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    BroadbandandNarrow-BandNoiseModelinginPowerlineCommunications

    CHAPTER·DECEMBER2015

    DOI:10.1002/047134608X.W8289

    READS

    11

    3AUTHORS,INCLUDING:

    ThokozaniShongwe

    UniversityofJohannesburg

    15PUBLICATIONS37CITATIONS

    SEEPROFILE

    A.J:HanVinck

    UniversityofDuisburg-Essen

    243PUBLICATIONS2,011CITATIONS

    SEEPROFILE

    Allin-textreferencesunderlinedinbluearelinkedtopublicationsonResearchGate,

    lettingyouaccessandreadthemimmediately.

    Availablefrom:ThokozaniShongwe

    Retrievedon:10January2016

    https://www.researchgate.net/publication/289531322_Broadband_and_Narrow-Band_Noise_Modeling_in_Powerline_Communications?enrichId=rgreq-1a11f4ba-347e-4f3d-9286-11409662a64a&enrichSource=Y292ZXJQYWdlOzI4OTUzMTMyMjtBUzozMTYzNTU5MjI4NTc5ODRAMTQ1MjQzNjUzNzA2Ng%3D%3D&el=1_x_2https://www.researchgate.net/publication/289531322_Broadband_and_Narrow-Band_Noise_Modeling_in_Powerline_Communications?enrichId=rgreq-1a11f4ba-347e-4f3d-9286-11409662a64a&enrichSource=Y292ZXJQYWdlOzI4OTUzMTMyMjtBUzozMTYzNTU5MjI4NTc5ODRAMTQ1MjQzNjUzNzA2Ng%3D%3D&el=1_x_3https://www.researchgate.net/?enrichId=rgreq-1a11f4ba-347e-4f3d-9286-11409662a64a&enrichSource=Y292ZXJQYWdlOzI4OTUzMTMyMjtBUzozMTYzNTU5MjI4NTc5ODRAMTQ1MjQzNjUzNzA2Ng%3D%3D&el=1_x_1https://www.researchgate.net/profile/Thokozani_Shongwe?enrichId=rgreq-1a11f4ba-347e-4f3d-9286-11409662a64a&enrichSource=Y292ZXJQYWdlOzI4OTUzMTMyMjtBUzozMTYzNTU5MjI4NTc5ODRAMTQ1MjQzNjUzNzA2Ng%3D%3D&el=1_x_4https://www.researchgate.net/profile/Thokozani_Shongwe?enrichId=rgreq-1a11f4ba-347e-4f3d-9286-11409662a64a&enrichSource=Y292ZXJQYWdlOzI4OTUzMTMyMjtBUzozMTYzNTU5MjI4NTc5ODRAMTQ1MjQzNjUzNzA2Ng%3D%3D&el=1_x_5https://www.researchgate.net/institution/University_of_Johannesburg?enrichId=rgreq-1a11f4ba-347e-4f3d-9286-11409662a64a&enrichSource=Y292ZXJQYWdlOzI4OTUzMTMyMjtBUzozMTYzNTU5MjI4NTc5ODRAMTQ1MjQzNjUzNzA2Ng%3D%3D&el=1_x_6https://www.researchgate.net/profile/Thokozani_Shongwe?enrichId=rgreq-1a11f4ba-347e-4f3d-9286-11409662a64a&enrichSource=Y292ZXJQYWdlOzI4OTUzMTMyMjtBUzozMTYzNTU5MjI4NTc5ODRAMTQ1MjQzNjUzNzA2Ng%3D%3D&el=1_x_7https://www.researchgate.net/profile/Aj_Vinck?enrichId=rgreq-1a11f4ba-347e-4f3d-9286-11409662a64a&enrichSource=Y292ZXJQYWdlOzI4OTUzMTMyMjtBUzozMTYzNTU5MjI4NTc5ODRAMTQ1MjQzNjUzNzA2Ng%3D%3D&el=1_x_4https://www.researchgate.net/profile/Aj_Vinck?enrichId=rgreq-1a11f4ba-347e-4f3d-9286-11409662a64a&enrichSource=Y292ZXJQYWdlOzI4OTUzMTMyMjtBUzozMTYzNTU5MjI4NTc5ODRAMTQ1MjQzNjUzNzA2Ng%3D%3D&el=1_x_5https://www.researchgate.net/institution/University_of_Duisburg-Essen?enrichId=rgreq-1a11f4ba-347e-4f3d-9286-11409662a64a&enrichSource=Y292ZXJQYWdlOzI4OTUzMTMyMjtBUzozMTYzNTU5MjI4NTc5ODRAMTQ1MjQzNjUzNzA2Ng%3D%3D&el=1_x_6https://www.researchgate.net/profile/Aj_Vinck?enrichId=rgreq-1a11f4ba-347e-4f3d-9286-11409662a64a&enrichSource=Y292ZXJQYWdlOzI4OTUzMTMyMjtBUzozMTYzNTU5MjI4NTc5ODRAMTQ1MjQzNjUzNzA2Ng%3D%3D&el=1_x_7

  • On Broadband and Narrow-band Noise Modellingfor OFDM systems in Powerline Communications

    Thokozani ShongweDepartment of Electrical and

    Electronic Engineering Technology,University of Johannesburg,

    P.O. Box 17011, Doornfontein, 2028,Johannesburg, South AfricaEmail: [email protected]

    A. J. Han VinckUniversity of Duisburg-Essen,

    Institute for Experimental Mathematics,Ellernstr. 29, 45326 Essen, Germany

    E-mail: [email protected]

    Hendrik C. FerreiraDepartment of Electrical and

    Electronic Engineering Science,University of Johannesburg,

    P.O. Box 524, Auckland Park, 2006,Johannesburg, South AfricaEmail: [email protected]

    Abstract—This article gives an overview of noise modellingof the two most problematic types of noise in powerline com-munications (PLC) namely, impulse noise (broadband noise) andfrequency disturbers (narrow-band interference). The models forthese two types of noise are studied for OFDM systems. The firstpart of the article looks at several impulse noise models in theliterature. A demonstration of the effects of impulse noise onOFDM, using the Middleton Class A model is given. Methods ofcombating impulse noise are also studied. In the second part ofthe article we focus on narrow-band interference (NBI) modellingand also show how NBI affects OFDM systems. The main focusof NBI is on presenting a detailed NBI model because NBImodelling has not been given much attention in the literature.In our discussion of the NBI model we show that, with someassumptions, the NBI model and the Middleton Class A modelcan be similar.

    Index Terms—Impulse noise models, Multi-carrier modulation,Single-carrier modulation, Bernoulli-Gaussian, Middleton ClassA, Symmetric alpha-stable distribution, Narrow-band interfer-ence, noise model, Powerline communications, OFDM.

    INTRODUCTION

    The article is divided into two chapters. The first chapterdiscusses broadband noise (impulse noise) and its modelling,and the second chapter discusses narrow-band noise and itsmodelling. In chapter one: this article gives an overview ofimpulse noise and its models, and points out some importantand interesting facts about the study of impulse noise whichare sometimes overlooked or not well understood. We discussthe different impulse noise models in the literature, focusing ontheir similarities and differences when applied in communica-tions systems. The impulse noise models discussed are mem-oryless (Middleton Class A, Bernoulli-Gaussian and Symmet-ric alpha-stable), and with memory (Markov-Middleton andMarkov-Gaussian). We then go further to give performancecomparisons in terms of bit error rates for some of the variantsof impulse noise models. We also compare the bit error rateperformance of single-carrier (SC) and multi-carrier (MC)communications systems operating under impulse noise. Itcan be seen that MC is not always better than SC underimpulse noise. Lastly, the known impulse noise mitigationschemes (clipping/nulling using thresholds, iterative based anderror control coding methods) are discussed. In chapter two:

    a narrow-band interference (NBI) model for the powerlinecommunications channel is presented. We give frequencydomain details and analysis of the NBI model specifically forOFDM systems; it can easily be adapted to model NBI forother communications systems. We also show that by makingthe same assumptions as in the Middleton class A model, ourNBI model becomes the Middleton Class A noise model.

    The work in this article is taken from our two papers[1] and [2] which directly resulted in chapters one and two,respectively.

    1. BROADBAND NOISE–IMPULSE NOISEI. INTRODUCTION: IMPULSE NOISE

    The effects of impulse noise are experienced by mostcommunications systems. There has been significant researchpertaining to impulse noise, which involve modelling of theimpulse noise phenomena and combating impulse noise incommunications systems. Research articles addressing im-pulse noise and its effects are found across different fieldsin communications, some of which are electromagnetic in-terference, wireless communication, and recently powerlinecommunication. We therefore see it necessary to bring acontribution which gives a general view of impulse noise incommunications systems, from the different research fields.The purpose of this chapter is to give an overview of impulsenoise, and point out some important and interesting facts aboutthe study of impulse noise which are sometimes overlooked.We begin by looking at the earliest work on impulse noisemodelling by Middleton [3, Chapter 11], in Section II. Thenwe go on to discuss the common impulse noise models inthe literature, in Section III, dividing them into those withmemory and without memory. We also look at the applicationof these models with single-carrier and multi-carrier systems.Lastly, we give an overview of the currently known methods ofcombating impulse noise, in Section IV. This chapter followsfrom the preliminary work in [4].

