A Mathematical Model of Noise in Narrowband
Power Line Communication Systems
Masaaki Katayama,Senior Member,IEEE,Takaya Yamazato,Member,IEEE,and Hiraku Okada,Member,IEEE
Abstract—This manuscript introduces a mathematically tractable and accurate model of narrowband power line noise based on experimental measurements.In this paper,the noise is expressed as a Gaussian process whose instantaneous variance is a periodic time function.With this assumption and representation, the cyclostationary features of power line noise can be described in close form.The periodic function that represents the variance is then approximated with a small number of parameters.The noise waveform generated with this model shows good agreement with that of actually measured noise.Noise waveforms generated by different models are also compared with that of the proposed model.
Index Terms—Colored noise,impulsive noise,narrowband power line communication(PLC),noise measurement,noise model,nonstationary noise.
I.I NTRODUCTION
P OWER-LINE COMMUNICATIONS(PLCs)can be clas-si?ed on the basis of their frequency bandwidth:PLCs can be either wideband(broadband),or narrowband systems.Wide-band PLC systems are recently calling attention for high-speed Internet access.The systems of this class use high-frequency (HF)band,or up to the low-end of very high-frequency(VHF) band.Because of this very wide frequency band,the wideband/ broadband PLC systems achieve10–100Mb/s without any new communication wiring effort.Though the use of the HF/VHF has advantageous features,it also contains dif?culties.In HF band,there are many radio users listening very weak signals, such as shortwave broadcasting listeners,radio amateurs,re-ceivers for emergency calls,and radio-astronomers[1].Because of the strong concern about interference from PLC systems, many countries do not allow the use of the HF band for PLC systems.
Narrowband systems use low or medium frequencies,be-tween3and148.5kHz in Europe under CENELEC and below 450kHz in Japan.The systems of this class have been used for many years,and there is no dif?culty concerning the reg-ulations.Though the allowed frequency range is relatively nar-rower than in wideband PLC systems,the range is still wide enough for some applications such as controls.For example,
Manuscript received April18,2005;revised January15,2006and February 21,2006.This paper was presented in part at the International Symposium on Power Line Communications,Vancouver,BC,Canada,2005.
M.Katayama and T.Yamazato are with the Division of Information and Communication Sciences,Ecotopia Science Institute,Nagoya University, Chikusa Nagoya464-8603,Japan(e-mail:katayama@nuee.nagoya-u.ac.jp; yamazato@nuee.nagoya-u.ac.jp).
H.Okada is with the Center for Transdisciplinary Research,Niigata Univer-sity,Niigata950-2181,Japan(e-mail:hiraku@ie.niigata-u.ac.jp).
Digital Object Identi?er10.1109/JSAC.2006.874408ECHONET consortium[2],which was established in1977,has de?ned speci?cations for narrowband PLC for control and mon-itor of home appliances.The control of power-grid systems is another important application,which can be realized by PLC systems with relatively low speeds[3],[4].In addition to PLC on mains power lines,there have been many applications with relatively low speeds on variety of power lines.The control of ground-lights of airport runways through0.4–6kV current loops[5]and communications of control data with40kb/s in streetcar/subway systems on750V DC networks[6]are inter-esting examples of them.
Although narrowband PLC systems have often been used for low-speed applications,they have been considered unreliable because of the higher noise levels present at lower frequen-cies.However,the low speed and low quality of narrowband PLC systems are not an ineluctable destiny.The reason of the insuf?cient performance of the conventional narrowband PLC systems is that they simply introduce relatively old-fashioned signaling schemes developed for an environment very different from power lines.Therefore,the employment of sophisticated schemes may result in much higher performance.
For the introduction of sophisticated communication schemes in narrowband PLC systems,the environment of power lines as communication channels should be known.One of the most pe-culiar features of power lines is strong and time-varying non-white noise.The power line noise at an outlet is the sum of noise waveforms produced and emitted to the lines from the appliances connected to the power line network.In the design and the analysis of conventional communication systems,sta-tionary additive white Gaussian noise(AWGN)is often used as a model of noise.In PLC systems,however,the statistical be-havior of this man-made noise is quite different from that of stationary AWGN.The non-Gaussian features of the noise are often believed to be the cause of the low quality of PLC systems. Actually,the fact that Gaussian distribution has the largest en-tropy implies that the communications under the non-Gaussian noise may achieve better performance than under AWGN,if the system is designed to adapt the noise statistics[7].Thus,narrow-band PLC systems still retain much capacity for performance improvement if the behavior of the noise is clari?ed and taken into account in the system design.
In order to study communications in a man-made impulsive noise environment,a simple model that expresses the noise behavior in closed-form equations is necessary.One of the most popular and important models of non-Gaussian noise is the probability density function(PDF)proposed by Middleton [8].According to this model,the PDF of impulsive noise can be expressed as a sum of Gaussian functions with different variances.With this model,various classes of impulsive noise
0733-8716/$20.00?2006IEEE
Fig.1.Noise waveform by a CRT color TV.
can be expressed by a simple function with a small number of parameters.The disadvantage of this model is that the model does not de?ne time-domain features.The PDF does not describe whether the noise waveform is peaky(impulsive)or smooth in the time-domain.
