Hindawi Publishing CorporationJournal of Computer Systems, Networks, and CommunicationsVolume 2010, Article ID 802826, 9 pagesdoi:10.1155/2010/802826

Research Article

Articulating Factors Defining RMS Delay Spread inLV PLC Networks

Sabih Guzelgoz, Hasan Basri Celebi, and Huseyin Arslan

Department of Electrical Engineering, University of South Florida, 4202 East Fowler Avenue, ENB-118, Tampa, FL 33620, USA

Correspondence should be addressed to Sabih Guzelgoz, sguzelgo@mail.usf.edu

Received 26 April 2010; Accepted 10 August 2010

Academic Editor: Abdallah Shami

Copyright © 2010 Sabih Guzelgoz et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Communication over the power line network (PLN) referred as power line communication (PLC) has a long history of narrowbandapplications. With the recent developments in the field of digital communications, current interest is to exploit this medium forwideband communications for several applications such as Internet access, home networking, and in-vehicle data communication.In line with this recently emerging interest which envisions the conversion of a power transmission network into a communicationnetwork, understanding the root-mean-squared (RMS) delay spread is essential for multipath PLC channels for the establishmentof reliable communication systems. In this paper, factors that play a role on the RMS delay spread value of low voltage (LV)PLC channels are articulated. Among these factors, dependency of the RMS delay spread on attenuation, loading, and physicalcharacteristics of the communication channel in the PLNs is investigated.

1. Introduction

Communication over the power line network (PLN) referredas power line communication (PLC) is recently gainingsignificant momentum for various communication appli-cations such as Internet, data and voice transmission [1].PLC is very promising for many communication applicationsin the sense that the communication medium is based onthe use of an existing infrastructure in a very extensivenetwork that virtually reaches anywhere in the world. Designpurpose of the PLNs is the transmission of power fromone point to another, and conveying the high frequencycommunication signals is not the actual reason behind itsexistence. Therefore, PLN is considered to be a very harshcommunication medium such that PLC systems must tacklevarious challenges such as noise and multipath in order toensure reliable communication.

Future PLC-based systems are envisioned to provide veryhigh data rates requiring wideband to support high-qualitymultimedia. The popularity of wideband PLC especiallyin low voltage (LV) networks for last-inch applicationsis growing [2, 3]. In wideband communication channels,multipath induced intersymbol interference (ISI) is one of

the phenomena that leads to performance degradation. Incommunications community, significance of ISI is quantifiedby a parameter called root-mean-squared (RMS) delayspread. In a nutshell, the RMS delay spread indicates thecapability of the communication channel of supporting highdata rate communications by implying the probability ofperformance degradation which may occur due to the ISIas a result of multipath signal propagation. Understandingthe RMS delay spread in PLC channels is crucial frommany aspects. For instance, the driving technology inPLC applications requiring wideband is overwhelminglyconsidered to be orthogonal frequency division multiplexing(OFDM) due to its inherent advantages [1]. Cyclic prefix isemployed in OFDM systems in order to combat multipathrelated ISI, and RMS delay spread of the communicationchannel is considered while determining the duration of thecyclic prefix [4]. The apparent relation between RMS delayspread and the cyclic prefix in OFDM systems is one of theconsiderations that favors the idea of looking into RMS delayspread of LV PLC channels in a more detailed way. In PLCsystems, the RMS delay spread depends on the followingfactors:

2 Journal of Computer Systems, Networks, and Communications

(1) frequency-distance-dependent cable attenuation,

(2) loading condition, that is the impedances of theelectrical loads appearing at termination points,

(3) physical characteristics of the operating PLCmedium.

The aim of this study is to investigate and explain statisticallythe impact of these factors on the RMS delay spread value ofLV PLC networks.

Based on extensive measurements, frequency-distancedependent attenuation in LV PLC networks is defined as [5]

A(f ,d

) = exp((−a0 − a1 f

k)d)

, (1)

where f and d correspond to frequency of the signal andthe distance covered, respectively. a0, a1, and k are all cable-dependent parameters and are mostly extracted by empiricalmeasurements [5]. a0, a1, and k are considered to be timeinvariant, that is, fixed parameters for a PLC network. Hence,attenuation defined by (1) does not cause any time variationin the RMS delay spread for a given topology. Unlike thefirst factor listed above, loading in LV PLC networks is timevarying. Branches in the network are terminated with variouselectrical loads with different impedance characteristics. Theloading condition of the network may change throughoutthe day. Hence, RMS delay spread of the LV PLC networksmay change significantly as the impedances seen at thetermination points change even if the considered networktopology is fixed. In addition, even if the loading conditionis not altered, the dependency of the load impedances onthe Alternating Current (AC) mains cycle [6] gives rise toa cyclic change in the RMS delay spread of PLC channels.Indeed, several studies performed earlier noticed this relationbetween RMS delay spread and loading by considering somespecific topologies [7, 8]. An excerpt taken from [8] says:

“It has been shown the maximum delay spreadoccurs for cases when the channel is terminatedeither in low impedances or high impedances”.

