A semi-hidden Fritchman Markov modeling of indoor CENELEC A narrowband power line noise based on signal level measurements

Int. J. Electron. Commun. (AEÜ) 74 (2017) 21–30

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International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.com/locate/aeue

Regular paper

A semi-hidden Fritchman Markov modeling of indoor CENELEC A narrowband power line noise based on signal level measurements Ayokunle D. Familua ⇑, Ling Cheng School of Electrical and Information Engineering, University of the Witwatersrand, 1 Jan Smuts Avenue, Braamfontein, Johannesburg, South Africa

a r t i c l e

i n f o

Article history: Received 4 July 2016 Accepted 23 January 2017

Keywords: Baum–Welch algorithm Maximum likelihood estimation Narrowband Power line communications Power line noise Semi-Hidden Fritchman Markov Model

a b s t r a c t To achieve error-free communication on the narrowband power line communication (NB-PLC) channel, there is need for constant measurement campaign and modeling of burst errors resulting from noise, perturbation and disturbances on the channel. This work thus reports a signal level measurement and FirstOrder Semi-Hidden Fritchman Markov modeling of the three major NB-PLC noise types for a typical South African residential and laboratory indoor low voltage environment. Two distinguishable disturbance scenarios: mildly disturbed and heavily disturbed were considered and Baum–Welch maximum likelihood estimation (MLE) algorithm is used to obtain the most probable model parameters that statistically depict the experimentally measured noise error sequences. Ó 2017 Elsevier GmbH. All rights reserved.

1. Introduction The pervasive PLC network offers a solution to the last-mile access communication problem. Its application includes, utilization in in-house automation and inter-connection of home appliances for a smart home. The in-home NB-PLC channel makes use of the existing power line networks (PLNs), originally conceptualized for powering end-user electrical and electronics appliances for data communication purposes. Hence, considering the pervasive or ubiquitous nature of power lines, huge cost of deploying new cables in buildings can be saved. The NB-PLC make use of the frequency band between 3 and 148.5 kHz (further categorized into bands A, B, C and D) for both low speed and high speed NB-PLC applications standardized by CENELEC European committee and up to 450 kHz standardized by ARIB for Japan [1,2]. NB-PLC is highly relevant for both low and high speed NB-PLC purposes. Its importance ranges from, its historical applications for automatic meter reading (power grid control), control of street light, airport runways ground-light control to its use in street car/subway systems [3,4]. PLC technology generally operates in a hostile environment inherited from the PLN. The effect of this hostile environment is the resulting performance degradation. Hence, despite the attractive advantages PLC technology has to offer, like every other communication technology, it must overcome the challenges posed by ⇑ Corresponding author. E-mail addresses: [email protected] (A.D. Familua), [email protected]. za (L. Cheng). http://dx.doi.org/10.1016/j.aeue.2017.01.015 1434-8411/Ó 2017 Elsevier GmbH. All rights reserved.

the hostile environment to ameliorate the overall system design and performance. The low voltage indoor CENELEC A band suffers the most noise impairment amidst the CENELEC category. Noise, perturbation and disturbances caused by the inherent attribute of the PLN itself, the intrinsic electrical attributes of the numerous electrical and electronic appliances connected onto the network and external disturbances poses a hostile environment which results into signal degradation. The introduction and conduction of noise harmonics is caused by the uncoordinated ‘‘ON” and ‘‘OFF” switching of end user appliances on the network, thus resulting in burst errors and consequently data loss or signal corruption at the receiver. Thus, quantifying, characterizing and modeling of these noise impairment, perturbations and disturbances is vital in improving the overall NB-PLC system design, and performance for a reliable communication. In this article, we thus report a signal level measurement and First-Order Semi-Hidden Fritchman Markov modeling (SHFMM) of the three major CENELEC A band NB-PLC noise captured in a typical South African residential and laboratory in-door environment. Error sequences for each noise type is obtained for two distinct scenarios: ‘‘mildly disturbed” and ‘‘heavily disturbed”. A three state SHFMM is assumed for modeling the three NB-PLC noise types, with Baum–Welch, an iterative parameter estimation algorithm used to obtain the most probable parameters that statistically depict the measured error sequences. The resulting statistical models, are a class of mathematical model which captures the information about the error distribution of the originally measured sequences for the three noise types experimentally obtained. These

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statistical modeling results are thus useful in designing and evaluating of the performance of modulation techniques (robust and adaptive) and FEC codes capable of mitigating errors on the burst error prone in-door NB-PLC CENELEC A channel. Moreover, these evaluations are utilized in optimizing communications on NB-PLC channel. The remaining part of this article is arranged as follows: Section 2 explains the three major PLC noise classifications. In Section 3, an extensive but concise description of the Fritchman model assumed in this work, it’s parameters and the reason behind the choice of this model is presented. Baum–Welch algorithm, a MLE iterative algorithm aimed at estimating the SHFMM parameters for the three noise types measured is discussed in Section 4. Section 5 discusses the experimental setup and methodology for capturing and measuring the three major noise types. In Section 6, the SHFMM statistical modeling results for the measured noise types are presented and analyzed. Section 7 gives a summary and conclusion of work done in this article.

