An Analysis of Transient Impulsive Noise in a Poisson Field of Interferers for Wireless Channel in Substation Environments

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Finally, in section V, computer simulations and measurements impulsive interference sources in substation environments. The are provided to demonstrate the efﬁciency of the analysis over ﬁrst order statistics of the interfering sources can be extended computer simulations and measurements results. by considering an additive background noise.

II. MATHEMATICAL FORMULATION OF MULTIPLE B. Interference sources in substation INTERFERENCE SOURCES IN SUBSTATIONS In substation environments, radiations from interferers re- A. Basic Poisson field of interferers ceived at the antenna are impulsive and are caused by par- Stochastic geometry approach is used to derive the first tial discharges mainly in air. They can be located in HV order of the characteristic function of interfering sources equipments when physical conditions are reached to discharge in presence of impulsive noise. Mathematical notations are such as presence of defects, high electric field, free electrons inspired by [24]. First, we need to define the statistics of etc, [25]. In presence of multiple interference sources in the the interference field. Let Π(r,t) be a random space-time vicinity of the antenna, a low density of space Poisson process field as a linear superposition of individual fields randomly Nr can be assumed where impulsive sources are randomly produced by activated sources (emission of radiations) in a located in the three dimensional space in far-field region. For domain of sources. We postulate the Poisson random field the activated impulsive sources, charge particles and currents r r r produced by a discharge radiate impulsive electromagnetic Π( ,t)= i δ i,ti where δ ,t is the Dirac measure on a finite space-time domain Λ 3 where the three dimensional radiations. The fields E and B can be obtained from retarded P ⊂ R × R Ψ r A space is considered. We note that = ,t i is a set of points potentials and V respectively the magnetic potential vector representing active sources in the space-time{ } domain Λ. Thus, and the scalar potential by : the interference field can be written as : ∂A E (6a) r Υr r r = V Π( ,t)= ,t|d( , 0) (1) −∇ − ∂t ψ∈Ψ B = A (6b) X ∇× Υ 2 where (Ω, , P ) is an ensemble of measurable random The retarded vector and scalar potentials satisfying the Lorenz ∈ L F r r waveforms of emitting sources, d( , 0) is a distance parameter gauge condition can be written as a wave equation : where r0 is the position of the antenna and t d(r, r0) is related to the delay of propagation induced by the| position of an 1 ∂V 1 interference source and the receiver. We now assume that 2 r (7a) V 2 = ρ( ,t) ∇ − c ∂t −ε0 these impulsive noise are separable functions such that the r r X∈N interference field can be written as : 1 ∂A 2A = µ J(r,t) (7b) ∇ − c2 ∂t − 0 r Υ Υ r∈Nr Π( ,t)= r t|d(r,r0) (2) X r∈Nr t∈Nt where ρ(r,t) and J(r,t) are charge density and current X X density respectively of an activated sources Si on the emitting where Nr is the point process related to emitting sources and ′ ′ element dv in v as depicted on Fig. 1. The sum r Nt is the point process of radiations in time domain. The i i ∈Nr Ψ represents the superposition of individual source randomly intensity of measure of the point process (B)= E [ (B)], P where B is a Borel set, has a density λ(ψ)=Z λ(r,t). By using located in the vicinity of the antenna R. The speed of Campbell’s theorem, the mean interference field is : light in the medium is represented by c, ε0 and µ0 are the permittivity and the permeability of the vacuum respectively. r Υ Each activated source Si radiates electromagnetic waves in E [Π( ,t)] = r,t|d(r,r0)λ(ψ)dψ (3) Λ the medium induced by retarded potentials. A receiver R can Z v v By assuming the ergodicity of the ensemble Υ such that : received these waves on the receiving element d R in R. By considering successive radiations from each source in the time domain, the Poisson process should be extended to space-time Υr r r = UrU r r (4) ,t|d( , 0) t|d( , 0) process. where Ur,t is the basic waveform of interference sources. From Solution of equations in (7) is given by : the Laplace functional of the equation (3), the first order of characteristic function Q(jξ) of the superposition of these ′ ′ 1 ρ(r ,t r r /c) ′ emitting radiations is given by : V (r,t)= −′ | − | dv (8a) ′ r r 4πε0 ′ v r ∈Nr Z | − | X ′ ′ µ0 J(r ,t r r /c) ′ Q(jξ) = exp 1 exp [ jξUr,t] λ(ψ)dψ (5) A(r,t)= −′ | − | dv (8b) ′ r r − Λ { − − } 4π ′ v Z r ∈Nr Z | − | The first order statistics such as moments, cumulants, X amplitude distribution and density can be derived from the The interference sources are independent such that radia- characteristic function which depends on the definition of the tions at the antenna surface is a superposition of independent basic waveform of interference sources. It will be specified for impulsive noise. Generally, the receiver has a directional 3

′ J r′ v3 3( 3, t) ′ S3 ′ receiver is distorted due to multipath propagation. Thus, the v1 ρ3(r3, t) ′ resulting impulsive noise is a random process where amplitude J1 r , t ′ S1 ( 1 ) r3 ′ ′ v r 4 envelope and instantaneous phase are random processes. The ρ1( 1, t) ′ ′ r1 J4(r4, t) r′ ′ propagation law may also induce the randomness of the 4 S4 ρ4(r4, t) O amplitude scale factor. In addition, a background noiseshould

′ be considered as combination of multiple independent in- r2 ′ |r3 − r| terference sources below impulsive interference sources e.g, ′ v2 ′ ′ J2(r2, t) |r1 − r| ambient noise from substations, thermal noise from receiver, ′ ρ2(r2, t) S2 r etc. r′ r | 4 − | We then fully write the random process X as a combination ′ t |r2 − r| of the shot-noise process It with an additive background noise nt such that :

Xt = It + nt (11)