Some other studies available in the literature implicitlyexpress this fact by calling the open circuit condition attermination points as the worst case scenario [9]. Findings ofthis study will naturally shed some light on these statementsextensively encountered in the PLC community throughthe establishment of a mathematical background while theimpact of loading is examined. In addition, PLC systemsare deployed on PLNs possessing different physical charac-teristics. Number of branching nodes between transmitterand receiver, length statistics of branches, and statisticsregarding the number of branches extending from eachbranching node are some of the characteristics which maysignificantly differ from network to network. Dependingupon the communication application and deployment con-dition, distance between transmitter and receiver is anotherparameter which may also change. These different attributesof the PLNs give rise to a change in the multipath profileof the communication channel between transmitter andreceiver leading to a change in the RMS delay spread values.

Revealing the relation between these physical characteristicsof PLNs and the RMS delay spread is another objective of thisstudy.

The remainder of the paper is organized as follows.Section 2 gives the analytical background for the multipathpropagation model in PLC channels and defines RMS delayspread. Section 3 discuses impact of attenuation and loadingon the RMS delay spread in PLC channels. Section 4provides the details on the relation between the physicalcharacteristics of PLC channels and RMS delay spread.Finally, the concluding remarks are given in Section 5.

2. PLC Multipath Channel Model andRMS Delay Spread

When a signal is transmitted on the power line conductors,the signal received at the receiver consists of attenuated,delayed, and phase-shifted replicas of the transmitted signal.If the total number of received signal replicas are consideredto be limited to N , a complete characterization of the PLCchannel can be given by its channel frequency response(CFR) as follows [5]:

H(f) =

N∑

i=0

⎡

⎣K∏

k=1

Γik

M∏

m=1

Tim

⎤

⎦A(f ,di

)exp

(− j2π f τi), (2)

where Γ and T correspond to the reflection and transmissioncoefficients along the propagation path, respectively, A( f ,di)means the frequency-distance dependent attenuation stem-ming from the physical characteristics of the cable used inthe network and given by (1), and exp(− j2π f τi) refers to thephase of the ith component due to the time delay. Finally, Kand M represent the number of reflection and transmissioncoefficients experienced by a propagating signal along aparticular path denoted by the subscript i. Multiplication ofreflection (Γ) and transmission (T) coefficients leads to aparameter which is called reflection factor in the literature.Reflection factor, denoted as |ri| exp( jθi) for a certain pathimplied by the subscript i, is usually but not necessarily acomplex number. It accounts for the losses inflicted uponthe transmit signal due to physical characteristics of thePLC environment. It is clear from (2) that the accuratecomputation of the reflection factors (reflection and trans-mission coefficients) is essential for a true characterizationof the PLC channel. In PLC systems, a transmit signalpropagating from one location to another suffers reflectionsat impedance discontinuities along its path to the receiver.Due to these impedance mismatches, some part of the signalis reflected back towards the source whereas some proceedsto the destination. Reflection coefficient Γ articulates theamplitude/phase ratio between the reflected signal and theincident signal whereas transmission coefficient T impliesthe amplitude/phase relation between the proceeding signaland the incident signal. Reflection and transmission coef-ficients at a particular impedance discontinuity for a cable

Journal of Computer Systems, Networks, and Communications 3

Incident signal

Incident signal

Delivered signal

Proceeding signalZ0

Z0ZD

Ztotal

Reflected signal

Reflected signal

Termination point

Figure 1: Reflection/transmission coefficients at branching andtermination.

with characteristic impedance Z0 are given by the followingequations [10]:

Γ = ZL − Z0

ZL + Z0, T = 1 + Γ, (3)

where ZL refers to the impedance that the signal sees ata discontinuity. Since branching and impedance appearingat the termination points are the main reason behind theimpedance discontinuity for homogeneous PLNs which areto be considered in our study, it is worth taking a closerlook at the calculation of the reflection and transmissioncoefficients at these instants as illustrated in Figure 1.