2. Power line communication channel noise categories In most communication systems, Additive White Gaussian noise (AWGN) model is often assumed. While AWGN is a suitable model for some communication channels, this model does not depict the attributes of the burst errors present on the NB-PLC channels. Several power electronics appliances, particularly those having switching circuits, injects periodic or random noise pulses onto the channel. The resulting noise is non-white, non-Gaussian and transient due to its impulsive nature, hence, it is been referred to as Non-AWGN, which is a deviation from the AWGN model [5,6]. In literature, PLN noise has been classified into three major kinds namely: impulse noise, background noise and narrowband noise discussed as follows [5–8]:

2.1. Background noise Background noise (BN) possess comparatively low power spectral density (PSD), and often result from the sum of various low power noise sources connected onto the channel. It is frequently identified by a constant envelope occurring over a prolonged duration [7]. This noise type includes; flickering noise, and thermal noise emanating from receivers’ front end amplifier. BN also emanates from universal motors often found in but not limited to end user gadgets such as fans, drilling tools and dryers. BN is also identified due to its non-white attribute, hence it possesses a frequency-dependent power spectral density and is always present on the NB-PLC channel. The power spectral density of this noise type decays as frequency increases, possessing a slope varying between 20 and 25 dB/decade in an indoor low voltage NB-PLC environment [7,8] and is principally present in narrowband frequencies than in broadband frequencies [8].

2.2. Narrowband Noise Narrowband noise (NBN) is typically limited to a certain frequency slot dependent on its source. It emanates primarily from signals (sinusoidal) having modulated amplitude and are radiated or conducted from both internal and external appliances onto the network, thus the power line acting as an antenna. In literatures, this noise has been found to originate from the horizontal retrace frequency of televisions [7]. Other NBN origins are spurious electromagnetic disturbances emanating from end user gadgets with in-built transmitters and receivers [7,8].

2.3. Impulse noise Impulse noise (IN) is transient in nature and it is described as the principal cause of burst errors on the PLC channel. Unlike NBN, impulse noise covers a wider part of the spectrum in use. It possesses high power spectral density and is distinguished by its inter-arrival time, duration and amplitude. It is significant to clarify that on a low voltage NB-PLC channel, two main classification of impulse noise exist: the ‘‘Periodic impulse noise” and ‘‘Aperiodic impulse noise” [7–9]. Aperiodic impulse noise also regarded as asynchronous impulsive noise, originates from arbitrary emission events or isolated activities at both homes and industrial sites. Classic aperiodic impulse noise emanates from switching transient such as: on and off switching, plugging and unplugging of appliances and co-existence issues that often occur due to uncoordinated PLC transmissions. This impulse noise type is predominant in the high frequency band ranging from several hundred kHz to 20 MHz [10,11]. Periodic impulse noise also referred to as ‘‘Cyclostationary impulsive noise” is sub-divided into two: Periodic synchronous impulse noise and Periodic asynchronous impulsive noise as discussed as follows:

2.3.1. Periodic synchronous impulse noise This IN waveform exhibit a train of impulses synchronous to the low voltage AC mains 50/60 Hz frequency. It comprises of a series of impulses that are isolated, with fairly large amplitude and duration. They originate from non-linear power electronic gadgets like; silicon controlled rectifier operations in power supplies, thyristors operation in light dimmer, laptops, desktop computers, LCD monitors and from a brush motor commutating effects [12].

2.3.2. Periodic asynchronous impulse noise This noise type is characterized by periodic noise impulses or trains of impulses which occurs with a frequency and repetition rates independent of mains frequency [12,13]. It has repetition rates between 50 and 200 kHz and is majorly injected by transient operations such as switching of relays that occurs in switch mode power supplies connected to the network [7,12]. The noise impulses typically possess much shorter durations and much lower amplitudes than those of the periodic synchronous impulse noise [12,14]. As aperiodic impulse noise is predominant on broadband power lines, recent indoor and outdoor noise measurements on both lowvoltage and medium-voltage PLNs established that ‘‘cyclostationary noise (both periodic synchronous and asynchronous to mains frequency impulse noise)” are the prevailing noise impairment present on the 3–500 kHz NB-PLC spectrum [1,2,15]. This noise kind possesses long noise bursts with periodically varying statistics and whose period is same as half the AC main’s cycle. In PLC systems, a typical periodic synchronous impulse noise is composed of noise bursts having high power which spans from 10% to 30% of the period [15], which is oftentimes a lot prolonged than the standardized duration of a typical OFDM symbol [11] and amounts to 833 ls–2.5 ms in the US FCC band [15,16]. A single cyclostationary noise burst could possibly lead to corruption of multiple successive OFDM symbols. For instance, for a PLC-G3 standard functioning in the 3–95 kHz CENELEC A band [1], the OFDM symbol duration is 695ls, hence, cyclostationary noise burst that lasts 30% of a period is bound to corrupt up to four successive OFDM symbols [16]. During certain period of the bursts, the noise power at particular frequency bands could rise to 30–50 dB greater than in the remaining period [15,16].