R A(r, t) V (r, t) C. Non-Gaussian noise process v R A common receiver design operates at a given carrier or center frequency. Therefore, the noise process has a resonant Fig. 1. Geometry of interfering sources and the receiver frequency such that impulsive noise is a transient signal with damped oscillation (see Fig. 2). It is seen that impulsive noise is distorted randomly due to constructive and destructive waves radiation pattern not necessarily omnidirectional. The antenna induced by the multipath channel related to the geometry of has an effective length related to the induced voltage or current interference source and the receiver. It is argued that Ut is at the terminals to the incident field E. The receiver may also generally non-stationary process where noise samples are non- have RF and IF (intermediate frequency) stages e.g, low noise i.i.d. amplifier and linear filters. As a result, the superposition of these radiations gives the typical waveform obtained from the 0.6 receiver R by : 0.4

0.2 ] V [ 0 It = aR(θ,ζ,t) E(θ,ζ,t)dvR t v ∗ X Z R (9) -0.2 = Ut -0.4

t∈Nt X -0.6 0 1 2 3 4 5 where aR(θ,ζ,t) is the aperture weighting function in sphe- Time [µs] rical coordinate system (r, θ, ζ) respectively represented by the radial distance, the polar and the azimuthal angles. The 0.6 aperture weighting function includes both the radiation pattern 0.4 of the antenna and the linear impulse response of ﬁlters. It can be seen as an impulse response of the receiver where the 0.2 ]

receiving ﬁeld is converted into a time waveform alone. The V

[ 0 convolution product operates for temporal impulse response. t U The antenna receives the electric ﬁeld E, induced by the -0.2 activated interferers, on the receiving element dvR in vR. -0.4 The resulting waveform It is a superposition of independent impulsive noise Ut produced by activated interference sources. -0.6

The process It is excited by a Poisson process Nt related to 4.3 4.32 4.34 4.36 4.38 the number of impulses in time domain. It is denoted as a Time [µs] shot-noise process [19], [26]. The typical impulsive noise U t Fig. 2. Example impulsive noise measured in a 735 kV substation after any RF and IF stages of (linear) ﬁltering is written as :

The receiver may have a local oscillator to recover any 1 u(θ, ζ) U = u(t)ejϕ(t) (10) desired signals in baseband. In this condition, signals can be t 4π r demodulated at the desired resonant frequency ω0. In complex where u(θ, ζ)/r is the amplitude scale factor induced by domain, the baseband representation is given by : geometryk of interferingk source and the receiver. u(t) is the amplitude envelope and ϕ(t) the instantaneous phase of the X = I ejϕI (t) + n ejϕn(t) e−jω0t (12) impulsive interference. In practice, receiving the signal at the t | t| | t| 4 where the instantaneous phase of any analytic signal is ex- A. An autoregressive process for impulsive noise waveform pressed as ϕ(t) = i ωit where the resonant frequency ω0 modelling exist in ϕ(t). Baseband signals may be more tractable for P We consider real-valued random process Ut, the impulse impulsive noise signal processing. Indeed, the power spectrum shape received at the antenna produced by partial discharges density can be estimated by using classical parametric spectral as a discrete-time series such as an AR(p) process model is density estimation such as Yule-Walker method. As a result, given by : these impulsive noise can be reproduced by using discrete-time series models such as autoregressive process. p Fig. 3 is an example of typical impulsive waveform and U = φ U + ε (13) psd measured in a 735 kV substation in baseband, demod- t i t−i t i=1 ulated at f0 = 800 MHz. It is seen that a second order of X AR process model gives suitable estimation of the decay of where amplitude at the past samples Ut−i are weighted by φi 1/f k of the noise process. The non-i.i.d of noise samples named AR(p) coefficients. εt is the innovation process that ∼in presence of an impulsive noise is induced by the decay leads to distortions of Ut. We assume a second order of the of the power spectral density. The innovation process should AR(2) process such that : be defined to compute distortions. The determination of first and second order statistics of the non-Gaussian process X t U = φ U + φ U + ε (14) strongly depends on the specification of the impulsive shapes t 1 t−1 2 t−2 t

Ut, see the characteristic function in the equation (5). The The AR coefficients φ1 and φ2 will be defined to ensure the basic waveform of the impulsive noise should take account stability of the process, i.e all its roots from the characteristic physical parameters such as the duration of radiations, the non- function lie outside the unit circle. Thus, the stationarity con- stationary behaviour of the impulsive noise Ut in which the ditions should be verified. The second order of the AR process amplitudes of the random process are non-i.i.d. model allows to determined roots and the autocorrelation

−100 function of the random process Ut easily.

0.6 −110

0.4 −120 B. Deﬁnition of the innovation process

0.2 −130

] The randomness of the process Ut is induced by the V [ 0 −140 t innovation process εt. The latter should take account many U ) [dBW/Hz]