2.1. Reflection/Transmission Coefficient at Branching. At abranching node, calculation of reflection/transmission coef-ficient is based on treating each branch extending from thenode as parallel connection. If a homogeneous network witha particular cable impedance Z0 is assumed, it is easy to showthat the impedance seen by the incident signal arriving at abranching node is given by the following expression:

Ztotal = Z0

n− 1, (4)

where n is the total number of branches extending froma node including the branch on which the incident signalpropagates (Note that 3 is the minimum value that n canassume to ensure branching keeping in mind that a nodeshould have at least one branch for arriving signal and onebranch for proceeding signal.). By using (3), it can be easilyshown that reflection and transmission coefficients are givenby

Γ = Ztotal − Z0

Ztotal + Z0= 2− n

n, T = Γ + 1 = 2

n, (5)

2.2. Reflection/Transmission Coefficient at Termination Points.In a similar fashion, reflection/transmission coefficient at atermination node can be computed as follows:

Γ = ZD − Z0

ZD + Z0, T = Γ + 1 = 2ZD

ZD + Z0, (6)

where ZD denotes the impedance seen by the incident signalat the termination point. Note that the incident wave isfully reflected in case the termination point is open or shortwith the same amplitude but 180◦ of phase difference. Incase a device is connected to the termination point, thenimpedance of the corresponding device must be taken intoconsideration while computing reflection/transmission coef-ficients. Signal passes through many impedance discontinuesand experiences multiple reflections on its way to the receiverin a PLC network. Reflection factor, that was previouslymentioned while introducing the multipath characteristicsof PLC channels, represents the total effect of all these reflec-tion/transmission coefficients on the propagating signal.

If inverse fast Fourier transform (IFFT) operation isapplied on CFR given by (2), channel impulse response(CIR) is obtained as follows:

h(τ) =R∑

j=0

hjδ(t − τj

). (7)

RMS delay spread is derived from the CIR and defined as [11]

τrms =

√√√√√√

∑Rj=0 τ

2j

∣∣∣hj

∣∣∣

2

∑Rj=0

∣∣∣hj

∣∣∣

2 −⎛

⎜⎝

∑Rj=0 τj

∣∣∣hj

∣∣∣

2

∑Rj=0

∣∣∣hj

∣∣∣

2

⎞

⎟⎠

2

, (8)

where R is the number of paths considered in the calculationand usually determined by thresholding as will be clear inthe forthcoming sections. It is worth mentioning that delayof the first arriving path, which is denoted as τ0, is aligned tozero before computation.

So far, signal propagation and multipath characteristicsof PLC channels as well as the computation of the RMS delayspread are elaborated. These concepts are important in asense that they form a basis for our future discussions. Fromthis point on, our discussion will be extended to the RMSdelay spread and the factors that characterize it in PLNs.

3. Impact of Attenuation and Loading onRMS Delay Spread

Our initial objective is to understand the impact of attenu-ation and loading on the RMS delay spread. T-network isthe main PLC topology in LV PLC networks that is usuallytaken into consideration. Multipath phenomenon was firstelaborated by considering a T-network topology in [5].Similarly, signal reflections are analyzed by considering thisfundamental topology structure in [12]. Its popularity inPLC channel analysis stems from the fact that it forms a basisfor more complex LV networks in a way that these complexnetworks can be decomposed into different connections ofT-network structure [5, 12]. Stemming from the fact that

4 Journal of Computer Systems, Networks, and Communications

Reflections

Electrical load

Transmitter Receiver

C

B

A D

d3

d1 d2

Figure 2: T-network topology.

our initial objective is to investigate the impact of attenuationand loading on the RMS delay spread, physical characteristicsof PLC channels will be isolated from our analysis andpopularly used T-network topology, being a more controlledenvironment, will be utilized in line with the previousstudies.

T-network topology is depicted in Figure 2. It is com-posed of three branches connected to node B with the lengthof d1, d2, and d3. A denotes the point where the signalis injected into the network, and D is the point wherethe signal is received. Consideration of the homogeneousnetwork structure in which all the branches have the samecharacteristic impedance Z0 is one of the assumptions madefor the ease of analysis. In addition, A and D are assumed tobe matched to Z0 for the sake of simplicity, hence B and C arethe only sources of reflection in the topology.