A.D. Familua, L. Cheng / Int. J. Electron. Commun. (AEÜ) 74 (2017) 21–30

3. Semi-Hidden Fritchman Markov Model 3.1. Semi-Hidden Fritchman Markov Model overview Discrete channel models chronicles the statistical error distributions resulting from noise impairments. The realization of a mathematically inclined statistical modeling of noise impairments is used to alleviate burst error effects on NB-PLC channels. The description of channel impairments behavior through mathematically inclined channel models helps in realizing and designing robust FEC codes capable of mitigating these noise impairments. Thus, our modeling attempt in this article is entirely centered on the SHFMM approach proposed by Fritchman [17,18]. Fig. 1 shows a distinct three state SHFMM with two good-states and a single bad-state utilized for modeling in this work. The choice of the SHFMM shown in Fig. 1 is simply because of its simplicity and the fact that the error-free run distribution (EFRD) denoted by Pr (0m—1) distinctively defines the one-errorstate, in essence, parameters of the noise models could be inferred from the EFRD and vice versa [19,20]. As shown in Fig. 1, f ð1Þ; f ð2Þ ¼ 0 depicts the error-free state conditional probability, while f ð3Þ ¼ 1 symbolizes the error state conditional probability. Furthermore, ‘‘0” is synonymous to no transmission error, while on the other hand, ‘‘1” implies a transmission error. The fundamental feature of the chosen single-error-state SHFMM as seen from Fig. 1, is that state transitions is impermissible between states in similar groups, in other words, transitioning between states are only permitted between the good states to the single bad state and not between the two good states [21]. Therefore, this model is accessible for real life channel modeling, as it affords us the luxury of modeling memory channels such as the PLC channel. The choice of one error state is because the measurement is not as bursty as to require another error state. 3.2. Semi-Hidden Fritchman Markov Model parameters defined The matrix representation of the state transition probability for the proposed SHFMM in Fig. 1 is denoted by P as follows.

2

p11 6 P ¼ 40 p31

0 p22 p32

3

2

3

0:9 0 0:1 7 7 6 0:8 0:2 5 p23 5 ¼ 4 0 0:1 0:7 0:2 p33 p13

ð1Þ

23

Note that entries p12 and p21 of P are zeros due to a unique SHFMM chosen. It is important to note also that the diagonal elements of P are also uniquely chosen [17]. The p-values in (1) are randomly chosen such that the probability of transitioning to a good state represented by matrix elements p11 ; p22 ; p31 and p32 is high, while the probability of transitioning to a bad state represented by matrix elements p13 ; p23 and p33 is low. The Baum–Welch algorithm is then used to iteratively obtain the most likely parameter estimates that produced the experimentally observed data (the measured error sequence) as seen from the re-estimated state transition probabilities in Tables 2–5. Similarly, the output symbol/error producing probability matrix denoted by B takes a binary form as follows [22].

 B¼

1 1 0 0 0



1

ð2Þ

The B matrix represents the output symbol or error generating for the Fritchman model structure proposed in Fig. 1. The first two columns of the B matrix in Eq. (2) represents the two good or errorfree states as shown in Fig. 1, while the third and final column represents the single bad or error state accordingly. On the other hand, the first row typifies no error, while row two typifies an error occurrence. Based on the structure of the assumed Fritchman model: The value of row 1 column 1 and row 1 column 2 is ‘‘1” (one- which signifies certainty) respectively and represents the probability of having no error in state one and two. The values of row 2 column 1 and row 2 column 2 is ‘‘0” (zero- which signifies impossibility) respectively and represents the probability of no error occurring in state one and two respectively. On the other hand, the value of row 1 column 3 is ‘‘0” (zero- signifying impossibility) and represents the probability that it is impossible not to have an error in state three. While, the value of row 2 column 3 is ‘‘1” (one- signifying certainty) representing the probability that there will always be an error in state three. The final model parameter is p, which symbolizes the initial or prior state probability, of being in any of the three states and is assumed for the proposed 3-state SHFMM as follows.

p ¼ ½ p1 p2 p3  ¼ ½ 0:4 0:4 0:2 

ð3Þ

A First-Order SHMM is adopted, hence, probability of an observation (error sequence ‘‘E”) at time ‘‘t” is dependent only on the previous state. 4. Baum–Welch algorithm implementation

Fig. 1. Typical three-state Semi-Hidden Fritchman Markov Model.

