# GMM-based Symbol Error Rate Prediction for Multicarrier Systems with
Impulsive Noise Suppression

N. Rozic, , P. Banelli, , D. Begusic, and J. Radic,

###### Abstract

Theoretical analysis of orthogonal frequency division multiplexing (OFDM)
systems equipped at the receiver by a non-linear impulsive noise suppressor is
a challenging topic in communication systems. Indeed, although an exact
closed-form expression for the output signal-to-noise ratio (SNR) of such OFDM
systems is available for widely used impulsive noise models, theoretical
analysis of the associated symbol error rate (SER) is still open. So far, the
analytical SER expressions available in the literature approximate the time-
domain impulsive noise, as Gaussian distributed in the discrete frequency
domain. Conversely, this work presents an accurate analysis of the distortion
noise at the nonlinearity output exploiting a Gaussian mixture model (GMM). By
using GMMs we unified the approach of SER prediction for unmitigated systems,
as well as for the mitigated ones, equipped by non-linear impulsive noise
suppressors, including blanking, clipping, clipping-blanking and attenuating
processors. Closed form expressions for the SER are derived both for non-
fading and frequency-selective Rayleigh and Rician fading channels affected by
impulsive noise which is represented by GMMs, thus including Bernoulli-
Gaussian (BG), Middleton Class-A, as well as (approximated) alpha-stable
noise. Theoretic SER performance are compared with simulations, showing very
good agreement for all the impulsive noise scenarios and the non-linear
suppressors.

Index Terms: symbol error rate, multicarrier systems, impulsive noise
mitigation, distortion noise distribution, Gaussian mixture approximation

##  I Introduction

Multicarrier modulations (MCM) are widely employed in most of the wired and
wireless communication systems. Among the several reasons for their widespread
use, one is certainly their higher resistance to impulsive noise (ImpN) with
respect to single carrier modulations (SCM) [1]. The evaluation of the output
SNR performance, and the associated SER, is an active research area for
communication engineers, especially in non-ideal conditions, such those faced
in ImpN environments. For instance, [2, 3, 4] investigated the effects of non-
linear distortions in OFDM systems. Furthermore, the effect of impulsive noise
has been investigated on the system performance of coded SCM [5], uncoded SCM
and MCM [1], as well as for coded MCM in power line communications (PLC) [6,
7], including Bernoulli-Gaussian (BG), Middleton Class-A [8, 9], and alpha-
stable noise models [10, 9]. Some fundamental results in the analysis of their
performance limits can be found in [11, 12, 13] where authors derived the
achievable pairwise error probability (PEP) bounds for OFDM systems affected
by impulsive noise represented by a GMM. More recently, performance analysis
for MCM/OFDM systems equipped with impulsive noise suppressors either for 4G
cellular systems (UMTS-LTE), broadcasting (DVBT), local area networks (WiFi),
or PLC, have been derived in [13, 14, 15, 16, 17, 18], respectively. Therein,
sample-based non-linear suppressors simply clip, or null, or both clip and
null, the complex envelope of the baseband received signal, when it overpasses
some given thresholds. Both the thresholds and the parameters of these simple
receivers are typically optimized to maximize the output SNR. More
specifically, ImpN mitigation relies on non-linear instantaneous
transformations of the signal, such as the optimal Bayesian estimator (OBE)
[19], and sub-optimal thresholding based approaches, such as blanking [20, 13,
19, 17], clipping-blanking [21, 14], or optimal Bayesian attenuating/clipping
devices [18]. Furthermore, the performance of space-time coded systems over
frequency-selective multiple-input multiple-output (MIMO) channels impaired by
Middleton Class-A impulsive noise, is evaluated in [22], [23].

Actually, the SER closed-form expressions available in the literature are
typically developed assuming that the distortion noise at the ImpN suppressor
output has a Gaussian distribution. However, this approximation is acceptable
only when the average number of noise impulses per OFDM/MCM frame is
sufficiently high to invoke the central-limit theorem [13, 15]. To overcome
this issue, the authors in [15] proposed a semianalytical SER analysis for
instantaneous ImpN removal in frequency-selective fading channels, which turns
out to be quite accurate. Similarly, authors in [24] presented a
semianalytical approach to evaluate the error probability of OFDM systems
subject to carrier frequency offset (CFO) in frequency-selective channels,
characterized by Rayleigh or Rician fading.

This work proposes a unified GMM-based analytical approach for both
unmitigated and mitigated OFDM systems affected by ImpN. For mitigated
systems, the residual distortion noise at the output of the non-linear
suppressor (NLS) is modeled with a $\mathcal{K}$-GMM, where $\mathcal{K}$ is
not directly related with $\mathit{K}$-GMM ImpN in the channel111in this work,
$\mathit{K}$ denotes the number of ImpN components in the actual channel,
while $\mathcal{K}$ rather denotes the number of GMM components used to
approximate the distortion noise at the ImpN suppressor output.. The proposed
framework addresses the SER analysis of the most interesting non-linear
impulsive noise suppressors such as the OBE, the Genie-aided detector (GAD),
as well as the thresholding-based Bayesian attenuating suppressor (BAS), thus
including as particular cases simple estimators, such as blanking, clipping,
and clipping-blanking. Specifically, we derive SER analytical expressions for
flat and frequency-selective channels, both in non-fading and Rayleigh/Rician
fading environments affected by $K$-GMM impulsive noises such as BG, Class-A,
and GMM-approximated alpha-stable noises. The theoretical SER performance has
been compared with the results obtained by simulation, showing very good
agreement for all the realistic impulsive noise scenarios. Although the paper
focuses on OFDM systems, the theoretical framework and conclusions can be
easily extended to any other MCM system.

The first essential and innovative contribution of this manuscript is the
novel GMM-based unified approach for prediction of the SER in multicarrier
(OFDM) systems, affected by arbitrarily distributed ImpN, with or without ImpN
mitigation. To the best of our knowledge, such generalized framework is new in
the scientific literature. Furthermore, in the presence of ImpN mitigation, we
propose a GMM approximation of the output distortion noise, which exploits a
novel component-by-component representation. This way, we derive accurate
closed-form analytical expressions for the associated SER, improving the
accurancy w.r.t. conventional approaches based on approximated assumptions of
Gaussian distribution for the residual noise or, equivalently, on sufficiently
long OFDM blocks to accommodate a sufficiently high number of ImpN events [13,
17].

The rest of the paper is organized as follows. Section II briefly summarizes
the system and noise models. Section III derives SER performance for
frequency-flat non-fading OFDM system affected by K-GMM ImpN, in the absence
of any impulse noise suppressor, where AWGN (1-GMM) and 2-GMM [1] are two
particular cases. Section IV extends the analysis to systems employing non-
threshold based suppressors, namely the GAD [25] and OBE [19]. Then, the SER
analysis for threshold-based ImpN suppression is derived in Section V. Section
VI extends the analysis to frequency-selective channels in the presence of
slow Rayleigh/Rician fading, when the system is equipped by ImpN NLSs.
Numerical and simulation results are presented in Section VII, including
comparison with some SER prediction alternatives, while Section VIII provides
some concluding remarks. Appendices A and B are dedicated to detailed proofs
and mathematical derivations.

##  II System and Noise Models

This work focuses on M-QAM multicarrier (OFDM) communications over frequency
flat and frequency-selective, possibly fading, channels, impaired by impulsive
noise. Thus, in this section we introduce the associated system and noise
models.

###  II-A System Model

Let’s define
$\mathbf{s}_{q}=\left[s_{q}[0],s_{q}[1],\cdots,s_{q}[L-1]\right]^{T}$ the qth
information symbol to be transmitted on $L$ orthogonal subcarriers of an OFDM
system, as shown in Fig. 1. Assuming the data
$\left\\{s_{q}\left[l\right]\right\\}_{l=0,...,L-1}$ on different subcarriers
are independent, with zero mean and same variance
$\sigma_{s}^{2}=E\left\\{\left|s_{q}\left[l\right]\right|^{2}\right\\}$, the
time-domain OFDM symbol
$\mathbf{x}_{q}=\left[x_{q}\left[0\right],x_{q}\left[1\right],\cdots,x_{q}\left[L-1\right]\right]^{T}$
is obtained by $\mathbf{x}_{q}=\mathbf{F}_{L}^{H}\mathbf{s}_{q}$, where
$\mathbf{F}_{L}$ is the L-point unitary DFT matrix [26, 27] and $(\cdot)^{H}$
is the Hermitian operator. The inter-symbol interference is avoided by adding
a cyclic prefix (CP) with length $L_{h}$ to the OFDM block, resulting in
$\mathbf{x}_{q}^{\textrm{cp}}=\left[x_{q}\left[L-L_{h}\right],\cdots,x_{q}\left[L-1\right],x_{q}\left[0\right],x_{q}\left[1\right],\cdots,x_{q}\left[L-1\right]\right]^{T}$.
Then, after D/A conversion and RF modulation, the OFDM block is transmitted
through a frequency-selective channel, with discrete-time impulse response
$\mathbf{h}_{q}=\left[h_{q}\left[0\right],h_{q}\left[1\right],\ldots,h_{q}\left[L_{h}-1\right],0,\ldots,0\right]^{T}$.
The frequency-flat channel is simply a special case when $h[0]=1$ and
$\left\\{h\left[l\right]=0\right\\}_{l=1,\cdots,L-1}$, and consequently (1)
simply reduces to $y\left[l\right]=x\left[l\right]+n[l]$. In this case, the
CP+ box in Fig. 1 at the transmitter, as well as the CP- and ”Equalizer” boxes
at the receiver, are omitted. Rayleigh fading is a typical channel model used
to mimic worst-case scenario in non-LoS conditions, assuming each $h_{q}[l]$
is distributed as a complex zero-mean Gaussian RV, with stationary variance
$\sigma_{l}^{2}=E\left\\{\left|h_{q}\left[l\right]\right|^{2}\right\\}$.
Conversely, in the presence of Line of Sight (LoS),
$E\left\\{h_{q}(0)\right\\}\neq 0$ makes the statistics of $|h_{q}[0]|$ a
Rician one. More general models include the Nakagami-n models [28, 29].

In this paper, we consider Rayleigh and Rice slow (quasi-static) fading, where
all the $L_{h}$ active paths are assumed as time-invariant during the
transmission of each OFDM block (block-fading model). The extension of the
proposed analysis to fast fading where the $L_{h}$ active paths are time-
variant within each OFDM symbol (double-selective channels), although quite
straightforward, it is out of the scope of this paper.

After CP removal, the received OFDM symbol
$\mathbf{y}_{q}=[y_{q}\left[0\right],y_{q}\left[1\right],\cdots,y_{q}\left[L-1\right]]^{T}$
is obtained by a cyclic convolution with the channel, as expressed by

$y_{q}\left[l\right]=h_{q}\left[l\right]\otimes_{L}x_{q}\left[l\right]+n_{q}[l],$
(1)

Figure 1: Block scheme of the baseband system model

which in the frequency domain corresponds to carrier by carrier multiplication
of the channel transfer function with the transmitted data, as expressed by

$\mathbf{y}_{q}^{(f)}=\mathbf{F}_{L}\mathbf{y}_{q}=\textrm{diag}\left(\mathbf{h}_{q}^{(f)}\right)\mathbf{s}_{q}+\mathbf{n}_{q}^{(f)},$
(2)

where $\mathbf{h}_{q}^{(f)}=\sqrt{L}\mathbf{F}_{L}\mathbf{h}_{q}$ and
$\mathbf{n}_{q}^{(f)}=\mathbf{F}_{L}\mathbf{n}_{q}$ are the frequency-domain
channel response and noise, respectively, during the qth OFDM block. Assuming
the block-fading channel paths are uncorrelated, the variance of the received
signal $y_{q}[l]$ is given by
$\sigma_{y_{q}}^{2}=\sum_{l=0}^{L-1}\left|h_{q}\left[l\right]\right|^{2}\sigma_{x}^{2}+\sigma_{n}^{2}$
where $\sigma_{x}^{2}=E\\{\left|x_{q}\left[l\right]\right|^{2}\\}$ and
$\sigma_{n}^{2}=E\\{\left|n_{q}\left[l\right]\right|^{2}\\}$ are the
transmitted OFDM signal and receiver noise variances, respectively. Zero-
forcing (ZF) equalization is applied in the frequency domain, simply dividing
$\mathbf{y}_{q}^{(f)}$ in (2) by $\mathbf{h}_{q}^{(f)}$, leading to

$\mathbf{y}_{q,e}^{(f)}=\mathbf{s}_{q}+\left(\textrm{diag}\left(\mathbf{h}_{q}^{(f)}\right)\right)^{-1}\mathbf{n}_{q}^{(f)}.$
(3)

As shown in Fig. 1, the instantaneous non-linear suppressor (NLS), which
mitigates the noise impulses in the time-domain before DFT processing, spreads
in the frequency-domain, over all the OFDM subcarriers, the distortion
introduced by the time-domain sparse noise impulses.

The NLS output, which can be formally expressed by

$\hat{x}_{q}[l]=g(y_{q}[l];\bm{\pi}_{q}),$ (4)

depends on the vector
$\bm{\pi}_{q}=\left\\{\sigma_{n},\sigma_{x},\Sigma_{l}|h_{q}[l]|^{2}\right\\}$,
which contains the set of parameters that influence the statistical property
of the qth OFDM block, such as the signal power $\sigma_{y_{q}}^{2}$, and the
noise power $\sigma_{n}^{2}$.