The power line noise is not white and it has a complicated power density spectrum(PDS).In[9],the bandwidth is divided into several subbands,and noise voltages are sampled at each subband to de?ne a PDF by a histogram of noise voltages.The noise waveform is then generated using the set of PDFs of all the subbands.In[10],the same concept is applied,and PDF for each subband is approximated by Nakagami-m distribution. With this model,the colored noise can be expressed by a set of closed form functions.
In many studies about power line noise,including those re-ferred to above,it is assumed that the noise is stationary and, therefore,only the time averaged power spectrum densities and the time-independent intensity distributions are taken into ac-count.The characteristics of the noise in power lines,however, are not stationary.
It is known that the power line noise has periodic features, which are synchronous to the mains voltage.This is because many appliances generate noise synchronously to the instanta-neous value of the mains voltage.Some switching devices turn on and off at a certain voltage of AC,and these switching op-erations also cause impulses synchronous to the mains voltage. Therefore,a model,which can describe the statistics of the in-stantaneous value of the noise,is still needed.Examples of snap-shots of noise from individual appliances are shown in Figs.1–3, which are taken at the same outlet where the appliances are con-
nected.In the?
gures,is1/2of the mains cycle duration,
which is1/120s in Nagoya,or western part of Japan.In addition to the behaviors of appliances,channel characteristics between the noise sources and a receiver may vary synchronously to the mains voltage[11].This channel?uctuation also causes in the periodic features of the noise.
The frequency of the periodic features of the noise is the same or twice the mains frequency.This is relatively slow compared with the data/packet rate of high-speed wideband systems.In the case of narrowband PLC systems,however,the symbol duration and thus packet length tend to be long because of the relatively narrow bandwidth,and thus the periodic features of noise
cannot Fig.2.Noise waveform by an inverter driven?uorescent lamp(30
W).
Fig.3.Noise waveform by a vacuum cleaner with brush motor(600W).
be ignored.Furthermore,these features can be used by com-munication systems to?nd/estimate time slots with low-noise level for adaptive signal transmission and maximum a poste-riori probability(MAP)detection with reliability of data based on the noise.
In[12],the power line noise is divided into four classes,i.e., (A)continuous colored noise;(B)continuous tone jammers;
(C)Periodic impulses synchronous to mains;and(D)impulsive noise asynchronous to mains.In[13],Zimmermann and Dostert represent the time variant features of noise with a partitioned Markov chain with multiple states.This model represents the last class,(D),of noise for wideband PLC systems.
In narrowband PLC systems,however,the continuous noise in classes(A)and(B)are dominant,together with periodic impulses of(C).For this reason,the authors have?rst proposed a mathematical model which represents the continuous noise with the periodic features[15].In addition,in[16],the model that includes stationary and impulsive components is proposed. Further considerations are made in[18].Based on these former studies,this paper proposes a mathematical model of the power line noise that can represent its nonstationary and nonwhite features by a simple PDF function with a small number of parameters.
In Section II,the setup to measure the power line noise at outlets is described and an example of measured noise wave-form is https://www.doczj.com/doc/9c9538760.html,ing this measurement as an example, Section III describes the proposed model of power line noise. The model de?nes power line noise as the Gaussian process whose variance is a time and frequency dependent function. Then,a simple approximation to express the variance function
KATAYAMA et al.:A MATHEMATICAL MODEL OF NOISE IN NARROWBAND PLC SYSTEMS
1269
Fig.4.Measurement system.
is proposed.Section IV describes the procedure to?nd the pa-rameters of the model from the measured noise waveform,and Section V illustrates an example waveform of noise generated by the model.In Section VI,another approximation for the vari-ance function is provided,and statistical comparisons are made. The?nal section concludes this paper with future problems.
II.N OISE M EASUREMENT
A.Measurement Setup
In many measurement reports on power line noise,spectrum analyzers are often used to show average or peak values of power line noise in the frequency domain.Some reports focus on noise waveforms in the time domain,but only for short time period.In contrast,in this study,the discussion is based on the measurement of the whole noise waveforms in a long observa-tion time.
The system for the noise measurement is shown in Fig.4.The system consists of a pickup circuit followed by an A/D converter (ADC)and a computer(PC),and the injection circuit with a waveform generator.The pickup and injection circuits are con-nected to an outlet and no other electrical appliance shares the outlet.The measurements are made in university laboratories, apartments,individual houses.Since the measured waveforms at different locations have similar features,a waveform taken at the laboratory of the authors is used as an example of discussion in this paper.
The pickup circuit is shown in Fig.5.In this?gure,the mains alternating current(AC)component is attenuated by the capaci-tors,common-mode noise component is removed by a radio-fre-quency transformer,and balance-to-unbalance transform of the circuit is performed by a balun.The noise component is obtained at the terminal labeled“noise”in the?gure.In order to eliminate aliasing effect,low-pass?lter(0–1.9MHz)is inserted before the ADC.ADC then samples the waveform with10M
[samples/s]Fig.5.Pickup circuit.