Baum–Welch MLE algorithm [22,23], an iterative algorithm is primarily utilized for the parameter estimation of the statistical SHFMM for three NB-PLC noise kinds obtained through signal level measurement [24]. The algorithm helps compute the parameters of the model that is most likely to have produced the measured error sequence, hence, the name maximum likelihood estimation algorithm. It is also used in other applications as recorded in the following literatures [25,26]. Fig. 2 shows the First-Order Baum– Welch algorithm flow chart for the parameter estimation of the SHFMM. Refer to [27] for a step-by-step procedures and mathematical equations for the First-Order SHMM used in this work. Note that ‘‘E”, with length T = 10,000 represents the measured error sequences. Find as follows the Baum–Welch training procedures.

Fig. 2. First-Order Baum–Welch algorithm flow chart.

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Note, that 81 initial model parameters are assumed and the most probable parameter is obtained and selected after training with the measured error sequences. Step 1 Initialization: assume initial full parameters C ¼ ðP; B; pÞ for the adopted SHFMM. Step 2 Based on assumed initial parameters and training sequence, compute both forward probabilities (a) and backward probabilities (b). Step 3 Making use of the computed forward probabilities (a) and backward probabilities (b), two intermediate variables are obtained in order to re-estimate the most probable SHFMM parameters. ct ðiÞ, the anticipated number of transitions made from state i, at time instant t, with the model C ¼ ðP; B; pÞ specified is first computed mathematically as follows [22,27].

ct ðiÞ ¼ Pr½st ¼ ijE; C ¼

at ðiÞbt ðiÞ Pr½EjC

;

i ¼ 1; 2;    ; N

ð4Þ

The second intermediate variables nt ði; jÞ symbolizing the probability of transitioning to state i at time instant t and state j at time instant t + 1, with the model C ¼ ðP; B; pÞ and E, the observation sequence specified is mathematically computed as follows [22,27].

nt ði; jÞ ¼ Pr½st ¼ Si ; stþ1 ¼ Sj jE; C ¼

at ðiÞpij bj ðEtþ1 Þbtþ1 ðjÞ P½EjC

ð5Þ

The following intermediate variables are also computed: the anticipated number of times transition to state i at time instant (t = 1) is made; ct ðjÞ the anticipated number of times transition originated from state j and, the expected number of observing Ot ¼ ek in state j.Step 4 The intermediate variables obtained in previous step is utilized in computing the re-estimated SHFMM parameters: b p ij ; b b j ðe Þ and

pi as follows [22,27]. T 1 X

b p ij ¼

ð6Þ

ct ðiÞ

t¼1

Computation of b b j ðek Þ, the re-estimated output symbol generation matrix is obtained as follows.

b b j ðek Þ ¼

X jEt ¼ eTk ct ðjÞ t¼1 T X

5. Experimental setup for the signal level noise measurement 5.1. Description of measurement sites and scenarios The experimental signal level noise measurement is carried out at both residential and laboratory sites in South Africa at different times of the day in 2015 and 2016. This is because noise impairments that occurs on PLC channels depends on place, time, topology of powerline and mains voltage, therefore, continuous measurement campaigns is vital for deriving statistical mathematical models. Appliances available at both locations includes but not limited to: incandescent lamps, electric cooker, washing machine, TV, pressing iron, shaving machine, oscilloscope, table power supply, tower computers with monitor, air conditioning unit, soldering Irons e.t.c. Two measurements scenario were also defined. In scenario I (mildly disturbed), measurement were carried out with the several electrical and electronic appliances at their normal position and mild disturbance introduced from some of these appliances, while for Scenario II (heavily disturbed), the electrical and electronics appliances are positioned closer to the power line outlet measuring point and all appliances are active on the power line as at the time of measurement, hence, a worse-case scenario of disturbance is experienced. 5.2. Experimental setup

k

nt ði; jÞ

t¼1 T1 X

[8,28,29], and gradient search method [30]. Refer to Appendix A of [27] for a more detailed Baum–Welch algorithm computation for a sample error sequence of length T = 4.

ð7Þ

ct ðjÞ

The signal level measurements of the three noise types were carried out with the setup shown in Fig. 3. Noise measurement is obtained from the PLC channel via a differential coupling mode capacitive coupler designed as a bandpass filter (allowing CENELEC band frequencies to pass). It also provides galvanic isolation to prevent damage of the mixed domain oscilloscope. A Mixed Domain Oscilloscope (MDO) is a digital oscilloscope with integrated spectrum analyzer, capable of allowing simultaneous viewing of signals in both time and frequency domain. Refer to [27] for a description of the coupling circuit. Fig. 4 shows the schematic of the experimental setup. Categorization of the different noise from the captured waveform is done based on the corresponding attributes of the noise to be categorized. The amplitude of a narrowband noise is constant

t¼1

The re-estimated prior state probability lows [22,27].

