are the samples (e.g. observations) and the m rows are the variables. We wish to linearly transform this matrix, X into another matrix, Y , also of dimension m  n so that for some m  m , P ,

Y  PX (1) This equation represents a change of basis. If we rewrite P as PT  p1T  pT2   pTm  , where pTi is the i th row vector of P , and X as X   x1  x 2  x n  , then Eq.1 can be interpreted in the following way,

Y  PX 

p   1     p2    x     1      p m 

x2  p m  (2)

and it can be expressed further as

Y  PX 

p x  1 1   p 2 x1       p m x1 

p1x2



p 2 x2 

 

pm x2 

   p2 xn      p m x n 

p1x n

(3)

We can note the form of each column of Y , y i as

yi 

p x   1 i     p2 xi            p m x i 

(4)

We recognize that each coefficient of y i is a dot product of xi with the corresponding row in P . In other words, p i x j is just the standard Euclidean inner (dot) product for pi  x j  R m . This tells us that the original data, X is

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being projected on to the rows of P . Thus, the rows of P , p1  p 2  p m  are a new basis for representing the columns of X .

Until now, there is no rule to define the transformation matrix P , however, the PCA method use some tricks. In such method, the matrix P is selected to be a matrix where each row pi is an eigenvector of XXT . For a positive definite or positive semi‐definite matrix, like covariance matrix XXT , we have the following results:

S X  XXT  EDET (5) where E is an m  m orthogonal matrix whose columns are the orthogonal eigenvectors of XXT , and D is a diagonal matrix which has the eigenvalues of XXT as its (diagonal) entries.

Therefore, by the mentioned selection,

P  ET (6) Substituting Eq.6 into the covariance matrix S Y  YYT , we have S Y   PX  PX   PXXT PT  ET  EDET  E  D (7) T

From Eq.7, we can see that the PCA method makes the fundamental assumption that the new variables in the new basis space p1  p 2  p m  , should be as uncorrelated as possible, but with powers as large as possible. This is equivalent to saying that the co‐variances of different variables in the covariance matrix S Y , should be as close to zero as possible, with variances as large as possible. It is evident that the choice of P diagonalizes the covariance matrix S Y . This is the goal of PCA. The Proposed Noise Mitigation Based on PCA Fundamentals of the Proposed Scheme From the definitions of PCA, we can see that the PCA can get the principal components of the data X , corresponding to the eigenvalues of the matrix XXT . Thus, its most common use is as the first step in trying to analyze large data sets, such as dimension reduction, blind source separation, and data compression. When applying it to de‐noising, the used data are often graphics, that is to see the data is with two dimensions at least.

Until now, there is no paper to discuss the application of PCA to seismic signals de‐noising. The reason is that signals with one dimension, such as seismic signal x  t  , is without eigenvectors and thus the transform matrix P can’t be formalized according to Eq.efEq.transform.

In the paper, we proposed a new scheme to mitigate random noises based on PCA in noisy seismic signals.

Suppose the noisy seismic signal y0  t  as:

y0  t   x  t   n0  t  (8) where x  t  is the seismic signal of interesting, and n0  t  is random noise at time t . The discrete versions of such signal are as follows with sample interval  :

y0  k   x  k   n0  k 

k  01 K  1 (9)

where K is the time length of signal in discrete version. Therefore, the discrete seismic signal can be expressed in the form of column vector:  y0  0    x  0   n0  0        y 1 x 1 n 1      0   (10)  y0   0            y0  K  1   x  K  1  n0  K  1 

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In order to use PCA, we generalized the column vector y 0 to a new data matrix Y first: Y   y 0

y1  y L 1  (11)

where L is the number of the generalization, and can also be called signal generalization dimension Dgeneralization in the paper. For the l th generalized noisy signal y l , it can be expressed as:  yl  0     y 1  yl   l       yl  K  1  (12) x  0   nl  0   n0  0      x 1  nl 1  n0 1         x  K  1  nl  K  1  n0  K  1 

where nl  k   k  01 K  1 is the random noise at t  (k  1) in the l th generalized noisy signal y l . Because

nl  k   k  01 K  1 is a random process, it has the following properties:

E  nl  k    0  2  for j  l and q  k (13) E  nl  k  n j  q     l 0 else 

where E   is mathematical expectation operator, and  l2 is the variance(or power) of the random noise nl  k  .