TABLE I
C IRCUIT E
LEMENTS
and passes the data to the computer.As described in the fol-lowing sections,the rest of processing is executed with digital signal processing programs in the PC.
The pickup circuit also outputs a down-transformed AC waveform with the peak-to-peak voltages of2[V]at the ter-minals labeled“AC.”This AC voltage waveform is used as a trigger to start the ADC.The circuit elements are shown in Table I.
In power line communication systems,line impedance is not stationary.It often?uctuates along with the mains AC voltage[11].This?uctuation of line impedance often causes the impedance mismatch between lines and a receiver,which decreases both the received noise and signal levels.In order to monitor and compensate the effects of the impedance mis-match,we introduce a reference tone,which is injected into the power lines from the same outlet where pickup circuit is con-nected.The injection circuit and the pickup circuit for“noise component”have the same con?guration and parameters.
B.Normalization of Noise Waveforms
The input to the ch.1of the ADC can be expressed
as
,
where denotes the reference tone,
while
is the noise component.According to the results of authors’ex-periments in several locations,in wideband(broadband)PLC, which uses whole HF band,the frequency dependence of line-impedance as a time function cannot be ignored.Thus,we have to use a multitone signal as a reference.In narrowband PLC, however,the?uctuation of line-impedance is almost frequency independent.In other words,the ratio of the voltage of reference signals with different frequencies is time independent.For this reason,the measurements described in this paper use a single tone with the
frequency kHz as the reference.
1270IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS,VOL.24,NO.7,JULY2006 Let us assume that the power of the reference tone is constant
for a short
duration.The reference tone for
this time duration can be denoted
as
(1)
where kHz
and are the frequency and the phase of
the reference tone.The
power is calculated
by
(2)
In the
measurement,is sampled with the sampling
rate
MHz.The sample
at
is
.The train of the samples is stored in PC and the refer-
ence tone and noise components are extracted by narrowband
(bandpass and band-rejection)?lters of the center
frequency,
which are realized by onboard digital signal processing.After
the extraction
of
and components,the power of
the reference tone in the vicinity
of is calculated
as
(3)
where.In the numerical example shown in this man-
uscript,
10s,and
thus.
In order to compensate the?uctuation of noise level caused
due to mismatch between?uctuating line-impedance and
pickup circuit,each noise
sample is divided by the
instantaneous level of the measured reference tone
component
.The resultant normalized noise
level,is ob-
tained as
follows:
(4)
Let us
have samples in the duration
of cycles of the
mains AC voltage,
i.e.,
where is the mains
cycle duration.In Nagoya,or in western Japan,the mains fre-
quency is60Hz
and
10.The
integer
is the largest integer no greater
than.Then,the
time-averaged variance
of is calculated
as
(5)
With this
value is normalized to have unity variance
as
(6)
Fig.6shows an example of the normalized noise
waveform
.From this?gure,we can observe that the noise has the
periodic features with the frequency
of Hz
.
Fig.6.A snapshot of a measured noise waveform(normalized).
III.N OISE M ODEL
A.Cyclostationary Gaussian Model
Fig.7shows an example of the cumulative probability dis-
tribution functions of a measured noise waveform[16].In the
?gure,solid lines indicate the distribution of the noise level(ab-
solute voltage)and dotted lines show Gaussian distribution of
the same power.Since the noise in power lines has periodic
features with the
frequency,in(A),we calculate the
cumulative probability distribution of the noise level sampled
with this
frequency,at the instance when the mains AC
voltage crosses zero.The distribution of the noise level taken
at the peaks of the mains absolute voltage is shown in(B).For
comparison,the distribution of the noise level taken at random
phase is also given in(C).
From(C),we can?nd that the distribution function of the am-
plitude of the noise in power lines is larger than that of Gaussian
when amplitude is small,though the Gaussian has larger values
of distribution function at larger amplitude region in all?gures.
This is the typical feature of the impulsive noise.In other words,
it can be concluded that the noise in power lines is impulsive if
the samples are taken at random.
On the contrary,in(A)and(B)of the?gure,in which the
noise is observed periodically at
every,the amplitude
distributions can be assumed as Gaussian.The authors have
con?rmed that this Gaussian distribution of periodically sam-
pled noise can be observed at any phase of the mains and in
many other power line noise waveforms.Considering this inter-
esting feature,the authors have proposed a model of power line
noise[16],which assumes that the noise is cyclostationary(pe-
riodically stationary)[17]additive Gaussian noise whose mean
is zero and the variance is synchronous to the AC voltage of
mains.With this assumption,the PDF of the noise at the in-
stance can be expressed
as
(7)
In this
equation,is the instantaneous vari-
ance of the noise,
where denotes ensemble average.Since
the
variance is a periodic time function of the freqency