−0.2 f −150

( random phenomena such as distortion of the impulsive shape U

S −160 −0.4 and the non-stationary behaviour of the process linked by the

−170 duration of radiations received at the antenna. The definition −0.6 Measured data AR(2) process −180 of ε is based on physical assumptions : −2 −1 0 t 4.3 4.32 4.34 4.36 4.38 10 10 10 Time [µs] Frequency [GHz] • When an interference source is activated, radiations re- (a) Impulsive noise in baseband (b) Power spectral density ceived at the antenna is a superposition of the constructive and destructive impulsive waves caused by multipath Fig. 3. Impulsive noise in baseband measured in a 735 kV substation effects. These can be seen as a wave distorted by an i.i.d random variable εt such that the equation (14) is satisfied. GENERAL IMPULSIVE NOISE WAVEFORM MODEL III. A • Reflectors in substations cause multiple delayed paths that USINGDISCRETE-TIME SERIES obey to the propagation law. In this condition, the ampli- In this section, a basic waveform of impulsive interference tude of the impulse received at the antenna is necessarily is specified. The impulse waveform at the receiver may depend decaying with respect to time until it vanishes below on RF and IF stages at the receiver where the resulting to the background noise i.e, εt has a time-dependent waveform has damped oscillations generally at the carrier parameter denoted by ϑt with the constraint that εt is frequency. The general impulsive noise waveform model help a function that decay over time or samples. to make some simple assumptions to derive first and second From these assumption, we can define εt as a white noise order statistics of the non-Gaussian process Xt. where the variance is a discrete-time function, i.e, heterosce- Discrete-time series models can compute these typical ran- dastic white noise process [27], [28] : dom waveforms observed from experimentations, i.e transient impulsive noise waveform with damped oscillation, damped εt = ϑtWt (15) exponential or a mixture of damped exponential oscillation. The amplitude at the present sample denoted by Ut depends where ϑt is the time-dependent standard deviation of the on amplitude at the past samples denoted by U where i> 0. white noise process W (0, 1). For those complex-valued t−i t ∼ N These are weighed by coefficients which give the behaviour of random process Ut, one can consider complex-valued white the obtained waveform Ut. The definition of these coefficients noise process Wt. The discrete-time function ϑt can be defined should be carefully defined for the stability of the process. To as a positive power-law or log-normal function. The latter make Ut as a random process, the innovation process have to takes account rise time and a fall time of impulsive noise. be a random variable to be defined. Hence we write ϑt as : 5

2 The roots depends on the value of the terms φ1 +4φ2. The 2 ϑ0 (log t µt) process has a stationary solution if and only if : ϑt = exp −2 (16) tσt√2π − 2σt ! φ φ < 1 (20a) 2 − 1 where σt is related to the time decay of the impulse. µt may φ + φ < 1 (20b) refer to the time where the envelope of an impulse is maximum  2 1  φ2 < 1 (20c) and may be related to the presence of the main path received | | These AR coefficients specify the behaviour of the wave- at the antenna. ϑ0 is a normalized parameter. It is convenient  that these parameters should be set such that the rise time and form of the impulsive noise Ut. They would help us to the decay time of an impulse are much shorter than the sample determine the problem statement for the determination of first size of the non-Gaussian process. and second order statistics of the non-Gaussian process Xt in A basic waveform of impulsive noise received at the an- section IV. tenna has been specified based on physical assumptions. The model produce impulsive noise waveforms where amplitude D. Power spectral density of Ut are distorted randomly by the innovation process as depicted By remembering that the innovation process is a heterosce- on Fig. 4 where dt is a time-increment defining a sample. The dastic white noise process, the power spectral density of Ut lined curve is the real-valued impulsive noise process where has a classical AR(2) psd form given by : amplitudes should decay with respect to time represented by S (f) the dashed curve. The process is non-stationary due to the S (f)= ε (21) U 1 φ ej2πf φ ej4πf 2 time-dependent of the standard deviation of the innovation | − 1 − 2 | process εt. where Sε(f) is the psd of εt. By using the equation (61) in Appendix A, we write the complete psd of Ut as :

2 σϑ U = P + SU (f)= (22) t Ut φiUt−i εt 1 φ ej2πf φ ej4πf 2 i | − 1 − 2 | 2 where σϑ is the variance of ϑt. The variance of the white noise 2 is σW = 1. Depending on the roots of the AR process, the dt t psd of Ut has different behaviour [30] : 2 • For real roots, i.e, the terms φ1 + 4φ2 0, if the characteristic equation has at least one real≥ roots close to the unit circle, then SU (f) will have peak at f =0 if φ1 is positive. The psd will have peak at f = 0.5 if φ1 is negative. 2 Fig. 4. Impulse waveform of impulsive noise distorted by a random • For complex roots, i.e, the terms φ1 +4φ2 < 0, if the innovation process roots are closed to the unit circle, a peak occurs near the resonant frequency at f0 given by :

1 −1 φ1 f0 = cos (23) C. Stationarity conditions 2π 2√ φ − 2 To ensure the stability of the process U , the stationarity t E. Autocorrelation function of U conditions should be veriﬁed. By using the AR(2) process t model, we use the Box-Jenkins modelling approach [29]. The impulse shape has a complex form due to the ran- The non-stationary process is differenced until stationary domness of the amplitude. It may useful to provide the is achieved. Hence, from the equation (14), we write the autocorrelation function (ACF) of the process Ut. From the difference-stationary process such that : roots of the characteristic equation on (18), the equation (17) is rewritten as :

Φ(L)Ut = εt (17) −1 −1 Ut = (1 G1L) (1 G2L) εt (24) 2 − − where Φ(L)=1 φ1L φ2L and L is the lag operator where G = 1/r and G = 1/r remembering that r are i − − 1 1 2 2 1,2 such that UtL = Ut−i. Characteristic equation of the AR(2) the roots of the quadratic equation (18). The autocorrelation process is given by : denoted by E [UtUt−k] ρk of the process Ut following a closed form solutions : ≡ 2 1 φ1L φ2L =0 (18) − − 2 k+1 2 k+1 (1−G2)G1 −(1−G1)G2 The quadratic equation (18) has two roots r and r where : when r1 = r2 1 1 (G1−G2)(1+G1G2) 6 ρ = (25) 2 k  φ1 φ +4φ2 k 1  (1+φ2)k φ r1,2 = ± (19)  1 1+ 1−φ φ when r1 = r2 p2φ2 2 2 −   6

