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Bernoulli-Gaussian Distribution with Memory As a Model for Power Line Communication Noise Victor Fernandes, Weiler A

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Bernoulli-Gaussian Distribution with Memory As a Model for Power Line Communication Noise Victor Fernandes, Weiler A XXXV SIMPOSIO´ BRASILEIRO DE TELECOMUNICAC¸ OES˜ E PROCESSAMENTO DE SINAIS - SBrT2017, 3-6 DE SETEMBRO DE 2017, SAO˜ PEDRO, SP Bernoulli-Gaussian Distribution with Memory as a Model for Power Line Communication Noise Victor Fernandes, Weiler A. Finamore, Mois´es V. Ribeiro, Ninoslav Marina, and Jovan Karamachoski Abstract— The adoption of Additive Bernoulli-Gaussian Noise very useful to come up with a as simple as possible models (ABGN) as an additive noise model for Power Line Commu- of PLC noise, similar to the additive white Gaussian noise nication (PLC) systems is the subject of this paper. ABGN is a (AWGN) model. model that for a fraction of time is a low power noise and for the remaining time is a high power (impulsive) noise. In this context, In this regard, we introduce the definition of a simple and we investigate the usefulness of the ABGN as a model for the useful mathematical model for the noise perturbing the data additive noise in PLC systems, using samples of noise registered transmission over PLC channel that would be a helpful tool. during a measurement campaign. A general procedure to find the With this goal in mind, an investigation of the usefulness of model parameters, namely, the noise power and the impulsiveness what we call additive Bernoulli-Gaussian noise (ABGN) as a factor is presented. A strategy to assess the noise memory, using LDPC codes, is also examined and we came to the conclusion that model for the noise perturbing the digital communication over ABGN with memory is a consistent model that can be used as for PLC channel is discussed. The ABGN is a discrete-time and the evaluation of the noise effect on the digital communication memoryless model that is equivalent to the continuous-time over PLC channel even when the PLC system uses the memory- Two-state, Middleton Class A model. Moreover, a comparison dependent (i.e. error control codes) consistency that do not hold performance analysis of the digital communication system when using the (memoryless) ABGN model. when the transmission is perturbed by the measured noise and Keywords— power line communication, power line noise model, by the noise generated according to ABGN model is discussed. Bernoulli-Gaussian noise, noise with memory, impulsive noise Due to the limitation of ABGN model to handle memory- sensitive procedures, we propose a more sophisticated model, I. INTRODUCTION which takes into account the memory that is intrinsic to the The Power Line Communication (PLC) channel is perturbed nature of the PLC noise. This model is so-called ABGN with by noise which is commonly considered to be a stochastic memory (ABGNM) and it is suitable for PLC systems that process. For a given percentage of the time it is in an impulsive make use of any memory-dependent tool (e.g., error control state (high or “strong” noise power) and for the remaining time codes). is on the background state (low or “weak” noise power). The Basedonthepresentedanalyses,wecameuptothe characterization of the noise perturbing PLC channel (hereafter conclusion that the ABGN model is a good model for the called PLC noise) has been the subject of studies and some PLC noise when the digital communication system does not models have been proposed in order to mimic the measured use any memory-sensitive procedure. However, it does not PLC noise behavior without the need to realize a measure mimic the system performance with measured noise when it campaign [1]–[3]. A widely accepted model is the Middleton has memory, which is a similar behavior of Middleton Class A Class A, which assumes that the noise impulsiveness level is model. Finally, we show that the ABGNM model can mimic classified in an infinite number of states [4]. As shown in [5], the measured PLC noise, resulting in a similar performance the Two-state, Middleton Class A model can be as good as the for both measured and synthetic noises even when the digital general Middleton Class A, agreeing with the fact that a two- communication system has any memory-dependent tool (e.g., state model offers a good balancing between the usefulness error control codes). and simplicity. By reviewing the literature, we note that the Middleton II. COMMUNICATION SYSTEM MODEL Class A model was not evaluated when memory-dependent A digital communication system in which a modulated tool is taken into account. This is an important issue because signal is transmitted through a medium is, in