Merging the Bernoulli-Gaussian and Symmetric Alpha-Stable Models for Impulsive Noises in Narrowband Power Line Channels

Merging the Bernoulli-Gaussian and Symmetric α -Stable Mo dels for Impulsiv e Noises in Narro wband P o we r Li ne Channels Bin Han a, ∗ , Y ang Lu c , Kai W an c , Hans D. Sc hotten a,b a Institute of Wir eless Commun ic ation (WiCon), T e chnische Universit¨ at Kaiserslautern, 67663 Kaiserslautern, Germany b R ese ar ch Gr oup Intel ligent Networks, German Re se ar ch Center for Artificial Intel ligenc e (DFKI GmbH), 67663 Kaiserslautern, Germany c Glob al Ener gy Inter c onn e ction R ese ar ch Institute (GEIRI), State Grid Corp or ation of China (SGCC), Beijing 102209, China Abstract T o mo del impulsiv e noise in p o w er line c ha nnels, b o th the Bernoulli-Gaussian mo del and the symmetric α -stable mo del are usually applied. T o w ards a merge of existing noise mea- suremen t databases and a simplification of comm unication system design, the compatibility b et w een the t wo mo dels is of in terest. In this pap er, w e sho w that they can be appro ximately con v erted to eac h other under certain constrains, although nev er generally unified. Based on this, w e pro p ose a fast mo del con v ersion. Keywor ds: impulsiv e noise, p ow er line comm unication, non- Gaussian mo del, stable random pro cess, non-stationary random pro cess 1. In tro duction Impulsiv e noise, whic h is generated b y nume rous electrical devices connecte d to the p ow er grid, ubiquitously exists in p o w er line c hannels. These noise p eaks with high amplitude can erase most comm unication signals transmitted ov er the p ow er line c hannel, and are pro v ed to significan tly impact t he p erfo rmance of p ow er line comm unication (PLC) systems. Mean while, the o ccurrence of suc h impulses is challenging to predict b ecause of its highly ∗ Corresp o nding author Email addr esses: bi nhan@e it.un i-kl.de (Bin Han), luya ng@ge iri.s gcc.com.cn (Y ang Lu), wankai @geir i.sgcc.com.cn (Kai W an), scho tten@e it.uni-kl.de (Hans D. Schotten) Pr eprint submitt e d to Physic al Communic ation January 9, 2019 non-stationary dynamics. An intensiv e in terest in modeling noises of this t yp e therefore arises, driv en b y the demand of impulsiv e noise mitigation for PLC. Since ov er tw o decades, differen t time-domain mo dels hav e b een prop osed or a do pted to c ha racterize them, including: • the Middleton’s Class-A (MCA) mo del [1], whic h c haracterizes the sparsit y of high- amplitude spik es in noise; • the Bernoulli-Gaussian (BG ) mo del [2], whic h considers the impulsiv e noise as a Bernoulli sequence mo dulated to a Ga ussian noise with On- Off-Keying; • the Symmetric α - Stable (S α S) mo del [3], whic h statistically describ es the distribution of noise amplitude; • the Mark ov-Middleton mo del [4] , whic h is an extended MCA mo del where the MCA parameters randomly switc h amo ng sev eral states; • the Mark o v-Gaussian mo del [5], which exten ds the BG mo del b y replacing the Bernoulli pro cess with a Mark o v pro cess. Comparativ e studies on t hese mo dels hav e b een rep ort ed [6, 7]. G enerally , the Mark ov - Middleton a nd Mark o v-Gaussian mo dels are enhanced v aria tions of the MCA and BG mo d- els, resp ectiv ely , whic h intro duce Marko v c hains to describ e the burst noise phenomenon. When ignoring noise bursts, the MCA mo del, the BG mo del and the S α S mo del a r e mainly used. As p oin ted out in [6 ], the BG mo del is usually preferred ov er the MCA mo del for its b etter tractability . Mean while, fo cusing on the statistics instead of the dynamics o f noise, the S α S mo del outp erforms the MCA mo del with its excellen t p erformance in fitting t he hea vy-tailed probability densit y function (PDF) of noise amplitude. Comparing the BG mo del with the S α S mo del, each side has its ow n pros and cons. On the one hand, the BG mo del is cost-friendly for implemen ta t io ns, a nd can b e easily extended to the Mark o v-Gaussian mo del to describ e burst noise. In contrast, the