c pi ¼ a1 ðiÞb1 ðiÞ

pi is also obtained as folð8Þ

Step 5 If convergence of the model parameters is achieved. The re-estimated parameters becomes the most probable Fritchman SHMM parameters that produced the observation sequence E i.e. b ¼ C. Other than that, we return to procedure two with the C b B; b p b = ð P; b Þ and replicate obtained re-estimated model parameters C procedure two to four until desired level of convergence is achieved. Lastly, a regenerated error vector (model generated error sequence) is obtained using the re-estimated SHFMM parameters and a plot of the error-free run distribution PrðOm j1Þ for both the original error sequences experimentally obtained and that of the model regenerated error sequences is plotted for comparison. It is vital to state that the regenerated sequences are not in any way a replica of the originally measured error sequence but a statistical equivalent of it. Other techniques for SHFMM parameter re-estimation includes: interval distribution curve fitting method

Fig. 3. Photograph of experimental setup for the signal level noise measurement.

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Fig. 4. Schematics of experimental setup for the signal level noise measurement.

in the time domain with its power spectral often localized well in a narrow region around a constant frequency [31,32]. Likewise, background noise possess low amplitude and in time domain they are often distributed over a comparably small, fixed amplitude interval [31,32]. Background noise also possess constant PSD but is acknowledged to increase with decreasing frequency. On the contrary, in time domain, impulse noise exhibits extremely dynamic amplitude whose modulus ranges from very low to high values and are often characterized by peak with short duration and high amplitudes. It possesses high PSD often assumed to spread over wide portions of the frequency spectrum [31,32].

Based on low noise floor (low amplitude) and high noise floor (high amplitude) a threshold is set in milli-Volts in-between the low and high amplitude to categorize between impulse noise and background noise based on their attributes. Since, impulse noise exhibits high amplitude and background noise on the contrary exhibits low amplitude. The region where the peak value or amplitude is greater that the set threshold (P peak > Pthreshold ) is considered to be dominated by impulse noise events, while the region where the amplitude is less than the set threshold (P peak < Pthreshold ) is dominated by background noise. Note that, Ppeak (Impulse noise)  Ppeak (Background noise). A more elaborate literature on the categorization of each noise type can be found in [31]. Hence, the occurrence of a particular noise is denoted by a ‘‘1” typifying that transmission error will occur at this time, while the nonoccurrence of such noise type is denoted by a ‘‘0” typifying an error-free transmission. Three error sequences are thus generated for the three major NB-PLC noise considered. 6. Measurement and modelling results analysis and discussion The measurement results, model results and analysis is presented in this section. Fig. 5 shows a spectrogram of a noise sample measured at one of the low voltage sites showing the cyclostationary features of the measure noise both in time and frequency. 6.1. Signal level measurement noise occurrences recorded Interesting observations were recorded for both measurement locations, taking into consideration both the inter-arrival time of measurements obtained and the amount of impulses recorded per chosen time frame. It can be observed from Tables 1 that on the average, more impulsive noise were recorded at the residential site when compared to measurements of noise impulses recorded for the laboratory site for both 2015 and 2016 measurement. Similarly, for both 2015 and 2016 measurement, it can also be observed that more noise impulses were recorded for the highly disturbed scenario (scenario II) when compared to the mildly disturbed scenario (scenario I). This is due to the heavy disturbances experienced and closeness of noise sources to the measurement point in Scenario II, as opposed to Scenario I where mild disturbances were recorded. This observation justifies the reason for intensive and continuous experimental measurement campaign from time to time so as to obtain a very rich set of PLC noise samples that represents the ever fluctuating, non-white, non-Gaussian and non-stable noise that inhibit transmission on the NB-PLC channel. 6.2. Re-estimated state transition probability matrix for the SHFMMs

Fig. 5. Spectrogram of a noise sample measured at one of the low voltage sites.

The re-estimated state transition probability matrix portrays the channel changeover from particular state to another dependent on the B matrix (output symbol probability generation matrix), often influenced by the power line channel status.

Table 1 Noise measurement taken in 2015 and 2016 (Scenario I and Scenario II). Type of Noise

NBN NBN BN BN IN IN

Location

Residential Laboratory Residential Laboratory Residential Laboratory

No. of Occurrences (2015)

No. of Occurrences (2016)

Scenario I

Scenario II

Scenario I

Scenario II

47 69 27 54 43 38

84 72 92 60 86 57

76 35 71 42 59 25

102 67 98 72 98 46

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A.D. Familua, L. Cheng / Int. J. Electron. Commun. (AEÜ) 74 (2017) 21–30

Tables 2 and 3 show the most probable estimated state transition probability values (amidst the 81 chosen initial parameters) for 2015 measured error sequences for Scenarios I and II respectively. Likewise, Tables 4 and 5 show the most probable estimated state transition probability values for 2016 measured error sequences for Scenarios I and II respectively. A close comparisons of Tables 2 and 3 shows dissimilar state transition probability values, which is as a result of non-uniformity in the error distribution for each noise models influenced by the fluctuating power line channel status. Similarly, comparing Tables 4 and 5, we can see that the state transition probability values are dissimilar, which can also be attributed to non-uniformity in the error distribution of the noise models influenced by the channel status.