###  II-B Impulsive Noise Models

This paper focuses on impulsive noise modeled by a $K$-GMM [30], whose
probability density function ($pdf$) $f_{N}(n)$ is expressed by

$f_{N}(n)=\sum_{k=0}^{K-1}p_{k}G(n,\sigma_{k}^{2}),$ (5)

where $\\{p_{k}\\}_{k=0,1,\cdots,K-1}$, with $\sum_{k=0}^{K-1}p_{k}=1$, are
the probabilities that an impulsive event with variance $\sigma_{k}^{2}$, is
generated according to the kth Gaussian $pdf$ $G(n,\sigma_{k}^{2})$. The
component $k=0$ represents the ”white” noise contribution, appearing with
probability $p_{0}$ and variance $\sigma_{0}^{2}$, while the other $K-1$
components represent the impulsive noise, which appears with probability
$p_{I}=\sum_{k=1}^{K-1}p_{k}=1-p_{0}$ and total variance
$\sigma_{I}^{2}=\sum_{k=1}^{K-1}p_{k}\sigma_{k}^{2}$.

The reason to choose model (5) are twofold. Firstly, $K$-GMM in (5) is a
mixture of Gaussian functions, thus ensuring analytical tractability and
elegance of calculation, and secondly, $K$-GMM can successfully fit any ImpN
distribution either exactly [31, 32], or approximately [33, 34].

In the following we will focus on widely used ImpN models that can be
described by (5), including the Bernoulli-Gaussian (BG) model, the Middleton’s
Class-A model, and the symmetric alpha-stable (S$\alpha$S) model. However, the
approach presented in this work can be obviously applied to any other noise
that is modelled as $K$-GMM.

####  II-B1 BG Impulsive Noise

The BG model [1, 13] corresponds to a 2-GMM ImpN, where a noise sample is
considered either thermal or impulsive with probabilities $p_{0}$ and
$p_{1}=1-p_{0}$, respectively. The noise $W\sim{\it G}(w,\sigma_{0}^{2})$ and
the impulse noise $I\sim{\it G}(i,\sigma_{1}^{2})$ are both zero-mean
independent Gaussian RVs, resulting in ImpN with a total variance
$\sigma_{N}^{2}=p_{0}\sigma_{0}^{2}+p_{1}\sigma_{1}^{2}$. AWGN is the even
simpler case with $K$=1 and $p_{0}=1$.

Actually, the 2-GMM can also well approximate Class-A ImpN for $A\ll 1$ and,
under proper conditions, can be used as a simple approximative model for the
distortion noise at the output of the NL ImpN suppressors, which works
noticeably better than the AWGN approximation in [14],[15].

####  II-B2 Class-A Impulsive Noise

The Middleton’s canonical Class-A model [8], [9] assumes ImpN with $K=\infty$,
as a sum of the two independent components $W$ and $I$, where $W$ is a
stationary Gaussian (background) noise with variance $\sigma_{W}^{2}$, and $I$
is the impulsive noise with variance $\sigma_{I}^{2}$ generated independently
by interfering sources (simultaneous, possibly infinite). Only a specific one
of the K components could be present in each noise sample with a probability
$p_{k}$ modeled according to a Poisson distribution, as expressed by

$p_{k}=e^{-A}A^{k}/k!,$ (6)

where $A$ is a canonical parameter, also known as the impulsiveness index.

From (6), when $k=0$ it follows $p_{0}=e^{-A}$, which represents the
probability that the channel experiences only the thermic noise $W$.
Otherwise, with probability $1-p_{0}$, the channel experiences an impulsive
noise, whose k-$th$ component variance is
$\sigma_{k}^{2}=\sigma_{W}^{2}+\sigma_{I}^{2}\frac{k}{A}$.

Obviously, a canonical Class-A model is generally compliant with (5). The
”white” noise component is defined by $p_{0}$ and $\sigma_{0}^{2}$, the
impulsive component by $p_{I}=1-p_{0}$ and
$\sigma_{I}^{2}=\sum_{k=1}^{\infty}p_{k}\sigma_{k}^{2}/(1-p_{0})$, and the
total ImpN variance by
$\sigma_{n}^{2}=\sum_{k=0}^{\infty}p_{k}\sigma_{k}^{2}=p_{0}\sigma_{0}^{2}+p_{I}\sigma_{I}^{2}$.

For realistic systems, the ${K}$-GMM in (5) with $K\geq 10$ is generally a
very good approximation of the Class-A model. However, instead of simple
truncation of the Poisson distribution as proposed in [20], [35], we prefer
using a more accurate approximation which preserves the first two moments,
which is obtained by fitting the approximated $\bar{p}_{k}$,
$\bar{\sigma}_{k}^{2}$ according to

$\bar{p}_{k}=p_{k}/\sum_{k=0}^{K-1}p_{k},\hskip
3.0pt\bar{\sigma}_{k}^{2}=\sigma_{k}^{2}/\sum_{k=0}^{K-1}p_{k}\sigma_{k}^{2},$
(7)

which keeps unchanged the signal-to-impulsive noise ratio (SIR) defined as

$\text{SIR}=\frac{\sigma_{x}^{2}}{\sigma_{I}^{2}},$ (8)

where $\sigma_{I}^{2}=\sum_{k=1}^{K}\bar{p}_{k}\bar{\sigma}_{k}^{2}/(1-p_{0})$
represents to the impulsive noise variance.

####  II-B3 $\alpha$-Stable Impulsive Noise

An $\alpha$-stable random variable models an ImpN with a so-called heavy-
tailed (algebraic) distribution. Generally, the $\alpha$-stable pdf is defined
by a set of four canonical parameters $(\alpha,\beta,\gamma,\mu)$ where the
characteristic exponent $\alpha$ controls the heaviness of the $pdf$ tails,
$\beta$ controls the skewness, while $\gamma=\sigma^{\alpha}$ measures the pdf
spread around its mean $\mu$. Unfortunately, an explicit analytical expression
for the $\alpha$-stable pdf does not exist, except for some specific values of
alpha including $\alpha=0.5$ (Cauchy pdf) and $\alpha=2$ (Gaussian pdf), which
is the pdf with minimal thickness. However, due to its generality,
$\alpha$-stable distributions have found wide interest in statistics and
communications [10, 36].

Indeed, also an $\alpha$-stable distribution can be analytically defined by an
infinite mixture of Gaussians [37], that can be actually approximated by a
finite scaled Gaussian mixture in practice [38]. An efficient approximation of
the standard symmetric $\alpha$-stable S$\alpha$S distribution
($\beta=0,\gamma=1,\mu=0$) by the $K$-GMM presented in [39] is realized
through computing the weighting factors $w_{k}$ and variances $\sigma_{k}^{2}$
by minimizing the relative entropy between the $K$-GMM pdf and the S$\alpha$S
pdf in the amplitude range $\pm A_{\rm{N}}$.

##  III SER Analysis for Frequency-flat OFDM Systems Without ImpN Suppression

Assuming that a $K$-GMM impulsive noise affects OFDM symbols in a frequency-
flat channel ($h[0]=1,h[l]=0\quad\forall l\geq 1$, in Fig.1), and the receiver
does not perform any kind of ImpN mitigation (NLS block in Fig.1 is omitted),
an exact analytical expression for the SER can be derived as a generalization
of the approach in [1].

###  III-A SER for K-GMM ImpN Without Suppression

Let’s consider a $K$-GMM ImpN with associated probabilities and variances
$\left\\{p_{k},\sigma_{k}^{2}=\gamma_{k}\sigma_{x}^{2}\right\\}_{k=1,\cdots,K-1}$
222$\gamma_{k}$ obviously denotes the relative variance
$\gamma_{k}=\frac{\sigma_{k}^{2}}{\sigma_{x}^{2}}$.. Let $K-1$ impulse
components affect $l\leq L$ samples of an $L$-sized OFDM block, and the rest
of $L-l$ samples are affected only by the Gaussian noise component with
variance $\sigma_{0}^{2}=\gamma_{0}\sigma_{x}^{2}$, whose probability is
$p_{0}=1-\sum_{k=1}^{K-1}p_{k}$. The probability that $l\leq L$ samples are
affected by an impulsive noise $I$, possibly composed by at most $K-1$
components, is given by the multinomial distribution [40]

$w_{\mathbf{l_{K-1}}}=w\left(l_{1},\cdots,l_{K-1}\right)=\frac{l!}{l_{1}!\cdots
l_{K-1}!}q_{1}^{l_{1}}\cdots q_{K-1}^{l_{K-1}}$ (9)

where $\mathbf{\text{l}}_{K-1}=[l_{1},\cdots,l_{K-1}]$,
$\sum_{k=1}^{K-1}l_{k}=l$, and $q_{k}=\frac{p_{k}}{1-p_{0}}$ with
$\sum_{k=1}^{K-1}q_{k}=1$.

Let’s express by $w(l)=\frac{L!}{(L-l)!l!}p_{0}^{L-l}p_{I}^{l}$ the binomial
distribution that defines the probability to experience an impulsive $l$-tuple
event characterized by (9) within the L-sized OFDM block.

Thus, generalizing the approach in [1] for 2-GMM, the SER for M-QAM square
constellation affected by $K$-GMM impulsive noise can be expressed by

$p_{s}^{(K-GMM)}=\sum_{l=0}^{L}w(l)\sum_{l_{1}=0}^{l}\sum_{l_{2}=0}^{l-l_{1}}\cdots\sum_{l_{K-1}=0}^{l-\sum_{m=1}^{K-2}l_{m}}w_{\mathbf{l_{K-1}}}p_{s;\mathbf{l}_{K}}^{(AWGN)}$
(10)

where
$p_{s,\mathbf{l}_{K}}^{(AWGN)}=4\frac{\sqrt{M}-1}{\sqrt{M}}B_{\mathbf{l}_{K}}\left(1-\frac{\sqrt{M}-1}{\sqrt{M}}B_{\mathbf{l}_{K}}\right)$,
with
$B_{\mathbf{l}_{K}}=Q\left(\sqrt{\frac{3\,\rho_{\mathbf{l}_{K}}}{M-1}}\right)$.
Here, $Q\left(x\right)$ denotes the Gaussian Q-function [41], and
$\rho_{\mathbf{l}_{K}}$ is the SNR 333note the equality of the output SNR
$\rho$ in the time- and the frequency domain. given by

$\rho_{\mathbf{l}_{K}}=\frac{\sigma_{x}^{2}}{\sigma_{\mathbf{l}_{K}}^{2}}=\frac{L}{\sum_{k=0}^{K-1}l_{k}\gamma_{k}}.$
(11)

It should be noted that the computational complexity of (10) is defined by the
number of the $K$-tuple combinations. Even for relatively low $K$, such as
$K=5$, the number of combinations is pretty high, and, as discussed in Sect.
IIB for Class A ImpN, an approximation with e.g. 4-GMM, can provide good SER
predictions with acceptable complexity, as shown in Sect. VII. In addition,
further reduction of the computational complexity can be reached using an
$L^{\prime}<L$ rather than $L$ in (10), for instance by considering only those
$l$-tuples whose $p_{s,\mathbf{l}_{K}}^{(AWGN)}$ is over $10^{-20}$. This
approach is particularly efficient for BG and Class-A ImpN at low $p_{I}$. On
the contrary, for relatively high $p_{I}$, the simulation results presented in
Sect. VII show that accurate SER prediction can be obtained even by using a
simple 1-GMM (AWGN) model. However, such conclusions do not hold true for an
arbitrarily distributed ImpN, including $S\alpha S$ ImpN.

###  III-B SER Prediction for 1-GMM (AWGN) Approximation

When $K=1$, expression (10) reduces to the well known SER formula for M-QAM
square constellations in AWGN channels [42], as expressed by

$p_{s}^{(AWGN)}(\rho)=4\frac{\sqrt{M}-1}{\sqrt{M}}B_{0}(\rho)\left(1-\frac{\sqrt{M}-1}{\sqrt{M}}B_{0}(\rho)\right),$
(12)

where $B_{0}(\rho)=Q\left(\sqrt{\frac{3\,\rho}{M-1}}\right)$, and
$\rho=\frac{\sigma_{x}^{2}}{\sigma_{n}^{2}}=\frac{1}{\sum_{k=0}^{K-1}p_{k}\gamma_{k}}$
denotes the average SNR.

Under general ImpN conditions, an AWGN model (formally 1-GMM) can actually be
employed as a good predictor of the SER, when the total ImpN distribution in
the frequency domain can be well approximated by a Gaussian one. As shown in
Fig. 4, this is the case for Class A ImpN when $A\geq 0.1$. However, for
S$\alpha$S ImpN Fig. 5 shows that the AWGN model (12) does not provide
accurate SER predictions even when the impulsive noise is highly probable.

###  III-C SER Prediction for 2-GMM Approximation

With $K=2$, expression (10) reduces to

$p_{s}^{(2-GMM)}=\sum_{l=0}^{L}w_{l}p_{s}^{(AWGN)}(\rho_{l})$ (13)

where $w_{l}=\frac{L!}{(L-l)!l!}p_{0}^{L-l}p_{1}^{l}$ with
$\rho_{l}=\frac{\sigma_{x}^{2}}{\sigma_{l}^{2}}=\frac{L}{(L-l)\gamma_{0}+l\gamma_{1}}$,
and $\gamma_{0}=\frac{\sigma_{0}^{2}}{\sigma_{x}^{2}}$,
$\gamma_{1}=\frac{\sigma_{1}^{2}}{\sigma_{x}^{2}}$.

Note that expression (13), which is here derived from (10) for a special case
$K=2$, corresponds to the BG ImpN case already considered in [1].

Under some conditions, as discussed in Sect. II, a 2-GMM is an accurate
approximation of a $K$-GMM ImpN, and thus can be used as a good SER predictor.
Particularly, a Class A ImpN with a relatively low impulse probability, e.g.,
when the weighting factors for $K>2$ are relatively low, can be safely
approximated by a two-component Gaussian mixture.