S α S mo del cannot mo del burst noise, and is expensiv e to compute due to t he lac k of generic close form PDF. On the other hand, the S α S mo del can accurately matc h the fa t -tailed distribution of impulsiv e 2 noises, a nd the parameters can b e consisten tly estimated from the amplitude statistics [8]. The BG mo del, in comparison, do es not guaran tee a go o d fit for the o v erall sample amplitude distribution, and its parameter estimation highly relies on the accuracy o f impulse detection and extraction [7]. In PLC system design, up on sp ecific requiremen ts of differen t a pplications, one or sev eral mo dels lis ted ab ov e can b e preferred o ve r the others and therefore fle xibly selec ted. F or example, when it is essen tial to consider impulsiv e noises with broader bandwidth than the signals, suc h like in cognitive PLC systems that flexibly select the w orking frequ ency range [9], the S α S mo del can b e preciser [6]. In contrast, for cost-critical narro wband PLC applications suc h as smart grid systems [1 0], the BG mo del can b e mo r e practical as a high computational effort is required for the S α S mo del. Ho w ev er, fro m the p ersp ectiv es of c hannel me asuremen t and system ev aluation, there is a solid demand of unification or con v ersion b et w een differen t noise mo dels. Fir st, field measu remen ts of p o w er line noise s are usually exp ensiv e in cost and effort, and the measured results are usually rep ort ed and a r c hiv ed in the form of estimated mo del parameters instead of raw data, e.g. as it is done in [1 1]. A unification among v a rious noise mo dels will enable to reuse the v alua ble data o f measuremen ts in differen t applications and thereb y greatly sav e the measuring cost. Moreo v er, in the ev aluation of PLC systems, to ensure the generality of r esults, it is often necessary to rep eat the test with noises generated by different mo dels, a s rep orted in [12]. A noise mo del unification also helps reduce suc h effort b y a significan t degree. F ortunat ely , the Marko v-Middleton and Mark o v-Gaussian mo dels are endogenously com- patible with the MCA and BG mo dels, resp ectiv ely . Mean while, the BG mo del has b een demonstrated as capable to approx imate t he MCA mo del with simple adjustmen ts [6]. Ho w- ev er, the compatibility b et w een the BG and S α S mo dels, to the b est of our kno wledge, has nev er b een thro ughly inv estigated y et. F o cusing on t his unsolv ed problem, in this pap er w e: 1. demonstrate the similar perfo rmance of these t w o mo dels in c har a cterizing impulsiv e p ow er line noises, 2. deriv e the quasi-stability o f BG noise in the con text of PLC, and 3. pro - p ose a fast and appro ximate p olynomial conv ersion from the the mo del to the S α S mo del. The remainder of this man uscript is org anized as follow s. W e review the BG and 3 S α S mo dels in Section 2. Then w e comparativ ely ev aluate b oth o f them w ith filed mea- suremen ts of p o w er line noise in Section 3. Subsequen tly , in Section 4 w e analyze the compatibilit y b etw een the t w o mo dels. Our results indicate that they cannot not b e gen- erally unified, but a re compatible with eac h ot her to a satisfactory degree, especially when the impulses a re sparse and limited in p o w er. Afterwards , in Section 5, we in v ok e existing signal pro cessing tec hniques to fit BG pro cesses with t he S α S mo del, in order t o build a p olynomial mo del of conv ersion b etw een BG and S α S parameters, and ev aluat e its fitting p erformance. The details of all exp erimen tal results are agg regated in Section 6. A t the end w e close this pap er with our conclusions in Section 7. 