The behaviour of the ACF ρk depends on the nature of the Hence, the Poisson process Nt in time domain may be inho- roots of the quadratic function : mogeneous and cyclostationary. However, the presence of the • For real roots, with the constraint that G and G < 1, three phase voltages the superposition of the activated sources | 1| | 2| the ACF ρk can be seen as mixture of damped exponen- may become homogeneous Poisson process. In this condition, tials or damped exponential oscillation that decay to zeros we assume the presence of interference sources driven by the when k increases. three phase of voltages independently. Therefore, a constant • For complex roots, the ACF is a damped sinusoidal density of the Poisson process λ(ψ)= λ is assumed. function where the explicit expression of ρk is given by Furthermore, we assume a large time observation to have a [30] : non-negligible number of impulsive noise. As a result, we may write the non-Gaussian noise process Xt as a superposition of k sin (2πkf + ς) 0 shot-noise process It produced by each individual interference φ2 (26) − sin (ς) source plus an additive background noise nt such that : p where f0 is the resonant frequency of the system and ς = (1 φ )/(1 + φ ) tan(2πf ). Ne − 2 2 · 0 r r r A general impulsive noise Ut has been specified by using Xt = K j It|d( j , 0) + nt (27) j=1 AR process model. If the stationarity condition of the process X is ensured, one can reproduce a complete random waveform where Krj is a random amplitude scale factor induced by the impulsive noise Ut with damped oscillation at a desired geometry of individual interference source and the antenna. resonant frequency by set the AR coefficient φ1 and φ2 such Ne is the number of the interference sources activated in the that the roots of the characteristic equation has complex roots. vicinity of the antenna. The impulsive noise can be represented in baseband where 2) The basic waveform of impulsive noise Ut: First order the psd SU (f) should have a peak at f =0. In this condition, statistics can be difficult to derived especially when Ut is a the roots of the characteristic equation have to be real and random process. Nevertheless, a suitable approximation can particularly the first AR coefficient has to be positive, φ1 > 0. be obtained by finding an equivalent deterministic of these We are now ready to derive the first and second order impulse shapes. In [20], [26] suggest that the equivalent statistics of the non-Gaussian noise process Xt based on function of Ut can be determined by using the expected value the equation (5) by using the basic waveform Ut of an E [Ut] denoted by γt. From the equation (17), we write the impulsive interference from AR process model. However, it equivalent impulsive noise function as : may be a non-trivial task since the impulsive noise shape has random amplitude. Some simplifying assumptions have to be Φ(L)γt = E [εt] (28) considered. The second order statistics is the power spectrum of the process. It is given by the Carson’s theorem [19], [23]. The innovation process is a heteroscedastic white noise. There- fore, E [εt]=0. Thus, the expected value of Ut is derived IV. FIRST AND SECOND ORDER STATISTICS OF THE from the second order of the difference equation (28). We see NON-GAUSSIAN PROCESS that the solution of this equation depends on the roots of the First and second order statistics are the first interest for characteristic function. implementation of threshold algorithms for signal detection • For real roots, a solution of the equation (28) is : and estimation. It may be difficult to provide exact analytical γ = K e−at e−bt (29) probability density function and power spectral densities when t − those results depend on the impulsive waveform of interference where b is related to the rise time, a is the fall time of sources. In presence of impulsive interference sources in sub- the discharge γt and K is a random amplitude factor. station environments, many random impulses can be observed • For complex roots, a solution of the equation (28) is for a given time observation where inter-arrival time, energy written as : of individual impulsive noise, and occurrences are randomly γ = K e−at e−bt cos(2πf t + ϕ) (30) distributed. t − 0 where the resonant frequency of the system is given the A. Problem statement equation (23) and ϕ an arbitrary phase. Here, b and a is We need to made some simplifying assumptions in terms not necessarily equal as find on the equation (29). of statistics from the basic Poisson field of interferers as well It is convenient for impulsive waveforms that a and b as in terms of basic waveforms of impulsive noise. are real numbers strictly positive. Moreover, we restrict the 1) The homogeneity of the Poisson field of interferers: mathematical development of first and second order statistics Interference sources are generally detected in presence of for baseband impulsive noise. Nevertheless, one can follow HV equipments under voltage and the Poisson process Nr in the same approach if the damped oscillation in the equation space domain can be homogeneous. However, most impulsive (30) need to be considered. Therefore, first and second order interferences are generated by AC voltages in substation envi- statistics are derived based on the waveform from the equation ronments. Thus, the interfering sources are activated whenever (29). One can recover impulsive noise with damped oscilla- the electric field reaches the dielectric strength of the air. tions by multiplying the impulse in baseband by a carrier wave 7

at the desired resonant frequency f0. We consider the basic We assume a non-negative impulsive shape such that b waveform γt as a continuous-time function and we assume higher than a. In this condition, for a ﬁnite value of m = that the shot-noise process It and the background noise nt are 1, 2, 3, 4, 5, 6 , the m-cumulants κm is a non-monotonic independent random processes. sequence{ if K} m is negative for odd values of m, i.e κ h i m is negative for odd values and κm is positive otherwise. The B. Moments and cumulants m-cumulants can be linked with the m-moments of the non- gaussian model. The skewness χ1 and the kurtosis χ2 are The description of the shape of amplitude distributions determined by the 3rd and the 4th standardized moments of and densities can be given, in some extent, by moments the non-Gaussian noise Xt respectively. They can be written and cumulants. For example, the skewness is a measure the in terms of cumulants such that : asymmetry of noise and the kurtosis, a measure of how outlier- prone the distribution is. We start by calculating the mth cumulant κ of the shot-noise process I . In equation (29), κ3 m t χ = (35a) we assume K as a random variable which assume positive and 1 3/2 κ2 negative values make distributions. Then, from the extension κ4 th χ2 = (35b) of the Campbell’s theorem, the m cumulant κm is given by 2 κ2 [19] :