Reflection and transmission coefficients at node B (Γb,Tb) and C (Γc, Tc) for this particular network topology aregiven by (5) and (6) for n = 3. Note that ZD in (6) refers toimpedance of the electrical load connected to node C.

Following the discussion on PLC channel and RMSdelay spread given in Section 2, it is now convenient toarticulate why the first two factors (attenuation and loading)that are listed in Section 1 play a role in the RMS delayspread of PLC channels. Recall that the RMS delay spreadof a communication channel is computed by aligning thefirst arriving path to zero delay. Upon this alignment, thenumber of paths to be included in the RMS delay spreadcomputation, which is R as can be seen in (8), is determinedby applying a threshold considering the maximum powervalue in the delay profile. With this threshold so determined,the paths with the power values below are considered to benoise and excluded from the analysis. With the explanationgiven, the impact of attenuation on RMS delay spread canbe explained by an example. Considering the T-networktopology given in Figure 2, assume that only two paths satisfythe threshold condition. So, these paths should naturallyfollow the following paths: A-B-D (at τ0) and A-B-C-B-D (atτ1). Assume that d3 is slowly increased. Note that this changein the topology does not lead to any change in either τ0 orin the power of the first arriving path since d1 + d2 staysthe same. However, this affects the second arriving path ina way that it arrives at a later delay with a smaller value ofpower due to the attenuation which increases with distance.The attenuation may reach at a point (as d3 → ∞) atwhich the threshold condition is not satisfied even for thesecond arriving path leading us to consider it as a part ofbackground noise in the computation of RMS delay spread.

Disappearance of a particular path from delay profile (orarriving at a larger delay) leads to a change in the RMS delayspread value as can be clearly seen from (8).

The second factor given in Section 1, which is the loading,determines Γ’s and T ’s in (2). Therefore, any change in theloading condition leads to a change in Γ’s and T ’s in thenetwork and results in a change in the RMS delay spread.Recall also that even if the loading condition is not altered,the dependency of the load impedances on the AC mainscycle [6] gives rise to a cyclic change in the RMS delay spreadof PLC channels. If the termA( f ,d) in (2) is ignored in orderto isolate ourselves from the effect of attenuation and solelyfocus on the impact of loading, (2) reduces to the followingform:

H(f) =

N∑

i=0

⎡

⎣K∏

k=1

Γik

M∏

m=1

Tim

⎤

⎦ exp(− j2π f τi

). (9)

If IFFT operation is applied to the CFR, CIR is obtained asfollows:

h(τ) =N∑

i=0

⎡

⎣K∏

k=1

Γik

M∏

m=1

Tim

⎤

⎦δ(t − τi). (10)

Note that the reflection factor of the direct path (A-B-D)is composed of only one term which is the transmissioncoefficient at B, namely, Tb. The reflection factors of otherpaths consist of Γc, Γb, and Tb. Based upon this observation,the CIR of T-network topology assumes the following form:

h(τ) = Tbδ(τ − τ0) +R∑

j=1

Tb

{2n

} j{2− nn

} j−1

(Γc)jδ(τ − τj

),

(11)

where R refers to the number of reflections consideredbetween B and C.

The graphical illustration of the amplitude of (11) isgiven in Figure 3. As can be seen, amplitude of the signalcomponents decreases with increasing delay. This is dueto the fact that power of the signal reduces as it passesthrough more reflection points on its way to the receivereven though the attenuation effect is not accounted for.Taking the attenuation into consideration, amplitude of eachcomponent should be even smaller. Therefore, the RMS delayspread values obtained from Figure 3 are pessimistic since theeffect of attenuation especially for late arriving paths is notaccounted for. However, this will not affect our conclusionssince our focus is on the impact of loading rather thanattenuation.

For the purpose of mathematical tractability, if two of thedelays (τ0 and τ1) along the delay axis are considered (R = 1)in Figure 3, using (8) and (11) the RMS delay spread τrms

takes the following form:

τrms = τ12n|Γc|

n2 + 4(Γc)2 = τ1

6|Γc|9 + 4(Γc)

2 . (12)

Journal of Computer Systems, Networks, and Communications 5

|h(τ)|

h0 = Tb

h1 = |Tb2nΓc|

· · ·

ττ1τ0 τ2 τR

hm = |Tb

(2n

)R(2−nn

)R−1

(Γc)R|

Set to 0 while calculatingRMS delay spread

Figure 3: Graphical representation of the CIR for T-networktopology.