##  IV SER Analysis for Frequency-flat OFDM System with ImpN Suppression

In order to extend the previous analysis to systems employing non-linear
suppression (NLS) of ImpN, as shown in Fig. 1, it is possible to approximate
the non-Gaussian distribution of the output distortion noise with a Gaussian
mixture (5). Intuitively, the distortion noise at the output of the NLS may
retain some of the features of the impulsive noise at the suppressor input,
because the impulsive noise suppressor can only mitigate noise impulses, and
does not eliminate them completely444due to the fact that ImpN is a random
variable. Actually, when the input noise is $K$-GMM, and the full noise state
information (NSI) is available, the optimal GAD suppressor is linear [18] and,
consequently, also the distribution of the distortion noise at the output of
the non-linear device is effectively modeled by a $K$-GMM, with the same
weighting factors, but different variances. This scenario is considered in
subsection B.

However, in realistic systems, NSI is not fully available and suppressors turn
to be non-linear, such as the OBE in [19]. In this case the output distortion
noise can not be exactly modeled by a GMM, because each single component is no
longer Gaussian distributed, and a $\mathcal{K}$-GMM can be used only as an
approximation. The use of the OBE is considered in subsection C, and a
threshold-based NLS in Section V.

###  IV-A SNR at the Output of the NLS

Considering the non-fading channel, the statistical parameters of the received
signal and of the channel noise are fixed over all the OFDM blocks. Then,
similarly to (4),555note that for a non-fading channels we can omit subscript
$q$ the output of a memory-free non-linear suppressing device
$g\left(y\right)$ is given by $\hat{x}=g\left(y;\bm{\pi}\right)$. By the
Bussgang’s theorem [43, 2, 3], where $y=x+n$, the output $\hat{x}$ can be
expressed as

$\hat{x}=g(x+n;\bm{\pi})=\alpha_{\bm{\pi}}x+d,$ (14)

where $\alpha_{\bm{\pi}}$ is the scaling factor of the information-bearing
signal $x$, which depends on the ”average” CSI contained in $\bm{\pi}$, and
$d$ is the distortion noise.

It follows from (14) that the output distortion noise is given by

$d=\hat{x}-\alpha_{\bm{\pi}}x.$ (15)

For any arbitrary non-linear device, with real or complex valued input, the
average output SNR $\rho$ can be computed by [17]

$\rho=\frac{E\left\\{\left|\alpha
x\right|^{2}\right\\}}{E\left\\{\left|d\right|^{2}\right\\}}=\frac{E\left\\{\left|\alpha
x\right|^{2}\right\\}}{E\left\\{\left|\hat{x}-\alpha
x\right|^{2}\right\\}}=\frac{\alpha_{\bm{\pi}}^{2}\sigma_{x}^{2}}{\sigma_{d\bm{\pi}}^{2}}$
(16)

where $\sigma_{d\bm{\pi}}^{2}$ is the variance of the output distortion noise
in (14).

Considering a $K$-GMM ImpN and the SER in (10) for the associated SNR (11), we
can use the same formula in the presence of ImpN mitigation, by expressing
(16), similarly to (11), as

$\rho_{\mathbf{l}_{K}}=\frac{\alpha_{\pi}^{2}\sigma_{x}^{2}}{\sigma_{\mathbf{l}_{K}}^{2}}=\frac{L\alpha_{\pi}^{2}}{\sum_{k=0}^{K-1}l_{k}\gamma_{k}}.$
(17)

###  IV-B SER Prediction for GAD

If the full NSI is available to the receiver, the genie-aided suppressor [25]
can optimally mitigate the received signal samples according to the actual
noise variance. The MMSE optimal genie-aided Bayesian estimator knows which
noise component $k\in\left\\{0,K-1\right\\}$ is active and simply computes the
output by $\hat{x}_{\text{GAD}}=\beta_{k}y_{k}$, where
$\beta_{k}=\frac{1}{1+\gamma_{k}}$ [19].

Obviously, the overall output noise is a nonlinearly attenuated version of the
input noise, while the useful output signal is attenuated by the factor
$\alpha_{\text{GAD}}=\sum_{k=0}^{K-1}{p_{k}}\beta_{k}$ [18] by means of the
Bussgang’ theorem. Given $\hat{x}$ and $\alpha_{\text{GAD}}$, the output SNR
is [18]

$\rho^{\text{GAD}}=\frac{\left[\sum_{k=0}^{K-1}p_{k}\beta_{k}\right]^{2}}{\sum_{k=0}^{K-1}p_{k}(1+\gamma_{k})\beta_{k}^{2}-\left[\sum_{k=0}^{K-1}p_{k}\beta_{k}\right]^{2}}.$
(18)

Given (18), the SER performance can be confidently computed either by (13) for
BG ImpN ($K=2$), or by (10) for $K\geq 3$.

Note that the SER granted by the GAD, which exploits full NSI at the receiver
(e.g., which noise component is active at any sample), is the best achievable
SER.

###  IV-C SER Prediction for OBE

When NSI is limited, i.e., the receiver does not know which noise component is
active at any time, the optimal Bayesian estimator (OBE) is obtained by a non-
linear signal attenuation, whose optimal parameters are exactly defined by
simple closed-form analytical expressions [17, 18]. However, the output
performance, including SNR and SER, can not be expressed in a closed-form
because the output noise distribution is not Gaussian, as well as each one of
its components.

As proved in [18], the output SNR, and consequently $\alpha_{\text{OBE}}$, can
be closely approximated with the output SNR, and $\alpha_{\text{BAS}}$,
characterizing the BAS output, when the number of thresholds $M_{T}$ is high
enough ($M_{T}\geq 10$). Therefore, in the analysis we can exploit the
following approximation

$\alpha_{\text{OBE}}\approx\alpha_{\text{BAS}}=2\sigma_{x}^{2}\sum_{m=0}^{M}\beta_{m}\sum_{k=0}^{K-1}p_{k}(a_{m,k}-a_{m+1,k}),$
(19)

where
$a_{m,k}=\left[1+A_{m}^{2}/2(1+\gamma_{k})\sigma_{x}^{2}\right]e^{-{A_{m}^{2}}/2(1+\gamma_{k})\sigma_{x}^{2}}$,
as detailed in [18].

Given $\rho_{\mathbf{l}_{K}}$ (17) and $\alpha_{\text{OBE}}$ (19), the SER for
the optimal Bayesian estimator directly follows from (10).

Note that when only limited NSI is available to the receiver, the SER
performance granted by the OBE, is the best acievable one [19].

##  V SER Analysis for Frequency-flat OFDM Systems with Threshold-based
Suppressors

This section analyses the SER performance for the ideal, frquency-flat channel
scenario ($h[0]=1,h[l]=0,\forall l\geq 1$), when the NLS block in Fig.1
performs ImpN mitigation by using a (sub)optimal single thresholding, based on
its (sub)optimal detection of the channel state (e.g. ”white noise” or
”impulse noise”).

Common thresholding suppression techniques include blanking (nulling) [13],
clipping-blanking [14], Bayesian attenuating or Bayesian clipping [18], and
their combinations with multiple thresholds. Such threshold-based suppressors
may be preferred to OBE due to their relatively low complexity and high
efficacy. We consider the simplest scenario where the receiver employs a
single threshold to perform a binary hypothesis test on the noise state, based
on the received signal envelope $|y|$. Specifically, the receiver assumes the
state ”signal+AWGN” when $|y|\leq A_{T}$, and conversely ”signal+impulse
noise” when $|y|>A_{T}$. As detailed in the following subsections, the
distribution of the output distortion noise for this single-threshold
suppressor can be generally approximated by $\mathcal{K}$-GMM.

###  V-A Mixture Model for the Distribution of the Distortion Noise

Since the $K$-GMM ImpN variable $N$ includes ”white” noise $W$ and impulsive
noise $I$ components, both mitigated by the single-threshold non-linear
suppressor, the noise at the suppressor output is composed of four possible
components, each belonging to one of the following events

$\begin{array}[]{l}{\;\left.\begin{array}[]{l}{y_{1}=x+w}\\\
{y_{2}=x+i}\end{array}\right\\}{\rm when}\;\left|y\right|\leq A_{T}}\\\
{\;\left.\begin{array}[]{l}{y_{3}=x+w}\\\ {y_{4}=x+i}\end{array}\right\\}{\rm
when}\;\left|y\right|>A_{T}},\end{array}$ (20)

where y is the complex base-band received signal sample at the input of the
non-linear impulsive noise suppressor, and $A_{T}$ is the single ImpN
detecting threshold.

Consequently, an adequate model is a four-component scaled mixture, whose
$pdf$ is expressed by

$f_{D}\left(d\right)=\sum_{k=1}^{4}w_{k}f_{D_{k}}\left(d\right),$ (21)

where $\left\\{f_{D_{k}(d)}\right\\}_{k=1,\cdots,4}$ and
$\left\\{w_{k}\right\\}_{k=1,\cdots,4}$ are the distortion noise $pdf$s and
the associated weighting factors of the different components in (20),
respectively.

Figure 2: Distortion noise distributions $f_{D_{k}}(d);k=1,..,4$ for BG ImpN
at SIR=-10 dB, $p_{1}=0.01$, SNR=25 dB Figure 3: Distortion noise
distributions $f_{D_{k}}(d);k=1,..,4$ for S$\alpha$S ImpN at $\alpha=1.2$,
SIR=-10 dB, $p_{1}=0.01$, SNR=25 dB

Given the threshold $A_{T}$, the weighting factors $w_{k}$ in (21) can be
easily computed from $w_{1}=$ Prob$\left\\{|y_{1}|\leqslant
A_{T}|N=W\right\\}$ $\times$ Prob$\left\\{N=W\right\\}$, etc., leading to

$\begin{array}[]{l}{w_{1}=p_{0}\int_{0}^{A_{T}}p_{\left|Y_{0}\right|}\left(y\right)dy=p_{0}\left(1-c_{0}\right)}\\\
{w_{2}=p_{1}\int_{0}^{A_{T}}p_{\left|Y_{I}\right|}\left(y\right)dy=p_{1}\left(1-c_{1}\right)}\\\
{w_{3}=p_{0}\int_{A_{T}}^{\infty}p_{\left|Y_{0}\right|}\left(y\right)dy=p_{0}c_{0}}\\\
{w_{4}=p_{1}\int_{A_{T}}^{\infty}p_{\left|Y_{I}\right|}\left(y\right)dy=p_{1}c_{1}},\end{array}$
(22)

where $c_{0}=\exp\left(-\frac{A_{T}^{2}}{\sigma_{y_{0}}^{2}}\right)$,
$c_{1}=\exp\left(-\frac{A_{T}^{2}}{\sigma_{y_{I}}^{2}}\right)$, and
$\sigma_{y_{0}}^{2}=2\left(\sigma_{x}^{2}+\sigma_{0}^{2}\right)$,
$\sigma_{y_{I}}^{2}=2\left(\sigma_{x}^{2}+\sigma_{I}^{2}\right)$.

The $pdf$s $\left\\{f_{D_{k}(d)}\right\\}_{k=1,\cdots,4}$, as well as their
variances $\left\\{\sigma_{D_{k}}^{2}\right\\}_{k=1,\cdots,4}$ are derived
analytically in Appendix A.

Figs. 2 and 3 show the shape of the analytical results of $f_{D_{k}}(d)$, when
either BG or S$\alpha$S ImpNs affect the OFDM receiver input. Note that the
component distributions in Figs. 2 and 3 are scaled with their weighting
factor $w_{k}$ in (22), in order to show their impact on the total $f_{D}(d)$
in (21).

###  V-B Gaussian Mixture Model for $f_{D}(d)$

The proposed approximation of the $f_{D}(d)$ by $\mathcal{K}$-GMM leverages on
a component-by-component Gaussian fitting, as detailed in Appendix B. The
closed-form expressions for the $\mathcal{K}$-GMM parameters, e.g., the
dimension $\mathcal{K}$, the variances
$\left\\{\sigma_{G_{k}}^{2}\right\\}_{k=1,\cdots,\mathcal{K}}$, and the
scaling factors $\left\\{w_{G_{k}}\right\\}_{k=1,\cdots,\mathcal{K}}$, allow
to compute the SER for any ImpN scenario, by (10) and (11), where
$\gamma_{k}=\frac{\sigma_{G_{k}}^{2}}{\sigma_{x}^{2}}$ and $w_{k}=w_{G_{k}}$,
for $k=1,\cdots,\mathcal{K}$. The results of the $\mathcal{K}$-GMM fitting of
$f_{D}(d)$ for any ImpN scenario show that an order
$\mathcal{K}\in\left\\{2,3,4\right\\}$ is typically enough, as verified in
Section VII, which shows quite accurate SER estimates.

It is important to stress that the GMM approximation of the time domain
distribution $f_{D}(d)$ clearly shows that the 1-GMM (AWGN) can not be
considered as a good fitting model for any ImpN scenario. Thus, as
demonstrated in Sect.VII, and confirmed by results available in the literature
[13], [15], conventional expressions for $M$-QAM error probability in AWGN
channels [42] can be effectively used for OFDM systems only under certain ImpN
scenarios. That is, only when the output distortion noise in the frequency
domain can be safely approximated as a Gaussian RV [13, 15], which is
approximately true when the average number of noise impulses per OFDM block is
relatively high, and the noise is BG or Class-A distributed. Conversely, when
an alpha-stable ImpN affects the OFDM channel, even with relatively high
average number of impulses per OFDM block, such AWGN approximation does not
generally hold true.

##  VI SER Performance of OFDM in Frequency-selective Fading Channels Affected
by ImpN

The SER performance analysis for fading systems affected by ImpN can be
developed based on the results obtained for the non-fading ones. For
frequency-selective block-fading channels the SER in (10) for OFDM systems
affected by $K$-GMM ImpN, can be still used for each fading realization and
each subcarrier, by replacing $p_{s}^{(AWGN)}(\rho_{l})$ with a proper
$p_{s}^{(fading)}(\rho_{l})$ accounting for the specific fading channel model.