    II. ON MIDDLETON NOISE MODELLING

    The phenomenon of impulse noise was first described indetail by Middleton [3, Chapter 11] in the 1960s, where he

    https://www.researchgate.net/publication/268223263_On_Impulse_Noise_and_its_Models?el=1_x_8&enrichId=rgreq-1a11f4ba-347e-4f3d-9286-11409662a64a&enrichSource=Y292ZXJQYWdlOzI4OTUzMTMyMjtBUzozMTYzNTU5MjI4NTc5ODRAMTQ1MjQzNjUzNzA2Ng==https://www.researchgate.net/publication/261048116_Narrow-band_interference_model_for_OFDM_systems_for_powerline_communications?el=1_x_8&enrichId=rgreq-1a11f4ba-347e-4f3d-9286-11409662a64a&enrichSource=Y292ZXJQYWdlOzI4OTUzMTMyMjtBUzozMTYzNTU5MjI4NTc5ODRAMTQ1MjQzNjUzNzA2Ng==https://www.researchgate.net/publication/268223263_On_Impulse_Noise_and_its_Models?el=1_x_8&enrichId=rgreq-1a11f4ba-347e-4f3d-9286-11409662a64a&enrichSource=Y292ZXJQYWdlOzI4OTUzMTMyMjtBUzozMTYzNTU5MjI4NTc5ODRAMTQ1MjQzNjUzNzA2Ng==

  • gave a model for impulse noise in communications systems.To obtain the model, Middleton [3, Chapter 11] describedimpulsive noise in a system as consisting of sequences ofpulses (or impulses), of varying duration and intensity, andwith the individual pulses occurring more or less randomin time. He went further to divide the origin of impulsenoise into two categories: (a) Man-made, which is inducedby other devices connected in a communications network and(b) naturally occurring, due to atmospheric phenomena andsolar static which is due to thunderstorms, sun spots etc.The man-made impulse noise was described as trains of non-overlapping pulses, such as those in pulse time modulation.The impulse noise due to natural phenomena was describedas the random superposition of the effects of the individualnatural phenomena [3, Chapter 11]. A model for such noiseis given in [5], [6] and [7] as

    n(t) =

    L∑i=1

    aiδ(t− ti), (1)

    where• δ(t − ti) - is the ith unit (ideal) impulse, described as a

    delta function.• ai - are statistically independent with identical probability

    density functions (PDFs),• ti - are independent random variables uniformly dis-

    tributed in the time period T0,• L - the number of impulses in any observation period T0,

    assumed to obey a Poisson distribution

    PT0(L) =(ηT0)

    Le−(ηT0)

    L!. (2)

    In (2), η is the average number of impulses per second, andT0 is the observation period of the impulses. Therefore, ηT0is the average number of impulses in the period T0. It can beseen that the noise model described by (1) is ideal because theimpulses are assumed to be delta functions, where aiδ(t− ti)describes the ith impulse of amplitude ai.

    The noise model described by (1) and (2) was originallyreferred to as the Poisson noise model in [3, Chapter 11],and was widely studied and applied in many systems [5]–[8]. Ziemer [5], utilised the Poisson impulse noise model tocalculate error probability characteristics of a matched filterreceiver operating in an additive combination of impulsiveand Gaussian noise. In [7], the Poisson noise model wasused to evaluate the performance of noncoherent M -ary digitalsystems, ASK, PSK and FSK.

    In his later work, Middleton [3] developed statistical noisemodels which catered for noise due to both man-made andnatural phenomena [9]–[11]. In [9] Middleton classified thenoise models into the following three categories: Class A – thenoise has narrower bandwidth than that of the receiver; ClassB – the noise has larger bandwidth than that of the receiver;Class C – the sum of Class A and Class B noise. The mostfamous of these noise models is the so-called Middleton ClassA noise model, which has been widely accepted to model the

    effects of impulse noise in communications systems. We will,in short, refer to the Middleton Class A model as Class Amodel.

    III. IMPULSE NOISE MODELS

    Following Middleton’s noise models [3, Chapter 11], manyauthors studied impulse noise modelling. In this section, wediscuss some impulse noise models found in the literature.In our discussion of the other impulse noise models wewill occasionally mention the Middleton Class A model forreference or comparison purposes as it is a very importantmodel in the study of impulse noise. To date, the followingnames appear in the literature for different impulse noisemodels:

    1) Impulse noise models without memory• Middleton Class A• Bernoulli-Gaussian• Symmetric Alpha-Stable distribution

    2) Impulse noise models with memory• Markov-Middleton• Markov-Gaussian.

    Impulse noise models without memory

    A. Middleton Class A

    The Class A noise model is still a form of the Poison noisemodel. However, unlike in (2), the impulse width/duration istaken into account in the Class A noise model as we shall seewhen studying the impulsive index A. We dedicate space todescribing the Class A noise model because it has becomethe cornerstone of impulse noise modelling and has beenextensively studied and utilised in the literature (see [12]–[19].) The Class A noise model gives the probability densityfunction (PDF) of a noise sample, say xk as follows:

    FM (xk) =

    ∞∑m=0

    PmN (xk; 0, σ2m), (3)

    where N (xk; 0, σ2m) represents a Gaussian PDF with meanµ = 0 and variance σ2m, from which the k

    th sample xk istaken.

    Pm =Ame−A

    m!(4)

    and

    σ2m = σ2I

    m

    A+ σ2g = σ

    2g

    ( mAΓ

    + 1), (5)

    where σ2I is the variance of the impulse noise and σ2g is the

    variance of the background noise (AWGN). The parameterΓ = σ2g/σ

    2I gives the Gaussian to impulse noise power ratio.

    We can see that the expressions in (2) and (4) are PoissonPDFs. The main difference in (4) is that the term (ηT0) hasbeen replaced by the parameter A (which does not implythat A = ηT0). The parameter A here represents the density

  • of impulses (of a certain width) in an observation period.Therefore

    A =ητ

    T0= η

    τ

    T0, (6)

    where η is the average number of impulses per second (asin (2)), τ is the average duration of each impulse (where allimpulses are taken to have the same duration) and τ/T0 isthe fraction of the period T0 occupied by an impulse of widthτ . We now talk of density of impulses instead of number ofimpulses as done in (2). In (4) we therefore have the densitiesof impulse noise occurring according to a Poisson distribution.

    The density is what has become accepted as “impulsiveindex”, A. The impulsive index is a parameter that is notwell explained in the literature. We therefore give some detailsabout the impulsive index, to enhance its understanding. It isworth stating that A ≤ 1, this follows from the definitionof impulsive index being a fraction of impulses in a givenobservation period T0. Therefore, for ητ > T0, the impulsiveindex is capped at 1 no matter how large ητ is, in theobservation period T0.

    1 2 η

    A = ητT0ητ

    T0

    . . .τ

    (a)

    T0 = 1

    ττ τ

    1 2 3

    A = ητT0 = 0.3τ = 0.1;η = 3;T0 = 1;

    (b)

    Fig. 1. Example of Impulsive index: (a) Impulsive index (density) of ηimpulses, each with width (duration) τ , occupying a given time period T0 and(b) impulsive index (density) of η = 3 impulses, each with width (duration)τ = 0.1s, occupying a given time period T0 = 1s.

    Fig. 1 shows a pictorial view of the impulsive index, A, andwhat it means. Fig. 1 (a) shows η impulses each of durationτ seconds, where the impulses occur in bursts (next to eachother). Fig. 1 (b) shows η = 3 impulses each of durationτ = 0.1 seconds, where the impulses do not necessarily occurin bursts. We also specify the period of observation as T0 = 1second in Fig. 1 (b), which is usually the case in the calculationof the impulsive index. The conclusion drawn from Fig. 1 isthat whether impulses occur in bursts or not, the calculationof the impulsive index follows the same procedure. In otherwords, the impulsive index is concerned with the density of

    impulses in a periodic of observation, not how the impulsesare distributed in that period or observation.