3 ∂m • The skewness value only depends on K . Indeed, if κm log [QI (s)] the skewness is , then the probability density ≡ ∂sm χ1 0 s=0 | | ≤ function of amplitude (pdf) can be left-skewed or right- m = λ γt dt (31) skewed, i.e presence of longer tail on the left or on the 3 ZR right. The pdf can be also symmetric if K = 0. m −at −bt m In practice, the asymmetry may be induced by random = λ K e e dt h i R+ − distortions of impulse shapes. Z 4 2 • In presence of impulsive noise, if K > 0 and κ >κ , where QI (s) is the first order moment generating function of 4 2 then the kurtosis is always χ > 0. Hence, the pdf can the shot-noise process. is the expectation taken over the 2 distribution of the randomh·i variable. From binomial formula, be leptokurtic, i.e a peak around the mean and long tail th at higher amplitude values. we write the m cumulant κm as : m m m −at m−k −bt k κm = λ K e e dt h i + k − ZR k=0 C. Moment generating function and characteristic function X (32) m m Km = λ ( 1)k h i The moment generating function of the non-Gaussian noise k − a(m k)+ bk is expressed by : kX=0 − A series can be identified with a binomial sequence. By using the ratio test, we prove that κ is convergent for infinite −sX m QX (s) E e value of m as seen in appendix B. Hence, κ is necessarily ≡ m = E e−sI e−sn finite. We extend to the non-Gaussian noise and by assuming (36) the independence between the shot-noise process and the = QI(s)Qn(s) background Gaussian noise. Hence, we have : ∂m κ log [Q (s)] where s and QI (s) and Qn(s) are the moment generating m ≡ ∂sm X ∈ R s=0 (33) function of the shot-noise It and the additive background Gau- ∂m ∂m = log [Q (s)] + log [Q (s)] ssian noise nt. The moment generating function of Gaussian ∂sm I ∂sm n s=0 noise Qn(s) is given by :

Finally, for an additive background noise in which an i.i.d 2 2 Gaussian noise of zero mean and variance σ2 is assumed, σ s n Q (s) = exp n (37) n (0, σ2 ). Thus, we have : n 2 t ∼ N n K (b a) The generating function of the shot-noise process is more κ1 = λh i − (34a) difficult to obtain. Nevertheless, a closed expression form of ab 2 the moment generating function can be provided in terms of 2 K2 κ = λ ( 1)k + σ2 (34b) cumulant and by using the series expansion. We emphasize 2 k − a(2 k)+ bk n k=0 that the cumulant κm is finite when m goes to infinite as we X − m m proved in Appendix B. Hence, we start by the series expansion m k K κm = λ ( 1) h i (34c) of the cumulant generating function of the shot-noise process k − a(m k)+ bk k=0 m>2 such that : X −

∞ m 1 κ s jξγt −jξx m fI (x)= exp λ 1 e dt e dξ log [QI (s)] = m! 2π ℜ R − R − m=1 Z Z (42) X ∞ m m ( 1)ksm Km Extended to the non-Gaussian noise, and by using the = λ − h i k m! a(m k)+ bk characteristic function of QX (jξ) based on the equation (39), m=1 k=0 − X X (38) we have :

Extended to the non-Gaussian noise, we link all values of the ∞ m 1 κm(jξ) cumulant κm as deﬁned in equations (34) to the moments f (x)= exp jξx dξ (43) X 2π ℜ m! − of the distribution. The cumulant generating function can be ( R m=1 ! ) Z X linked by the moment generating function QX (s) where the The pdf fX (x) should be bounded in [0, 1] x . We may cumulant generating function is the logarithm of the moment rewrite the pdf by setting : ∀ ∈ R generating function :

σ2 = κ (44a) ∞ m 2 µms Q (s)=1+ x κ1 X m! ν = − (44b) m=1 σ X (39) ∞ κ sm From the equation (43), we rewrite the pdf as : = exp m m! m=1 ! X 2 2 ∞ m 1 −jξσν− ξ σ κm(jξ) where µm are moments of the distribution of the non-Gaussian f (x)= e 2 exp dξ X 2π ℜ m! noise such that : (ZR m=3 ! ) X (45) The expression of the pdf is complex to derived. An approxi-

E [Xt]= µ1 = κ1 (40a) mation approach can help to achieve tractable forms. 2 2) Series approximation of pdf: The complex form of pdf E Xt = µ2 = κ2 + µ1κ1 (40b) 3 in equation (43) may be approximated by series approximation E Xt = µ3 = κ3 +2κ2µ1 + κ1µ2 (40c) [19], [31]. From the equation (42) it is convenient to deﬁne : (40d) ··· 1 T/2 H(jξ)= ejξγt dt (46) The cumulant generating function can be seen as an entire T function for complex values of s where the function converges Z−T/2 everywhere in the complex plane, (see Appendix B). Thus, the where H(jξ) is seen as the characteristic function of γt. In characteristic function QX (jξ) can be expressed by replacing this condition, the pdf of the shot-noise process in the equation s = jξ in the equation (39). (42) can be written as :

1 D. Amplitude probability distribution and density of non- f (x)= exp(λT H(jξ) λT jξx) dξ (47) I 2π ℜ − − Gaussian noise process ZR 1) General expression of pdf: The general expression of Thus, from the equation (45), we consider that : probability density function of amplitude of Xt can be given 2 2 by the convolution product of the shot-noise process I and 1 m −jξσν− ξ σ m −1 m t (jξσ) e 2 dξ = ( 1) σ Θ (ν) (48) the background Gaussian noise n due to the independence of 2π − t ZR It and nt. We write the pdf fX (x) as : where :

m fX (x)= fI+n(x) ∂ 2 Θ(m)(ν)=(2π)−1/2 e−ν /2 (49) ∂νm = fI (u)fn(x u)du − (41) Hence, the pdf of instantaneous amplitude of X is given ZR t asymptotically by collecting terms according to power of = Q (jξ)Q (jξ)e−jξxdξ I n λ−1/2 [19], [22], [32] : ZR The pdf and the characteristic function of the background Gaussian noise is well known but pdf of the shot-noise ∞ m −4 nt −λT (λT ) −1 (0) κ3σ (3) fX (x) e σ Θ (ν) Θ (ν) process It has complex form. A general form of the pdf can ≈ m! − 3! m be obtained by the inverse of the Fourier transform of the X κ σ−5 κ2σ−7 characteristic function of the shot-noise process [19], [20] such + 4 Θ(4)(ν)+ 3 Θ(6)(ν) + (50) 4! 72 ··· that : 9