6.3. Error probabilities comparison for the SHFMMs The fitness of a model can be ascertained by the close agreement between the error probabilities of the original sequence and the model generated sequence. A close agreement shows that the model is a best fit model. It should be noted that the regenerated sequence does not reproduce the original error sequence gotten from measurement, but it gives a statistically regenerated sequence that captures the distribution of errors in the measured error sequence. Table 6 shows the error probabilities (original sequence vs. model regenerated sequence) for the two scenarios (for 2015 noise models). Similarly, Table 7 shows the error probabilities for the two scenarios (for 2016 noise models). A close

Table 2 Re-estimated state transition probability for 2015 noise models (Scenario I). Residential

Laboratory

P

NBN

BN

IN

NBN

BN

IN

p11 p13 p22 p23 p31 p32 p33

0.9852 0.0148 0.8049 0.1951 0.6673 0.2091 0.1236

0.9946 0.0054 0.8656 0.1344 0.2815 0.5711 0.1474

0.9946 0.0054 0.8678 0.1322 0.4763 0.3966 0.1271

0.9867 0.0133 0.7945 0.2055 0.7955 0.0885 0.1160

0.9963 0.0037 0.9594 0.0406 0.1866 0.7253 0.0881

0.9968 0.0032 0.7835 0.2165 0.4656 0.4013 0.1331

Table 3 Re-estimated state transition probability for 2015 noise models (Scenario II). Residential

Laboratory

P

NBN

BN

IN

NBN

BN

IN

p11 p13 p22 p23 p31 p32 p33

0.9856 0.0144 0.8148 0.1852 0.3959 0.4319 0.1721

0.9873 0.0127 0.8179 0.1821 0.3554 0.5248 0.1198

0.9873 0.0127 0.9380 0.0620 0.2735 0.5049 0.2216

0.9878 0.0122 0.9092 0.0908 0.6945 0.2396 0.0658

0.9893 0.0107 0.8010 0.1990 0.4739 0.3447 0.1814

0.9948 0.0052 0.9811 0.0189 0.2264 0.7151 0.0585

Table 4 Re-estimated state transition probability for 2016 noise models (Scenario I). Residential

Laboratory

P

NBN

BN

IN

NBN

BN

IN

p11 p13 p22 p23 p31 p32 p33

0.9927 0.0073 0.8937 0.1063 0.2893 0.5964 0.1143

0.9910 0.0090 0.8704 0.1296 0.4733 0.3800 0.1467

0.9934 0.0066 0.8947 0.1053 0.3913 0.4808 0.1279

0.9954 0.0046 0.8917 0.1083 0.3927 0.4573 0.1500

0.9958 0.0042 0.8255 0.1745 0.4296 0.3966 0.1738

0.9967 0.0033 0.9786 0.0214 0.1425 0.6661 0.1914

Table 5 Re-estimated state transition probability for 2016 noise models (Scenario II) Residential

Laboratory

P

NBN

BN

IN

NBN

BN

IN

p11 p13 p22 p23 p31 p32 p33

0.9901 0.0099 0.8620 0.1380 0.3570 0.4983 0.1446

0.9924 0.0076 0.8388 0.1612 0.3251 0.5193 0.1556

0.9966 0.0034 0.9667 0.0333 0.0640 0.8328 0.1032

0.9851 0.0149 0.8508 0.1492 0.5023 0.3962 0.1015

0.9867 0.0133 0.7555 0.2445 0.5519 0.3363 0.1118

0.9905 0.0095 0.8109 0.1891 0.6008 0.3356 0.0636

27

A.D. Familua, L. Cheng / Int. J. Electron. Commun. (AEÜ) 74 (2017) 21–30 Table 6 Error probabilities ((Pe ) – original sequence and (P e ) – model regenerated sequence) for the 2015 noise models (Scenario I and Scenario II). Residential

P e (Scenario I) P e (Scenario I) P e (Scenario II) P e (Scenario II)

Laboratory

NBN

BN

IN

NBN

BN

IN

0.0221 0.0219

0.0215 0.0211

0.0163 0.0159

0.0150 0.0147

0.0098 0.0094

0.0081 0.0080

0.0306 0.0302

0.0299 0.0296

0.0280 0.0278

0.0169 0.0167

0.0195 0.0192

0.0114 0.0112

Table 7 Error probabilities ((Pe ) – original sequence and (P e ) – model regenerated sequence) for the 2016 noise models – (Scenario I and Scenario II). Residential

P e (Scenario I) P e (Scenario I) P e (Scenario II) P e (Scenario II)

Laboratory

NBN

BN

IN

NBN

BN

IN

0.0241 0.0239

0.0225 0.0223

0.0182 0.0180

0.0104 0.0103

0.0124 0.0122

0.0062 0.0060

0.0410 0.0408

0.0369 0.0366

0.0310 0.0307

0.0239 0.0237

0.0265 0.0263

0.0227 0.0225

comparison between Scenarios I and II in Table 6 and similarly, Table 7 shows a little deviation but close agreement between the error probabilities of the originally measured sequence and the model obtained sequence. Otherwise, both 2015 and 2016 noise models for the two scenarios considered shows that close agreement exist between the error probabilities of the originally measured sequence and the model obtained sequence. This establishes the fact that the estimated model parameters is best fit and that the re-estimated parameters are the most likely/ probable model parameters that generated the measured error sequences.