    B. Bernoulli-Gaussian

    The Middleton Class A noise model has already beenexplained. It can be seen that the PDF of the Class A noisemodel in (3) is a sum of different zero mean Gaussian PDFswith different variances σ2m, where the PDFs are weighted bythe Poisson PDF Pm. This summing of weighted GaussianPDFs is generally referred to as a Gaussian mixture. Anotherpopular impulse noise model, which is a Gaussian mixtureaccording to the Bernoulli distribution, exists in the literatureand is called the Bernoulli-Gaussian noise model (can befound in [20]–[23].) This noise model is described by thefollowing PDF:

    FBG(xk) = (1− p)N (xk; 0, σ2g) + pN (xk; 0, σ2g + σ2I ). (7)The Bernoulli-Gaussian noise model has similarities to the

    Class A noise model. To show the similarities, we use thechannel models in Fig. 2. Fig. 2 (a) is a two-state repre-sentation of the Class A noise model. In the two-state ClassA noise model the probability of being in the impulse noisestate is A and the average impulse noise variance is σ2I . Forthe two-state Class A noise model, the probability of impulsenoise occurrence is found to be A by finding the weightedsum of the probabilities (mPm) of the Class A noise model:∑∞m=1mPm = A. Fig. 2 (b) is a representation the Bernoulli-

    Gaussian noise model. The models in Fig. 2 look very similar,with the only difference being that in Fig. 2 (b) it is explicitlystated that the noise sample added to the data symbol Dk, ineither of the two states, is Gaussian distributed. Whereas inFig. 2 (a), only the state with variance σ2g can have a Gaussiandistribution. However, the state with impulse noise does notnecessarily have a Gaussian distribution.

    It should be noted that in the impulse noise model in Fig. 2,for the states with impulse noise, the impulse noise varianceσ2I is divided by the probability of entering into that state(A or p), such that the impulse noise variance in the system(total number of time samples) becomes σ2I . To explain this,let us use the Class A noise model as follows: in the ClassA noise model which is defined by (3)–(5), it can be seenthat the impulse noise variance of the state m is (σ2Im)/A asshown in (5). This variance of state m occurs with probabilityPm (see (4)), hence the average impulse noise variance of theClass A noise model is

    ∞∑m=0

    Pmσ2Im

    A=σ2IA

    ∞∑m=0

    mPm =σ2IA×A = σ2I . (8)

    From a simulation point of view, we explain the divisionof σ2I by A as follows: let us assume a transmission of Nsymbols. For the impulse noise variance, in the vector of lengthN , to be approximately σ2I , each symbol has to be affectedby impulse noise variance σ2I/A. It can easily be shown thatthis situation will result in the impulse noise of σ2I over the

  • Dk D̃k

    σ2g

    σ2g + σ2I/A

    A

    1− A

    (a)

    Dk D̃k

    N (0, σ2g)

    N (0, σ2g + σ2I/p)p

    1− p

    (b)

    Fig. 2. (a) Two-state Class A noise model and (b) Bernoulli-Gaussian noisemodel.

    N symbols, as follows: we know that impulse noise occurswith probability A, and for N symbols (assuming very largeN ) we have approximately AN symbols affected by impulsenoise of variance σ2I/A. This gives the impulse noise variancein N samples as σ2N = AN×σ2I/A = Nσ2I . Then the averageimpulse noise variance is σ2N/N = σ

    2I , which is the result in

    (8). For the two models in Fig. 2 to be more similar, set A = p.The Bernoulli-Gaussian noise model has been widely

    adopted in the literature, and some researchers prefer toemploy it over the Class A noise model because it is moretractable than the Class A noise model. The Class A modelhas the advantage of having its parameters directly related tothe physical channel. If so desired the Class A model can beadjusted to approximate the Bernoulli-Gaussian, hence givingthe Bernoulli-Gaussian model the advantages of the Class Amodel as well.

    The Class A model can also be simplified, and be mademore manageable. It was shown in [13] that the PDF of theClass A noise model in (3) can be approximated by the firstfew terms of the summation and still be sufficiently accurate.Truncating (3) to the first K terms results in the approximationPDF (normalised), which is

    FM,K(xk) =

    K−1∑m=0

    P ′mN (xk; 0, σ2m), (9)

    where

    P ′m =Pm∑K−1m=0 Pm

    .

    The model in (9) allowed Vastola [13] to design a threshold

    detector with a simpler structure, which would not have beenthe case using the model in (3) which has infinite terms. Itwas also shown in [13] that the first two or three terms aregood enough in (9) to approximate the PDF in (3). In [19],the first four terms were used to approximate the PDF of theClass A model. In our simulations, we shall use up to the firstfive terms of (9), and such a model is shown in Fig. 3.

    We now give some results showing the bit error rate (BER)versus SNR when using the model in (9) for different Kvalues. Such results are shown in Figs. 4 and 5, where BPSKmodulation is used and K = 2, 3 and 5. Figs. 4 and 5show the effect of different values of A and Γ on the model.In each figure, we use a theoretical BER curve for BPSKas a reference curve against which all curves are compared.The expression for the theoretical BER curve for BPSK isgiven by (10), where Eb is the signal’s bit energy. Whileour results are limited to BPSK for demonstration purposes,MPSK and MQAM modulations can also be used to obtainBER performances. The general expression for the theoreticalBER curve for MPSK is given by (11), where M is the orderof the PSK modulation.

    Dk

    P ′4

    2σ2IA + σ

    2g

    P ′0

    P ′3

    D̃k

    σ2g

    σ2IA + σ

    2g

    4σ2IA + σ

    2g

    3σ2IA + σ

    2g

    P ′1

    P ′2

    Fig. 3. Five-term, K = 5, approximation of the Class A model.

    Pe,BPSK =(1−A)

    2Q

    (√Ebσ2g

    )

    +A

    2Q

    (√Eb

    σ2g(1 + 1/AT )

    ). (10)

    Pe,MPSK ≈(1−A)B Q

    (√BEbσ2g

    sin( πM

    ))

    +A

    BQ(√

    BEbσ2g(1 + 1/AT )

    sin( πM

    )),(11)

    where B = log2(M).It can be observed in Figs. 4 and 5 that the model in (9)

    approximates the Class A model in (3) better for low valuesof A (see Fig. 4), such that even for two terms, K = 2, weget a very good approximation of the theoretical BER curve.For high values of A (see Fig. 5), however, we require more

  • 0 5 10 15 20 25 30 35 4010

    −5

    10−4

    10−3

    10−2

    10−1

    100

    Eb/2σ2g, dB (SNR)

    Bit

    Err

    or R

    ate

    A/2

    −10log10

    (AΓ)

    AWGN + Impulse noise, K=2

    AWGN, theory

    AWGN + Impulse noise, theory

    AWGN + Impulse noise, K=3

    AWGN + Impulse noise, K=5

    (a) A = 0.01, Γ = 0.1.

    0 5 10 15 20 25 30 35 40 45 5010

    −5

    10−4

    10−3

    10−2

    10−1

    100

    Eb/2σ2g, dB (SNR)

    Bit

    Err

    or R

    ate

    A/2

    AWGN, theory

    AWGN + Impulse noise, K=5

    AWGN + Impulse noise, K=3

    AWGN + Impulse noise, theory

    AWGN + Impulse noise, K=2

    −10log10

    (AΓ)

    (b) A = 0.01, Γ = 0.01.

    Fig. 4. Bit error rate results using the impulse noise model shown in Fig.3, with K = 2, 3 and 5. BPSK was used for the modulation.

    terms in (9) to approximate the results of the model in (3),at least for the part of the curve influenced by A (the errorfloor).

    Fig. 5 shows that the results of the K = 5 channel modelclosely approximate the effect of A on the BER curve betterthan when K = 2. This is obviously due to the fact that themore terms (higher K values), the better the approximation ofthe Class A PDF’s probability of impulse noise. However, theK = 2 channel model results show a better approximationof the impulse noise power (1/(AΓ), which is observedaround a BER of 10−5) compared to when K = 5. This isbecause of the m parameter in the term σ2Im/A in (5), whichinfluences the impulse noise power. Using more terms in (9)to approximate the results of the model in (3) is more effectivein estimating the effect of A in the BERs, but not the impulsenoise power.

    C. Symmetric alpha(α)-stable distribution for impulse noisemodelling

    The impulse noise models discussed so far (the MiddletonClass A and the Bernoulli-Gaussian) are by far the mostwidely used in the literature to model impulse noise. There is

    0 5 10 15 20 25 30 35 4010

    −5

    10−4

    10−3

    10−2

    10−1

    100

    Eb/2σ2g, dB (SNR)

    Bit

    Err

    or R

    ate

    AWGN, theoryAWGN + Impulse noise, theoryAWGN + Impulse noise, K=5AWGN + Impulse noise, K=2AWGN + Impulse noise, K=3

    (a) A = 0.3, Γ = 0.1.

    0 5 10 15 20 25 30 35 4010

    −5

    10−4

    10−3

    10−2

    10−1

    100

    Eb/2σ2g, dB (SNR)

    Bit

    Err

    or R

    ate

    AWGN, theoryAWGN + Impulse noise, theoryAWGN + Impulse noise, K=5AWGN + Impulse noise, K=2AWGN + Impulse noise, K=3

    (b) A = 0.3, Γ = 0.01.

    Fig. 5. Bit error rate results using the impulse noise model shown in Fig.3, with K = 2, 3 and 5. BPSK was used for the modulation.

    another impulse noise model that is becoming more commonin the literature, and that is the symmetric α-stable (SαS)distribution. This section is therefore a short note on symmetricα-stable distributions used to model impulse noise.