Note that (12) depends on two parameters which are τ1 andthe reflection coefficient Γc. When τ0 is set to 0, time delay τ1

for the second arriving path is expressed as

τ1 = 2d3

v= 2d3

√εr

c0, (13)

where εr is the dielectric constant of the insulation materialand c0 is the speed of light in vacuum.

Referring back to (3), Γc being the reflection coefficientat node C depends on the impedance of the electrical deviceZD connected to the network which is regarded as a randomvariable (RV) in this study for generalization so that variousloading conditions can be taken into account. If C is assumedto be left open (ZD = ∞), Γc becomes 1. Γc becomes −1 ifa short circuit assumption (ZD = 0) is considered at nodeC. These two scenarios correspond to two extreme cases.Therefore, it is expected that any electrical device connectedto node C yields a reflection coefficient between the valuesgenerated by these extreme cases,−1 and 1. These two valuesare also the maximum and minimum values of Γc duringan AC cycle duration even if the electrical load connected tonode C is unchanged.

In order to understand the impact of loading on the RMSdelay spread τrms, we have to derive the probability densityfunction (PDF) of the variable

ψ = 6|Γc|9 + 4(Γc)

2 . (14)

Before proceeding with the statistical characterization of ψ,some important observations should be made regarding itsbehavior.

Lemma 1. ψ has the following characteristics:

(i) monotonically increasing for 0 < Γc ≤ 1,

(ii) monotonically decreasing for −1 ≤ Γc < 0,

(iii) has a critical point at Γc = 0.

Proof. ψ takes the following form within this interval 0 <Γc ≤ 1:

ψ = 6Γc9 + 4(Γc)

2 . (15)

The derivative of ψ with respect to Γc is given by

dψ

dΓc= 54− 24(Γc)

2

(9 + 4(Γc)

2)2 , (16)

Since dψ/dΓc > 0 for 0 < Γc ≤ 1, ψ is a monotonicallyincreasing function within this interval.

Similarly, the derivative of ψ with respect to Γc within theinterval −1 ≤ Γc < 0 is given by

dψ

dΓc= −54 + 24(Γc)

2

(9 + 4(Γc)

2)2 . (17)

Since dψ/dΓc < 0 for −1 ≤ Γc < 0, ψ is a monotonicallydecreasing function within this interval.

Equating (16) and (17) shows that ψ has a critical pointat Γc = 0, where the function may have a minimum ormaximum. However, it is easily seen from the first andsecond claims that Γc = 0 minimizes ψ. It is also worthmentioning that |Γc| = 1 maximizes ψ considering itsbehavior on the intervals analyzed.

An alternative insight into the lemma can be givenin this way. Γc = 0 corresponds to the case in whichthe impedance of the electrical device connected at nodeC matches the impedance of the cable Z0. This caseimplies that no reflection occurs at C leaving only onepropagation path between A and D. Disappearance of theother signal components along the delay axis in Figure 3gives rise to minimum value of the RMS delay spread,hence ψ. Other values of Γc /= 0 lead to the reception ofseveral signal components from other propagation pathsalong with the direct path. This phenomenon leads to anincrease in the value of the RMS delay spread, hence ψ.First and the second claims are details of this observation.Claims derived from our analytical analysis also shed somelight on the statements that are encountered frequently inthe literature as mentioned in Section 1. As empiricallyconcluded in the previous studies and mathematically shownhere, open/short circuit condition at the termination pointleads to the maximum value of RMS delay spread whereasits minimum value is obtained if the termination pointsare matched to the network cable impedance. This alsovalidates the reason why open/short circuit condition attermination points of LV PLC channels is considered as theworst case scenario from the perspective of multipath signalpropagation.

Referring back to the statistics of ψ, several PDFs canbe offered in order to characterize the statistical behaviorof Γc. Extracting the statistics of device impedances that arewidely used in LV networks and building a statistical modelwould be very desirable. Since no such study is available inthe literature, Γc is assumed to be uniformly distributed over[−1,1]. By using the fundamental theorem for functions ofone random variable [13] and employing change of variables

6 Journal of Computer Systems, Networks, and Communications

0 0.1 0.2 0.3 0.4 0.50.005

0.01

0.015

0.02

0.025

0.03

f ψ(ψ

)

SimulationTheory

ψ = τrms/τ1

Figure 4: PDF of the RMS delay spread of T-network topologywhen node C is randomly loaded.

