Coherence Time and Sparsity of Brazilian Outdoor

PLC Channels: A Preliminary Analysis

Antonio Angelo Missiaggia Picorone

UFJF, CEMIG and CES/JF

Juiz de Fora, Minas Gerais, Brazil

Email: picorone@ieee.org

Raimundo Sampaio Neto

Pontifıcia Universidade Catolica

Rio de Janeiro, Rio de Janeiro, Brazil

Email: raimundo@cetuc.puc-rj.br

Moises Vidal Ribeiro

UFJF and Smarti9

Juiz de Fora, Minas Gerais, Brazil

Email: mribeiro@ieee.org

Abstract—This work aims at offering an initial analysis ofcoherence time and sparsity of power line communication (PLC)channels, which were measured in the outdoor and low-voltageelectric distribution networks in Brazil. In this regard, a proce-dure for estimating coherence time and sparsity of PLC channelsis described. In the sequel, estimates of coherence time andsparsity of several measures of Brazilian PLC channels arereported. The attained results confirm that only the coefficientsof channel impulse responses with the largest amplitudes arerelevant to estimate the coherence time. Moreover, they alsoindicate that the sparsity of PLC channel is a random variablethat deserves more investigation and analysis.

Keywords—powerline communication, channel estimation,compressive sensing, digital communication.

I. INTRODUCTION

The eletric power distribution networks have lately been

seen as a promising option for being a data communication

medium, especially with the possibility of using it to meet

the demands of smart grid data communication [1]. The

technology that explores this opportunity is called Power

line Communication (PLC). It is well-known that such net-

works does not provide point-to-point connection between the

transmitter and the receiver, but a bus consisting of several

derivations, with loads that introduce changes in the system

transfer function. Usually, the loads connected in this bus

have a dynamic behavior, some being randomly connected

and disconnected to and from the bus, others synchronously

switching in or off with the main frequency. This dynamic

results in time-varying impedances at the access points besides

impedances mismatches in the branching points. Thus, the

signal does not propagate only directly from the transmitter to

the receiver, since signal reflections arise at points where there

are impedance mismatches, featuring multiple paths spreads

[2], [3].

Usually, the received signal has a longer duration than the

transmitted signal due to different delays in the signal paths.

This phenomenon is known as dispersion time [4]. Moreover,

the received signal can have a bandwidth greater than the

bandwidth of the transmitted signal, due to different values of

frequency shifts originated from Doppler effect of multipath

components. This second phenomenon is known as frequency

dispersion [4]. For all this, the modeling of the electric power

distribution network as a data communication medium can be

classified as time and frequency selective (doubly selective)

[5].

To design efficient PLC systems or to know, a priori,

which PLC system design will perform better on a specific

electric power distribution network, it is important to know the

variability of parameters that model these grids as data commu-

nication medium. One of the parameters used to describe this

variability is the so called coherence time. The knowledge of

the coherence time of PLC channels is important to the design

of efficient PLC systems, since it can prevent unnecessary

estimates performed within the coherence time and, therefore,

maximize the PLC system throughput. The typical coherence

time in indoor electric power grids are reported as not less than

600µs [6], although little is said about outdoor electric power

networks. Additionally, the sparsity of a channel is becoming

relevant because compressing sensing techniques can be used

to characterize PLC channels and to estimate their lengths of

cyclic prefix. Despite of their importance, sparse representation

and coherence time of outdoor and low-voltage PLC channels

have not been addressed in the literature.

In this regards, this contribution aims at offering an initial

analysis of coherence time and sparsity of PLC channels,

which were measured in the outdoor and low-voltage electric

distribution networks in Juiz de Fora city, Brazil [7], [8].

To do so, procedures to characterize the sparsity and the

coherence time of PLC channels are described and evaluated.

The reported results confirm that only the coefficients with the

largest amplitudes are relevant to estimate the coherence time.

Furthermore, they suggest that a straight line is the curve that

best represents the evolution of the correlation between the

coefficients of the channel impulse response (CIR).

This paper is organized as follows: Section II discusses a

brief review of the coherence time to analyze PLC channels.

Section III presents the procedure proposed to estimate the

coherence time of PLC channel. It also suggests a strategy to

represent measured outdoor PLC channel as sparse. The results

obtained in the analysis of the coherence time are shown in

Section IV. Conclusions about this work and some proposals

for future investigations are presented in Section V.