The computed forward and backward probability variables if not properly scaled has the tendency of exponentially gravitating towards zero resulting in numerical underflow. Hence, proper scaling of these variables is vital. Refer to [22,27] for an elaborate discussion on proper scaling of these variables. Note that, PrðEjCÞ is mathematically obtained based on the scaling factor C t as represented in Eq. (9). For observation sequence where T is large, C t is very small and is usually obtained mathematically by using a logarithmic computation of the values of C t as shown in Eq. (10) and accordingly, Eq. (10) symbolizes the log-likelihood ratio.

Pr½EjC ¼ 6.4. Comparison of the log-likelihood ratio plots for the SHFMMs The log-likelihood ratio is also used to access the fitness of the SHFMMs. In other words, the log-likelihood ratio establishes the probability of how probable it is that the observed data (error sequence) generated the SHFMM parameter estimates. The purpose of any given model is to realize model parameters which maximizes likelihood function value. The value of the loglikelihood ratio is constantly negative. Note that high loglikelihood ratios (close to zero) depicts a better fitting model.

T Y

Ct

ð9Þ

t¼1

log10 Pr½EjC ¼

T X log10 C t

ð10Þ

t¼1

Fig. 6a and Fig. 6b show the log-likelihood ratio plot for Scenario I and Scenario II respectively for the 2015 noise model. Likewise, Fig. 7a and Fig. 7b show the log-likelihood ratio plot for Scenario I and Scenario II respectively for the 2016 noise model. It can be deduced that the derived models for each noise type gives best fit models as the log-likelihood values are relatively

Fig. 6. Log-likelihood ratio plot for Scenario I and Scenario II (2015).

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Fig. 7. Log-likelihood ratio plot for Scenario I and Scenario II (2016).

close to zero. A close comparison between Fig. 6a and Fig. 6b show that Fig. 6a has better log-likelihood ratio value when compared to Fig. 6a. This is because the probabilities of errors for the error sequences obtained in the mildly disturbed scenario in Fig. 6a are lesser than the error probabilities of the error sequence obtained for the heavily disturbed scenario in Fig. 6b, that is, error sequence in Fig. 6b has the highest probability. And based on the trend recorded the lower the error probability the closer to zero the log-likelihood ratio values and vice versa. Similar trend was recorded for Fig. 7a and Fig. 7b, with Fig. 7a showing a better log-likelihood ratio value when compared to Fig. 7a which can also be attributed to the fact that lesser error probabilities were recorded for Fig. 7a, as opposed to larger error probabilities recorded for Fig. 7b error sequences. 6.5. Error-free run distribution plots These plots is another statistical measure of the fitness of the model. Denoted by Prð0m j1Þ, it depicts the probability of having series of m successive error-free runs after the occurrence of an error. Where m indicates the maximum length of intervals of the EFRD. The EFRD plots shows both the plot for the original measured sequences and the model regenerated sequences. It is noteworthy to state that the error-free run for the model regenerated sequence does not reproduce the measured error sequence, but gives a

statistical version of it. Fig. 8a and Fig. 8b show the error-free run distribution plot for Scenario I and II respectively for the 2015 noise models. Similarly, Fig. 9a and Fig. 9b show the error-free run distribution plot for Scenario I and II respectively for the 2016 noise models. A close comparison between Fig. 8a and Fig. 8b show that higher maximum length of intervals m for the error-free run distribution were recorded for Fig. 8a due to the low error probabilities as opposed to lesser maximum length of intervals m recorded for Fig. 8b due to high error probabilities in the sequence. This invariably means that the higher the probability of error, the lesser the maximum length of interval recorded for a sequence and vice versa. Likewise, a comparison of Fig. 9a and Fig. 9b show similar trend, with higher maximum length of intervals m recorded for the error-free run distribution plot of Fig. 9a due to the low error probabilities as opposed to lesser maximum length of intervals m recorded for Fig. 9b due to high error probabilities in the sequence. It is evident that the EFRD (Prð0m j1Þ) is a monotonically decreasing function of m in such a way that Prð00 j1Þ ¼ 1 and Prð0m j1Þ ! 0, which implies that it consistently decreases and never increases in value [33], as seen in Figs. 8a-9b. 6.6. Chi-squared test and mean square error for the models Apart from the popular error-free run distribution (EFRD) metric used in validating the accuracy of models like Fritchman model,

Fig. 8. Error-free run distribution plot for Scenario I and Scenario II (2015).