2. Bernoulli-Gaussian Mo del and S α S Mo del 2.1. Bernoul li-Gaussian Mo del The BG mo del describ es a sampled impulsiv e noise as n I ( k ) = σ 2 I n G ( k ) φ ( k ) (1) where k ∈ Z is the sample index, σ I is the standard deviation of the impulsiv e noise amplitude and n G ( k ) is a normalized white Ga ussian noise with a unit y pow er. φ ( k ) is a Bernoulli pro cess that describ es the o ccurrence of impulses: φ ( k ) =      1 z ( k ) ≤ p 0 otherw ise , (2) where z ( k ) ∼ U (0 , 1) and p ∈ [0 , 1] is the impuls e probability . In practice, it is usual to consider the mixture of impulsiv e noise and Gaussian bac kground noise as n BG ( k ) = σ 2 B n 0 ( k ) + σ 2 I n 1 ( k ) φ ( k ) , (3) where σ B is the standard deviation of the bac kground noise amplitude, n 0 ( k ) and n 1 ( k ) are t w o indep enden t G aussian noises with unit y p o w er. Th us, the PDF of n BG is f n BG ( x ) = 1 − p p 2 π σ 2 B e − x 2 2 σ 2 B + p p 2 π ( σ 2 B + σ 2 I ) e − x 2 2( σ 2 B + σ 2 I ) . (4) 4 2.2. Symmetric α -Stable Mo del A random v ariable X is called stable if and only if ∃ a ∈ R + , b ∈ R + , c ∈ R + , d ∈ R : aX 1 + bX 2 D = cX + d, (5) where X 1 and X 2 are t w o indep ende nt copies of X and A D = B denotes that A and B o b ey the same statistical distribution. Esp ecially , the distribution is called strictly stable if (5) holds for d = 0 [13]. The definition ab ov e is prov ed to ha v e the following equiv alence: X is α - stable if and only if ∃ (0 < α ≤ 2 , − 1 ≤ β ≤ 1) , γ 6 = 0 , δ ∈ R that X D = γ Z + δ, (6) where Z is a random v aria ble with characteris tic f unction ϕ ( t ) =      e j δt − γ | t | α [ 1 − iβ tan πα 2 sign( t ) ] α 6 = 1 e j δt − γ | t | [ 1+ iβ 2 π sign( t ) log | t | ] α = 1 . (7) Usually we refer to α a s the index o f stabilit y or characteristic exponent, β as the ske wness parameter, γ as the scale parameter and δ as the lo cation parameter [14]. Esp ecially , when β = 0, the PDF of X is symmetric about γ , and X is called sym metric α -Stable (S α S). Some sp ecial cases of α - stable distribution ha v e simple expressions of PDF, and hav e b een w ell- studied, including α = 2, β = 0 (Gaussian); α = 1, β = 0 (Cauc hy ) and α = 0 . 5, β = 1 (L ´ evy). Ho w ev er, field measuremen ts ha v e pro v ed that wh en applied on PLC noises, the α -stable mo del usually has parameters α ∈ (1 . 5 , 2), β ≈ 0 [15], whic h is a S α S case without an y closed-form presen tatio n of PDF. 3. T est with Field Measuremen t s T o ev aluate the p erformance of b oth mo dels, w e test them with filed measuremen ts. The ra w data w ere captured in 2012 at the B-phase live wire on the s econdary side of a lo w- v o ltage transformer, whic h w as lo cated in the p o w er distribution ro o m of a urban residen tial district in China. Th e measuremen t w as executed twice, once on Septem b er 9 th at 18:17 5 and the other on Septem b er 14 th at 01 :08, eac h la sting 200 ms with t he sampling rat e of 80 MSPS. Instead o f working with the raw measuremen t, we do wn-sample the data to 2 MSPS with a Butterw orth an ti-aliasing filter, due to tw o reasons: 1. The simple BG mo del is supp o sed to b e applied on impulsiv e noises in underspread c hannels, where the impulse width is significantly shorter than the sampling interv a l, so that the m ulti-path c hannel fading can b e ignored [16]. Under a v ery high sampling rate suc h as 80 MSPS, suc h appro ximation do es not hold an y more, and the noise m ust b e first de-con v oluted from a n unkno wn observ ation mat rix b efore fitted with the BG mo del, whic h would complicate the task. 2. Through the do wnsampling, the data size and hence t he computatio nal cost are re- duced. Mean while, b oth the BG and S α S mo dels a re consisten t to down sampling, so that their p erformance will not b e impacted. As indicated in [7 ], high-p ow ered nar ro wband in terferers in PLC systems a r e usually am- plitude mo dulated b y p erio dical en velopes sync hronous to the mains voltage, and can th us exhibit deterministic impulsiv e b ehav ior. Therefore, they may significan tly in terfere the analysis of sto c hastic impulsiv e comp onents in noise. Here w e inv ok e the Narro wband Re- gression metho d [17] to cancel p erio dically fluctuating narro wband in terferers from the do wn- sampled measuremen ts. Then we in vok e the blind BG impulse detector rep orted in [16] on the cleaned results to