II. COHERENCE TIME OF PLC CHANNEL

Both time and frequency dispersions of a channel cause vari-

ations in the received signal in the time domain. The coherence

time Tc is defined as the time separation for which the samples

of the same channel can be considered uncorrelated in the time

domain. In other words, Tc is a measure of the time interval

for which the channel can be considered nearly invariant in

time domain [9].

Due to signal reflections, transmitted signals often appear on

the receiver with different amplitudes and phases. Considering

that there are effectively L echoes of the transmitted signal

arriving at the receiver, the CIR at instant t due to a unit

impulse δ(·) applied at the instant τ can be expressed as:

h(t, τ) =

L∑

l=1

αl(t)δ(t− τ − ξl). (1)

2014 18th IEEE International Symposium on Power Line Communications and Its Applications

978-1-4799-4980-9/14/$31.00 ©2014 IEEE 1

If p(t) is the input to this channel its output is given by

po(t) =

∫ +∞

−∞

h(t, τ)p(τ)dτ =

L∑

l=1

αl(t)p(t− ξl). (2)

Therefore, ξl in (1) denotes individual time delay and αl(t)is a zero mean complex gain that incorporates the attenuation

factor and phase shift of the lth echo. In a time invariant linear

channel the complex gains do not change with time. Thus, the

coherence time of the channel modeled by (1) is related to the

coherence time of the gains {αl(t)}.

When the complex processes αl(t), l = 1, 2, . . . , L, are

uncorrelated and wide-sense stationary, that is:

E[αl(t1)α∗

k(t2)] = 0, l 6= k (3)

and

E[αl(t)α∗

l (t+∆)] = Rαl(∆t), l = 1, 2, . . . , L, (4)

then, the channel characterized by (1) is known as a WSSUS

(wide-sense stationary uncorrelated scattering) channel.

The correlation index between samples of αl(t) taken ∆ttime units apart is given by

ραl=

Rαl(∆t)

Pl, l = 1, 2, . . . , L, (5)

where Pl = Rαl= E[|α2

l (t)|] is the average power of the lthpath.

The correlation index of a WSSUS channel described by

(1) and (4) results as the weighted average of the individual

correlation index ραl(∆t):

ρh(∆t) =

∑Ll=1 Plραl

(∆t)∑L

l=1 Pl

; 0 ≤ |ρh(∆t)| ≤ 1. (6)

Hence the coherence time, Tc, of the channel can be

obtained via

|ρh(∆t)| ≥ β, (7)

where 0 < β < 1 refers to the minimum correlation index

admitted to characterize the channel as time-invariant during

the time interval Tc. From that point, the coherence time for

the correlation index β is denoted by T βc .

Section III presents the treatment given to the set of mea-

sured PLC channels obtained by sounding approach based on

orthogonal frequency division multiplexing (OFDM) technique

[10].

III. ADJUSTMENTS IN THE MEASURED CIR

Consider that one new CIR measurement is obtained every

(N+Lcp)Ts seconds, where N represents the number of sam-

ples taken in each one of the M measurements, Lcp denotes the

length of the cyclic prefix, Ts = 1/2B, the sampling period,

and B, the frequency band considered. Suppose also that the

channel is linear and invariant during each CIR measurement.

Thus, the nth normalized CIR is given by

h[n] =1

||h[n]||2[h1[n] h2[n] h3[n] . . . hN [n]]T ,

n = 1, 2, . . . ,M, (8)

where [·]T denotes matrix transpose ||·||p is the ℓp-norm, given

by

||s||p = p

√

√

√

√

N∑

i=1

|si|p. (9)

One can get an approximation h[n] of h[n], by keeping only

the more significant coefficients. Here we kept the Nt,n < Nfirst coefficients of h[n], so that

Nt,n = min

{

Nt :

Nt∑

l=1

|hl[n]|2 ≥ Kt

}

, (10)

where 0 < Kt ≤ 1. As it can be seen, Kt truncates the

measured CIR based on cumulative energy of their coefficients.

