In this section, we will validate the statistical model of Middleton Class A with the measured data of partial discharges obtained in Section II. Here, we do not focus on the frequency of the noise. We first present a brief overview of the Middleton Class A model. Then, we focus on the validation of this model with the measured data and the extension of Middleton Class A model for multiantenna system.
A. Middleton Class A model
Middleton Class A model refers to Narrowband Noise where interference spectrum is narrower than the receiver bandwidth. In this model, the received interference is assumed to be a process having two components [4, 5]:
X\left(t\right)={X}_{P}\left(t\right)+{X}_{G}\left(t\right)
(2)
where X_{
P
}(t) and X_{
G
}(t) are independent processes. They represent the nonGaussian (impulsive) and Gaussian components, respectively. The probability density function (PDF) of X(t) is given in [4]:
\begin{array}{c}{f}_{P+G}\left(x\right)={e}^{A}\sum _{m=0}^{\infty}\frac{{A}^{m}}{m!\sqrt{2\pi {\sigma}_{m}^{2}}}{e}^{\frac{{x}^{2}}{2{\sigma}_{m}^{2}}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{with}}\phantom{\rule{1em}{0ex}}{\sigma}_{m}^{2}=\frac{\frac{m}{A}+\Gamma}{1+\Gamma}\end{array}
(3)
Note that f is a weighted sum of zeromean Gaussians with increasing variance. A and Γ are the basic parameters of the model. Let us consider their definitions and physical significance:

1)
A is the Overlap Index or Nonstructure Index.
where v is the average number of emission events impinging on the receiver per second and T_{
s
}is the mean duration of a typical interfering source emission. The smaller A is, the fewer the number of emission (events) and/or their durations. Therefore, the (instantaneous) noise properties are dominated by the waveform characteristics of individual events. As A is made larger, the noise becomes less structured, i.e., the statistics of the instantaneous amplitude approach the Gaussian distribution (according to central limit theory [5]). Hence, A is a measure of the nonGaussian nature of the noise input to the receiver.

2)
Γ is called the Gaussian factor. It is the ratio of powers in the Gaussian and nonGaussian components
\Gamma =\frac{\left({X}_{G}\right)}{\left({X}_{P}\right)}
(5)
In general, A ∈ [10^{4}, 1] and Γ ∈ [10^{6}, 1] [19]. By adjusting the parameters A and Γ, the density in (3) can be made to fit a great variety of nonGaussian noise densities.
B. Validation of Middleton Class A model for partial discharge
We validated the Middleton Class A model with the measured datasets presented previously by the following procedure of Figure 4. From the measured noise, we used the method of moments [20] to estimate the parameters A and Γ of Middleton Class A model:
{A}_{\mathsf{\text{est}}}=\frac{9{\left({e}_{4}2{e}_{2}^{2}\right)}^{3}}{2{\left({e}_{6}+12{e}_{2}^{3}9{e}_{2}{e}_{4}\right)}^{2}}
(6)
{\Gamma}_{\mathsf{\text{est}}}=\frac{2{e}_{2}\left({e}_{6}+12{e}_{2}^{3}9{e}_{2}{e}_{4}\right)}{3{\left({e}_{4}2{e}_{2}^{2}\right)}^{3}}
(7)
where e_{2}, e_{4}, and e_{6} are the second, the fourth, and the sixth order moments of the envelope data respectively. These estimated parameters will then be used to generate the noise. In the procedure for validation, three statistical methods are used to compare measured and simulated noises:

1)
The probability density function (PDF) is estimated from measured data by using kernel density estimators [21].

2)
The complementary cumulative distribution function (CCDF) gives the probability that the random variable is above a particular level and is defined as:
\mathsf{\text{CCDF}}\left(X\right)=P\left(X>x\right)=\underset{x}{\overset{\infty}{\int}}\mathsf{\text{PDF}}\left(u\right)\mathsf{\text{d}}u=1\mathsf{\text{CDE}}\left(x\right)
(8)
where CDF is the cumulative distribution function.