distinguish the spik es f rom the bac kground noise. An instance of noise prepro cessing result is depicted in Fig. 1. F or the BG mo del, the impulse ratio p a nd the Gaussian parameters ( σ 2 1 , σ 2 2 ) can b e easily estimated from the labeled data. F or the S α S mo del, w e applied McCullo c h’s metho d [18] for parameter estimation. Then, based o n the estimated mo del parameters, we n umerically generate BG and S α S noise sequences , eac h with a length o f 5 × 10 5 samples. Subsequen tly , w e compare the empirical PDFs of measured and sim ulated no ise amplitudes. T o ev aluate the fitnesses of b oth mo dels, w e further calculate the w eigh ted ro o t mean square error ( R MSE) for eac h 6 0 20 40 60 80 100 120 140 160 180 200 Time / ms -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Amplitude / V Downsampled Measurement After NBN Removal Detected Impulses Figure 1: A s a mple fra gment of the noise measur ement with pr epro cess ing r esults. of them: ǫ model = s Z + ∞ −∞ f meas ( x )[ f model ( x ) − f meas ( x )] 2 d x, (8) where f meas and f model are the empirical PDFs of measured and sim ulated noise amplitudes, resp ectiv ely . The results, whic h are detailed in Section 6.1, demonstrate tha t b oth mo dels exhibit fitting p erformance to a similar degree of satisfaction. 4. Stabilit y of B ernoulli-Gaussia n Pr o cesses T o w ards a unification b etw een the BG and S α S mo dels, the first question is: ar e BG pr o c e sses stable? T o answ er this, w e test if the BG mo del defined in (3) fulfills the require- men t of stabilit y defined in (5). Consider tw o idenp enden t and iden tically distributed (i.i.d.) v ariables X and Y whic h ar e generated according to (3), and their sum W : w ( k ) = x ( k ) + y ( k ) , ∀ k ∈ Z . (9) The PDF of W will then b e f W ( w ) = + ∞ Z −∞ f Y ( w − x ) f X ( x )d x. (10) 7 As b oth X and Y hav e t he same PDF as giv en in (4), w e ha v e 1 f W ( w ) = + ∞ Z −∞ f BG ( w − x ) f BG ( x )d x = 1 − 2 p + p 2 p 4 π σ 2 B e − w 2 4 σ 2 B + p 2 p 4 π ( σ 2 B + σ 2 I ) e − w 2 4( σ 2 B + σ 2 I ) + p − p 2 p 2 π (2 σ 2 B + σ 2 I ) e − (2 σ 2 B + σ 2 I − 1) w 2 2( σ 2 B + σ 2 I )(2 σ 2 B + σ 2 I ) + e − (2 σ 2 B + σ 2 I − 1) w 2 2 σ 2 B (2 σ 2 B + σ 2 I ) ! . (11) Clearly , ( 11) differs in form from (4), so that w e know Bernoulli-Ga ussian pro cesses are not gener al ly stable . Only in the following three sp ecial cases, W is quasi-stable as it appro ximately approache s to a Gaussian pro cess: lim p → 0 f W  w √ 2  = lim p → 0 f BG ( w ) ≈ f B ( w ); (12) lim p → 1 f W  w √ 2  = lim p → 1 f BG ( w ) ≈ f I ( w ); (13) lim σ 2 I − σ 2 B → 0 f W  w √ 2  = f BG ( w ) ≈ f B ( w ) ≈ f I ( w ) , (14) where f B ( w ) and f I ( w ) are the PD Fs of n B ( k ) and n I ( k ), resp ectiv ely . The approxim ation (12) is v alid in the context of PLC, where p is sufficien tly lo w although σ 2 I is significan t ly higher than σ 2 B , as it will be demonstrated and discus sed with details in Section 6.2. 5. Estimating S α S P arameters of BG Pro c esses So far, w e hav e demonstrated a certain but limited compatibility b etw een the BG and S α S mo dels for p o w er line noises. Subsequen t ly , driv en b y the interes t in the p erformance of approximating BG mo dels with S α S mo dels, w e at tempt to apply the S α S mo del on BG pro cesses. Our metho do logy can b e summarized as f ollo ws. First, w e generate a BG noise with parameters ( p, σ B = 1 , σ I ), and nor ma lize it to n BG with unit y p o w er. Then w e apply Mc- Collo c h’s S α S parameter estimator [18] on it. Subseq uen tly , w e call Chambers’ metho d [19 ] 1 F o r the detailed deriv ation see Appendix. 