−1/2 where the first term is o(λ ) which is the normal distribu- autocorrelation function of the non-Gaussian noise process Xt tion, the second term is o(λ−1) and terms within brackets is is given by [19], [20], [37] : o(λ−3/2). The approximation is based on the Edgeworth series. By considering only the first term and linking the standard 2 2 E [XtXt+τ ]= E It + λE [γtγt+τ ]+ σnδ(τ) (55) deviation σ with the increment m such that σm = g(σ, m) a function of σ and m in the equation (50), one can find the The Carson’s theorem allow to express the power spectral Middleton Class A [22] such that : density with these terms if the integral of the autocorrelation function of γt is finite or equivalently, the integral of the psd ∞ m −λT (λT ) −x2/2σ2 is finite [19], [20]. We assume that 2 . One can find f (x) e e m (51) K < X ≈ m! ∞ m the integral of psd of γt is finite, R Sγ (f)df < f . X In this condition, extended to the non-Gaussian ∞ process, ∀ ∈ th Re However, the Edgeworth series expansion is often inaccurate resulting expression of the psd of RX is given by : in the far tail of distribution [33], [34]. t 3) Convergence to α-stable distribution: The non-Gaussian λ K2 (b a)2 noise process can be seen as a sum of independent processes S (f)= E I2 δ(f)+ − + σ2 (56) X t (a2 + ω2)(b2 + ω2) n where the shot noise It is written as a sum of independent processes Ut such that : where ω = 2πf. It is seen that the psd of the non-Gaussian noise process has a decay of 1/f k. Ut,1 + Ut,2 + + Ut,m ∼ Xt = ··· + nt (52) In this section, the first and second order statistics can be dm derived from the basic waveform of the impulsive noise. We where dm is a sequence of positive real numbers strictly proved that we can have high value of kurtosis in which positive. By definition, the process Xt is stable [35]. The the distribution is leptokurtic and also be asymmetric as random process Ut is impulsive noise where its distribution discussed. Furthermore, we proved that amplitude distributions is f (u) u −α−1 where α is the characteristic exponent. and densities of the non-Gaussian process can be approximated |U| ∼ | | From [21], [26], in absolute values, one can find that the by classical non-Gaussian pdf forms such as Middleton class distribution of basic waveforms write in equations (29) and A or α-stable. The power spectral density can also be derived (30) are γ −α−1 where 0 < α < 2. As a result, the where a decay of 1/f k is observed induced by waveforms ∼ | | ∼ random process Xt has a α-stable distribution such that the of impulsive noise. characteristic function is [35], [36] : V. RESULTS AND DISCUSSION α In this section, we discuss about the validity of our theore- QX (jξ) = exp jξµ σξ (1 jβ sign(ξ)η) (53) { − | | − } tical model when the electromagnetic environment has impul- where µ is a location parameter real value, σ 0 is a scale sive interferers where the resulting waveforms at a receiving factor, β is the skewness parameter where 1 ≥ β 1, η = point is a succession of independent impulsive noise. tan(πα/2) if α =1 and η = log ξ if α =1−. ≤ ≤ We will start by specifying waveforms of impulsive noise We emphasize6 that the pdf of the| | non-Gaussian noise is a using discrete-time series. We will define coefficients in which fat-tailed distribution with high value of kurtosis and it can the stationarity condition is ensured for the random impulsive be also asymmetric as argued. The energy and the duration waveform process Ut, see equation (20). Thus, a non-Gaussian of impulsive noise determine, to some extent, the “fatness” noise process Xt can be fully simulated where a succession of of the tail of the distribution. These parameters increase random impulsive noise is excited by Poisson point process. the probability of amplitude values higher than its standard Additionally, a background noise nt below the shot-noise deviation. The Midlleton Class A can approach the non- process It is considered. In this condition, the first order can be Gaussian noise however, it may be inaccurate in the tail of derived where empirical amplitude distribution and density can distribution. The α-stable can provide a suitable approximation be provided. We show how classical non-Gaussian noise model of the amplitude distribution for those random processes which such as Midlleton class A and α-stable amplitude distribution admit a power law decay of x −α−1 on distribution. The and density can be appropriated vis-a-vis the simulation results two approximations will be compared∼ | | in the section V. The as well as vis-a-vis real situations in substation environments. tail distribution is given by :

x A. Impulsive waveforms modelling F¯ (x)= P (X > x)=1 f (u)du (54) X − X In section III, a complete random impulsive noise can be Z−∞ computed based on discrete-time series such as AR process. According to the equation (14), a second order of the AR pro- E. Second order statistics : Power spectral density of Xt cess is used. We only restrict the discussion where impulsive k The power spectral density of the shot-noise process It can noise are in baseband, i.e, a decay of 1/f with a peak be given in terms of the rate λ and the Fourier transform of the at f = 0. In this condition, we restrict∼ AR coefﬁcients such impulse response of the associated linear ﬁlter by the Carson’s that their roots is real values and the stationarity condition is theorem [19], [23]. The power spectral density is given by the ensured, see equations (18) and (20). The innovation process 10

induced by εt is a heteroscedastic white noise process where 1) Simulation setup: We are now ready to simulate a non- the time-dependent standard deviation is given by the equation Gaussian noise process Xt in presence of non negligible (16). Parameters of the latter must be set such that the rise number of impulsive noise. First of all, the electromagnetic time and the decay time of a random impulsive waveform Ut environment should be specified by assuming : are much shorter than the sample size of the non-Gaussian • A homogeneous random space-time Poisson field of process. Many random impulsive noise can be simulated as interference sources where the density is an arbitrary depicted on Fig. 5 where parameters are set as follows : constant positive value λ(ψ)= λ< 1. • All radiations from interference sources emit impulsive TABLE I noise such that parameters set on the table I are satisfied. IMPULSIVENOISESHAPEPARAMETERS 2 • The energy denoted by U of each impulse is ran- k tk AR coeff. φ ϑt std. of εt domly distributed induced by charges and currents of partial discharge sources [38], [39]. We choose an ex- φ1 φ2 µt σt ponential law where the energy in average, denoted by 1.2 −0.3 7.0 2.25 2 Ut , is above the background noise such that the variancek k ratio between the background noise the shot- noise process is :