    SαS distributions are used in modelling phenomena encoun-tered in practice. These phenomena do not follow the Gaussiandistribution, instead their probability distributions may exhibitfat tails when compared to the Gaussian distribution tails[24, Chapter 1]. While stable distributions date back to the1920s (see [25]), their usage in practical applications hadbeen limited until recently because of their lack of closed-form expressions except for a few (Levy, Cauchy and Gaussiandistributions) [24, Chapter 1]. Nowadays powerful computerprocessors have made it possible to compute stable distri-butions despite the lack of closed form expressions. Thishas led to the increasing usage of stable distributions inmodelling. Impulse noise is one phenomenon encountered incommunication systems which has a probability distributionwith fat tails [26]. SαS distributions are therefore consideredappropriate for modelling impulse noise [26]–[29].

  • In this section we give examples of the PDFs of the SαSmodel for impulse noise, the Class A model and the Bernoulli-Gaussian model. This is meant to show how the SαS modelcompares, in terms of the PDFs, with the other impulse noisemodels already discussed.

    The SαS distributions are characterised by the followingparameters:• α: is the characteristic exponent, and describes the tail of

    the distribution (0 < α ≤ 2).• β: describes the skewness of the distribution (−1 ≤ β ≤

    1); if the distribution is right-skewed (β > 0) or left-skewed (β < 0).

    • γ: is the scaling parameter (γ > 0).• δ: is a real number that gives the location of the distribu-

    tion. This number tells us where the distribution is locatedon the x-axis (when the x-axis is used to represent thevalue of the random variable).

    The parameters α and β describe the shape of the distribution;while γ and δ can be thought of as similar to the variance andthe mean in a Gaussian distribution, respectively, care shouldbe taken when using these parameters as variance and mean.

    When modelling impulse noise using the SαS distribution,the noise is thought of as broadband noise, i.e, the bandwidthof the noise is larger than that of the receiver [30]. Hence, theSαS distribution can be used in the place of the MiddletonClass B noise model which requires more (six) parametersto be defined compared to the four parameters required todescribe the SαS distribution.

    Fig. 6 shows PDFs of the SαS models for different valuesof α while the other parameters are kept fixed (β = 0 andδ = 0). The parameter γ is set at γ = 1 for all PDFs exceptfor the α = 2 PDF where γ = 1/

    √2. This case of α = 2,

    γ = 1/√

    2, β = 0 and δ = 0 results in the normal distributionas seen in Fig. 6. It should be noted that the SαS distributionof α = 2, β = 0, δ = 0 and γ > 0 is generally the Gaussiandistribution; making the Gaussian distribution a special caseof the SαS distributions. Our main aim of presenting Fig. 6 isto show the change of the tails of the SαS PDFs with changein the parameter α and show that the tails are fatter than thatof the Gaussian distribution for α < 2.

    It can be seen in Figs. 6 and 7 that the PDFs of the impulsenoise models (Bernoulli-Gaussian, Middleton Class A andSαS) have fat tails. For the Bernoulli-Gaussian and MiddletonClass A PDFs, the tails are controlled by the probabilities ofimpulse noise p and A, respectively; when the probability ofimpulse noise (p or A) increases, the PDFs tails get fatter. Forthe SαS PDF, the tails are controlled by the parameter α; withlow values of α (α < 2) giving PDFs with fatter tails.

    Impulse noise models with memoryThrough measurements in a practical communications chan-

    nel, Zimmermann and Dostert [31] showed that impulse noisesamples sometimes occur in bursts, hence presenting a channelwith memory. They further proposed a statistical impulse noisemodel, based on a partitioned Markov chain, that takes intoaccount the memory nature of impulse noise. Following the

    −5 −4 −3 −2 −1 0 1 2 3 4 50

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    α=0.5

    α=2, γ=1/sqrt(2) − Gaussian

    α=1.5

    α=1

    (a) SαS distributions

    2.5 3 3.5 4 4.5

    0

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    Gaussian, α=2, γ=1/sqrt(2)

    α=0.5

    α=1.5

    α=1

    (b) Tails of the distributions

    Fig. 6. SαS distributions of different values of α while β = 0, γ = 1and δ = 0. The normal distribution is also included as a SαS distribution ofα = 2, β = 0, γ = 1/

    √2 and δ = 0.

    work in [31], other authors studied impulse noise with memoryas seen in [32], [33], [19] and [34]. In [32], a two-layer two-state Markov model is used to describe bursty impulse noise.The first layer uses a two-state Markov chain to describe theoccurrence of impulses and the second layer uses anothertwo-state Markov chain to describe the behaviour of a singleimpulse. To model impulse noise with memory, Markov chainsare invariably used by most authors in the literature. Thetwo models, Markov-Middleton [19] and Markov-Gaussian[33] are modifications of the Class A and Bernoulli-Gaussianmodels, respectively, by including Markov chains. Havingdiscussed the impulse noise models without memory, thereis no need for a lengthy discussion about the impulse noisemodels with memory. This is because the impulse noisemodels with memory are founded on those models withoutmemory. In Fig. 8 we show Markov-Middleton models, whichmeans Class A model with memory. These models in Fig. 8 arean adaptation of the model shown in Fig. 3. The model in Fig.8 (a) is a “direct” adaptation of the one in Fig. 3, with all theparameters unchanged except for the introduction of memory.However, the model in Fig. 8 (b) [19] allows for all states to beconnected such that it is possible to move from one bad state

  • 1.5 2 2.5 3 3.5 4 4.5 5−0.02

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    Class A: Γ=0.1, A=0.3

    Class A: Γ=0.01, A=0.3

    Class A: Γ=0.01, A=0.1

    Class A: Γ=0.01, A=0.01

    Class A: Γ=0.1, A=0.01

    Class A: Γ=0.1, A=0.1

    (a) Tails of the PDFs of the Class A model

    2.5 3 3.5 4 4.5 5

    0

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    Bernoulli−Gaussian: Γ=0.01, A=0.01

    Bernoulli−Gaussian: Γ=0.1, p=0.3

    Bernoulli−Gaussian: Γ=0.01, p=0.3

    Bernoulli−Gaussian: Γ=0.01, p=0.1

    Bernoulli−Gaussian: Γ=0.1, p=0.1

    Bernoulli−Gaussian: Γ=0.1, p=0.01

    (b) Tails of the PDFs of the Bernoulli-Gaussian model

    Fig. 7. Tails of the PDFs of the Class model for different values of A andΓ; tails of the PDFs of the Bernoulli-Gaussian model for different values ofp and Γ.

    (state with impulse noise) to another bad state, which was notpossible with the models in Fig. 3 and Fig. 8 (a). With thismodification, in Fig. 8 (b), comes a new parameter x, whichis independent of the Class A model parameters A, Γ and σ2I .The parameter x describes the time correlation between noisesamples. The transition state in Fig. 8 (b) has no time duration,it facilitates the connection of the other states. It was shownin [19] that the PDF of their model in Fig. 8 (b) is equivalentto that of Class A model shown in (9).

    A Note on Multi-carrier and Single-carrier modulation withImpulse noise

    Many authors may correctly argue that the short fall ofthe Class A and Bernoulli-Gaussian noise models is that theydo not take into account the bursty nature of impulse noise.However, for MC modulation it does not matter whether thenoise model employed has memory or is memoryless. Thisis because in MC modulation, the transform (DFT) spreadsthe time domain impulse noise on all the subcarriers inthe frequency domain such that it becomes irrelevant howthe noise occurred (in bursts or randomly). This is wellexplained by Suraweera and Armstrong [36], who showed thatthe degradation caused by impulse noise in OFDM systems

    2

    3

    4

    0

    1

    P ′0

    1− P ′0

    2σ2IA

    σ2IA

    4σ2IA

    3σ2IA

    P ′1

    P ′4

    P ′3

    P ′2

    1− P ′0

    1− P ′0

    1− P ′0

    (a)

    P ′0

    2σ2IA

    σ2IA

    4σ2IA

    3σ2IA

    P ′1P ′4 P′3 P

    ′2

    xxx xx

    234 1 0

    Transition state

    (b)

    Fig. 8. Markov-Middleton impulse noise models with five terms: (a) isadapted from [35] and (b) is adapted from [19]

    depends only on the total noise energy within one OFDMsymbol period, not on the detailed distribution of the noiseenergy within the symbol. When it comes to SC modulation,however, it may be important to distinguish impulse noise withand without memory.

    Here we employ the two-state Class A memoryless modelin Fig. 2 (a), with the PDF of the state with impulse noise andAWGN being Gaussian. This makes the model equivalent tothe Bernoulli-Gaussian in Fig. 2 (b). In this two-state ClassA model, ignoring the effect of the background noise for amoment, we know that the average impulse noise power isσ2I = σ

    2g/Γ. The impulse noise power affecting a symbol is

    σ̄2I = σ2I/A = σ

    2g/AΓ. For discussions and analysis, we will

    be using the impulse noise power σ̄2I = σ2g/AΓ.