8 to sim ulate a S α S noise sequence n S α S with the estimated parameters ( ˆ α, ˆ γ ). Afterw a rds, w e ev aluat e the fitting p erformance with the Kullbac k-Leibler div ergence [20]: D KL ( φ BG | φ S α S ) = + ∞ Z −∞ φ BG ( x ) ln φ BG ( x ) φ S α S ( x ) d x, (15) where φ BG and φ S α S denote the empirical PDF of n BG and n S α S , resp ectiv ely . This dive r- gence, also kno wn as the relative entrop y , is normalized to the range [0 , 1] and measures the degree that φ BG div erges from φ S α S . D KL ( φ BG | φ S α S ) = 0 indicates that b oth the distributions are highly similar, if not the same, while D KL ( φ BG | φ S α S ) = 1 denotes a minimal similarit y b et w een the distributions. By rep eating this pro cess with differen t Bernoulli-Gaussian pa- rameters, w e are able to in v estigate the dep endencies of ( ˆ α, ˆ γ ) on ( p, σ 2 I /σ 2 B ). Due to the lac k of closed-form densit y functions, most con ven tional analytic metho ds of statistics cannot b e applied to estimate S α S parameters for cases where α > 1. Nev erthe- less, a v a r iety of numeric al tec hniques hav e b een dev elop ed for this task. According to [14], classical appro ac hes can b e generally classified into f our categories: metho ds of maxim um lik eliho o d [21, 22], metho ds of sample fra ctiles [18, 8], metho ds of sample c haracteristic func- tions [23, 24]. Besides, some recen t metho ds hav e b een dev elop ed based on other principles e.g. negativ e-or der momen ts [25], extreme order statistics [8] and p oin t pro cess [2 6 ]. As w e ha v e derive d in Section 4, BG pro cesses are not strictly stable. This disqualifies the deplo ymen t of some metho ds listed ab ov e on data g enerated by the BG mo del, esp ecially the metho ds that require segmen tat io n o f t he sample data. F or instance, according to our exp eriment, the extreme-order-statistics-based estimators in [8] significantly depend on the sample size, and fail t o conv erge, as shown in Fig. 2. Nev ertheless, our exp erimen t s ha v e prov ed that at least the regressiv e metho ds of Koutrouvelis [2 7, 23] and McCullo ch [18] can b e applied, whic h r eturn similar results. W e pro vide sample outputs obtained by McCulloch’s estimators in Section 6.3. 9 0 1 2 3 4 5 Sample Size 10 6 0 2 4 6 8 Estimated S S Parameters Figure 2: Tsihrintzis ’s extre me-order- statistic-based estimators in [8] of S α S mo del fail to conv erge when applied on Be r noulli-Gaussia n distributed data. The seg mentation parameter o f the a ppr oach is s e t to j √ N k , where N is the sample size. (a) Estimated S α S parameter ˆ α , with the p oly- nomially fitted surface (b) Estimated S α S parameter ˆ γ , with the p oly- nomially fitted sur face 0 0.002 0.004 0.006 0.008 0.01 10 12 14 16 18 20 22 24 26 28 30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 (c) The Ku llb ac k-Leibler div ergence D KL ( φ BG | φ S α S ) Figure 3: Applying the S α S mo del on Ber noulli-Gaussia n noise s with differen t specific a tions. Under each sp ecification, 5 × 10 6 samples of noise are genera ted. 10 6. Results and Discussion 6.1. Fitness T est of Mo dels on Fi eld Me a sur ements The fitting results of BG and S α S mo dels on field measuremen ts men tioned in Section 3 are illustrated in Fig. 4, with the R MSEs listed in T ab. 1. It can b e summarized that: 1. Both BG and S α S mo dels are able to effectiv ely describe the amplitude distribution of PLC noises, although not p erfectly . 2. When applied on the same PLC noise, the S α S mo del give s a wider main lob e in PDF than the measuremen t, while the BG mo del has a narrow er main lob e than the measuremen t. 3. F or b ot h mo dels, the fitting p erformance v aries with the noise scenario. -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Amplitude / V 10 -3 10 -2 10 -1 10 0 10 1 10 2 Probability Density Measurement Estimated BG model Estimated S S model (a) Noise measur ed on September 9 th at 18:17 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Amplitude / V 10 -3 10 -2 10 -1 10 0 10 1 10 2 Probability Density Measurement Estimated BG model Estimated S S model (b) Noise measured on Septem b er 14 th at 01:08 Figure 4: Fitting the noise statistics with BG and S α S mo dels. 6.2. Stability T est of B G D istribution T o ve rify (12)–(14), we conducted numeric al sim ulations: three i.i.d. BG random v a ri- ables X, Y , Z w ere generated, and another v ariable V w as obtained by V = ( X + Y ) × q V ar( Z ) V ar( X + Y ) . Then we compared the PDFs of V and Z under different BG mo del sp ecifica- tions. First, the deviations w ere fixed to σ B = 1 and σ I = 50, and t he test was executed 11 T a ble 1 : W eigh ted RMSEs of the BG and S α S mo dels on PLC noise meas urements. ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ Fitting Mo del Meas. Time Sept. 9, 18:17 Sept. 14 , 0 1:08 BG 0.0044 0.0019 S α S 0.0039 0.0134 -20 -15 -10 -5 0 5 10 15 20 Amplitude 10 -4 10 -3 10 -2 10 -1 10 0 Probability Density f Z , p=0.01 f V , p=0.01 f Z , p=0.05 f V , p=0.05 f Z , p=0.1 f V , p=0.1 f Z , p=0.5 f V , p=0.5 f Z , p=0.9 f V , p=0.9 f Z , p=0.95 f V , p=0.95 f Z , p=0.99 f V , p=0.99 (a) BG distrib ution with d ifferen t impulse r atios. σ B = 1, σ I = 50. -20 -15 -10 -5 0 5 10 15 20 Amplitude 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 Probability Density f Z , I =1 f V , I =1 f Z , I =5 f V , I =5 f Z , I =10 f V , I =10 f Z , I =50 f V , I =50 (b) BG d istribution with different impu lse devi- ations. p = 0 . 1, σ B = 1. Figure 5: T esting the stability of B G distribution with resp ect to the mo del parameter s. 12 for differen t v a lues of p . T he results are shown in Fig. 5( a). It can b e observ ed that the PDFs of V and Z match each other w ell when the v alue of p is sufficien tly high or low, but deviate from each other as p approache s to 0 . 5, whic h matches our theory . Subsequen tly , w e set p = 0 . 1 , σ 2 B = 1, and rep eated the test for differen t v alues o f σ I . The results in Fig. 5(b) sho w that V ha s its distribution mor e similar to Z under lo w er impulse p ow er, as w e hav e exp ected. In the con text of PLC, according to [28], ov er the broad band up to 20 MHz , the impulse p ow er is usually b y 10 dB to 30 dB, i.e., 10 to 1000 t imes higher than that of the back ground noise. Mean while, the impulse probabilit y remains b elow 0 . 35% ev en under the heavies t disturbance, and falls do wn to 0 . 00135% under the w eak disturbance. Comparing these to the r esult in Fig. 5(a), it is reasonable to consider the BG mo del as quasi-stable when applied on p ow er line noises. 6.3. S α S Par ameter Estimation of BG Pr o c esses Setting σ B = 1 with different v alues of 0 . 01% ≤ p ≤ 1% and 10 dB ≤ σ 2 I σ 2 B ≤ 30 dB, we applied McCullo c h’s estimators on randomly generated BG pro cesses, and got the results in F ig. 3. W e can observ e from the Kullba ck-Leibler div ergence that the conv ersion can success fully pr ovide a precise appro ximation when either the impulse-to-back ground p ow er ratio σ 2 I σ 2 B or the impulse ratio p is limited. In the cases with extremely in tensiv e impulses where b oth σ 2 I σ 2 B and p are hig h, the fitness of mo del conv ersion sinks dramatically . W e also fitted (2,2)- p olynomial surfaces for ˆ α and ˆ γ to supp ort fast and approximate conv ersions from p o w er-normalized BG mo del to S α S mo del, the results are listed in T ab. 2. 7. Conclusion In this pap er, w e ha ve studied the compatibilit y b etw een t w o widely-use d mo dels for impulsiv e p o w er line noises: the BG mo del and the S α S mo del. With field measureme n t test, w e ha v e prov ed tha t they give differen t but similarly acceptable results when used to mo del p ow er line noise. Then we ha v e pro v ed that the BG distribution is neither strictly stable nor strictly fat-tailed, so that no analytical unification b etw een BG and S α S mo dels is feasible. 13 T a ble 2 : (2,2)-p o lynomial surface fitting re s ults o f ˆ α and ˆ γ as function o f p and σ I σ B . ˆ α ˆ γ c 00 2.005 0.5779 c 10 -1.457 6.256 c 01 − 5 . 575 × 10 − 4 0.01707 c 20 -40.36 2123 c 11 -0.1128 -2.43 c 02 1 . 426 × 10 − 5 − 5 . 249 × 10 − 4 RMSE 1 . 7 1 5 × 1 0 − 3 2 . 