Indeed, following the system model presented in Fig. 1, and assuming that
$h[l],\forall l$ are independent Gaussian RVs, all the $h^{(f)}[l]$ of the
frequency domain channel response will be equally distributed.666this is
simplifying assumption where all subcarriers are active (no guard-bands). In
general, we can compute $SER_{l}$ per each subcarrier and then take an
average.

Consequently, assuming the same signal constellation over all the sub-
carriers, the SNR $\rho_{q}[l]$ on the $q$th block will be the same for any
carrier $l=1,\dots,L$. Thus, to simplify notation, in the following we omit
the subcarrier index $l$ and we use the $\rho_{q}$ to denote the SNR in the
$q$th block for any subcarrier.

The slow (quasi-static) frequency-selective block fading channel, will make
the signal power $\sigma_{x_{q}}^{2}$ fluctuating from an OFDM block to
another, according to the qth realization
$\\{h_{q}\left[l\right]\\}_{l=0,\ldots,L-1}$ of the channel. Thus, if all the
L subcarriers transmit independent data $s_{q}\left[l\right]$777digital
systems regularly use scrambling before modulation [42], the time-domain
signal power $\sigma_{x}^{2}$ is stationary (after CP removal), and the
conditional $pdf$ $f_{x_{q}|\mathbf{h}_{q}}\left(x,\mathbf{h}_{q}\right)$ of
the useful component $x_{q}\left[l\right]$ for a given channel
$\mathbf{h}_{q}$ is a Gaussian function $G(x,0,\sigma_{x_{q}}^{2})$, where
$\sigma_{x_{q}}^{2}=E\\{|x_{q}\left[l\right]|^{2}\\}=g_{q}\sigma_{x}^{2}$, and
$g_{q}=\sum_{l=0}^{L-1}{|h_{q}[l]|^{2}}$ is the power attenuation induced by
the channel during the $q$-th OFDM block. Assuming that $g_{q}$ varies in a
block-fading fashion, while the noise variance $\sigma_{n}^{2}$ is fixed, the
received signal power is expressed by
$\sigma_{y_{q}}^{2}=g_{q}\sigma_{x}^{2}+\sigma_{n}^{2}$.

Consequently, in order to compute the average SER for slow (block) fading
channels equipped with NLS, it is necessary to develop the expression for the
conditional distribution of the signal-to-noise ratio
$\rho_{q}=\sigma_{x_{q}}^{2}/\sigma_{n}^{2}$. Let $f(\rho_{q};\bar{\rho}_{q})$
denote the required $pdf$ of $\rho_{q}$ for the $q$th block when the average
signal-to-noise ratio is $\bar{\rho}_{q}$, and $p_{s}(\rho_{q})$ denote the
conditional SER, then the unconditional SER follows from

$p_{s}^{(fading)}=\int_{0}^{\infty}f\left(\rho_{q};\bar{\rho}_{q}\right)p^{(AWGN)}_{s}(\rho_{q})d\rho_{q},$
(23)

under the assumption that the overall noise effect can be modelled as
Gaussian.

Specifically, considering a single threshold ImpN suppression, the optimal
threshold $A_{T_{q}}$, as well as the associated attenuating parameters
$\beta_{0_{q}}$ and $\beta_{1_{q}}$, should adapt their values to maximize the
system performance [18].

###  VI-A Rayleigh Fading

Rayleigh fading represents a multipath channel, constituted by a number of
simultaneous time-aligned weak reflected components without any direct line-
of-sight (LOS) path.

Let $\bar{\rho}_{q}$ denote the average SNR during the $q$-th OFDM block at
the NLS output888Note that, the average signal-to-noise $\bar{\rho_{q}}$ can
be simply computed from (16). Then, the instantaneous value $\rho_{q}$ for the
$q$th OFDM block is exponentially distributed as [44, 45]

$f\left(\rho_{q};\bar{\rho}_{q}\right)=\frac{1}{\bar{\rho}_{q}}e^{-\rho_{q}/\bar{\rho}_{q}},\quad\bar{\rho}_{q}\geq
0.$ (24)

Actually, in Rayleigh fading scenarios, as shown in [15], the total output
noise can be well approximated by a Gaussian $pdf$. Thus, given (24), the
unconditional SER $p_{s}^{Ray}(\bar{\rho}_{q})$ can be computed from (23) by

$p_{s}^{Ray}(\bar{\rho}_{q})=\int_{0}^{\infty}\frac{1}{\bar{\rho}_{q}}e^{-\rho_{q}/\bar{\rho}_{q}}p_{s}^{(AWGN)}\left(\rho_{q}\right)d\rho_{q},$
(25)

where $p^{(AWGN)}(\rho_{q})$ is expressed by (12) [45] for square $M$-QAM
constellation.

By exploiting in (25) the alternative $Q$-function formulation [46]

$Q(x)=\frac{1}{\pi}\int_{0}^{\pi/2}e^{-\frac{x^{2}}{2\sin^{2}\phi}}d\phi,\quad
x\geq 0,$ (26)

the average SER has the following closed-form expression

$\begin{array}[]{l}{p_{s}^{(Ray)}(\bar{\rho}_{q})=1-\frac{1}{M}-2\frac{\sqrt{M}-1}{M}\frac{a}{\sqrt{1+a^{2}}}}\\\
{\quad\quad-\frac{4}{\pi}\left(\frac{\sqrt{M}-1}{\sqrt{M}}\right)^{2}\frac{a}{\sqrt{1+a^{2}}}\tan^{-1}\frac{a}{\sqrt{1+a^{2}}}},\end{array}$
(27)

where $a=\sqrt{\frac{3\bar{\rho_{q}}}{2\left(M-1\right)}}$, and
$\bar{\rho}_{q}$.

Thus, exploiting the good Gaussian approximation for the noise in Rayleigh
fading scenario, the noticeably simpler expression (27) can be effectively
used, rather than (10), to get a good SER approximation. This is due to the
fact that the fading-compensated AWGN noise dominates the suppressed ImpN, as
it can be easily verified by comparing the distortion noise powers with that
ones obtained in flat channels.

###  VI-B Rician Fading

Differently to Rayleigh fading, in Rician fading the received signal includes
a direct LoS component, together with a number of Gaussian, typically weaker,
reflected components. Considering a slow (quasi-static) frequency-selective
block fading channel, the SNR per $q$th block is distributed according to a
noncentral chi-square $pdf$, as expressed by [44, 45]

$f\left(\rho_{q};\bar{\rho}_{q};K_{r}\right)=\frac{K_{1}e^{-K_{r}}}{\bar{\rho}_{q}}e^{-K_{1}\frac{{\rho_{q}}}{\bar{\rho}_{q}}}I_{0}\left(\sqrt{4K_{r}K_{1}\frac{\rho_{q}}{\bar{\rho}_{q}}}\right),$
(28)

where $K_{1}=(1+K_{r})$, and $K_{r}$ is the ratio of the LoS path with the
scattered powers. When the channel power is normalized to 1,
$K_{r}=\bar{h}_{0}^{2}/(1-\bar{h}_{0}^{2})$, where $\bar{h}_{0}$ is the
average LoS path gain.

As it happens with Rayleigh fading channels, if we consider SER results for
$K_{r}\leq 10$ and typical ImpN parameters, the fading interference dominates
the distortion noise caused by the (sub)optimal NLS and consequently, the
total output noise can be considered approximately Gaussian. Thus, rather than
using (10), also in this case it is possible to derive a much simpler
expression to closely approximate the SER.

Specifically, when $K_{r}\leq 10$, substituting (23) in (28), the
unconditional SER can be approximated by

$p_{s}^{(Rice)}(\bar{\rho_{q}},K_{r})=\int_{0}^{\infty}f\left(\rho_{q};\bar{\rho_{q}};K_{r}\right)p_{s}^{(AWGN)}(\rho_{q})d\rho_{q},$
(29)

which can be conveniently expressed in the following equivalent form [47, 44]

$p_{s}^{(Rice)}(\bar{\rho_{q}},K_{r})=\frac{e^{-K_{r}}}{\pi}\left[aI_{1}(\bar{\rho_{q}},K_{r})+bI_{2}(\bar{\rho_{q}},K_{r})\right],$
(30)

where the parameters $\left(a,b,c\right)$ depend on the modulation and the
signal constellation, $I_{1}(\rho,K_{r})$ and $I_{2}(\rho,K_{r})$ are given by

$\begin{array}[]{l}{I_{1}(\rho,K_{r})=\int_{0}^{\pi/2}h_{\rho,K_{r}}\left(\theta\right)e^{K_{r}h_{\rho,K_{r}}\left(\theta\right)}d\theta}\\\
{I_{2}(\rho,K_{r})=\int_{0}^{\pi/4}h_{\rho,K_{r}}\left(\theta\right)e^{K_{r}h_{\rho,K_{r}}\left(\theta\right)}d\theta,}\end{array}$
(31)

and $h_{\rho,K_{r}}\left(\theta\right)$ is a function of the SNR and Rician
factor, as expressed by

$h_{\rho,K_{r}}\left(\theta\right)=\frac{\left(K_{r}+1\right)\cos^{2}\theta}{\left(K_{r}+1\right)\cos^{2}\theta+\bar{\rho}c^{2}}\
.$ (32)

Specifically, for 4-QAM and 16-QAM the parameters $\left(a,b,c\right)$ are
expressed by [47]

4-QAM: $a=b=1$, $c=\sin\left(\pi/4\right)$,
16-QAM: $a=3/4,\;b=9/4$, $c=\sqrt{0.1}$.

Unfortunately, closed-form analytical solutions of the integrals in (31) are
not available. Thus in order to obtain a closed form expression, we need to
resort to an approximate solution by using the middle Riemann sum [48],
obtaining

$\begin{array}[]{l}{I_{1}(\rho,K_{r})=\beta\sum_{m=0}^{M_{0}-1}h_{\rho,K_{r}}\left(m\beta\right)e^{K_{r}h_{\rho,K_{r}}\left(m\beta\right)}}\\\
{I_{2}(\rho,K_{r})=\beta\sum_{m=0}^{M_{0}/2-1}h_{\rho,K_{r}}\left(m\beta\right)e^{K_{r}h_{\rho,K_{r}}\left(m\beta\right)},}\end{array}$
(33)

where $M_{0}$ is the number of samples999as experienced with a number of
simulations with different parameters, good results can be obtained by only
$M_{0}=11$ with middle point $\rho=\bar{\rho}$. used in the numerical method,
and $\beta=\frac{\pi}{2M_{0}}$.

Thus, the average SER in Rician fading channels (30) is approximated by

$\begin{array}[]{l}{p_{s}^{(Rice)}(\bar{\rho_{q}},K_{r})=}\\\
{\frac{e^{-K_{r}}}{\pi}\left[a\beta\sum_{m=0}^{M_{0}-1}h_{\bar{\rho_{q}},K_{r}}\left(m\beta\right)e^{K_{r}h_{\bar{\rho_{q}},K_{r}\left(m\beta\right)}}\right.}\\\
{+\left.b\beta\sum_{m=0}^{M_{\theta}/2-1}h_{\bar{\rho_{q}},K_{r}}\left(m\beta\right)e^{K_{r}h_{\bar{\rho_{q}},K_{r}}\left(m\beta\right)}\right]},\end{array}$
(34)

where the average SNR $\bar{\rho}$ in (32) is still given by (16).

However, when $K_{r}>10$, the SER results computed from (34) are not
acceptable and, similarly to (10), it is necessary to resort to more complex
expression making use of (34) for $p_{s,\mathbf{l}_{K}}^{(Rice)}$, rather than
$p_{s,\mathbf{l}_{K}}^{(AWGN)}$, leading to

$p_{s}^{(Rice_{K-GMM})}=\sum_{l=0}^{L}w(l)\sum_{l_{1}=0}^{l}\sum_{l_{2}=0}^{l-l_{1}}\cdots\sum_{l_{K-1}=0}^{l-\sum_{m=1}^{K-2}l_{m}}w_{\mathbf{l_{K-1}}}p_{s;\mathbf{l}_{K}}^{(Rice)}.$
(35)

##  VII Numerical and Simulation Results

Numerical and simulation results presented in this section consider the SER
performance of OFDM with 4-QAM and 16-QAM constellations employing $L$=256
subcarriers over nonfading and slow-fading frequency-selective Rayleigh/Rician
fading channels, affected by BG, Class-A, or general $K$-GMM, including
S$\alpha$S ImpN.

The SER performance are analyzed for a wide range of parameters of practical
interest, including the SIR, the impulse probability $p_{I}$, as well as the
”white” SNR defined by the signal to ”white” noise ratio
$\sigma_{x}^{2}/\sigma_{w}^{2}$. The presented results include SER performance
of unmitigated systems (Sect. III), mitigated nonfading systems with GAD and
OBE (Sect. IV), single-threshold mitigated nonfading systems (Sect. V) and
frequency-selective Rayleigh and Rician fading channels (Sect. VI).

###  VII-A SER Results for Unmitigated Non-fading Systems

Fig. 4 shows the SER versus SIR for a 4-QAM OFDM system with $L$=256
subcarriers affected by Class-A ImpN, accurately approximated by a 4-GMM by
using (7). Numerical results computed by (12), (13) and (10) for AWGN, 2-GMM
and 4-GMM, respectively, are compared with simulation results. The SER
performance are analyzed for a wide range of signal to impulse noise ratio SIR
$\in[-60\div 0]$ dB and for a wide range of the impulsiveness index
$A\in\left\\{1,0.1,0.01,0.001,0.0001\right\\}$, at SNR=25dB. Evidently, the
results obtained by the 4-GMM (10), which is an approximate model of a Class-A
ImpN, predict the SER accurately in the whole range of ImpN parameters.
However, it is also interesting to note that the simple model 2-GMM (13) is a
good SER predictor when $A\leq 0.01$. On the other side, the AWGN equivalent
model (12) is acceptable only when $A\geq 0.1$.