Y = |Γc|, the PDF of (14) can be expressed as follows:

fψ(ψ) =

(36ψ2 +

(3− 3

√1− 4ψ2

)2)2

24ψ2

(36ψ2 −

(3− 3

√1− 4ψ2

)2) , 0 < ψ <

613.

(18)

Integrating (18) leads to the cumulative distribution func-tion (CDF) of ψ. CDF can be calculated by changing the

variable cos(θ) =√

1− 4ψ2. After a series of cumbersomemathematical operations that are skipped here for the sakeof brevity, CDF is computed as

Fψ(ψ) = 3

2tan

(arc sin

(2ψ)

2

)

, 0 < ψ <6

13. (19)

Figures 4 and 5 show the PDF and CDF of ψ. It is clearly seenthat analytical derivations and simulation results are in goodagreement. Curves are also seen to be in good agreement withthe lemma provided. For instance, plugging |Γc| = 1 into(14) for the purpose of maximization yields 6/13. Due to themaximization, ψ must never exceed 6/13 and it is seen fromFigure 5 that this observation is satisfied.

Impact of attenuation and loading on the RMS delayspread is detailed in this section. Subsequently, impact of thephysical characteristics of the PLC operating environmentwill be our focus.

4. Impact of the Physical Characteristics ofthe PLC Channel on RMS Delay Spread

PLC systems may be deployed on LV PLC environmentshaving entirely different physical characteristics. Amongthese characteristics, impact of the following items on theRMS delay spread will be addressed in this section:

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

SimulationTheory

Pro

b[ψ≤a

bsci

ssa]

ψ = τrms/τ1

Figure 5: CDF of the RMS delay spread of T-network topologywhen node C is randomly loaded.

(i) number of branching nodes between transmitter andreceiver,

(ii) distance between transmitter and receiver,

(iii) length statistics of branches.

It is obvious that the analysis that we plan to perform inthis section requires the establishment of more complicatedPLC networks than the T-network topology utilized inSection 3. Modeling PLC systems and building simulationtechniques for them have been the focus of several studiesearlier in the literature. The model which considers the PLCchannel as a multipath communication environment wasfirst introduced in [5] as mentioned earlier. Based uponthis multipath consideration, analytical calculation of themultipath components by describing the PLC channel via aset of matrices is proposed in [14, 15]. PLC models that arebased on treating the transmission line as a two-port deviceare available in the literature as well [16, 17]. A channelmodel and a simulation platform along with the results ofvarious channel measurement campaigns are discussed in[18, 19]. A statistical PLC channel characterization regardingattenuation, multipath related parameters, and so forth, ispresented in [20, 21]. In our analysis, the matrix-based PLCsimulation technique proposed in [14] will be consideredas the basis. However, matrices introduced in [14] are tobe modified in a way that the simulation module letsus easily generate PLC networks with different physicalcharacteristics. In line with the procedure described in [14],generated PLC network topology, which is illustrated inFigure 6, is mapped into a single matrix as follows:

C[(t+b)×(t+b)] =[

0[t×t] CT[t×b]

CTT[b×t] CC[b×b]

]

, (20)

where t and b correspond to the number of terminationpoints and branching points (referred to as internal nodes

Journal of Computer Systems, Networks, and Communications 7

T1T1

Tx

Tn1

Tnb

lC1Tx

lC1T1

lC1C2

lCbTnblCbT1

lCb−1Cb ReceiverTransmitter

· · ·· · ·

· · ·Rx

C1

lC1Tn1

lCbRxCb

Figure 6: Graphical illustration of the PLC network topology considered in the study.

5 6 7 8 9 10 11 12 13

×10−7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

τrms (seconds)

b = 4b = 5

b = 6

Pro

b[τ

rms≤a

bsci

ssa]

Figure 7: Dependency of RMS delay spread (τrms) on the numberof nodes (b) between transmitter and receiver. Separation distancebetween transmitter and receiver is 150 m. Branch lengths areassumed to be uniformly distributed over [10 m–30 m].