433 × 10 − 2 Fitting mo del: f ( x, y ) = c 00 + c 10 x + c 01 y + c 20 x 2 + c 11 xy + c 02 y 2 , where x = p , y = 20 log 10 ( σ I /σ B ) Nev ertheless, when the impulses are sparse and not extremely strong in p ow er, whic h is the common case of p o w er line noises, BG pro cesses can b e appro ximately considered as quasi- stable, so that a n approx imate a nd empirical mo del con v ersion is p o ssible. Based o n this result, we ha v e prop osed a fast (2,2 ) - p olynomial conv ersion from BG mo del to S α S mo del. This fast conv ersion can b e applied to merge reference p ow er line noise scenarios based on differen t mo dels, and hence to simplify the p erformance ev aluation o f PLC sys tems. App endix A. The PDF of Sum of Two I.I.D. BG N oises The detailed deriv ation of (11) follo ws b elo w: 14 f W ( w ) = + ∞ Z −∞ f BG ( w − x ) f BG ( x )d x = + ∞ Z −∞ " 1 − p p 2 π σ 2 B e − ( w − x ) 2 2 σ 2 B + p p 2 π ( σ 2 B + σ 2 I ) e − ( w − x ) 2 2( σ 2 B + σ 2 I ) # × " 1 − p p 2 π σ 2 B e − x 2 2 σ 2 B + p p 2 π ( σ 2 B + σ 2 I ) e − x 2 2( σ 2 B + σ 2 I ) # d x = + ∞ Z −∞ " (1 − p ) 2 2 π σ 2 B e − ( w − x ) 2 + x 2 2 σ 2 B + p 2 2 π ( σ 2 B + σ 2 I ) e − ( w − x ) 2 + x 2 2( σ 2 B + σ 2 I ) + p (1 − p ) 2 π σ B p σ 2 B + σ 2 I e − ( w − x ) 2 2( σ 2 B + σ 2 I ) − x 2 2 σ 2 B + e − x 2 2( σ 2 B + σ 2 I ) − ( w − x ) 2 2 σ 2 B !# d x = + ∞ Z −∞ (1 − p ) 2 2 π σ 2 B e − 2 x 2 − 2 wx + w 2 2 σ 2 B d x + + ∞ Z −∞ p 2 2 π ( σ 2 B + σ 2 I ) e − 2 x 2 − 2 wx + w 2 2( σ 2 B + σ 2 I ) d x + + ∞ Z −∞ p (1 − p ) 2 π σ B p σ 2 B + σ 2 I e − (2 σ 2 B + σ 2 I ) x 2 − 2 σ 2 B wx + σ 2 B w 2 2 σ 2 B ( σ 2 B + σ 2 I ) +e − (2 σ 2 B + σ 2 I ) x 2 − 2( σ 2 B + σ 2 I ) wx +( σ 2 B + σ 2 I ) w 2 2 σ 2 B ( σ 2 B + σ 2 I ) ! d x. (A.1) Let σ 1 = p σ 2 B + σ 2 I , σ 2 = p 2 σ 2 B + σ 2 I = p σ 2 B + σ 2 1 : 15 f W ( w ) = (1 − p ) 2 2 π σ 2 B + ∞ Z −∞ e − 2 x 2 − 2 wx + w 2 2 σ 2 B d x + p 2 2 π σ 2 1 + ∞ Z −∞ e − 2 x 2 − 2 wx + w 2 2 σ 2 1 d x + p (1 − p ) 2 π σ B σ 1 + ∞ Z −∞ e − σ 2 2 x 2 − 2 σ 2 B wx + σ 2 B w 2 2 σ 2 B σ 2 1 + e − σ 2 2 x 2 − 2 σ 2 1 wx + σ 2 1 w 2 2 σ 2 B σ 2 1 ! d x = 1 − 2 p + p 2 p 4 π σ 2 B e − w 2 4 σ 2 B + ∞ Z −∞ 1 p 2 π σ 2 B e −  √ 2 x − √ 2 2 w  2 2 σ 2 B d √ 2 x − √ 2 2 w ! + p 2 p 4 π σ 2 1 e − w 2 4 σ 2 1 + ∞ Z −∞ 1 p 2 π σ 2 1 e −  √ 2 x − √ 2 2 w  2 2 σ 2 1 d √ 2 x − √ 2 2 w ! + p − p 2 p 2 π σ 2 2 e − ( σ 2 2 − 1) w 2 2 σ 2 1 σ 2 2 + ∞ Z −∞ 1 p 2 π σ 2 B σ 2 1 e − ( σ 2 x − σ B σ 2 w ) 2 2 σ 2 B σ 2 1 d  σ 2 x − σ B σ 2 w  + p − p 2 p 2 π σ 2 2 e − ( σ 2 2 − 1) w 2 2 σ 2 B σ 2 2 + ∞ Z −∞ 1 p 2 π σ 2 B σ 2 1 e − ( σ 2 x − σ 1 σ 2 w ) 2 2 σ 2 B σ 2 1 d  σ 2 x − σ 1 σ 2 w  . (A.2) Note that all the in tegration terms in the last step of (A.2) follo w the form of normal distribution. Hence, all the in tegrations ov er ( −∞ , ∞ ) ha v e the v alue of 1, and w e get: f W ( w ) = 1 − 2 p + p 2 p 4 π σ 2 B e − w 2 4 σ 2 B + p 2 p 4 π σ 2 1 e − w 2 4 σ 2 1 + p − p 2 p 2 π σ 2 2 e − ( σ 2 2 − 1) w 2 2 σ 2 1 σ 2 2 + e − ( σ 2 2 − 1) w 2 2 σ 2 B σ 2 2 ! . (A.3) References [1] D. Middleto n, Pro cedures for Deter mining the Parameters of the First-O r der Canonical Mo dels of Class A and Class B E lectromag netic Interference, T ransactions on Elec tr omagnetic Compatibility (3 ) (1 9 79) 190–2 08. [2] M. Ghos h, Analysis of the E ffect of Impulse Noise o n Multicarrie r a nd Single Car r ier QAM Sy s tems, IEEE T ransactions on Communications 44 (2) (1 996) 1 45–1 47. [3] G. Laguna-Sanchez, M. Lopez - Guerrero , An Exp erimental Study of the Effect of Human Activit y o n the Alpha-Stable Character istics of the Po wer-Line Noise, in: 18th IEEE International Symp osium on Po wer Line Communications a nd its Applications (ISPLC), IEEE , 2 014, pp. 6 – 11. 16 [4] G. Ndo, F. La b eau, M. K a ssouf, A Markov-Middleton Mo del for Bur s ty Impulsive Noise: Mo deling and Rece iver Design, IEEE T ransactions o n Pow er Delivery 28 (4) (20 13) 23 17–2 325. [5] D. F ertonani, G. Colavolpe, On Reliable Communications over C ha nnels Impaired by Bursty Impulse Noise, IEEE T ransactions o n Communications 57 (7). [6] T. Shongwe, A. J. H. Vinck, H. C. F er reira, On Impulse Noise a nd Its Mo dels, in: 18 th IEE E Inter- national Symposium on Po wer L ine Comm unications and its Applications (ISPLC), IEEE, 2014, pp. 12–17 . [7] B. Han, V. Stoica, C. Kais er, N. Otterbach, K . Dostert, Noise Char acterizatio n and