Fig. 5 shows the SER results versus SIR for 4-QAM OFDM systems, with L=256
subcarriers, affected by S$\alpha$S ImpN at $\alpha=1.2$. This S$\alpha$S
distribution is accurately approximated by a 16-GMM [39]. Numerical results,
computed by (12) for the AWGN model and by (10) for the approximate
4-GMM101010Note that the exact ImpN model with K=16 would require a relatively
high number of calculations in (10). are compared with the simulation results.
The SER performance are analysed in the wide range SIR $\in[-60\div 0]$ dB and
for a wide range of impulsiveness index
$A\in\left\\{1,0.1,0.01,0.001,0.0001\right\\}$, at SNR=25dB. Similarly to the
discussion for Class-A ImpN in Fig. 4, expression (10) with a 4-GMM model,
which is an accurate approximation of the exact 16-GMM ImpN, predicts quite
well the SER in the whole range of ImpN parameters. Note that the noticeable
deviation in the results due to an uneven approximation of the S$\alpha$S
distribution by only 4-component GMM at different SIR111111Note that the SER
results, obtained by approximative 4-GMM following (7), as shown in Fig. 5,
are far to be acceptable, thus a direct 4-GMM approximation of S$\alpha$S ImpN
is performed [39]..

Figure 4: SER versus SIR for 4-QAM OFDM system with $L$=256 subcarriers
affected by Class-A ImpN, restricted to 4 components by using (7), at
$A\in\left\\{1,0.1,0.01,0.001,0.0001\right\\}$ and SNR = 25 dB. Figure 5: SER
versus SIR for 4-QAM OFDM system with $L$=256 subcarriers affected by
S$\alpha$S ImpN approximated by 4-GMM at
$p_{1}\in\left\\{0.0001,\,0.001,\,0.01,\,0.1,\,0.3\right\\}$ and SNR = 25 dB.
Dashed line with black markers shows the SER results obtained by using (7).

###  VII-B SER Results for Optimally Mitigated non-fading Systems: GAD and OBE
Suppressors

Fig. 6 compares numerical results for the SER obtained by OBE and GAD ImpN
suppressors. Both of them rely on the optimal Bayesian estimation, however OBE
exploits a limited NSI, while the GAD exploits the full NSI. The BG ImpN is
considered at SNR=25 dB and at impulse noise probabilities
$p_{I}\in\left\\{0.1,{\rm\;}0.05,{\rm\;}0.01,{\rm\;}0.001\right\\}$ for SIR
$\in[-60\div 0]$ dB. In practice, the SER results for the OBE are obtained
approximately by computing the SER for the ten-threshold BAS (see [18]), while
SER results for the GAD are computed directly from (10), as explained in Sect.
IVB. The results for the OBE represent the maximal system performance for a
realistic scenario, when the impulse noise positions are not known to the
receiver, while SER results for the GAD are the overall best system
performance in a genie-aided scenario, which assumes that the impulse
positions are exactly known to the receiver. The gain of exploiting such a
full NSI is evident, and interestingly, the gain is the highest for those ImpN
parameters of highest practical interest in real systems.

Figure 6: The comparison of OBE and GAD suppressors; SER versus SIR at SNR =
25 dB and $p_{1}\in\left\\{0.001,\,0.01,\,0.05,\,0.1\right\\}$ for 4-QAM OFDM
with 256 subcarriers affected by BG ImpN.

###  VII-C SER Results for Threshold-based Mitigated Systems

Fig. 7 compares simulation results with the SER obtained by the AWGN model
(12), for 4-QAM OFDM systems with L=256 subcarriers, affected by a BG ImpN,
whose “white” noise SNR $\in[10\div 30]$ dB, an impulse component that
produces SIR=-20 dB, and impulsive probability
$p_{1}\in\left\\{0.01,\,0.02,\,0.05,\,0.1\right\\}$. Evidently, the AWGN
equivalent model approximates very well the simulation results only for
$p_{I}\geq 0.1$121212results for $p_{I}>0.1$ are not shown, but anticipated
from the results up to $p_{I}=0.1$. Conversely, when $p_{I}<0.1$, such an
approximated SER is acceptable only when SNR$<$10dB.

Figure 7: SER versus SNR for SIR=-20 dB and
$p_{1}\in\left\\{0.01,\,0.02,\,0.05,\,0.1\right\\}$ for 4-QAM OFDM with 256
subcarriers affected by BG ImpN.

Fig. 8 compares the SER results obtained by AWGN (12), and $\mathcal{K}$-GMM
models (10) with those obtained through simulation, for a 4-QAM OFDM system
equipped with an (sub)optimal single-threshold NLS [18], and L=256
subcarriers. The ImpN is a BG with a “white” noise component at SNR=25 dB and
SIR$\in[-60\div 0]$ dB, with probability
$p_{I}\in\left\\{0.001,{\rm\;}0.01,{\rm\;}0.05,{\rm\;}0.1\right\\}$.
Evidently, the $\mathcal{K}$-GMM model (10) approximates the simulation
results very well for every set of ImpN parameters131313Note that all
parameters of $\mathcal{K}$-GMM are computed as detailed in Appendix B.
However, for relatively high impulse probability ($p_{I}\geq 0.05$), the
results obtained by the AWGN model are acceptable as well. Conversely, for
relatively low impulse probability ($p_{I}<0.05$) the SER predictions obtained
by the AWGN model, are not accurate and become quite unacceptable for very low
impulse probabilities ($p_{I}\leq 0.01$). This is expected since for $L$=256
the average number of impulses per frame for $p_{I}<0.01$ is really low (e.g.,
2 or 3 on average), and the distribution of the output distortion noise in the
frequency domain is far from a Gaussian one.

Figure 8: SER versus SIR at SNR = 25 dB and
$p_{1}\in\left\\{0.001,\,0.01,\,0.05,\,0.1\right\\}$ for 4-QAM OFDM with 256
subcarriers affected by BG ImpN. Figure 9: SER versus SIR at SNR = 25 dB for
16-QAM OFDM with 256 subcarriers
$p_{1}\in\left\\{0.001,\,0.01,\,0.05,\,0.1\right\\}$ with BG ImpN.

Fig. 9, similarly to Fig. 8, compares the SER results obtained by simulation
and by (12) for AWGN and by (10) for $\mathcal{K}$-GMM when the OFDM system
uses square 16-QAM constellation and $L=256$ subcarriers. The impulsive noise
is also BG with the same parameters used in Fig. 8, and the overall discussion
about its results can be confirmed for 16-QAM OFDM system.

Fig. 10 shows the SER results for the symmetric alpha stable (S$\alpha$S)
ImpN, with $\alpha=1.2$ and impulse noise probabilities
$p_{I}\in\left\\{0.1,{\rm\;}0.01\right\\}$. Evidently, equation (10) for the
$\mathcal{K}$-GMM model approximates simulated SER results very well, for
every set of impulsive noise parameters. In this case, differently from BG and
Class-A cases, the AWGN-based equation (12) does not provide good SER
predictions, in particular neither for $p_{I}=0.1$ or for relatively high SIR
(i.e., SIR$\geq-20$ dB). This somewhat unexpected outcome can be explained
since, for a 16-GMM fitted to the S$\alpha$S pdf, the contribution of each
component to the total SER is significant. Consequently, there is no component
whose contribution to the total SER can be neglected. Under such condition,
even for relatively high average number of impulses per frame, and relatively
high SIR, each component among the $K-1$ impulsive ones, noticeably
contributes to the total SER141414These results point out to the so-called
weakness of the central limit theorem. Consequently, the noise distribution in
the frequency domain can not be approximated by a Gaussian pdf.

Figure 10: SER versus SIR for 4-QAM OFDM with 256 subcarriers at SNR = 25 dB
and $p_{I}\in\left\\{0.1,\,0.01\right\\}$ with S$\alpha$S ImpN at
$\alpha=1.2$.

###  VII-D SER Results for Frequency-selective Fading Mitigated Systems

Fig. 11 shows SER versus SIR for SNR=25dB and impulse noise probabilities
$p_{1}\in\left\\{0.0,\,0.0001,\,0.001,\,0.01,\,0.1\right\\}$ for slow Rayleigh
fading channels, equipped by (sub)optimal thresholding-based ImpN mitigation.
The considered system is a 4-QAM OFDM with $L$=256 subcarriers, affected by BG
ImpN. The comparison of numerical results obtained by (27) with simulation
results shows good agreement for every SIR.

Fig. 12 shows SER results versus SNR spanning from 0dB to 60dB for a fixed
SIR=-10dB, and
$p_{1}\in\left\\{0.0,\,0.00001,\,0.0001,\,0.001,\,0.01,\,0.1\right\\}$. It is
evident that the previous comments can be generalized, leading to conclude
that also in the case of Rayleigh fading channels, the relatively simple
equation (27), which assumes Gaussian noise in the frequency domain, offers
accurate SER predictions, even though the channel is actually affected by
ImpN. This result is explained by the fact that, in the time domain, the
output Gaussian noise dominantly contributes to the distribution of the
distortion noise at the output of an (sub)optimal ImpN suppressor.

Figure 11: SER versus SIR for SNR=25dB and
$p_{1}\in\left\\{0.0,\,0.0001,\,0.001,\,0.01,\,0.1\right\\}$ for OFDM with 256
subcarriers over slow Rayleigh fading channel affected by BG ImpN. Figure 12:
SER versus SNR for OFDM with 256 subcarriers over slow Rayleigh fading channel
affected by BG ImpN at SIR=-10dB and
$p_{1}\in\left\\{0.0,\,0.00001,\,0.0001,\,0.001,\,0.01,\,0.1\right\\}$.

Fig. 13 shows the SER versus SIR a for 4-QAM OFDM system with $L$=256
subcarriers equipped, by the Bayesian attenuating ImpN suppressor [18], at
$p_{1}=0.01$ and SNR=25dB, over the Rician slow fading channel with
$K_{r}\in\left\\{0,5,10,100,\infty\right\\}$. It is useful to remind that
Rician factor $K_{r}=0$ corresponds to Rayleigh fading, and $K_{r}=\infty$
corresponds to the AWGN (nonfading) channel. Specifically, Fig. 13 compares
the SER computed by (34) (named Rice-W) and (35) (named Rice-$K$-GMM)151515due
to simplicity, here we use $K$ instead of $\mathcal{K}$ with the simulation
results, and let us to conclude that for $K_{r}\leq 10$ the simpler Rice-W
model works quite well, while for $K_{r}>10$ the Rice-W model underestimates
the SER. Conversely, the Rice-$K$-GMM model (35) works quite well every time.
Obviously, for higher Rician factors $K_{r}$, the Rice-$K$-GMM model
noticeably outperforms in accuracy the Rice-W model, and when $K_{r}$
approaches infinity it approaches the performance shown in Fig. 12, which
correspond to the non-fading channel.

Figure 13: SER versus SIR for SNR=25dB and $p_{1}=0.01$ for OFDM with 256
subcarriers over Rician fading channel at
$K_{r}\in\left\\{0,5,10,100,\infty\right\\}$ affected by BG ImpN.

##  VIII Conclusion

We presented a unified GMM-based analytical approach for SER performance
evaluation of OFDM systems affected by impulsive noise in nonfading and
frequency selective Rayleigh/Rice fading channels. The analysis includes both
unmitigated and mitigated systems affected by ImpN modeled with a $K$-GMM,
such as BG (2-GMM), Class-A, as well as impulsive noise with alpha-stable
distributions.

For unmitigated non-fading systems it has been shown that the SER can be
exactly predicted for any $K$-GMM ImpN. However, at relatively high $K$ (e.g.
$K\geq 5$), due to a high number of combinations in the multinomial
distribution, the computation complexity becomes highly time consuming. In
such cases we have shown that, an accurate $\mathcal{K}$-GMM ($\mathcal{K}<K$)
approximation of the noise distribution, based on first two moments matching,
leads to accurate SER estimations.

For mitigated systems, the distortion noise at the output of the NLS is
generally non-Gaussian and it can be also effectively approximated by a
$\mathcal{K}$-GMM. Specifically, for threshold-based mitigated systems, the
NLS output distortion noise can be safely approximated with at most
$\mathcal{K}=4$ Gaussian components. The proposed unified approach is based on
the $\mathcal{K}$-GMM approximation of the time domain distortion noise
distribution $f_{D}(d)$, by performing a novel component-by-component Gaussian
fitting method presented in Appendix B. Based on the developed closed-form
expressions which exploit the mixture parameters of the fitted
$\mathcal{K}$-GMM, the SER can be simply computed for any ImpN scenario,
exploiting the expressions derived for unmitigated systems in Sect. III. The
use of appropriate values of $\mathcal{K}$ for different systems and different
scenarios guarantees the minimization of the computation complexity.

Although, we have shown results for BG, Class-A, and alpha-stable ImpNs with
the Bayesian non-linear ImpN suppressors, the developed closed-form SER
expressions can also be used for other ImpNs and other NLSs, including
thresholding-based NLS, such as blanking, clipping, and their combination, as
well as the multi-thresholded NLS. The approach presented in this work can be
extended to nonsymetric $pdf$ $f_{D}(d)$ of the distortion noise, will be the
subject of our future work.

##  Appendix A Closed-form Expression for Distortion Noise Distribution with
Single-threshold Suppressor

Given the Bussgang’s theorem (14), the distortion noise can be calculated from
(15).