its simplest the noise model must be able to capture the behavior of PLC form, modeled by a channel, where the input signal at the noise when error control code is used. Moreover, it would be receiver is y(t)=s(t)+w(t). This corresponds to modeling a transmission medium by the channel in which s(t) is the Victor Fernandes is with Universidade Federal de Juiz de Fora (UFJF), Juiz channel input (modulated signal), w(t) represents the noise de Fora - MG, Brazil. E-mail: [email protected]. Weiler A. Finamore is with Universidade Federal de Juiz de Fora (UFJF), perturbing the transmission, and y(t) is the channel output. In Juiz de Fora - MG, Brazil and University of Information Sciences and Tech- this section, we focus on a digital communication system with nology (UIST), Ohrid, Republic of Macedonia. E-mail: fi[email protected]. a binary phase shift keying (BPSK) modulator, transmitting a Mois´es V. Ribeiro is with Universidade FederaldeJuizdeFora(UFJF), th Juiz de Fora - MG, Brazil and Smarti9 Ltda. E-mail: [email protected]. string of bits, which its i bit is represented by {ui} (i ∈ Z Ninoslav Marina and Jovan Karamachoski are with University of Infor- and ui ∈{0, 1}). The modulated signal is expressed by mation Sciences and Technology (UIST), Ohrid, Republic of Macedonia. E- ∞ mails: [email protected], [email protected]. This work was partially supported by CNPq, CAPES, FAPEMIG, FINEP, s(t)= (1 − 2ui)p(t − iT )cos2πfct, (1) INERGE, and Smarti9 Ltda. i=−∞ 328 XXXV SIMPOSIO´ BRASILEIRO DE TELECOMUNICAC¸ OES˜ E PROCESSAMENTO DE SINAIS - SBrT2017, 3-6 DE SETEMBRO DE 2017, SAO˜ PEDRO, SP in which the formatting pulse is p(t) with energy equal to one, B. Bernoulli-Gaussian Noise T f is the signaling interval, and c is the modulating frequency. The ABGN is a discrete-time and memoryless model that is equivalent to the continuous-time Two-state Middleton Class A A. White Gaussian Noise model. In this model, besides the background noise power, characterized by N0, two other parameters: α (which char- y t s t w t The channel output (receiver input) ( )= ( )+ ( ), acterizes how much stronger than the background noise the representing the input signal added to the noise, is the ob- impulsive noise is) and p (which characterizes the percentage servation that will be used by the receiver to estimate the of time that only background noise is present) are used when information that has been sent. The noise perturbing the digital specifying the ABGN(N0,p,α) model. communication is, in many situations, modeled as a zero A digital communication system in which a modulated W t mean stochastic process ( ) with power spectral density signal is transmitted over a PLC channel is, in its simplest SW (f)=N0/2, rendering a value (at the decider input) s t 2 2 form, modeled as having the signal ( ), at the channel input σ = N0/2 σ with variance 0 . This is the simple AWGN( 0) and yPLC(t)=s(t)+wPLC(t) is the channel output, where model which have been extensively used to model different wPLC(t) is the PLC additive noise. E communication channels. The energy-per-bit b at the receiver As discussed in [7], the received and filtered PLC noise E /N input and the ratio b 0, of the energy-per-bit and the noise can be modeled as a stochastic process V (t). At the receiver parameter (a parameter of interest to characterize the state of end (at the input of the decider), at a given time instant t1, the system), can be easily calculated from (1). V (t1) can be a zero mean “weak” Gaussian rv with variance In such digital communication systems, assuming perfect 2 σ0 and, at a distinct time instant t2, V (t2) can be a zero mean synchronization, the continuous signal y(t) is processed by 2 2 2 “strong” Gaussian rv with variance σ1 = α σ0 ,(α>1). the receiver as follows: V (t1) and V (t2) are independent rvs. The noise component {V } σ2,p,α yo(t)=y(t)2 cos 2πfct sequence n , which can be modeled as an ABGN( 0 ), is defined as follows [8]: r(t)=yo(t) ∗ p(T − t) r(t)=([s(t)+w(t)]2 cos 2πfct) ∗ p(T − t) 2 th Definition #1 (ABGN(σ ,p,α)): Let {Un} be the n ∞ 0 sample of a sequence of Bernoulli rv with P[Un =1]=p r(t)= (1 − 2ui)q(t − iT )+2w(t)cos2πfct ∗ p(T − t) i=−∞ (and, of course, P[Un =0]=1− p)and{Wn} be a sequence σ2 rn = r(t)|t=nT of i.i.d. Gaussian rv with zero mean and variance 0 .The th ∞ sequence of rvs, {Vn}, with the n sample given by = (1 − 2un)δn−i + wn. (2) Vn = UnWn + α(1 − Un)Wn, (4) i=−∞ ∗ p(t) ∗ p(−t)=q(t) 2 where denotes linear convolution, in which α>1, is an ABGN(σ0,p,α). fulfill the 1st criterion of Nyquist for having free intersymbol δ w nth interference, n is the impulse sequence, and n is the Every sample Vn = αWn corresponding to Un =0,is z − u w discrete-time noise sample.
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