} 5 1 | 10 1 t, 2

0 t, U E n U 5 t | ℜ{ −5 Γ= < 1 (57) 0 2 0 500 1000 1500 2000 E [I ] 0 500 1000 1500 2000 t

} 5 | 2 10

2 In this condition, we set parameters as follows : the density t,

0 t, U

U 5 |

ℜ{ −5 of the Poisson ﬁeld interferers λ(ψ) is homogeneous and 0 0 500 1000 1500 2000 0 500 1000 1500 2000 constant and set to λ = λtλr where the average interference

} 5 | 3 10 r 3 sources is λ = 5 per unit volume and the average radiation t,

0 t, U

U 5 |

ℜ{ −5 emissions per source is λt = 5 per sample or per unit time. 0 0 500 1000 1500 2000 0 500 1000 1500 2000 Sample Sample The energy is random variable exponentially distributed where the average value is 2 . The variance ratio between (a) Amplitude (b) Envelope Ut = 10 the background noisek thek shot-noise process is Γ=0.1. 20 2) Measurement setup: Measurement campaign is made 10 in a 735 kV substation. The measurement setup includes a 0 ] wideband antenna (0.8 to 3 GHz), RF and IF stages such as

−10 high pass ﬁlter, ampliﬁer, etc. For data acquisition, we use an oscilloscope to capture waveforms in presence of impulsive

dB/Hz −20

)[ noise. The sample rate is 10 Gs/s for an observation time

f −30 (

U at 5 µs. Details about parameters of the environment during

S S (f) −40 Ut,1 the measurement campaign and the measurement setup is SUt,2 (f) −50 SUt,3 (f) given in [40]. The obtained waveforms contain background psd of AR(2) −60 noise including wireless communications from cellular or −3 −2 −1 10 10 10 Normalized frequency Hz communication in the ISM band. We demodulate the received (c) Power spectral densities signals at the resonant frequency, f0 = 800 MHz, to obtain waveforms in baseband. Fig. 5. Example of random impulsive noise Ut Results of the non-Gaussian noise process from computer simulation and measurement campaign in a 735 kV substation We see that the random process can generate many im- is provided on Fig. 6. Impulsive noise sample above back- pulsive noise with random amplitudes but the power spectral ground noise are produced by impulsive interference sources. densities have same behaviour, i.e a decay of 1/f k closed to a Lorentzian form. The desired rise time∼ and the decay 20 0.6 time deﬁning the duration of the impulse can be deﬁned by 15 0.4 setting parameters of the time-dependent standard deviation of 10 ] 0.2

5 V [ the heteroscedastic process. One can be able to compute many } t } X random impulsive waveforms with many behaviours based on 0 t 0 X

ℜ{ −5 −0.2 AR process as long as the stationarity condition is ensured. ℜ{ −10 −0.4 −15

−20 −0.6 B. First and second order statistics of non-Gaussian noise 0 1 2 3 4 5 0 1 2 3 4 5 4 µ process Sample x 10 Time [ s] (a) Amplitude Computer simulation (b) Amplitude S. 735 kV [40] Computer simulations and measurements are provided to validate the analysis. Fig. 6. Example of non-Gaussian process Xt 11

0 3) Amplitude probability distributions and densities: Am- 10

−1 plitude distribution and density of non-Gaussian process are 10

−2 depicted on Fig. 7 and 8 which correspond respectively to 10 −2 ) 10 ) x x samples from the model and from the measurement campaign ( ( X −3 X ¯ in a 735 kV substation. It is seen that the presence of impulsive F 10 f −4 noise has an inﬂuence in terms of amplitude distribution 10 −4 pdf and density. Indeed, low probability of high amplitude can ccdf 10

−5 be observed on the tail of the distribution such that we −6 10 10 Empirical Empirical have fat-tailed distributions. This is due to high amplitude of α-stable α-stable Class A −6 Class A 10 −2 −1 0 impulsive noise especially when those amplitude distributions −0.5 0 0.5 10 10 10 Ampltitude x [V] Ampltitude x are asymptotically power law distributions. Classical non-Gaussian noise distributions such as Middle- (a) Probability density (b) Tail distribution ton Class A and α-stable distributions have these behaviours, Fig. 8. Amplitude distribution and density of non-Gaussian noise measured i.e, leptokurtic distributions and may exhibit an asymmetry as in 735 kV substation seen on Fig. 8. These two distributions are compared based TABLE II on empirical data provided by the model and measurements. COMPARISONOFAMPLITUDEDISTRIBUTIONANDDENSITY Parameters of these distributions has been estimated from empirical data based on [41]–[44]. In this condition, amplitude Simulation S. 735 kV distribution and density has been superposed in order to Model α-stable Class A α-stable Class A discuss about the quality of the ﬁt. KL 0.0037 0.1957 0.0111 0.15 2 · 10−6 4 47 · 10−4 4 16 · 10−6 2 37 · 10−4 0 MSE . . . 10

−1 10 −1 10

−2 10

) red curve. On the psd obtained in the 735 kV substation, ) x −2 ( x 10

( wireless communications and harmonics can be observed at −3 X ¯ X 10 F f 1.5, 2.5, 6 GHz for example. Harmonics are caused by −3 10 −4

pdf interleaving artefacts and clock feedthrough from scope. 10 ccdf

−4 −5 10 10 30 −90 Empirical Empirical α α-stable -stable 20 −100 Class A Class A −6 10 0 1 −20 −10 0 10 20 10 10 10 −110 Ampltitude x Ampltitude x ]