Let the impulsive noise suppressor perform (sub)optimal Bayesian attenuation
of the received samples as $\hat{X}=\beta_{\Omega}Y$ [18]. With $Y=X+N$, the
time-domain distortion noise $D$ in (15) can be expressed by

$D=\left(\beta_{\Omega}-\alpha\right)X+\beta_{\Omega}N.$ (36)

where $X$ and $N$, and consequently $Y$, are complex-valued variables, and
$\beta_{\Omega}$ denotes the attenuation factor performed in the signal
amplitude range $\Omega$ (see Fig. 14).

Figure 14: Complex-valued signals and thresholding

Assuming independent $X$ and $N$, the distortion noise distribution
$f_{D_{\Omega}}\left(d\right)$ can be calculated by

$f_{D_{\Omega}}\left(d\right)=f_{\left(\beta_{\Omega}-\alpha\right)X}\left(d\right)*f_{\beta_{\Omega}N}\left(d\right)$
(37)

where $(\beta-\alpha)X$ and $\beta N$ are the (sub)optimally attenuated signal
and noise, respectively, and $*$ denotes the convolution operator.

Considering the single-threshold ImpN suppressor, the total region
$\Omega=\Omega_{0}\cup\Omega_{1}$ is split into two disjoint subregions
$\Omega_{0}$, $\Omega_{1}$ where $\Omega_{0}\triangleq\left\\{y:|y|\leq
A_{T}\right\\}$, and $\Omega_{1}\triangleq\left\\{y:|y|>A_{T}\right\\}$. Then,
applying (37) to each subregion, we write

$\begin{array}[]{l}{f_{D_{\Omega_{0}}}\left(d\right)=f_{\left(\beta_{\Omega_{0}}-\alpha\right)X_{\Omega_{0}}}\left(d\right)*f_{\beta_{\Omega_{0}}N_{\Omega_{0}}}\left(d\right)}\\\
{f_{D_{\Omega_{1}}}\left(d\right)=f_{\left(\beta_{\Omega_{1}}-\alpha\right)X_{\Omega_{1}}}\left(d\right)*f_{\beta_{\Omega_{1}}N_{\Omega_{1}}}\left(d\right)},\end{array}$
(38)

where $X_{\Omega_{0}}$ and $N_{\Omega_{0}}$ denote the signal and noise RVs
below the threshold $A_{T}$, and $X_{\Omega_{1}}$ and $N_{\Omega_{1}}$ denote
the signal and noise RVs above the threshold $A_{T}$, respectively.

Evidently, the distribution of the distortion noise (38) at the output of the
threshold-based ImpN suppressor is expressed by the $pdf$
$f_{(\beta-\alpha)X_{\Omega}}(x)$ and $f_{\beta N_{\Omega}}(n)$ of the
attenuated signal and noise, respectively, evaluated below and above the
threshold. In the following we firstly develop the closed-form expressions for
the distributions of the non-attenuated variables $X_{\Omega}$ and
$N_{\Omega}$, and then take into account the attenuating factors in the
convolution (38) and compute the $pdf$ $f_{D_{\Omega_{0}}}(d)$,
$f_{D_{\Omega_{1}}}(d)$ of the distortion noise.

Due to thresholding of the received signal amplitude
$\left|y\right|=\sqrt{y_{r}^{2}+y_{i}^{2}}$, the detected signal component
$X_{r}$ is related to both the received signal components $Y_{r}$ and $Y_{i}$.
Thus, we write

$\begin{array}[]{l}{f_{X_{\Omega_{0}}}\left(x\right)=f_{X_{r}}\left(x_{r}\right)\left|{}_{|y|\leq
A_{T}}\right.}\\\
{\quad\quad=\frac{1}{B}{\displaystyle\iint\nolimits_{\Omega_{0}}}f_{X_{r}Y_{r}Y_{i}}\left(x_{r},y_{r},y_{i}\right)dy_{r}dy_{i},}\end{array}$
(39)

where $B=P\left(\left|y\right|\leq A_{T}\right)$.

Assuming that $y_{i}$ is independent of $x_{r}$, where,

$f_{X_{r}Y_{r}Y_{i}}\left(x_{r},y_{r},y_{i}\right)=f_{X_{r}}\left(x_{r}\right)f_{Y_{r}Y_{i}|X_{r}}\left(y_{r},y_{i}|x_{r}\right)$

and by chain rule,

$f_{Y_{r}Y_{i}|X_{r}}\left(y_{r},y_{i}|x_{r}\right)=f_{Y_{r}|X_{r}}\left(y_{r}|x_{r}\right)f_{Y_{i}|Y_{r}}\left(y_{i}|y_{r}\right),$

which substituted in (39) leads to

$\begin{array}[]{l}{f_{X_{\Omega_{0}}}\left(x\right)=\frac{f_{X_{r}}\left(x_{r}\right)}{B}}\\\
{\quad\times{\displaystyle\iint\nolimits_{\Omega_{0}}}f_{Y_{r}|X_{r}}\left(y_{r}|x_{r}\right)f_{Y_{i}|Y_{r}}\left(y_{i}|y_{r}\right.)dy_{i}dy_{r}.}\end{array}$
(40)

The conditional density $f_{Y_{r}|X_{r}}\left(y_{r}|x_{r}\right)$ is obviously
imposed by the noise density $f_{N_{r}}\left(n_{r}\right)$, which let us to
write

$f_{X_{\Omega_{0}}}\left(x\right)=\frac{f_{X_{r}}\left(x_{r}\right)}{B}\iint\nolimits_{\Omega_{0}}f_{N_{r}}\left(n_{r}\right)f_{Y_{i}|Y_{r}}\left(y_{i}|y_{r}\right)dy_{i}dn_{r}.$
(41)

As illustrated in Fig. 14, it is clear that the integration region for
$n_{r}=y_{r}-x_{r}$ is defined by $\pm A_{T}-x_{r}$, and for $y_{i}$ the
integration region is defined by $\pm\sqrt{A_{T}^{2}-y_{r}^{2}}$. Thus

$\begin{array}[]{l}{f_{X_{\Omega_{0}}}\left(x\right)=}\\\
{=\frac{f_{X_{r}}\left(x_{r}\right)}{B}{\displaystyle\int}_{-A_{T}-x_{r}}^{A_{T}-x_{r}}f_{N_{r}}\left(n_{r}\right){\displaystyle\int}_{-\sqrt{A_{T}^{2}-y_{r}^{2}}}^{\sqrt{A_{T}^{2}-y_{r}^{2}}}f_{Y_{i}}\left(y_{i}\right)dy_{i}dn_{r}}.\end{array}$
(42)

The $pdf$s $f_{X}\left(x\right)$, $f_{N}\left(n\right)$ and
$f_{Y}\left(y\right)$ are Gaussians, while $f_{\left|Y\right|}\left(y\right)$
is Rayleigh, and consequently

$\int_{-\sqrt{A_{T}^{2}-y^{2}}}^{\sqrt{A_{T}^{2}-y^{2}}}f_{Y}\left(y\right)dy=erf\left(\sqrt{\frac{A_{T}^{2}-y^{2}}{2\sigma_{y}^{2}}}\right),$
(43)

while

$\begin{array}[]{l}{B=P\left(\left|y\right|\leq
A_{T}\right)={\displaystyle\int}_{0}^{A_{T}}\frac{\left|y\right|}{\sigma_{y}^{2}}\exp\left(-\frac{\left|y\right|^{2}}{2\sigma_{y}^{2}}\right)dy}\\\
{\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad=1-\exp\left(-\frac{A_{T}^{2}}{2\sigma_{y}^{2}}\right)}.\end{array}$
(44)

Thus exploiting (43) and (44), and with $y_{r}=x_{r}+n_{r}$, Eq. (42)
becomes161616in the following, due to the same distributions of real and
imaginary parts, we simply use symbols $X$, $N$ and $Y$ without denoting real
or imaginary.

$\begin{array}[]{l}{f_{X_{\Omega_{0}}}\left(x\right)=\frac{\exp\left(-\frac{x^{2}}{2\sigma_{x}^{2}}\right)}{\left(1-\exp\left(-\frac{A_{T}^{2}}{2\sigma_{y}^{2}}\right)\right)\sqrt{2\pi\sigma_{x}^{2}}}}\\\
{\quad\quad\times{\displaystyle\int\limits_{-A_{T}-x}^{A_{T}-x}}\frac{\exp\left(-\frac{n^{2}}{2\sigma_{n}^{2}}\right)}{\sqrt{2\pi\sigma_{n}^{2}}}{\kern
1.0pt}erf\left(\sqrt{\frac{A_{T}^{2}-\left(x+n\right)^{2}}{2\sigma_{y}^{2}}}\right)dn}.\end{array}$
(45)

Unfortunately, the integral in (45) can not be analytically solved. Therefore,
similarly to the derivation of Eq. (34), we resort to an accurately
approximated numerical solution based on the middle Riemann sum. By using
$M_{d}$ equidistant points $d_{i}$ in the numerical integration within the
region $\mp 10\sigma_{y}$, we get

$\begin{array}[]{l}{f_{X_{\Omega_{0}}}(d_{i})=\frac{h^{2}}{1-\exp\left(-\frac{A_{T}^{2}}{2\sigma_{y}^{2}}\right)}\frac{\exp\left(-\frac{d_{i}^{2}}{2\sigma_{x}^{2}}\right)}{\sqrt{2\pi\sigma_{x}^{2}}}}\\\
{\quad\times{\displaystyle\sum}_{j=-M_{d}-i}^{M_{d}-i}\frac{\exp\left(-\frac{d_{i}^{2}}{2\sigma_{n}^{2}}\right)}{\sqrt{2\pi\sigma_{n}^{2}}}\,erf\left(\sqrt{\frac{A_{T}^{2}-\left(d_{i}+d_{j}\right)^{2}}{2\sigma_{y}^{2}}}\right),}\end{array}$
(46)

where $M_{d}$ should be relatively high to obtain good results (e.g.
$M_{d}=5\cdot 10^{3}$).

Given (45), the conditional $pdf$ $f_{X_{\Omega_{1}}}(x)$, due to a mutual
exclusivity of events $|y|\leq A_{T}$ and $|y|>A_{T}$, can be simply computed
from

$f_{X_{\Omega_{1}}}(x)=\frac{f_{X}\left(x\right)-f_{X_{\Omega_{0}}}(x)P\left(\left|y\right|\leq
A_{T}\right)}{1-P\left(\left|y\right|\leq A_{T}\right)}.$ (47)

Given (47), it is straightforward to compute $f_{X_{\Omega_{0}}}(d_{i})$ at
discrete values $x_{i}=d_{i}$, which is omitted due to lack of space.

Expressions (46) and (47) calculate accurately $pdf$s below and above the
threshold $A_{T}$ for the signal $X$. In the same way, it is straightforward
to write the corresponding expressions $f_{N_{\Omega_{0}}}(d_{i})$ and
$f_{N_{\Omega_{1}}}(d_{i})$ for the noise $N$ by replacing $x$ and
$\sigma_{x}^{2}$ with $n$ and $\sigma_{n}^{2}$ in (46) and (47).

Given the closed form expressions for signal and noise $pdf$s
$f_{X_{\Omega_{0}}}(d_{i})$, $f_{N_{\Omega_{0}}}(d_{i})$ and
$f_{X_{\Omega_{1}}}(d_{i})$, $f_{N_{\Omega_{1}}}(d_{i})$, below and above the
threshold, respectively, we can compute (38) by a discrete convolution as

$\begin{array}[]{l}{f_{D_{\Omega_{0}}}(i)=\sum_{m=-M_{d}}^{M_{d}}f_{\left(\beta_{\Omega_{0}}-\alpha\right)X_{\Omega_{0}}}(i-m)f_{\beta_{\Omega_{0}}N_{\Omega_{0}}}(m)}\\\
{f_{D_{\Omega_{1}}}(i)=\sum_{m=-M_{d}}^{M_{d}}f_{\left(\beta_{\Omega_{1}}-\alpha\right)X_{\Omega_{1}}}(i-m)f_{\beta_{\Omega_{1}}N_{\Omega_{1}}}(m).}\end{array}$
(48)

Finally, assuming that the noise $N$ represents either white noise $W$ with
variance $\sigma_{w}^{2}$ and probability $p_{0}$, or impulse noise $I$ with
variance $\sigma_{I}^{2}$ and probability $1-p_{0}$, a four-component mixture
(21) is completely defined by (21) and (48), where it holds

$\begin{array}[]{l}{f_{D_{1}}\left(d\right)=f_{D_{\Omega_{0}}}\left(d\right);\;{\rm
for}\;N=W}\\\ {f_{D_{2}}\left(d\right)=f_{D_{\Omega_{0}}}\left(d\right);\;{\rm
for}\;N=I}\\\ {f_{D_{3}}\left(d\right)=f_{D_{\Omega_{1}}}\left(d\right);\;{\rm
for}\;N=W}\\\ {f_{D_{4}}\left(d\right)=f_{D_{\Omega_{1}}}\left(d\right);\;{\rm
for}\;N=I.}\end{array}$ (49)

Considering $f_{D_{1}}(d)$ as the ”white” distortion noise component, the
”impulsive” part $f_{D_{I}}(d)$ is expressed by

$f_{D_{I}}\left(d\right)=\frac{1}{1-w_{D_{1}}}\sum_{k=2}^{K}w_{G_{k}}f_{G_{k}}\left(d\right).$
(50)

Given (49), the variances $\sigma_{D_{k}}^{2}$ and weighting factors
$w_{D_{k}}$ of the $k$-th component can be accurately calculated from

$\begin{array}[]{l}{\sigma_{D_{k}}^{2}=\sum_{i=-M_{d}/2}^{M_{d}/2}d_{i}^{2}f_{D_{k}}\left(d_{i}\right),}\\\
{w_{D_{k}}=w_{k};\quad k=1,\cdots,4},\end{array}$ (51)

where $\\{w_{k}\\}_{k=1,\cdots,4}$ are given by (22).