3)
The KullbackLeibler divergence (KL) is a relative entropic criterion, and it measures the dissimilarity between two probability distributions P and Q, where (KL) = 0 indicates that P = Q [22, 23].
Figures 5 and 6 show both PDF and CCDF for two measured noises (generator bar and TeslaCoil), respectively. The estimated parameters for the two measured noises are (A_{est} = 0.0280, Γ_{est} = 0.3978) for generator bar and (A_{est} = 0.3575, Γ_{est} = 0.1194) for TeslaCoil. We denote Middleton1 and Middleton2 the estimated Middleton class A noise calculated using the estimated parameters (A_{est} = 0.0280, Γ_{est} = 0.3978) and (A_{est} = 0.3575, Γ_{est} = 0.1194), respectively. The PDF and CCDF of the estimated Middleton class A noises and the Gaussian noise are also presented on Figures 5 and 6. These figures show that the PDF and CCDF of the estimated Middleton class A noises (Middleton1 and Middleton2) are more close to the measured noises than the Gaussian case. Table 1 confirms these observations by presenting the KL divergences of the measured noises and the two models of noise (Middleton and Gaussian). So, the KL divergence of Measured noise1 density is 0.04 from the estimated Middleton Class A density and 0.3 from the Gaussian density. For Measured noise2 density, the KL divergence is 0.02 from the estimated Middleton Class A density and 0.27 from the Gaussian density. These results confirm that the measured impulsive noise is better modeled by the Middleton Class A model as compared to Gaussian noise. Hence, we can use the Middleton Class A as an approximated model for impulsive noise in highvoltage substation. Therefore, we evaluate the performance of wireless communication in this environment using the estimated parameters of the measured noises.
C. Extension of Middleton Class A model for multiantenna systems
In order to evaluate the performances of MIMO systems under the impulsive noise, an extension of the Middleton model is derived. Middleton Class A model was derived for singleantenna systems. For a twoantenna system, we considered a bivariate Middleton Class A model used in [7]. This model is limited to n_{
r
}= 2 antennas. Thus, we derive an extension for n_{
r
}≥ 2. We can write (3) as:
f\left(x\right)=\sum _{m=0}^{\infty}{a}_{m}g\left(x,\mu ,{\sigma}_{m}^{2}\right)
(9)
where {a}_{m}=\frac{{e}^{A}{A}^{m}}{m!},\mu =0 and g\left(x,{\sigma}_{m}^{2}\right)=\frac{1}{\sqrt{2\pi {\sigma}_{m}^{2}}}{e}^{\frac{{x}^{2}}{2{\sigma}_{m}^{2}}}. The density of Middleton Class A can be approximated by the twoterm model [19]:
f\left(x\right)={e}^{A}g\left(x,{\sigma}_{0}^{2}\right)+\left(1{e}^{A}\right)g\left(x,{\sigma}_{1}^{2}\right)
(10)
Let x = [x_{1}, x_{2}, x_{3}, ..., x_{
k
}] be a vector of k = n_{
r
}random variables, each variable has a Middleton Class A density function and x_{
k
}is the noise observation at the k th antenna. Then, the multivariate density of x can be written as [19]:
{f}_{x}\left(x\right)=\sum _{m=0}^{\infty}{a}_{m}g\left(\mathsf{\text{x}},{K}_{m}\right)
(11)
where a_{
m
}is as in (9), K_{
m
}is the covariance matrix that represents the spatial correlation in the noise and g is a multivariate Gaussian function:
g\left(\mathsf{\text{x}},{K}_{m}\right)=\frac{1}{{\left(2\pi \right)}^{\frac{{n}_{r}}{2}}{\left{K}_{m}\right}^{\frac{1}{2}}}{e}^{\frac{{\mathsf{\text{x}}}^{T}{K}_{m}^{1}\mathsf{\text{x}}}{2}}
(12)
where . denotes the determinant. From (11) and (12), we obtain:
{f}_{x}\left(x\right)=\sum _{m=0}^{\infty}\frac{{a}_{m}}{{\left(2\pi \right)}^{\frac{{n}_{r}}{2}}{\left{K}_{m}\right}^{\frac{1}{2}}}{e}^{\frac{{\mathsf{\text{x}}}^{T}{K}_{m}^{1}\mathsf{\text{x}}}{2}}
(13)
Equation (13) represents a general extension of Middleton Class A model for multiantenna systems. We can use the approximation as in (10). Then, we obtain an approximate version of the extension:
\begin{array}{lll}\hfill {f}_{x}\left(x\right)& =\frac{{e}^{A}}{{\left(2\pi \right)}^{\frac{{n}_{r}}{2}}{\left{K}_{0}\right}^{\frac{1}{2}}}{e}^{\frac{{\mathsf{\text{x}}}^{T}{K}_{0}^{1}\mathsf{\text{x}}}{2}}\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ \phantom{\rule{1em}{0ex}}+\frac{1{e}^{A}}{{\left(2\pi \right)}^{\frac{{n}_{r}}{2}}{\left{K}_{1}\right}^{\frac{1}{2}}}{e}^{\frac{{\mathsf{\text{x}}}^{T}{K}_{1}^{1}\mathsf{\text{x}}}{2}}\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}
(14)
where K_{
m
}is n_{
r
}× n_{
r
}covariance matrix and is defined as:
{K}_{m}=\left(\begin{array}{ccc}\hfill \mathsf{\text{Var}}{\left({x}_{1}\right)}_{m}\hfill & \hfill \dots \hfill & \hfill \mathsf{\text{Cov}}{\left({x}_{1},{x}_{k}\right)}_{m}\hfill \\ \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill \mathsf{\text{Cov}}{\left({x}_{k},{x}_{1}\right)}_{m}\hfill & \hfill \dots \hfill & \hfill \mathsf{\text{Var}}{\left({x}_{k}\right)}_{m}\hfill \end{array}\right)
(15)
where \left\{\begin{array}{c}\hfill \mathsf{\text{Var}}{\left({x}_{k}\right)}_{m}=\frac{\frac{m}{A}+{\Gamma}_{k}}{1+{\Gamma}_{k}}={\sigma}_{km}^{2}\hfill \\ \hfill \mathsf{\text{Cov}}{\left({x}_{i},{x}_{j}\right)}_{m}={\rho}_{ij}{\sigma}_{im}{\sigma}_{jm}\hfill \end{array}\right..
Γ_{
k
}is the Gaussian factor at the k th antenna and ρ_{
ij
}is the correlation coefficient between the noise observations at i and j antennas, 1 ≤ ρ ≤ 1. Finally, we can write K_{
m
}as
{K}_{m}=\left(\begin{array}{ccc}\hfill {\sigma}_{1m}^{2}\hfill & \hfill \dots \hfill & \hfill {\rho}_{1k}{\sigma}_{1m}{\sigma}_{km}\hfill \\ \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill {\rho}_{k1}{\sigma}_{km}{\sigma}_{1m}\hfill & \hfill \dots \hfill & \hfill {\sigma}_{km}^{2}\hfill \end{array}\right)
(16)