    Given a fixed impulse noise power σ̄2I = σ2g/AΓ, we vary

    impulse noise probability A and the impulse noise strength Γsuch that σ̄2I remains the same. This means that if we lower Aby a certain amount, we have to increase Γ by the same amountsuch that the product AΓ is unchanged. This we do in order tokeep σ̄2I the same, while observing the effect of changing the

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    ate

    128−OFDM, A=0.01, Γ=0.01SC, A=0.1, Γ=0.001 SC, A=0.01, Γ=0.01 SC, A=0.001, Γ=0.1 SC, A=0.0001, Γ=1

    Fig. 9. Comparison of MC and SC modulation in a channel with AWGNand impulse noise of variance σ̄2I = σ

    2g/AΓ = 1/(0.01× 0.01) = 104. σ̄2I

    is fixed at 104 while different values of A (10−1, 10−2, 10−3 and 10−4)are seen to influence the performance of SC modulation.

    probability of impulse noise A on the performance of Single-Carrier and Multi-Carrier Modulation. It is interesting to notethat for very low A, SC modulation performs better in thelow SNR region compared to MC modulation. However, SCmodulation gives an error floor, while MC modulation doesnot. This behaviour is seen in Fig. 9.

    Two important conclusions can be drawn from the behaviourobserved in Fig. 9:

    • For very low A, very few symbols are affected in SCmodulation, hence the low probability of error in SC nomatter the strength (or average variance) of the impulsenoise. However, with MC modulation, what matters is theaverage impulse noise variance in the system because thenoise power is spread on all subcarriers causing everysymbol to be affected by the impulse noise.

    • MC modulation has the benefit of eventually outperform-ing SC modulation as the SNR increases. This is becausewith MC modulation, the factor 1/A does not affect theSNR requirement like in SC modulation. We show thisindependence from A in MC modulation in Fig. 10.

    From the two points above, we can say that MC modulation’sperformance is independent on the probability of impulse noiseoccurrence A, while SC modulation’s performance shows astrong dependence on A. Therefore, one has to carefullychoose between MC and SC modulation depending on theprobability of impulse noise that can be tolerated in thecommunication. By this we mean that if, for example in Fig.9, A = 10−4 and communication is acceptable at probabilityof error of 10−4, then SC modulation will be the best choiceover MC modulation because it will only give an error floorjust below A, at A(M − 1)/M . Ghosh [20] also mentionedthat there are conditions where SC modulation performs betterthan MC modulation. It was also shown in [21], using theBernoulli-Gaussian noise model, that the impact of impulsenoise on the information rate of SC schemes is negligible aslong as the occurrence of an impulse noise event is sufficiently

    0 5 10 15 20 25 30 35 4010

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    A=0.001, Γ=0.1A=0.01, Γ=0.01A=0.1, Γ=0.001A=0.001, Γ=0.01A=0.01, Γ=0.1AWGN, theory

    A=0.001, Γ=0.1

    A=0.01, Γ=0.1A=0.01, Γ=0.01

    A=0.001, Γ=0.01 A=0.1, Γ=0.001

    AWGN, theory

    −10log10

    (0.01)

    −10log10

    (0.1)

    −10log10

    (0.001)

    Fig. 10. The results show that with MC modulation the SNR requirementis σ2I = 1/Γ instead of 1/AΓ, even though a symbol is affected by impulsenoise of variance σ2I/A . BPSK OFDM was used, with a DFT size of 10000.

    small (i.e. very low p in (7).)

    IV. COMBATING IMPULSE NOISE

    Several techniques for combating impulse noise have beenpresented in the literature. We shall discuss these techniquesin light of MC modulation, OFDM. These techniques fall intothe following three broad categories:

    1) Clipping and Nulling (or Blanking):With clipping or nulling, a threshold Th is used to detectimpulse noise in the received signal vector r beforedemodulation. Clipping and nulling differ in the actiontaken when impulse noise is detected in r. If a sampleof r, rk is detected to be corrupted with impulse noise,its magnitude is clipped/limited according to Th = Tclip(Clipping), or set to zero (Nulling) according to Th =Tnull. Given the received sample rk, then the resultingsample r̃k, from the clipping technique, is given by

    r̃k =

    {rk, for |rk| ≤ Tclip

    Tclipej arg(rk), for |rk| > Tclip ,

    and the resulting sample r̃k, from the nulling technique,is given by

    r̃k =

    {rk, for |rk| ≤ Tnull0, for |rk| > Tnull ,

    where usually Tclip < Tnull.Zhidkov [37] gave performance analysis and optimiza-tion of blanking (or nulling) for OFDM receivers inthe presence of impulse noise, as well as a compari-son of clipping, blanking, and combined clipping andblanking in [38]. In [39], the authors advocated for theclipping technique to combat impulse noise in digitaltelevision systems using OFDM. The clipping technique,in OFDM, is also seen in [40], where the focus is onderiving and utilising a clipping threshold that does notrequire the a priori knowledge of the PDF of impulse

  • noise. Recently, Papilaya and Vinck [41] proposed toinclude an additional action (with its threshold) to theclipping and nulling actions, which is termed replace-ment. This was done for an OFDM system. The replace-ment action uses a replacement threshold (Trep) whichfalls in between the clipping and nulling thresholds, andreplaces impulse corrupted samples with the averagemagnitude of the noiseless OFDM samples.

    2) Iterative:With the iterative technique, the idea is to estimatethe impulse noise as accurately as possible and thensubtract the noise from the received vector r. The noiseestimation can be done in the time and/or frequencydomains. For good iterative methods, the more iterationsthe better the estimate of the impulse noise. There isof course a limit to the number of iterations, abovewhich there is little or no improvement in the technique.One of the earliest works on the iterative technique tosuppress impulse noise, in OFDM, was by Häring andVinck [15]. A block diagram of a receiver performingthe iterative technique in [15] is shown in Fig. 11 (a).Another application of the iterative technique againstimpulse noise, in OFDM, is found in [42], where theiterative algorithm is applied in the frequency domainafter demodulation and channel equalization.

    3) Error correcting coding:Error correcting coding has become a necessary part ofany communications system in order to correct errorscaused by channel noise. In impulse noise environments,error correcting codes are employed to correct errorscaused by impulse noise. Most research on using errorcorrecting codes to combat impulse noise effects in MCsystems tend to lean towards convolutional coding [43][44], Turbo coding [45] [46] and low density parity-check coding [47] [48] or codes that are iterativelydecoded [17].

    The first two techniques of combating impulse noise, clip-ping and/or nulling and the iterative technique, are termed pre-processing because they process the received vector before thedemodulator processing. Error correcting coding (decoding)is not a pre-processing technique, it is implemented after thedemodulator to correct errors caused by the impulse noise.It can be used alone or together with the pre-processingtechniques.

    It has become common practice to implement a combinationof the three impulse noise combating techniques above in onesystem in order to combat impulse noise. Mengi and Vinck[49] employed an impulse noise suppression scheme whichcombined the iterative technique, and the clipping and nullingtechniques, in OFDM. In [50], the iterative and blankingtechniques are used together in OFDM. In [40], clipping anderror correcting coding are used to combat impulse noiseeffects in OFDM.

    Most impulse noise mitigation schemes have been appliedon the memoryless impulse noise models. However, we seevery few attempts, in the literature, at combating impulse noise

    with memory. For an example, in [34] they employed theMarkov-Gaussian model for impulse noise and used convo-lutional error correcting coding.

    We give examples of block diagrams of two OFDM systemsto illustrate the clipping and/or nulling and iterative techniquesin combating impulse noise (see Fig. 11). The figure showsthree important points about the clipping and/or nulling andthe iterative techniques. Firstly, in Fig. 11 (a) the iterativeprocess is performed to get a good estimate of the noise vectorn(l), which is ñ(l), where l represents the lth iteration. ñ(l) isfound from n(l) using a threshold T as follows: any samplesof n(l) that exceed the threshold T are passed on as samplesof ñ(l); any samples of n(l) not exceeding the threshold Tare made zero in the corresponding positions in ñ(l). Using agood estimate of ñ(l), a more accurate estimate of the desiredsignal vector S(l) can be obtained. It should be noted that n(l)

    is found by subtracting the estimated desired signal vector ofthe lth iteration, s(l) or S(l), from the received vector r or R.Secondly, it should be noted from both Fig. 11 (a) and Fig. 11(b) that the vector n(l) can be obtained by doing the subtractionof the desired signal vector from the received signal, either inthe time or frequency domain. Thirdly, Fig. 11 (b) shows thecombination of clipping and/or nulling and iterative technique.