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Fig. 9. Error-free run distribution plot for Scenario I and Scenario II (2016).

Table 8 Chi-Square and MSE for 2015 Models (Scenario I and Scenario II). Chi-Square ðv2 Þ

Noise Types

BN Lab BN Res IN Lab IN Res NBN Lab NBN Res

MSE

Scenario I

Scenario II

Scenario I

Scenario II

4.8654e04 3.9518e04 5.0256e04 4.8651e04 3.2109e04 3.5906e04

7.2289e04 5.2603e04 4.3812e04 8.0959e04 4.4762e04 6.9497e04

5.5633e07 3.3296e07 3.9024e07 2.1978e07 7.9156e07 1.6484e06

1.0638e06 1.6064e06 1.0152e06 1.5915e06 8.5653e07 2.2222e06

Table 9 Chi-Square and MSE for 2016 Models (Scenario I and Scenario II). Chi-Square ðv2 Þ

Noise Types

BN Lab BN Res IN Lab IN Res NBN Lab NBN Res

MSE

Scenario I

Scenario II

Scenario I

Scenario II

7.0851e04 6.4335e04 7.2179e04 5.8976e04 6.4132e04 1.2000e03

3.5233e04 5.1787e04 4.8119e04 5.8078e04 4.7711e04 5.5178e04

6.3052e07 9.1912e07 4.7847e07 8.5324e07 6.3291e07 1.4199e06

1.3889e06 5.6980e07 1.2195e06 5.0916e07 1.8605e06 8.1633e07

the Mean Square Error (MSE) and the Chi-Square test ðv2 Þ are another popular metrics used to carry out the fitness or accuracy check of SHMMs in other to ascertain the closeness between the measured original error sequences and the SHFMM statistically re-generated error sequences. Table 8 and Table 9 show tables of the computed Chi-Square and MSE values for the most probable parameter (1) that produced the 2015 and 2016 measurements respectively. The smaller the v2 and MSE values, the fitter the model (indicating close agreement between original sequence and model generated sequence), hence, the estimated SHFMM parameters with the best (smallest) v2 and MSE values is chosen and tabulated in Table 8 and Table 9. This ascertains that the models are best fit model as both the distribution of the original sequence and model re-generated sequence are close match as shown in Fig. 8a-9b.

prevalent on the in–home CENELEC A-Band PLC channel is achieved. The necessity for precise and mathematically inclined statistical model based on experimental noise measurement and not simulation based statistical channel model motivated the work carried out in this article. An experimental signal level measurement and modeling of the three major NB-PLC noise is carried out for mildly and heavily disturbed scenarios at both residential and laboratory indoor environment. The resulting mathematically inclined statistical NB-PLC channel models typifies the statistical distribution of errors obtainable on the NB-PLC channel and the behavior of the ever fluctuating fading prone NB-PLC channel. Thus, these statistical models are vital in facilitating the design and evaluation of robust modulation and coding techniques for PLC noise mitigation to achieve reliable communication and reduce performance degradation in NB-PLC system.

7. Conclusion

Acknowledgement

In this article, measurement and a First-Order Semi-Hidden Fritchman Markov modeling of background noise, narrowband noise and impulse noise, the three major NB-PLC noise types

The financial assistance of the Centre for Telecommunication Access and Services (CeTAS), the University of the Witwatersrand, Johannesburg, South Africa is hereby acknowledged.

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Ayokunle Damilola Familua received his B. Eng Electrical and Electronics Engineering degree from the Federal University of Technology Akure, Nigeria in 2005, MSc (Engineering) degree in Electrical from the School of Electrical and Information Engineering, University of Witwatersrand, Johannesburg, South Africa in 2013. He is presently studying towards his PhD degree in Electrical (Telecommunications major) at the same institution. Ayokunle is a recipient of the best student paper award in the 2015 IEEE ISPLC conference held in Austin, Texas. His area of research interest includes: Digital communication, especially Power Line Communications, Visible Light Communications, Mobile Communications and Machine Learning algorithms for channel modeling.

Ling Cheng received the degree B. Eng. Electronics and Information (cum laude) from Huazhong University of Science and Technology (HUST) in 1995, M. Ing. Electrical and Electronics (cum laude) in 2005, and D. Ing. Electrical and Electronics in 2011 from University of Johannesburg (UJ). His research interests are in Digital Communications, especially Power Line Communications and Information Theory, especially coding techniques. In 2010, he joined University of the Witwatersrand where he was promoted to Associate Professor in 2015. He has served as the Secretary of IEEE South African Information Theory Chapter since 2012. He has been a visiting professor at four universities and the principal advisor for over twenty full research post-graduate students. He has published more than 50 research papers in journals and conference proceedings. He was awarded the Chancellors medal in 2005. The IEEE ISPLC 2015 best student paper award was made to his Ph.D. student in Austin.