Note that in this Appendix we derived closed-form expressions for
$f_{D_{k}}\left(d\right)$, $k=1,\cdots,4$ for 2-GMM ImpN (denoted as BG
model). For Class A and generally K-GMM ImpNs, the approach can be easily
extended by taking into account all the impulsive components with the
corresponding probabilities (weighting factors)
$\\{p_{k}\\}_{k=0,1,\cdots,K-1}$ and variances
$\\{\sigma_{k}^{2}\\}_{k=0,1,\cdots,K-1}$. Due to a limited space, the details
are omitted.

##  Appendix B The Proposed GMM Approximation of $f_{D}(d)$

As discussed in Section V, the distortion noise $pdf$ $f_{D}(d)$ is generally
nonGaussian. However, due to its symmetry, it can be approximated by a mixture
of zero-mean Gaussian $pdf$s. Due to the fact that, we are interested in the
closed-form expressions for variances and scaling factors of normal random
variables, rather than resorting to iterative algorithms, we build the GMM by
a novel component-by-component approach,

Assuming that $f_{D}(d)$ is a Gaussian mixture, its variance distribution
among the potential components $f_{D_{k}}(d)$, can be calculated from two
neighbour samples $d_{i}$ and $d_{i+1}$ of
$\left\\{f_{D}(d_{i})\right\\}_{i=1,\cdots,M_{d}}$ by

$\sigma_{f}^{2}(d_{i})=0.5\frac{d_{i+1}^{2}-d_{i}^{2}}{\log f_{D}(d_{i})-\log
f_{D}(d_{i+1})}.$ (52)

Expectedly, the function $\sigma_{f}^{2}(d_{i})$ in (52) is symmetric likewise
the $f_{D}(d)$. Fig. 15 illustrates171717Note that the red line is obtained
when $f_{D_{3}}(d)$ in (49) is omitted in order to eliminate ”bursts” in
$\sigma_{f}^{2}(d_{i})$ (blue line), since the contribution of $f_{D_{3}}(d)$
to $\sigma_{f}^{2}(d_{i})$ can be safely neglected. the variance function
$\left\\{\sigma_{f}^{2}(d_{i})\right\\}_{i=1,\cdots,M_{d}}$, which we exploit
to estimate the Gaussian components i.e., their variances $\sigma_{G_{k}}^{2}$
and scaling weights $w_{G_{k}}$.

Given the variances
$\left\\{\sigma_{G_{k}}^{2}\right\\}_{k=1,\cdots,\mathcal{K}}$ and the scaling
weights $\left\\{w_{G_{k}}\right\\}_{k=1,\cdots,\mathcal{K}}$, the total
distortion noise distribution $f_{D}(d)$ is approximated by a
$\mathcal{K}$-GMM as

$f_{D}\left(d\right)\approx
f_{K-GMM}\left(d\right)=\sum_{k=1}^{\mathcal{K}}w_{G_{k}}f_{G_{k}}\left(d\right).$
(53)

Figure 15: Examples of Gaussian variance function $\sigma_{f}^{2}(d_{i})$
obtained by (52); the thin blue line (denoted by $K=3$) shows results obtained
with included $f_{D_{3}}(d)$, and the red tick line when $f_{D_{3}}(d)$ is
excluded. Value $d_{max}$ is the noise amplitude when $f_{D_{I}}(d)$ is lower
than the numerical approximation tolerance, e.g. $10^{-15}$.

###  B-A Component-by-component GMM Approximation of $f_{D}(d)$

Approximation of the total distortion noise distribution $f_{D}(d)$ (53) is
performed in a component-by-component way through at most four steps, as
presented in the following.

\- step 1.

Assuming relatively low probability of impulse noise, the ”white” noise
distribution $f_{D_{1}}(d)$ in (49) can be well approximated by Gaussian pdf
(see Figs. 2 and 3). Thus, the closed-form expression for the mixture
parameters of the first Gaussian component $f_{G_{1}}(d)$ are simply expressed
by

$\sigma_{G_{1}}^{2}=\sigma_{D_{1}}^{2},\quad w_{G_{1}}=w_{D_{1}},$ (54)

where $\sigma_{D_{1}}^{2}$ and $w_{D_{1}}$ are given by (51) for $k=1$.

Then the ”impulsive” distribution $f_{D_{I}}(d)$ expressed by (50) is
separately approximated by a GMM, as detailed by the following steps. As shown
in Figs. 2-3, for any ImpN scenario, at most three Gaussians are necessary to
approximate $f_{D_{I}}(d)$.

\- step 2.

Let’s compute the point $d_{G_{2}}$ given by

$d_{G_{2}}=\mathop{\arg}\limits_{d}\left\\{f_{D_{I_{1}}}(d)-c_{0}f_{D}(d)\right\\},$
(55)

where the residual distribution $f_{D_{I_{1}}}(d)$ given by

$f_{D_{I_{1}}}(d)=f_{D}(d)-w_{G_{1}}f_{G_{1}}(d)$ (56)

falls to $c_{0}f_{D}(d)$, when we assume that an additional Gaussian component
is needed for proper fitting181818numerical results show that the fitting
quality is quite robust to factor $c_{0}=0.9$. Specifically, for GMM fitting
to almost GMM $pdf$s, factor $c_{0}$ should be close to one e.g. 0.99.
Generally, an optimal $c_{0}$ would undergo an optimization method, which
however in this work is omitted, because we are interested in closed-form
solutions..

Given the Gaussian variance function expressed by (52), the variance
$\sigma_{G_{2}}^{2}$ of the second Gaussian component $f_{G_{2}}(d)$ can be
computed from

$\sigma_{G_{2}}^{2}=\begin{cases}\sigma_{D_{I}}^{2}&\quad\text{if
}d_{G_{2}}\geqq d_{max}\\\ \sigma_{f}^{2}(d_{G_{2}})&\quad\text{if
}d_{G_{2}}<d_{max}\end{cases}$ (57)

where $d_{max}$ is the noise amplitude when $f_{D_{I}}(d)$ is lower than the
numerical approximation tolerance, e.g., $10^{-15}$ (see Fig. 15).

If (57) results in $\sigma_{G_{2}}^{2}=\sigma_{D_{I}}^{2}$ (case $K$=2 in Fig.
15), $f_{D_{I}}(d)$ is approximated by a 1-GMM, and thus the total $f_{D}(d)$
can be well approximated by $\mathcal{K}=2$, resulting in a 2-GMM with the
following mixture parameters

$\begin{array}[]{l}{\sigma_{G_{1}}^{2}=\sigma_{D_{1}}^{2},\quad\sigma_{G_{2}}^{2}=\sigma_{D}^{2}-w_{D_{1}}\sigma_{D_{1}}^{2}},\\\
{w_{G_{1}}=w_{D_{1}},\quad w_{G_{2}}=1-w_{D_{1}}}.\end{array}$ (58)

Otherwise, if (57) results in $\sigma_{G_{2}}^{2}=\sigma_{f}^{2}(d_{G_{2}})$
with $d_{G_{2}}<d_{max}$ (case $\mathcal{K}$=3 in Fig. 15), the variance and
scaling factor of the second Gaussian component are expressed by

$\sigma_{G_{2}}^{2}=\sigma_{f}^{2}(d_{G_{2}}),\quad
w_{G_{2}}=\frac{f_{D_{I}}(d=0)}{f_{G_{2}}(d=0)}.$ (59)

\- step 3.

Given $\sigma_{G_{2}}^{2}$ and $w_{G_{2}}$ in (59), the residual distribution
$f_{D_{I_{2}}}(d)$ is given by

$f_{D_{I_{2}}}(d)=f_{D_{I_{1}}}(d)-w_{G_{2}}f_{G_{2}}(d).$ (60)

Given the residual distribution $f_{D_{I_{2}}}(d)$ (60), we compute the point
$d_{G_{3}}$ where $f_{D_{I_{2}}}(d)$ falls to $c_{0}f_{D_{I}}(d)$, in order to
estimate where an additional Gaussian component would be beneficial to the
distribution. Thus, we obtain

$d_{G_{3}}=\mathop{\arg}\limits_{d}\left\\{f_{D_{I_{2}}}(d)-c_{0}f_{D_{I}}(d)\right\\}.$
(61)

Figure 16: Example of $f_{D}(d)$ and its approximation by 3-GMM, at SIR=-30dB,
SNR=25dB and $p_{I}=0.001$, for OFDM with 256 subcarriers and 4QAM affected by
S$\alpha$S ImpN at $\alpha=1.2$.

Given $d_{G_{3}}$ in (61), we can compute the variance and scaling factor of
the third Gaussian component by191919slightly improved estimate of
$\sigma_{G_{3}}^{2}$ is obtained by averaging with
$\sum_{i=i_{3+1}}^{2i_{3}}\sigma_{f}^{2}(d_{i})$, where
$i_{3}=\mathop{\arg}\limits_{i}\left\\{d_{G_{3}}\right\\}.$

$\sigma_{G_{3}}^{2}=\sigma_{f}^{2}(d_{G_{3}}),\quad
w_{G_{3}}=\frac{f_{D_{I}}(d=d_{G_{3}})}{f_{G_{2}}(d=d_{G_{3}})}.$ (62)

Given (58) and (62), $f_{D_{I}}(d)$ can be approximated by a 2-GMM, and thus
the total $f_{D}(d)$ can be well approximated by $\mathcal{K}=3$, i.e., a
3-GMM (Fig. 16) with the following mixture parameters

$\begin{array}[]{l}{\sigma_{G_{1}}^{2}=\sigma_{D_{1}}^{2},\quad\sigma_{G_{2}}^{2}=\sigma_{f}^{2}(d_{G_{2}}),\quad\sigma_{G_{3}}^{2}=\sigma_{f}^{2}(d_{G_{3}})},\\\
{w_{G_{1}}=w_{D_{1}},\quad
w_{G_{2}}=\frac{f_{D_{I}}(d=0)}{f_{G_{2}}(d=0)},\quad
w_{G_{3}}=\frac{f_{D_{I}}(d=d_{G_{3}})}{f_{G_{2}}(d=d_{G_{3}})}}.\end{array}$
(63)

\- step 4.

Given $\sigma_{G_{3}}^{2}$ and $w_{G_{3}}$, the residual distribution is
obtained by

$f_{D_{I_{3}}}(d)=f_{D_{I_{2}}}(d)-w_{G_{3}}f_{G_{3}}(d).$ (64)

Given the residual distribution $f_{D_{I_{3}}}(d)$ (64), we compute the point
$d_{G_{4}}$ where $f_{D_{I_{3}}}(d)$ falls to $c_{0}f_{D_{I}}(d)$ in order to
estimate where an additional Gaussian component would be beneficial to the
distribution approximation. Thus, we compute

$d_{G_{4}}=\mathop{\arg}\limits_{d}\left\\{f_{D_{I_{3}}}(d)-c_{0}f_{D_{I}}(d)\right\\}$
(65)

and, if $d_{G_{4}}=d_{max}$, we conclude that an additional Gaussian component
is not needed and the GMM model is completed by (63), approximating $f_{D}(d)$
by a 3-GMM.

Otherwise, if $d_{G_{4}}>d_{max}$, the additional Gaussian component is
considered (case $\mathcal{K}=4$ in Fig. 15), with a variance and scaling
factor expressed by

$\sigma_{G_{4}}^{2}=\sigma_{f}^{2}(d_{G_{4}}),\quad
w_{G_{4}}=\frac{f_{D_{I}}(d=d_{G_{4}})}{f_{G_{2}}(d=d_{G_{4}})}.$ (66)

Finally, given (63) and (66), the total $f_{D}(d)$ is approximated by
$\mathcal{K}=4$, i.e. 4-GMM with mixture parameters expressed by

$\begin{array}[]{l}{\sigma_{G_{1}}^{2}=\sigma_{D_{1}}^{2},\sigma_{G_{2}}^{2}=\sigma_{f}^{2}(d_{2}),\sigma_{G_{3}}^{2}=\sigma_{f}^{2}(d_{G_{3}}),\sigma_{G_{4}}^{2}=\sigma_{f}^{2}(d_{G_{4}})}\\\
{w_{G_{1}}=w_{D_{1}},\quad
w_{G_{2}}=\frac{f_{D_{I}}(d=0)}{f_{G_{2}}(d=0)},\quad
w_{G_{3}}=\frac{f_{D_{I}}(d=d_{G_{3}})}{f_{G_{2}}(d=d_{G_{3}})}},\\\
{w_{G_{4}}=\frac{f_{D_{I}}(d=d_{G_{4}})}{f_{G_{2}}(d=d_{G_{4}})}}.\end{array}$
(67)

Evidently, expressions (54), (58), (63), and (67), are the closed-form
formulas to compute the variances and the weighting factors for
$\mathcal{K}\in\\{1,2,3,4\\}$ in (53), to approximate the distortion noise
distribution $f_{D}(d)$ by a proper $\mathcal{K}$-GMM.