    The combination of clipping and/or nulling and iterativetechnique in Fig. 11 (b) was shown in [49] to give betterperformance than the iterative technique alone in Fig. 11 (a).In the system in Fig. 11 (b), the clipping and/or nulling is usedfor the first iteration to significantly improve the estimation ofS(l) in the first iteration. This clipping and/or nulling in thefirst iteration is the reason for the better performance deliveredby the system in Fig. 11 (b).

    The impulse noise combating techniques discussed havebeen in light of MC modulation, OFDM. To combat impulsenoise in SC modulation, the same techniques can be employed.However, for SC modulation we only see the importance ofthe impulse noise combating techniques when they are usedtogether with error correcting coding. For example, clippingand/or nulling the impulse noise affected transmitted samplesin SC modulation has no effect on the BER performancewithout coding. This is obviously because the demodulator inSC modulation does not discriminate between high and lowamplitude noise when making a decision on the transmittedsymbol. The benefit of clipping and/or nulling impulse noiseaffected samples, in SC modulation is observed when perform-ing soft-decision decoding. Impulse noise in SC modulationhas the same effect as narrow-band interference (NBI) inOFDM, considering the NBI model in [51]. Therefore, thesame techniques of handling NBI in OFDM found in [51],clipping and/or nulling and error correcting coding, can beused to handle impulse noise in SC modulation. It was shownin [51] that the best technique that gives optimal performancewhen using a convolutional decoder, is to combine nulling andconvolutional soft-decision decoding. Combining clipping andconvolutional soft-decision decoding gives suboptimal perfor-mance. Impulse noise in SC modulation can be handled exactlythe same way. If the impulse noise has memory, the classical

  • IDFT

    DFTDFT

    DFT > T

    r r(l) R(l) S(l)

    S(l)

    −N (l)

    R

    n(l)ñ(l)

    −ML

    Detection

    Iterative process

    (a)

    r R(l) S(l)

    S(l)

    s(l)n(l)ñ(l)

    −ML

    DetectionDFT

    Clip and Null

    IDFT> T−

    (b)

    Fig. 11. (a) Iterative impulse noise suppression [15] and (b) Iterative impulsenoise suppression with clipping and nulling [49].

    interleaver can be employed to randomise the occurrence oferrors, hence improving the decoded BER performance.

    Focusing on the PLC channel, Vinck [52] proposed the useof a combination of permutation codes and MFSK, to combatimpulse noise, as well as other noise types in the PLC channel.This technique was applied in SC modulation, MFSK. Lateron, the idea of employing permutation codes with MFSK, in[52], to combat noise typical for a PLC channel (includingimpulse noise) was taken further in [53] and [54], wherepowerful permutation codes (called permutation trellis codes)were constructed and used not only to combat impulse noise.

    Other impulse noise combating techniques

    Another technique used to combat impulse noise in OFDMsystems is compressed (or compressive) sensing (CS). Withcompressed sensing, the idea is to reconstruct a digitizedsignal using a few of its samples. CS works well with sparsesignals. Using CS, the impulse noise in an OFDM signal canbe estimated by using pilot subcarriers with their values setto zero. We see the idea of using CS to estimate and cancelimpulse noise in OFDM in [55]. In [56] the authors proposedchannel estimation working in conjunction with compressedsensing to combat impulse noise for OFDM based power linecommunication systems. While most research on combatingimpulse noise focuses on non-bursty impulse noise, Lampe[57] proposed a CS based impulse noise mitigation techniquefor OFDM that can detect bursty impulse noise. After detectingthe impulse noise positions in the OFDM samples, impulsenoise cancellation or suppression was applied.

    Another technique for combating impulse noise in OFDM,which is similar to compressed sensing, is using the sim-ilarity between the DFT (in OFDM) and error correctingcodes (particularly Bose-Chaudhuri-Hocquengem and Reed-Solomon codes). This idea, of using the similarities betweenthe DFT and error correcting codes to combat impulse noise,dates back to the 80s where we see Wolf [58] showing thatthe DFT sequence carries redundant information which canbe used to detect and correct errors. Wolf [58] compared theDFT to BCH codes. In [59] the authors show that the OFDMmodulator is similar to a Reed-Solomon (RS) encoder andthese similarities can be used in OFDM to cancel impulsenoise effects. While the scheme in [59] could correct avery limited number of impulse errors because the limitationimposed by the amount of redundancy, an improved schemewith better correcting capabilities was proposed by Mengi andVinck [60]. The Mengi and Vinck [60] scheme also usedOFDM as a RS code, where they proposed to observe notonly the subcarriers containing the redundancy symbols butthe subcarriers containing information symbols as well.Thatway their scheme could correct more impulse errors.

    V. CONCLUSION: IMPULSE NOISE

    Our conclusion is mainly a summary of the interesting factsabout impulse noise models. We have also included Table I tosummarise some of the important features of the noise models.It should be noted that in Table I the bandwidth of the noise isnarrow or broad in reference to the bandwidth of the receiver.

    TABLE ISUMMARY OF THE FEATURES OF THE IMPULSE NOISE MODELS.

    Class A Bernoulli-Gaussian SαSNoise notBandwidth Narrow-band specified Broad-bandClosed-form doesexpression exists exists not existsPDF exhibits exhibits exhibits

    fat tails fat tails fat tailsBursty as as does notnoise Markov- Markov- modelmodelling Middleton Gaussian bursty noise

    In this chapter we have discussed some important im-pulse noise models found in the literature. The noise modelsare divided into those without memory (Middleton Class Aand Bernoulli-Gaussian) and those with memory (Markov-Middleton and Markov-Gaussian). We went further to lookat the approximation of the PDF of the Middleton Class Amodel with five terms. We also showed that the Bernoulli-Gaussian model has similarities with the Middleton ClassA, and it can be approximated with the Middleton ClassA model. We then showed Bit error rate simulation resultsof the approximation of the Middleton Class A with fiveterms. Using the Middleton Class A model with five termswe showed equivalent Markov-Middleton models. In additionto the Middleton Class A and Bernoulli-Gaussian models,we also discussed the symmetric alpha(α)-stable distribution

  • used to model impulse noise. The Symmetric alpha(α)-stabledistribution as an impulse noise model was compared withthe PDFs of the Middleton Class A and Bernoulli-Gaussianmodels. All the three models had PDFs that exhibit fat tails.We also showed that single-carrier modulation performs betterthan multi-carrier modulation under low probability of impulsenoise occurrence. With OFDM transmission, it is irrelevantwhether the noise occurred in bursts or randomly, what mattersis the total noise energy within one OFDM symbol period.Lastly, we discussed impulse noise mitigation schemes: clip-ping, nulling, iterative and error correcting coding.

    2. NARROW-BAND INTERFERENCE

    I. INTRODUCTION: NARROW-BAND INTERFERENCE

    It is widely accepted that the power line communicationschannel is characterised by three main noise types namely,coloured background, impulse and narrow-band interference(NBI). Of these three noise types, the narrow-band interferencehas not been given much attention and as a result it lacks awell defined model. We find some good efforts in describingor modelling narrow-band interference in [61]–[62]. Galdaand Rohling [61] gave a brief NBI model description, wherethe bandwidth of the interfering signal was considered tobe small compared to the OFDM subcarrier spacing, hencethe interfering signal could be modelled as a single-tone.The frequency and phase of the single-tone interfering signalwere assumed to be stochastic with a uniform distribution.The power of the interfering signal was a fixed parameter. In[63], the interfering signal was considered to be a stochasticprocess in the time domain. The authors in [64] and [65] gavedescriptions of interference models together with expressionsfor the interfering signal. With the aid of the expressions, theyexplained the conditions under which an interfering signalaffects only one subcarrier or several adjacent subcarriers.The work [61]–[65] mostly described the signal processing ofthe NBI which was sufficient for the purposes of that work.Unfortunately, not much statistical detail of the models wasgiven. The contribution of this chapter is therefore to givea detailed and well-defined narrow-band interference modelapplicable to an OFDM system, which is an extension of ourprevious NBI model in [62]. We will focus to the OFDMsystem when used in the powerline communications (PLC)channel.

    Any communications system, with a given finite bandwidth,is susceptible to frequency interference. The frequency inter-ference can originate from several sources and appear on thefrequency spectrum of the system as interfering signals. Wewill call an interfering signal, an interferer. In the powerlinecommunications, the sources of frequency interference canbe classified into two main classes: 1. interference due toelectrical devices connected in the same PLC network as thetransmitter of the desired signal, and 2. interference due toradio broadcasters. The devices connected to the PLC net-work cause interference at their switching frequency and theyusually affect PLC transmission taking place in frequencies uphundreds of kilohertz. Radio broadcasters commonly operatein the megahertz region of the frequency spectrum and willinterfere with PLC transmission occurring in the megahertzregion. In this work, we shall focus on interferers with narrowbandwidth and occurring from independent sources. This viewis somewhat generic to the two classes of sources of frequencyinterference in PLC already mentioned.