This step is called signal whitening in the paper. Signal whitening can represent the original signal y0  k  with one dimension as a data set Y , where all of the new signals(corresponding to column vectors) are uncorrelated with the original signal y0  k  . Thus, the l th column of the data set Y is with the seismic signal of interesting

x  k  and the random noise n  k   n0  k   nl  k  .

In the next step, PCA is applied to data set Y , and we have

Z  PY (14) with

P

p  1   p2       p K

        

(15)

where, each row vector p k is the k th eigenvector of YY T . At the same time, p k is placed in descending order of the corresponding eigenvalues of YYathbfT . Thus, p1 is corresponding to the largest eigenvalue of YY T .

In the third step, we select the previous Dreco row vectors of transform matrix P to formalize a new transform matrix Preco :

Preco 

                

               reco 

p1

p2 

(16)

pD

In this matrix, the row vectors correspond to Dreco largest eigenvalues in the original matrix P . Such

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Dreco largest eigenvalues are considered as the powers of the generalized signal of interesting,

x  k   k  01 K  1 . After getting transforming matrix Preco , we have

Z reco  Preco Y (17) In order to get the seismic signals of interesting, we apply the inverse matrix of Preco to Z reco : T X  Preco Zreco (18)

Applications of the Proposed Method As discussed previous, the received OFDM signals are mixed with strong interferences which can be modelled as the Doppler signals. It is very difficult to separate them in time domain or in frequency domain because they overlapped in these domains.

In the paper, we proposed a method, based on the signals’ PCA transform, to separate the Doppler signals and the OFDM signals with help of the new order selection scheme, as discussed in Section II. In order to evaluate the performance with respect to bit‐error‐rate (BER) of the proposed method, we applied it to the simulation data and measurements data respectively. For simulations data, the used block diagram of the OFDM system is shown in Fig.1, and the simulated OFDM signal x(t ) can be expressed as follows:  N 1

x t     X l k  e l 0 k 0

j 2 f k  t  lTsym 

(19)

where Tsym is the l th OFDM symbol X l duration, and N is the number of the sub‐carrier in OFDM system with sub‐carrier frequency f k  Tk . sym

With respect to the simulation parameters, we adopted IEEE P1901.2 standard. The frequency range is from 359  4875kHz , and the sampling frequency is set to 12 MHz . We used 256 sub‐carriers to transmit the data.

Thirty bits are inserted into the OFDM symbol as cyclic prefix with the duration of 25 s , while 22 bits as the effective cyclic prefix with the duration of 183 s . The sub‐carrier spacing is 46875kHz , therefore, the OFDM symbol duration is 2371 s . The modulation scheme is QAM with constellation size 16 . Under such simulation environment, the maximum data rate can reach at 2076kb  s for non‐coherent modulations.

The performance with respect to BER is presented in Fig. 1. From the simulation results, we can see that the effects by the Doppler interferences are mitigated significantly. For the lower SNRs, the improvement of BER can be in 1 2 orders, while it can reach at 3 orders in the relative high SNR cases. For example, if the SNR is about 8dB , the BERs are about 10 2 and 10 5 for without interference mitigation and with mitigation, respectively. If the SNR is about 18dB , the BERs are about 10 4 and 10 6 , respectively.

FIGURE 1. BER PERFORMANCES FOR THE PROPOSED METHOD

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We also applied the proposed scheme to the measurements data. The BER is improved from about 10 3 to about 10 5 . The improvement is significantly.

Conclusions In the project the authors involved, we are encountered by the heavy random noises with the overlapping frequencies which the PLC transceiver works on. The interferences are so strong enough to make the PLC systems fail to work. In order to mitigate such interferences, we introduced the PCA transform to the OFDM signals processing. The scheme consists of four steps. In the first step, the PLC signal with one dimension is generalized to a data set with multiple dimensions (degrees). In order to de‐correlate the new data set, generalized signals are whitened with Gaussian noises in the second step. For the uncorrelated data set, the PCA technology is applied to cancel noises in the transformed domain. As the last step, the PLC signals of interesting are recovered by applying the inverse transform matrix. The results from the simulated seismic signals and the measurements data verify the validity of the proposed scheme.

ACKNOWLEDGMENT

For processing PLC‐OFDM signals, we used the software Smart Signal Processing developed and sold by Prof.

Ming‐Yue ZHAI from North China Electric Power University.

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