] 0 −120 (a) Probability density (b) Tail distribution −10 −130 dB/Hz dBW/Hz

Fig. 7. Amplitude distribution and density of non-Gaussian noise process )[ −20 −140 )[ f ( f ( X

−30 X S −150 From the proposed model and measurements, it can be seen S −40 −160 that those distributions can fit empirical data with more or −50 −170 Empirical Empirical less accuracy. The quality of the fit is determined by the Burg method Burg method −60 −180 −4 −3 −2 −1 6 7 8 9 10 10 10 10 10 10 10 10 Kullback-Liebler. It is used to measured the divergence of the Normalized frequency [Hz] Frequency [Hz] amplitude density (pdf). Mean square error is used to compare (a) Computer simulation (b) S. 735 kV tail distributions (ccdf). Results are set on the table II. It is seen that the α-stable distribution fits well the empirical data better Fig. 9. Power spectral density of Xt than the Middleton Class A. See the KL divergence value and the MSE value of α-stable compared to Middleton Class A whatever samples from the model and measurements on the VI. CONCLUSION table II. This may be explained by the approximation based In this paper we develop a non-Gaussian noise model in on the Edgeworth series expansion where only the first term presence of transient impulsive noise in substation environ- on the equation (50) is used. Therefore, a lack of accuracy is ments. We use Poisson field of interferers in which impulsive observed in the far tail of distribution. The α-stable distribution transient interference sources are space-time Poisson process. converge to the empirical data due to the definition given in the Based on stochastic geometry, first and second order statistics equation (52) and by arguing that impulsive noise waveforms can be derived. In presence of impulsive noise, it is proved that Ut in absolute value can be seen as power law distributions the amplitude distribution and density can be approximated −α−1 f|U|(u) u . by classical non-Gaussian noise such as Middleton Class A 4) Power∼ | | spectral densities: Second order statistics is and α-stable distributions. It is seen that the latter is a better presented on Fig. 9. The psd of the non-Gaussian noise is approximation than the Middleton Class A due to its approxi- estimated and smoothed by using parametric method such as mation by using Edgeworth series expansion. Basic impulsive Bug’s method [45], [46] to observe the decay of 1/f k waveform is specified by using discrete-time series where induced by transient impulsive noise. It is represented∼ by the the innovation process is heteroscedastic to ensure both the 12 the randomness and the transient behaviour of the impulsive interference sources. It is proved that the non-Gaussian noise z L = lim k+1 < 1 (63) can be expressed as a second order statistics where the power k→∞ z k spectral density is a decay of 1/f k. Computer simulation ∼ and experimental data are provided to show the validity of the For the binomial coefficients we have : analysis. In Future works we will discuss about the reproducibility of m the model vis-a-vis the experimentations. Physical parameters =1 (64a) 0 such as number of activated interfering sources in the envi- m m m k ronment, the variance ratio between background noise and the = − (64b) k +1 k k +1 shot-noise process and duration of impulsive noise have to be estimated from the desired environment and validated in terms In this condition, based on the equation (32), we write L such of first and second order statistics. that :

APPENDIX A POWERSPECTRUMDENSITYOFHETEROSCEDASTIC ( 1)(m k) am + (b a)(k + 1) L = lim − − − (65) PROCESS εt k→∞ k +1 am + (b a)k − From the innovation process ε defined in equation (15), we t We define higher than . Moreover, since and positive start by calculating the autocorrelation function as : b a k m integers, m is necessarily infinite, we determine≤ that L =0 < 1. It is proved that zk is convergent and thus, the series κm is E [εtεt−k]= E [ϑtϑt−kWtWt−k] (58) convergent.

The white noise process Wt is i.i.d such that : B. The radius of convergence of the cumulant generating function E [WtWt−k]=0 (59) From the power series expansion of the cumulant generating for all values of k = 0. By assuming that ϑ and W are 6 t t function is given by the equation (38), where s is complex, independent variables, we write the equation (58) as : the radius of convergence of the power series can∈C be discussed. A power series converge for some values of the variable s and E [εtεt−k]= E [ϑtϑt] E [WtWt] may diverge for others. Thus, the radius of the convergence (60) 2 can be calculated from Cauchy-Hadamard theorem’s : = E [ϑtϑt] σW δ(0) 2 1 where is the variance of the white noise. The Fourier −1 m σW r = lim sup κm (66) m→∞ transform of the autocorrelation function of εt allow us to | | write as discrete convolution product between the variance of The radius of the convergence can be calculated from the the white noise and the psd of ϑt denoted by Sϑ(f) : ratio test of κm/m! :

2 S (f)= S (f) σ −1 κm+1 ε ϑ ∗ W r = lim m→∞ (m + 1)κ 2 m = σW Sϑ(f)df (61) m+1 R m+1 m+1 Z2 2 k [a (m +1 b)+ bk] K = σ σ k=0 − W ϑ = lim m→∞ P m 2 m m where σϑ is the variance of ϑt. (m + 1) k [a(m b)+ bk] K k=0 − h i 1 P APPENDIX B lim =0 ∼ m→∞ m +1 ABOUTTHECONVERGENCEOFTHECUMULANT κm AND (67) THE CUMULANT GENERATING FUNCTION A. Convergence of mth cumulant κ where r is the radius of convergence which is inﬁnite, r , m → ∞ We postulate that k m where k and m are positive integer i.e the cumulant generating function converges everywhere in ≤ the complex plane. Therefore, it is an entire function. and m> 0. From the equation (32), we denote (zk) a sequence of non-zero real values sequence as : m Km APPENDIX C z = λ ( 1)k h i (62) k k − a(m k)+ bk POWERSPECTRALDENSITYOF γt − By using the ratio test, the convergence of the series κm is The power spectral density of γt is given by the Wiener- ensured if and only if : Khinchine theorem : 13

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