    Now, we assume that interferers originate from differentindependent sources, and we also assume that a very largenumber of interferers rarely occurs within a given frequencyband W . We therefore model the probability of a certainnumber m of interferers on the system’s spectrum as a Poisson

  • distribution

    PW (m) =(βW )me−(βW )

    m!, (12)

    where m = 0, 1, . . .∞. In general, β in (12) is a quantityindicating the average number of occurrences of certain events,defined over a specified observation bandwidth W . The quan-tities β and W here are equivalent to the one used in (2),which are η and T0, respectively. In our model, we define theaverage fraction of bandwidth occupied by NBI in a systembandwidth W as

    λ =βΩ̄

    W= β

    Ω̄

    W,

    where β is the average number of interferers, each of averagebandwidth Ω̄, and λ is equivalent to the impulsive index Adefined in (6). Then we obtain

    Pm =λme−λ

    m!, (13)

    where Pm then is the probability that there are m interfererson the frequency band W , which is equivalent to the Pm in(4). The next task is then to find the power of the interferer(s)that affects the system as NBI, and to also approximate theprobability distribution of the power of this interference (noise)in the system.

    We specify our system of interest in this article as theOFDM (Orthogonal Frequency Division Multiplexing) system,and we shall present our NBI model for this system.

    Our NBI model defines the interferers in the frequencydomain, with the assumption that they correspond to realsignals in the time domain. The effect of the interferers on thesystem is also completely described in the frequency domain.

    OFDM Signal Generation Overview:

    OFDM is a multicarrier transmission scheme, where datais carried on several subcarriers which are orthogonal to eachother to avoid mutual interference. In the OFDM system ofinterest, an IDFT (inverse discrete Fourier transform) takes inas input, data symbols Dk, from a phase-shift-keying (PSK)modulation scheme and produces a discrete sequence in thetime domain, dn. The relationship between Dk and dn isrepresented by

    dn =1√N

    N−1∑k=0

    Dkej2πnk/N , (14)

    where N is the number of subcarriers used to carry data. dn isthe complex baseband transmit signal from the output of theIDFT normalized by the factor 1√

    N.

    II. NBI POWERLet us define an arbitrary interferer x(n), of bandwidth Ω,

    as a discrete-time signal which is a sum of arbitrary single-tone signals as

    x(n) =

    Ω∑i=0

    Aiej(2πfin+φi), (15)

    where Ai, fi and φi are the corresponding amplitudes, fre-quencies and phases of the different arbitrary signals, respec-tively. At the receiver the interferer goes through the N-pointDFT and appears on the OFDM spectrum. The result of thisoperation is described by

    X(ω) =

    N−1∑n=0

    x(n)e−jωn, (16)

    where ω = r 2πN , for r = 0 . . . N − 1. X(ω) is the ampli-tude spectrum of x(n) after the DFT. We can represent thecontinuous amplitude spectrum of X(ω), in frequency f , asX(f).

    If the interferer is a single-tone then it can be described bythe ith component in (15) as

    xi(n) = Aiej(2πfin+φi) (17)

    and its amplitude spectrum is given by

    Xi(f) = Aiejφiej(N−1)(πfi−πf)

    sinN(πfi − πf)sin (πfi − πf)

    1. (18)

    From here onwards, we will refer to the power or amplitudespectrum of a time domain signal (or interferer) after window-ing and DFT, simply as power or amplitude spectrum of thesignal (interferer).

    Now, returning to the amplitude spectrum in (16) which hasa bandwidth Ω. We are interested in the effect of the powerspectral density (PSD) of the interferer x(n) on the OFDMspectrum and hence, how the interferer affects the OFDMsignal as noise in the frequency domain.

    In Fig. 12 we show an arbitrary power spectrum of theinterferer x(n) denoted by |X(f)|2; together with that of twoconsecutive subcarriers Sc and Sc+1, centred at fc and fc+1,respectively.

    f

    PSD

    . . . . . .

    Sc

    Sc+1

    |X(f)|2 |X(fc)|2

    fc fc+1

    Fig. 12. The effect of the power spectrum |X(f)|2 of an arbitrary interfererx(n), on the OFDM spectrum showing two neighbouring subcarriers Sc andSc+1. The Centre frequencies of Sc and Sc+1 are fc and fc+1, respectively.|X(fc)|2 is the power contribution of the spectrum |X(f)|2, of the interferer,on Sc.

    1The derivation of the equation can be found in the Appendix

  • In Fig. 12 we also show some power contribution of theinterferer on subcarrier Sc as |X(fc)|2. This power contribu-tion on a subcarrier(s) is the interferer’s power that affectsthe subcarrier as noise or effective noise power. The powercontribution |X(fc)|2 of the interferer on a subcarrier(s) canvary depending on the position of the interferer’s PSD onthe OFDM spectrum. We will shortly explain this powercontribution in relation to the position of a given interfereron a subcarrier.

    Focusing on one subcarrier Sc, we are interested in findingthe power contribution of the interferer x(n) on the subcarrier,in the frequency domain. The interferer contributes its powerto the subcarrier centred at fc when its power spectrum isevaluated at fc. This evaluation of the interferer’s PSD at fcresults in the power contributed being |X(fc)|2, as alreadystated.

    It is sufficient to show the power contribution of theinterferer(s) on one subcarrier, because the analysis of thepower contribution on every other subcarrier follows in thesame manner. As such, the analysis of the power contributionon one subcarrier can be applied to all subcarriers.

    It should be noted that when the bandwidth of the interferer,Ω is larger than the subcarrier spacing, the interferer may con-tribute power to several adjacent subcarriers. The contributionof the interferer’s power to one subcarrier can be analysedwithout affecting the contribution to another subcarrier. Thisis in agreement with our argument that analysing the inter-ferer’s power contribution on one subcarrier is enough to giveus an understanding of the frequency interference on everysubcarrier.

    Now, let us have an interferer of fixed average amplitudeand bandwidth Ω (0 < Ω < W ). This interferer can belocated anywhere along the OFDM frequency spectrum W , itsposition is therefore unknown (variable y). We are interestedin finding the noise power contribution of this interferer on aparticular subcarrier with a centre frequency fc. The powercontributions of the interferer at fc are the instantaneouspoints on the interferer’s PSD when evaluated at fc. Let theposition of the interferer, around the subcarrier, have a uniformprobability distribution. Since the interferer has bandwidth Ω,then the probability distribution of its power contributionson the subcarrier at fc, is P = 2Ω/W . Therefore P is adiscrete uniform probability distribution. The factor of 2 isdue to the fact that the interferer’s PSD can be on either sideof fc, in position, and still contribute power at fc. So, eachpower contribution of the interferer on a subcarrier has equalprobability P , where each power contribution correspondsto the instantaneous location/position of the interferer. Thisenables us to calculate the average power contribution of theinterferer in question on a subcarrier, which will be the sumof all the possible power contributions weighted by the theirprobabilities P . We denote this average power contribution ofa single interferer on subcarrier centred at fc by χ̄, and it is

    defined as

    χ̄ =

    ∫ Ω0

    P |Xy(fc)|2dy

    =2Ω

    W

    ∫ Ω0

    |Xy(fc)|2dy, (19)

    where y are the different frequency values (positions on thespectrum) the interferer can assume, which is a variable overthe bandwidth of the interferer; |Xy(fc)| is the amplitudespectrum of the interferer at position y evaluated at thesubcarrier centre frequency fc. The term

    ∫ Ω0|Xy(fc)|2dy in

    (19) gives the total power of the interferer, and χ̄ gives theaverage effective power, which is the power contributed by theinterferer on the subcarrier.

    To make the analysis simpler for any given number ofinterferers m, we assume they have an average bandwidthΩ = Ω̄ and the same amplitude, and hence the same averageeffective power χ̄ as calculated in (19). Therefore the powercontribution, due to the m interferers, is the sum of the averageeffective power of the individual interferers,

    σ2m = mχ̄. (20)

    We call σ2m the effective narrow-band interference (NBI)power, given m interferers. The total average effective NBIpower in the system is

    σ2 =

    ∞∑m=1

    σ2mPm

    = χ̄

    ∞∑m=1

    mPm

    = χ̄λ, (21)

    where Pm is as defined in (13) and χ̄ as defined in (19).Note: If the positions of the interferers are mostly static on

    the frequency band of interest, as might be the case with thePLC channel, the model will still apply, but with the followingchanges on the average effective power χ̄ in (19). The positionof an interferer will no longer be described by a probabilitydistribution because the interferer’s position is fixed (static),and its power that affects a subcarrier centred at fc will be oneof the |Xy(fc)|2, for y = 0 . . .Ω. We can define the averageeffective power, χ̄, as the average of all the values |Xy(fc)|2,for y = 0 . . .Ω.

    χ̄ = E{|Xy(fc)|2

    }, (22)

    where E {.} means the expectation. It should be noted thatwhether the positions of the interferers are static or changing,that has no bearing on the probability of having interferers inthe system, Pm.