##  Appendix C An Example of the Closed-form Approximation of a Given 4-GMM
$pdf$

This Appendix illustrates an example of the closed-form approximation of a
4-GMM $pdf$ $f_{Z}(z)$, with
$\sigma_{k}^{2}\in\left\\{0.0032,\,10,\,100,\,1000\right\\}$ and
$p_{k}\in\left\\{0.90009,\,0.09,\,0.0099,\,0.00001\right\\}$, by
$\mathcal{K}$-GMM which leverages on a component-by-component Gaussian
fitting, as detailed in Appendix B. Assuming that $f_{Z}(z)$ is belongs to the
family of Gaussian $pdf$s, the variance distribution can be calculated from
two neighbour samples $z_{i}$ and $z_{i+1}$ of
$\left\\{f_{Z}(z_{i})\right\\}_{i=1,\cdots,M_{z}}$ by

$\sigma_{f}^{2}(z_{i})=0.5\frac{z_{i+1}^{2}-z_{i}^{2}}{\log f_{Z}(z_{i})-\log
f_{Z}(z_{i+1})}.$ (68)

Fig. 17 shows the right-side part of the variance function computed for
$M_{z}$ points, thus
$\left\\{\sigma_{f}^{2}(z_{i})\right\\}_{i=1,\cdots,M_{z}}$, which we exploit
to estimate the Gaussian components i.e., their variances $\sigma_{G_{k}}^{2}$
and scaling weights $w_{G_{k}}$ for $k=1,2,3,4$. The left-side of the variance
function $\sigma_{f}^{2}(z_{i})$ can be also computed by (68), however it is
expectedly symmetric one, likewise to the itself $f_{Z}(z)$.

Figure 17: Variance function $\sigma_{f}^{2}(z_{i})$ obtained by (68) for the
4-GMM example, as the support vector used to estimate the Gaussian components.

Fig. 18 shows the intermediate results obtained by means of (56), (60) and
(64), based on which are computed variances and scaling factors by means of
(54), (58), (63) and (67).

Fig. 19 compares the original 4-GMM and the fitted 4-GMM $pdf$s, also showing
the four estimated weighted Gaussian components $w_{G_{k}}f_{G_{k}}(z)$ for
$k=1,2,3,4$.

Figure 18: Plot of the components used to approximate of $f_{Z}(z)$ by a
$\mathcal{K}$-GMM, by means of (55),(56); (60),(61); (64),(65), as detailed in
Appendix B. Figure 19: Comparison of $f_{Z}(z)$ and its 4-GMM approximation
obtained by steps 1 to 4 presented in Appendix B.

The results in Fig. 19 show that the given $pdf$ can be well approximated by
$\mathcal{K}$-GMM built following the closed-form algorithm presented in
Appendix B. Note, that the better fitting results can be expected only by
performing an iterative optimization algorithm, which however is out of the
scope of this work. Actually, GMM approximation of the distortion noise $pdf$
at the output of the NLS, considered in this work, is used to obtain the GMM-
based closed-form formula for SER prediction.

## References

  * [1] M. Ghosh, “Analysis of the effect of impulse noise on multicarrier and single carrier QAM systems,” _IEEE Trans. Commun._ , vol. 44, no. 2, pp. 145–147, 1996.
  * [2] R. Price, “A useful theorem for nonlinear devices having Gaussian inputs,” _IEEE Trans. Inf. Theory_ , vol. 4, no. 8, pp. 69–72, 1958.
  * [3] J. Minkoff, “The role of AM-to-PM conversion in memoryless nonlinear systems,” _IEEE Trans. on Communications_ , vol. 33, no. 2, pp. 139–144, 1985.
  * [4] P. Banelli and S. Capobardi, “Theoretical analysis of performance of OFDM signals in non-linear AWGN channels,” _IEEE Trans. on Communications_ , vol. 48, no. 3, pp. 430–441, 2000.
  * [5] S. Miyamoto, M. Katayama, and N. Morinaga, “Performance analysis of QAM systems under Class-A noise environment,” _IEEE Trans. on Electromag. Comp._ , vol. 37, no. 2, pp. 260–267, 1995.
  * [6] Y. H. Ma, P. So, and E. Gunawan, “Performance analysis of OFDM systems for broadband power line communications under impulsive noise and multipath effects,” _IEEE Trans. Power Delivery_ , vol. 20, no. 2, pp. 674–682, 2005\.
  * [7] P. Amirshahi, S. M. Navidpour, and M. Kavehrad, “Performance analysis of uncoded and coded OFDM broadband transmission over low voltage power-line channels with impulsive noise,” _IEEE Trans. on Power Delivering_ , vol. 21, no. 4, pp. 1927–1934, 2006.
  * [8] D. Middleton, “Statistical-physical models for electromagnetic interference,” _IEEE Trans. Electromagn. Compat._ , vol. 19, no. 4, pp. 106–127, 1977.
  * [9] ——, “Non-gaussian noise models in signal processing for telecommunications: New methods and results for Class-A and Class-B noise models,” _IEEE Trans. Inf. Theory_ , vol. 45, no. 4, pp. 1129–1149, 1999.
  * [10] M. Shao and C. Nikias, “Signal processing with fractional lower order moments: stable processes and their applications,” _Proceedings of the IEEE_ , vol. 81, no. 7, pp. 986 –1010, July 1993.
  * [11] J. S. Seo, S. J. Cho, and K. Feher, “Impact of non-Gaussian impulsive noise on the performance of high-level QAM,” _IEEE Transactions Electromag. Compatibility_ , vol. 31, no. 2, pp. 177–180, 1989.
  * [12] J. Häring and A. J. H. Vinck, “Coding and signal space diversity for a class of fading and impulsive noise channels,” _IEEE Trans. Inform. Theory_ , vol. 50, no. 5, pp. 887–895, 2004.
  * [13] S. Zhidkov, “Performance analysis and optimization of OFDM receiver with blanking nonlinearity in impulsive noise environment,,” _IEEE Trans. Veh. Techn._ , vol. 55, no. 1, pp. 234–242, Jan. 2006.
  * [14] ——, “Analysis and comparison of several simple impulsive noise mitigation schemes for OFDM receivers,” _IEEE Trans. Commun._ , vol. 56, no. 1, pp. 5–9, 2008.
  * [15] P. Banelli and L. Rugini, “Impulsive noise mitigation for wireless OFDM,” in _Proc. IEEE Int. Workshop on Sig. Proc. Adv. in Wireless Comm., (SPAWC)_ , Stockholm, Sweden, June-July 2015, pp. 346–350.
  * [16] D. Darsena, G. Gelli, F. Melito, and F. Verde, “ICI-free equalization in OFDM systems with blanking preprocessing at the receiver for impulsive noise mitigation,” _IEEE Signal Proc. Lett._ , vol. 22, no. 9, p. 1321, 2015\.
  * [17] L. Rugini and P. Banelli, “On the equivalence of maximum SNR and MMSE estimation: Applications to additive non-Gaussian channels and quantized observations,” _IEEE Trans. on Signal Processing_ , vol. 64, no. 23, pp. 6190–6199, 2016.
  * [18] N. Rožić, P. Banelli, D. Begu$\check{\rm{s}}$ić, and J. Radić, “Multiple threshold-based estimators for impulsive noise suppression in multicarrier communications,” _IEEE Trans. on Sign. Processing_ , vol. 66, no. 6, pp. 1619–1633, 2018.
  * [19] P. Banelli, “Bayesian estimation of a gaussian source in Middleton Class-A impulsive noise,” _IEEE Signal Proc. Lett._ , vol. 20, no. 10, pp. 956–959, 2013.
  * [20] K. S. Vastola, “Threshold detection in narrow-band non-Gaussian noise,” _IEEE Trans.Commun._ , vol. 32, no. 2, pp. 134–139, Feb. 1984.
  * [21] H. A. Suraweera, C. Chai, J. Shentu, and J. Armstrong, “Analysis of impulse noise mitigation techniques for digital television systems,” in _in Proc. 8th Int. OFDM Workshop_ , Lisbon, Portugal, Sept. 2003 2003, pp. 172–176.
  * [22] P. Gao and C. Tepedelenlioglu, “Space-time coding over fading channels with impulsive noise,” _IEEE Trans. Wireless Commun._ , vol. 6, no. 1, pp. 220–229, 2011.
  * [23] R. Savoia and F. Verde, “Performance analysis of distributed space-time block coding schemes in Middleton Class-A noise,” _IEEE Trans. Veh. Techn._ , vol. 62, no. 6, pp. 2579–2595, 2013.
  * [24] L. Rugini and P. Banelli, “BER of OFDM systems impaired by carrier frequency offset in multipath fading channels,” _IEEE Trans. on Wireless Comm._ , vol. 4, no. 5, pp. 2279–2288, 2005.
  * [25] H. B. Eriksson, P. Odling, T. Koski, and P. Borjesson, “A genue-aided detector with a probabilistic description of the side information,” in _in Proc. of the IEEE International Symposium on Information Theory_ , Whistler, BC, Canada, Canada, 17-22 Sept. 1995, pp. 332–332.
  * [26] Z. Wang and G. B. Giannakis, “Wireless multicarrier communications,” _IEEE Signal Processing Magazine_ , vol. 17, no. 3, pp. 29–48, 2000\.
  * [27] P. Banelli and L. Rugini, “OFDM and multicarrier signal processing,” in _Academic Press Library in Signal Processing: Vol.2 - Communications and Radar Signal Processing_. Elsevier, 2014, vol. 2, pp. 187–293.
  * [28] S. O. Rice, “Statistical properties of a sine wave plus random noise,” _Bell Syst. Techn. J._ , vol. 27, pp. 109–157, 1948.
  * [29] M. Nakagami, _The m-distribution, a general formula of intensity distribution of rapid fading_. in Statistical Methods in Radio Wave Propagation, New York: Pergamon, 1960, vol. 2.
  * [30] S. A. Kassam, _Signal Detection in Non-Gaussian Noise_ , 1st ed. Philadelphia, USA: New York, Berlin, Dowden and Culver, Inc, 1988.
  * [31] T. Y. Al-Naffouri, A. A. Quadeer, and G. Caire, “Impulse noise estimation and removal for OFDM systems,” _IEEE Trans. on Communications_ , vol. 62, no. 3, pp. 976–989, 2014.
  * [32] S. Hur, S. Baek, B. Kim, Y. Chang, A. F. Molisch, T. S. Rappaport, K. Haneda, and J. Park, “Proposal on millimeter-wave channel modeling for 5G cellular system,” _IEEE J. of Select. Topics in Signal Process._ , vol. 10, no. 3, pp. 454–469, March 2016.
  * [33] K. L. Blackard, T. S. Rappaport, and C. W. Bostian, “Measurement and models of radio frequency impulsive noise for indoor wireless communications,” _IEEE J. Sel. Areas Commun._ , vol. 11, no. 9, pp. 991–1001, Sept. 1993.
  * [34] M. G. Sanchez and et al., “Impulsive noise measurements and characterization in a UHF digital TV channel,” _IEEE Trans. Electromagn. Compat._ , vol. 41, no. 2, pp. 124–136, May 1999.
  * [35] M. N. J.Lin and B. L. Evans, “Impulsive noise mitigation in powerline communications using sparse Bayesian learning,” _IEEE J. Sel. Areas Commun._ , vol. 31, no. 7, pp. 1172–1183, July 2013.
  * [36] W. Feller, _An Introduction to Probability Theory and its Application, Vol. II_. New York: John Wiley & Sons, Inc., 1971.
  * [37] H. N. Hamdan and J. P. Nolan, “Estimating the parameters of infinite scale mixtures of normals,,” in _Proceedings of the 36th Symposium on the Interface: Computing Science and Statistics_ , 2004.
  * [38] E. E. Kuruoglu, C. Molina, and W. J. Fitzgerald, “Approximation of alpha-stable probability densities using finite Gaussian mixtures,” in _Proc. EUSIPCO 1998_ , Rhodes, Greece, Sept. 1998.
  * [39] N. Rozic, M. Russo, M. Saric, J. Radic, and D. Begusic, “Short report on approximation of the symmetric alpha-stable density by GMM, p023-02311924-060212, http://marjan.fesb.hr/ radic/technicalreports/sas.pdf,” University of Split, FESB, Tech. Rep., 2013.
  * [40] M. Evans, N. Hastings, and B. Peacock, _Statistical Distributions_ , 3rd ed. Wiley, 2000.
  * [41] M. K. Simon, _Probability Distributions Involving Gaussian Random Variables: A Handbook for Engineers, Scientists and Mathematicians_. Secaucus, NJ, USA: Springer-Verlag New York, Inc., 2006.
  * [42] J. Proakis, _Digital Communications_ , 5th ed. New York, NY, USA: New York: Mc. Graw-Hill, 2008.
  * [43] J. J. Bussgang, _Cross correlation function of amplitude-distorted Gaussian signals, Tech. Rep. 216_. M.I.T Cambridge, March 1952.
  * [44] M.-S. Alouini and A. Goldsmith, “A unified approach for calculating error rates of linearly modulated signals over generalized fading channels,” _IEEE Trans. Commun._ , vol. 47, no. 92, pp. 1324–1334, 1999.
  * [45] R. Y. Yen, H.-Y. Liu, K.-F. Yang, M.-C. Hong, and S.-C. Nan, “Performance of OFDM QAM over frequency-selective fading channels,” in _in Proc. IEEE 2007 IFIP International Conference on Wireless and Optical Communications Networks (WOCN 2007.)_ , Singapore, Singaporel, 2-4 July 2007 2007, pp. 1–5.
  * [46] J. W. Craig, “A new, simple, and exact result for calculating the probability of error for two-dimensional signal constellations,” in _in Proc. of the IEEE Military Communications Conf. (MILCOM 91)_ , vol. 1, McLean, VA,, october 1991, pp. 571–575.
  * [47] M. G. Shayesteh and A. Aghamohammadi, “On the error probability of linearly modulated signals on frequency-flat Ricean, Rayleigh, and AWGN channels,” _IEEE Trans. Commun._ , vol. 43, no. 2/3/4, pp. 1454–1466, 19959\.
  * [48] D. Hughes-Hallet and W. G. McCullum, _Calculus_ , 4th ed. Wiley, 2005.