When considering the M channel measurements, the average

value of Nt,n can be obtained by

Nt =1

M

M∑

n=1

Nt,n. (11)

The average energy accumulated up to the lth coefficient of

the vector h[n] is given by

El =1

M

M∑

n=1

l∑

k=1

|hk[n]|2. (12)

Even with the CIR length limited to only Nt,n samples,

it can be considered in many cases that the CIR energy is

concentrated in only a few coefficients. Hence, the original

measured PLC channel has been represented as a sparse

channel. The sparse PLC channel is characterized by having

most of its coefficients equal to zero. The sparse level of a

vector v = [v1 v2 . . . vi]T can be defined as [11]:

S = ♯{i : vi 6= 0}, (13)

in which ♯{·} denotes the set cardinality. To achieve a sparse

representation of h[n] one must consider only the represen-

tative coefficients of each PLC channel realization. This can

be accomplished by zeroing the coefficients whose amplitudes

are smaller than a certain value. Therefore, a new vector is

obtained and given by:

h[n] = [h1[n] h2[n] h3[n] . . . hNt,n[n]]T , (14)

where,

hl[n] =

{

hl[n], if |hl[n]| ≥ Ks maxl{|hl[n]|}0, otherwise

,(15)

and 0 < Ks ≤ 1. As a result, different levels of sparsity may

result from distinct measured PLC channel, while maintain-

ing constant Ks. These sparse representations obtained from

PLC channels can be quite interesting to be used to deploy

compressive sensing techniques [12].

However, depending on the characteristics of the measured

channels and the choices of parameters Kt and Ks, the sparsity

variations may pose a problem for the estimation of the

coherence time, since distinct set of coefficients with rele-

vant amplitudes may occur in different measures. Moreover,

coefficients with small amplitudes are not relevant to estimate

the coherence time of the PLC channel, since they carry little

energy. These facts may cause a distortion in the estimates of

the coherence time. To overcome this effect, only the more

relevant coefficients are considered to evaluate T βc . In this

regards, let El be the mean energy of hl[n]

El =1

M

M∑

n=1

|hl[n]|2. (16)

We then order the coefficients of the vector h[n] in (14) in

descending order, such that, E(1) ≥ E(2) ≥ E(3) ≥ . . . ≥

2

E(Nt,n) and neglect the coefficients with small mean energy.

The vector containing only the most relevant coefficients for

calculating T βc is given by:

hc[n] = [h(1)[n] h(2)[n] h(3)[n] . . . h(Nc)[n]]T , (17)

where

Nc = max{

l : E(l) ≥ KcE(1)

}

. (18)

IV. RESULTS

The variability analysis of the measured PLC channel was

performed on the frequency band from 1.7 up to 100 MHz,

adopting Fs = 1/Ts = 200 MHz.

Table I presents the basic parameters of the PLC channel

estimation algorithm adopted in the measurement campaign

[7], [8], [10].

TABLE IPARAMETERS FOR THE ACQUISITION OF h[n].

Description Variable Value

Number of CIR obtained M 756

Samples for each measurement N 4098

Samples for cyclic prefix Lcp 512

Sampling frequency Fs 200 MHz

PLC channel bandwidth Bw 100 MHz

Table II summarizes the values adopted in the various

parameters to adjust the CIR measures.

TABLE IIVALUES OF PARAMETERS ADOPTED TO PROCESS THE CIR MEASURES

Description Variable Value

Measured CIR energy % (truncated CIR) Kt 0.9

Coefficients amplitude % (sparse CIR) Ks 0.05

Selecting the most relevant factors Kc 0.1

Minimum coherence index β 0.9

Figure 1 depicts, at the top, a PLC channel measure, and

at the bottom, a zoom of the first 2.5µs of its CIR. It can be

0 2 4 6 8 10 12 14 16 18 20−40

−20

0

20

40

0 0.5 1 1.5 2−40

−20

0

20

40

t/µs

t/µs

h(t)

mV

h(t)

mV

Fig. 1. (a) Measured CIR and (b) Beginning of the measured CIR.

observed from this figure that the most significant coefficients

are concentrated at the beginning of CIR which, in this case,

corresponds to a time interval of about 2.5µs. Furthermore, the

figure also suggests that in this interval, the small amplitudes of

the measured CIR coefficients may be considered background

noise originated from the electric power distribution network

and the data acquisition system, since their values are much

smaller than those found in the interval corresponding 2.5µs.

Figure 2 highlights the percentage of stored average energy

in each N -length CIR coefficient of M channel measures. This

0 100 200 300 400 500 600 700 800 900 10000.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

←− (251, 0.9)

N

EN

Fig. 2. Stored average energy in each CIR coefficient.

result suggests that, on the average, 90% of the measured

CIR energy of the PLC channel is concentrated in the first

251 coefficients. Thus, taking into account, Nt = 251 and

Kt = 0.9, equation (10) yields a first approximation to h[n],i.e., the truncated version h[n] containing only the Nt,n initial

coefficients of any measured PLC channel.