Emulation for Low-V oltage Pow er Line Channels acro ss Narr owband and Broa dband, Digital Signa l Pr o cessing 6 9 (2017) 259–2 74. [8] G. A. Tsihrintzis, C. L. Nikias, F ast Estimation of the Parameters o f Alpha-Stable Impulsive Int erfer- ence, IE EE T ransa ctions on Signal Pr o cessing 44 (6) (1996) 149 2–15 03. [9] W. Liu, G. Bumiller, H. Gao, On (Pow er -) Line Defined PLC System, in: 20 14 18th IE EE International Sympo sium o n Po wer Line Communications and its Applicatio ns (ISPLC), IEE E, 20 14, pp. 8 1–86 . [10] S. Galli, A. Scaglione, Z. W ang, F or the Gr id and thro ug h the Grid: The Role of Po wer Line Commu- nications in the Smart Grid, Pro ceedings o f the IEEE 99 (6 ) (2011) 998– 1027. [11] J. A. Cort´ es, L. Diez, F. J. Ca nete, J . J. Sanchez-Martinez, Analy sis of the Indo or Broa dba nd Po wer- Line Nois e Scena rio, IEEE T ransactions on e lectromagnetic compatibility 52 (4) (20 10) 84 9–85 8 . [12] J. Lin, M. Nassar, B. L. Ev ans, Non-pa rametric Impulsive Nois e Mitiga tion in OFDM Systems Using Sparse Bay esian Learning , in: 2 011 IEEE Glo ba l T elecomm unications Co nference (GLOBECO M), IEEE, 2011, pp. 1–5 . [13] J. P . Nolan, Stable Distributions - Mo dels for Heavy T ailed Data, B ir khauser, B oston, 201 7, in pr ogress , Chapter 1 online at http://fs2.american.edu/jpnola n/ www/stable/ s table.html. [14] C. L. Nikias , M. Shao , Sig nal Pr o cessing with Alpha-Stable Distributions a nd Applications, Wiley- Int erscienc e , New Y ork, NY, USA, 1995 . [15] G. Laguna-Sa nchez, M. Lop ez-Guerr ero, On the Use of Alpha-Sta ble Distributions in Noise Mo deling for PLC, IEEE T ransactions o n Pow er Delivery 30 (4) (2015 ) 18 63–18 70. [16] B. Han, H. D. Schotten, A F ast Blind Impulse Detector for B ernoulli-Gaus s ian Noise in Underspr e ad Channel, in: 20 18 IE EE International Co nference on Communications (ICC), IEE E, 2018, pp. 1 –6. [17] B. Han, C. K aiser, K. Dostert, A Nov el Appro ach of Canceling Cy clostationar y Noises in Low-V oltage Po werline Co mm unications, in: International Conference on Communications (ICC), IEEE, 2015. [18] J. H. McCullo ch, Simple Co nsistent Estima tors of Stable Distribution Parameters, Co mmu nications in Statistics-Simulation and Computation 15 (4) (19 86) 11 09–1 136. [19] J. M. Cham b ers, C. L. Mallows, B. Stuck, A Metho d fo r Sim ulating Stable Random V aria bles, Journal 17 of the Amer ican Statistical Asso cia tion 71 (35 4) (1 9 76) 3 40–34 4. [20] S. Kullback, Informatio n Theo r y and Sta tistics, Courier Corp o ration, 199 7. [21] B. W. Bror sen, S. R. Y ang, Max imum Likeliho o d E stimates of Symmetr ic Stable Distribution Param- eters, Communications in Statistics-Simulation and Co mputation 1 9 (4) (1 990) 145 9–146 4. [22] J. P . Nolan, Max im um Likeliho o d Estimation and Diag nostics for Stable Distr ibutio ns , L´ evy Pro cess es: Theory a nd Applicatio ns (2001 ) 379– 400. [23] I. A. Koutr ouvelis, An Iterative Pro cedure for the Estimatio n of the Parameter s o f Stable Laws: An Iterative P ro cedure for the Estimation, Commun ications in Statistics-Simulation and Co mputation 10 (1) (1 981) 1 7–28 . [24] S. M. Kogon, D. B. Williams, Character istic F unction Based Estimation of Sta ble Distribution Param- eters, A Prac tical Guide to Heavy T ailed Da ta (1 998) 311 – 335. [25] X. Ma, C. L. Nikias, O n Blind Channel Identification for Impulsive Signa l Environment s, in: 199 5 Int ernatio na l Conference on Aco ustics, Sp eech, and Signal P ro cessing (ICASSP), V ol. 3, IEE E, 1995 , pp. 1992 –1995 . [26] F. Marohn, Estimating the Index of a Stable Law via the Pot-Method, Statistics & Probability Le tters 41 (4) (1 999) 4 13–4 23. [27] I. A. Koutrouvelis, Regr ession-Type Estimation of the Parameters of Stable Laws, Journal of the American Statistical Asso ciation 75 (37 2) (198 0) 9 18–9 28. [28] M. Zimmermann, K. Dostert, An Analysis of the Bro adband Noise Scenario in Po werline Netw o rks, in: Int ernatio na l Symp osium on Pow erline Commun ications a nd its Applica tions (ISPLC), IEEE , 200 0, pp. 5–7. 18