    A second issue to be noted is that when the averagebandwidth of the interferers Ω̄ gets larger, there is likely goingto be more adjacent subcarriers affected by an interferer asNBI. Remembering from Section I that λ ∝ Ω̄, then as Ω̄ getslarger so does λ. This means that the probability of having

  • NBI in the system increases in proportion to an increasedΩ̄. As stated, interferers with larger bandwidth may affectadjacent subcarriers and this will likely result in a burst oferrors. Whether adjacent or random subcarriers are affected,the average number of errors that will occur in the systemdoes not change. This burst error phenomenon can easily betaken into account by using models that consider channels withmemory for example, a Gilbert-Elliot model.

    Similarities to Middleton’s Class A Noise Model:

    Just as in the Class A noise model discussed in SectionIII-A, the NBI model in the frequency domain can be seen asan infinite number of parallel channels each with effective NBIpower σ2m (m = 0 . . .∞) and additive white Gaussian noise ofvariance σ2g , where each channel is selected with probabilityPm prior to transmission. This channel model is shown in Fig.13, where a symbol Dk is affected by AWGN of variance σ2gand NBI of variance σ2m.

    Dkσ21

    Pm,m = ∞ σ2∞

    σ2g

    σ20Pm,m = 0

    Pm,m = 1

    .........

    D̃k

    Fig. 13. Narrow-band interference model including additive white Gaussiannoise (AWGN). A symbol Dk affected by AWGN of variance σ2g . Thenit enters one of infinite parallel channels with probability Pm, and in thatchannel the symbol is affected by NBI of power σ2m such that the output datasymbol D̃k is the original Dk plus noise of variance σ2g + σ

    2m.

    Now, let us make the assumption that the NBI amplitudedue to m interferers is a Gaussian random variable, and cantake any value nk, with mean µ and variance σ2m. Thenthis interference (noise) due to m interferers has a Gaussiandistribution with a PDF (probability density function) definedas P(nk|m) = N (nk;µ, σ2m).

    P(nk|m) =1√

    2πσ2mexp−(nk − µ)2

    2σ2m. (23)

    Then the probability distribution of the NBI sample nk is

    P(nk) =∞∑m=1

    P(nk|m)Pm

    =

    ∞∑m=1

    PmN (nk;µ, σ2m) (24)

    With the results of Equation (23) and Equation (24), wehave arrived at the same result of Middleton Class A noisemodel as shown in (3), where it was assumed that the noisehas a Gaussian distribution.

    III. HOW THE NBI MODEL IS APPLIEDTo demonstrate the application of our NBI model, we shall

    use an example. Firstly, we modify the infinite-parallel-channelmodel in Fig. 13 into an easy to use two-parallel-channel (two-state) model as shown in Fig. 14. The model shown in Fig.14 is similar to the models shown in Fig. 2.

    Dk D̃k

    σ2g

    σ2g +σ2

    λ

    λ

    1− λ

    Fig. 14. Narrow-band interference model including additive white Gaussiannoise (AWGN). A symbol Dk either enters a channel with AWGN (varianceσ2g ), with probability 1− λ, or enters a channel with noise σ2g + σ2/λ, withprobability λ. σ2 is the total average effective NBI power in the system asdefined in (21).

    A transmitted symbol from any modulation is affected byAWGN of variance σ2g , with probability 1 − λ. The symbolis affected by AWGN and NBI of average power σ2g + σ

    2/λ,with probability λ. That is, a symbol “chooses” one of thechannels in Fig. 14 according to the entrance probabilities λor 1 − λ. σ2 is the total average effective NBI power in thesystem as defined in (21).

    Now, let λ = 10−2 and σ2/λ = χ̄ = 10. For AWGN,σ2g = 1. Having specified the variances of the AWGN and theNBI, we now want to specify the noise samples of the AWGNand that of the NBI, which we call ng and nN , respectively.We focus on nN because ng is known, it is AWGN.

    Let us consider two distributions for the NBI sample nN ,which are the uniform distribution and Gaussian distribution.This means that the noise sample nN can be taken from eithera uniform or a Gaussian distribution.

    Case A: NBI has a uniform distribution.For a uniform distribution with limits a and b:• nN = a+ (b− a)Ru.• Ru, a function that generates a random number from a

    standard uniform distribution on the open interval (0, 1).• the variance σ2/λ = χ̄ is given by (b− a)2/12.• since we know the variance we need to specify a and b,

    let a = −b, then we have b = √χ̄√

    3 =√

    10√

    3.• nN = −

    √10√

    3 + 2√

    10√

    3Ru.

    Case B: NBI has a Gaussian distribution with mean µ = 0.• nN =

    √µ+√χ̄Rg .

    • Rg , a function that generates a random number from astandard normal distribution.

    • nN =√

    10Rg .

    The symbol is affected by noise as follows:

  • • with probability λ: D̃k = ng + nN +Dk• with probability 1− λ: D̃k = ng +Dk,

    where ng = Rg because σ2g = 1.

    If Dk is complex valued, then nN and ng have to becomplex too. For example, in [62] we generated each NBIsample nN as a complex random value using the functionRg , such that the sample was

    √σ2

    λ Rrg + j

    √σ2

    λ Rcg , where R

    rg

    and Rcg indicate that the value generated by Rg is differentfor the real and complex components of the sample nN . Thismeans that even though Rrg and R

    cg are each generated from

    a standard normal distribution, they are independent. As such,the NBI generated in [62] can be viewed as a random phasorthat can rotate in any direction, with the real and imaginarycomponents each having a magnitude determined by Rg and√

    σ2

    λ .

    Fig. 15. BPSK–256OFDM modulation bit error rate performance in thepresence of NBI with σ2/λ = 10 and λ = 10−2, and AWGN with σ2g = 1.

    In Figs. 15 and 16 we give the simulation results of CaseB, where the system is OFDM with N = 256 subcarriers, andBPSK is used as the modulation. In Figs. 15, we set λ = 10−2

    and σ2/λ = 10. The probability of error caused by NBI fora BPSK modulation will be 0.5× λ = 0.5× 10−2. The errorfloor in Fig. 15 confirms this estimate of the probability oferror. The Signal to Noise Ratio (SNR) is Eb/2σ2g in Fig. 15,which does not take into account the NBI power, hence thepersistent error floor. The role of Fig. 15 is to indicate theprobability of error caused by NBI, which is achieved by notincluding the NBI power in the SNR. We address the issue ofthe effect of the NBI power in the SNR in Fig. 16.

    Fig. 16 shows the simulation result with similar parametersto that of Fig. 15, except that now the SNR includes the NBIpower and is Eb/2(σ2g + σ

    2/λ). Also in Fig. 16, we have setλ = 10−2 and σ2/λ = 100. The SNR gap between the graphof AWGN only and that of AWGN + NBI confirms the NBIpower, σ2/λ = 100.

    IV. CONCLUSION: NARROW-BAND INTERFERENCEWe have given a narrow-band interference model which is

    applicable to the PLC channel when an OFDM system isused. In the model we gave the probability with which this

    Fig. 16. BPSK–256OFDM modulation bit error rate performance in thepresence of NBI with σ2/λ = 100 and λ = 10−2, and AWGN with σ2g = 1.

    NBI power affects data. We also showed how to calculatethe average effective power of the narrow-band interference,from a number of interferers, that affects the OFDM system.The average effective NBI power can be modelled with anappropriate distribution; in this paper we demonstrated the useof two distributions which were the uniform and Gaussiandistribution, and gave numerical results for the Gaussiandistribution case.

    CONCLUSION

    The article has presented a comprehensive study of noisemodels for impulse noise and narrow-band interference inOFDM systems. As can be seen, impulse/impulsive noisereceived more attention from researchers than narrowbandinterference. This is because researchers view impulse noiseas more destructive than narrow-band interference in OFDMsystems. It has been shown that with some assumptions, theNBI model (which is in the frequency domain) can be similarto the Middleton Class A noise model (which is in the timedomain).

    APPENDIX

    Let xi(n) denote a discrete time signal,

    xi(n) = Aiej(ωin+φi), (25)

    where ωi = 2πfi, Ai is the amplitude and φi phase of thesignal. To simplify the expression we set Ai = 1 since it isa constant that does not play any role in the finding of theFourier transform of xi(n).

    The Fourier transform of xi(n), Xi(ω) from an N -pointDFT, is

  • Xi(ω) =

    N−1∑n=0

    ej(ωin+φi)e−jωn

    = ejφiN−1∑n=0

    ejn(ωi−ω)

    = ejφi1− ejN(ωi−ω)1− ej(ωi−ω)

    = ej((N−1

    2 )ωi+φi)e−j(N−1

    2 )ωsin N2 (ωi − ω)sin 12 (ωi − ω)

    = ejφiejN−1

    2 (ωi−ω)sin N2 (ωi − ω)sin 12 (ωi − ω)

    ,

    where ω = r 2πN , for r = 0 . . . N − 1.

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