in [14]), respectively. CC is the submatrix which describesthe interconnections among branching nodes. CT showsthe connections between branching nodes and terminationpoints. The corresponding length of each interconnectionand impedances at termination points are kept in separatematrices. In order to isolate our analysis from the impactof impedance variation that was discussed in Section 3and focus solely on the impact of physical characteristicsof the environment, it is assumed that the terminationpoints are open circuit. In addition, number of branchesextending from each branching node is considered to beuniformly distributed over [3,6] in the simulations. Similarto the analysis performed in Section 3, transmitter andreceiver are also assumed to be matched to the characteristicimpedance of the homogeneous PLC network. Impact ofphysical attributes is statistically investigated by generating20000 realizations of PLC network for each case takeninto consideration. PLC topologies with different physicalattributes are generated by manipulating the values of t,b, and the length matrix whose elements are composedof the values li j shown in Figure 6. Note that a change

4 5 6 7 8 9 10 11 12×10−7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Tx − Rx distance = 100 mTx − Rx distance = 150 m

Tx − Rx distance = 200 m

τrms(seconds)

Pro

b[τ

rms≤a

bsci

ssa]

Figure 8: Dependency of RMS delay spread (τrms) on the separationdistance (d) between transmitter and receiver. Number of nodesbetween transmitter and receiver is 4. Branch lengths are assumedto be uniformly distributed over [10 m–30 m].

in the topology gives rise to a change in the values of tand b which results in a change in the dimensions of thesubmatrices denoted asCC andCT . For each realization, CIRwas calculated by taking the IFFT of CFR given by (2). AfterCIR is obtained, procedure described in Section 2 is followedwith the threshold value of 20 dB while computing the RMSdelay spread.

The impact of number of nodes between transmitterand receiver on the RMS delay spread can be seen inFigure 7. While deriving this figure, transmitter-receiverseparation distance and length statistics of the branchesare considered to be 150 m and U[10 m–30 m] (U refersto uniform distribution), respectively. Upon the analysisperformed, it is concluded that an increase in the number ofnodes while keeping all the other effective physical attributesof the PLC network the same gives rise to an increase inits RMS delay spread value. This behavior can be relatedto the multipath components arriving at larger delays asb is increased. This relation was previously noticed in [7]by considering some specific PLC network topologies. Ourfindings verify the results of this earlier study by taking

8 Journal of Computer Systems, Networks, and Communications

0.5 1 1.5 2 2.5×10−6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τrms(seconds)

Pro

b[τ

rms≤a

bsci

ssa]

lCiTj = U[10 m–50 m]lCiTj = U[10m–30m]

lCiTj = U[10m–70m]

Figure 9: Dependency of RMS delay spread (τrms) on the lengthstatistics of branches. Number of nodes between transmitter andreceiver is 4. Separation distance between transmitter and receiveris 150 m.

more general PLC network scenarios into account. Figure 8shows the impact of transmitter-receiver separation distanceon τrms. Similar to the previous case analyzed, increasingseparation distance between transmitter and receiver leadsto the reception of multipath components at larger delaysleading to an increase in τrms values. Finally, the impact ofbranch length statistics is seen in Figure 9. Among all ofthese aforementioned factors, change in the length statisticsof branches seems to yield the most drastic change in τrms

values. This is expected since expanding the range of valuesthat branch lengths may assume increases the probabilitythat some multipath components arrive at larger delays. Ifthe effect of attenuation is not significant within the lengthrange considered, it should naturally result in higher τrms

values as can be seen in the figure.

5. Conclusion

Trend in PLC indicates that the future applications willneed more bandwidth and higher data rates to addressuser requirements. This growing interest into PLC entailsa good understanding of its channel characteristics. RMSdelay spread which is frequently used in communicationscommunity while quantifying multipath characteristics ofcommunication channels was the focus of this study. Factors,which have a significant impact on the RMS delay spread,peculiar to PLC channels were elaborated in a detailed way.Among these factors, impact of attenuation and loadingwas discussed by exploiting T-network being the mostfundamental LV PLC topology that is frequently used inthe literature. The relation between attenuation and loadingto the RMS delay spread was explicitly demonstrated andcompared with the results already present in the literature.

Next, impact of the physical characteristics of PLC channelson the RMS delay spread was studied. Number of nodesbetween transmitter and receiver, transmitter-receiver sep-aration distance, and length statistics of branches are iden-tified as the physical characteristics which may significantlychange from one PLC network to another. A matrix-basedsimulation technique was employed while generating PLCnetwork topologies with distinct physical characteristics. Foreach attribute examined, statistics regarding the RMS delayspread were presented by observing corresponding CDFcurves. Based upon the results of extensive simulations,important conclusions regarding the relation between theseattributes and the RMS delay spread were drawn.

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