Figure 3 illustrates consecutive measurements of the h[n]amplitudes. Note from this figure that the majority of the

100 200 300 400 500 600 700

50

100

150

200

250

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

h[n] (mV)

n

coef

fici

ent

index

ofh[n

]

Fig. 3. Consecutive measurements of the amplitudes of h[n] (Nt coeffi-cients).

coefficients of h[n] have amplitudes close to zero. These fact

distort the estimates of the coherence time of the PLC channel.

To deal with this situation, h[n] was approximated by a sparse

version, adopting Ks = 0.05 in (15). Figure 4 illustrates the

consecutive measurements of the h[n] amplitudes when the

value Ks = 0.05 is adopted.

Figure 5 illustrates sparsity variation of the sparse version

measured of PLC channel when (13) is applied. It can be seen

from this figure that the sparsity of measured PLC channels is

not constant. In fact, for thus campaign, the sparsity assumes

values between 20 and 28, with mean µS ≈ 24 and variance

σ2S ≈ 3.

Figure 6 portrays the correlation evolution in each h[n]coefficient considering the measured PLC channel as WSSUS.

3

100 200 300 400 500 600 700

50

100

150

200

250

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

h[n] (mV)

n

coef

fici

ent

index

ofh[n

]

Fig. 4. Consecutive measurements of the amplitudes of h[n] for Ks = 0.05.

0 100 200 300 400 500 600 70019

20

21

22

23

24

25

26

27

28

29

n

S-s

par

se

Fig. 5. Sparsity of the measured PLC channel (h[n]).

The low correlation coefficients in this one can be justified by

100 200 300 400 500 600 700

5

10

15

20

25

30

35

40

45

50

55

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ρh[∆n]

n

coef

fici

ent

index

ofh[n

]

Fig. 6. Evolution of the correlation in each coefficient h[n].

the low values of amplitude for some of coefficients of h[n].Consequently, in order to estimate the coherence time of the

PLC channel, more relevant coefficients were selected in h[n]from (17). Figure 7 shows the evolution of the 23 correlation

coefficients of h[n], when it is adopts Kc = 0.1. This figure

suggests that the selected coefficients have a similar correlation

100 200 300 400 500 600 700

2

4

6

8

10

12

14

16

18

20

22

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ρhc[∆n]

n

coef

fici

ent

index

ofh

c[n

]

Fig. 7. Evolution of the 23 largest correlation coefficients h[n] (Kc = 0.1)

when considering all M measured CIR.

Finally, to obtain a coherence time estimate of the measured

PLC channel, the value β = 0.9 was considered in (7). Figure 8

shows the correlation curves overlap of the that 23 coefficients

of hc[n]. In this case, the curve that represents the correla-

0 100 200 300 400 500 600 700 8000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

← (75,0.9)

more relevant coefficients

average

n

ρhc[∆

n]

Fig. 8. Correlation curves of the 23 largest coefficients of the PLC channel.

tion index evolution along the measurements approximates a

straight line and has an abscissa equal to 75 for β = 0.9,

i.e., it takes Mc = 75 measures of the PLC channel to reach

a correlation equal to 0.9. Applying this result in (7), and

considering the time to obtain each CIR (NTs), the coherence

time estimated from the measurement campaign is:

T 0.9c = Mc(N + Lcp)Ts = 1.729ms. (19)

V. CONCLUSION

This contribution presented an initial analysis of coherence

time and sparsity of Brazilian outdoor and low-voltage PLC

channels and procedures to characterize both sparsity and

coherence time parameters. The numerical results, which are

based on measured PLC channels, confirm that only the

coefficients with the largest amplitudes are relevant to estimate

the coherence time of PLC channel. Based on this initial

investigation, the coherence time was found to be equal to

1.729 ms, which is longer than the reported coherence time for

indoor and low-voltage PLC channel, see [6]. Also, it suggests

4

that a straight line is the curve that best represents the evolution

of the correlation between the coefficients of a PLC channel.

The analysis of the sparsity of PLC channels show that it is

a random variable and because of that representative database

of measured PLC channels is require to carry out its statistical

characterization.

At the moment, a new measurement campaign is being

carried out to allow the introduction of statistical models for

both coherence time and sparsity parameters.

ACKNOWLEDGEMENTS

This work was supported by CNPq, CAPES, FAPEMIG,

FINEP, P&D ANEEL-CEMIG, INERGE and Smarti9.

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