IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 3, JULY 2005
1869
A Novel Approach to the Modeling of
the Indoor Power Line Channel—Part II:
Transfer Function and Its Properties
Stefano Galli, Senior Member, IEEE, and Thomas Banwell, Member, IEEE
Abstract—In Part I of this work, we introduced multiconductortransmission-line (MTL) theory to model the indoor power-line
(PL) channel. We have also shown that the proposed MTL approach can also be used to take into consideration both the topology
of the link and particular wiring practices such as bonding. In
this contribution, we continue our bottom-up approach to indoor
PL channel modeling and we show that the circuit model found
in Part I can be represented in terms of cascaded two-port networks (2PNs) coupled through a single modal transformer. Once
the equivalent 2PN representation is obtained, it is possible to represent the whole PL link by means of transmission (ABCD) matrices only. The results presented here allow us to reveal that the PL
channel is a more deterministic media than commonly believed and
also allows us to unveil interesting properties of the PL channel,
such as symmetry, that were previously unknown.
Index Terms—Communication channel, modeling, multiconductor transmission line, power cables, power-line communications, transfer function, transmission-line discontinuities, two-port
circuits, wire communication cable.
I. INTRODUCTION
A
CIRCUIT model of an indoor PL link was derived using
an MTL theory approach in Part I of this work [1]. This
approach allowed us to include mode coupling effects due to
particular wiring practices, something usually not included in
previously published models. One of the main results of Part I
of this work was to point out that traditional modeling of the
PL channel based on two-conductor transmission lines (TLs) is
unable to fully describe the underlying physics of signal propagation along PL cables when wiring practices commonly used
in residential and business premises (e.g., ground bonding) (see
Section II of [1]) are included in the network topology. In fact,
in this case, non-negligible mode coupling arises and it is impossible to account for it using models based on two-conductor
models only. In Part I, it was also shown that if mode coupling is
included in the analysis, then the circuit model that represents
the PL link is composed of two nearly independent networks
of simple (two-conductor) TLs coupled through a 2:1 modal
transformer.
Manuscript received April 23, 2004. Paper no. TPWRD-00199-2004.
S. Galli is with Telcordia Technologies, Piscataway, NJ 08854 USA (e-mail:
sgalli@research.telcordia.com).
T. Banwell is with Telcordia Technologies, Red Bank, NJ 07701 USA (e-mail:
bct@research.telcordia.com).
Digital Object Identifier 10.1109/TPWRD.2005.848732
In the present contribution, we will show how the two coupled
circuit models obtained in Part I of this work (the differential and
the companion models) can be represented in terms of cascaded
2PNs coupled through a transformer. Once the equivalent 2PN
representation is obtained, it is possible to represent the PL link
by means of transmission (ABCD) matrices, only. The possibility of representing a PL link as a cascaded 2PN allows the
definition of a model in the frequency domain that takes into
consideration both the particular topology of the link and the
wiring practices, thus allowing us to describe a priori all of the
echoes traveling along the line in the frequency domain instead
of in the time domain. Besides the improved accuracy, this is a
more practical approach than the previously proposed ones for
several reasons: there is no need for a preliminary channel measurement for fitting the parameters of the multipath model; resonant effects due to parasitic capacitances and inductances are
now explicitly taken into account by including cable characteristics; since the frequency-domain representation of the propagating signals contains the composite of all the signals reflected
by the discontinuities, the computational cost in the multipath
model of independently describing each path in the time domain disappears; particular wiring and grounding practices may
be taken into account.
Several new results are presented here. First, we give a procedure for computing analytically and a priori the transfer function of an indoor PL link. Second, we unveil some deterministic
aspects of the PL channel that allow us to isolate specific resonant modes. Third, we prove mathematically and confirm experimentally an interesting symmetry property of the PL channel.
In fact, as will be shown in Section IV, the indoor PL channel,
regardless of its topology, exhibits the same transfer function
when driven from either side, if and only if the source and load
impedances are the same. It is worth pointing out that this symmetry property cannot be proven simply by resorting to the Reciprocity Theorem (see the considerations made in Section IV).
The paper is organized as follows. An overview on the 2PN
modeling of a TL by means of ABCD matrices is given in Section II. In Section III, a method for the computation of the equivalent cascaded 2PN circuit of a PL link is provided. The symmetry property of the PL channel is proven in Section IV. Experimental results confirming the theoretical analysis are given
in Section V. Examples of how to exploit the determinism in the
PL channel for robust modem design are provided in Section VI
and, finally, concluding remarks are drawn in Section VII.
0885-8977/$20.00 © 2005 IEEE
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In particular, expressing the transfer function as the ratio of
the voltage on the load to the source voltage, we obtain the following relationship:
(5)
Fig. 1.
B. ABCD Coefficients of Transmission Lines
Generic two-port network (2PN).
In the case of a uniform two-conductor TL, the ABCD coefficients and the corresponding forward transmission matrix
take on the following expression:
II. TRANSMISSION-LINE MODELING VIA
TRANSMISSION MATRICES
A general result of TL theory is that every uniform TL can
be modeled as a two-port network (2PN), thus allowing us to
replace a distributed parameter circuit with a single lumped network. In TL theory, a common way to represent a 2PN is to use
the transmission matrix , also known as the ABCD matrix [2].
The ABCD modeling described here will be extensively used in
Section III for reducing the circuit model derived in [1] into a
single 2PN for each path and, in Section IV, to prove the symmetry property of the PL channel.
A. Transmission Matrix of a Two-Port Network
The relationship between current and voltage (in the frequency domain) at the two ports of a 2PN is given by the
following expression (Fig. 1)1
(1)
The ABCD coefficients, which are complex functions of frequency, fully characterize the electrical properties of a 2PN and
are defined as follows:
(2)
as functions of
To express
invert the ABCD matrix in (1)
, it is necessary to
(6)
where
, and
are the length, the propagation constant, and
the characteristic impedance of the cable, respectively. The elein (6) satisfy the following properties:
ments of
for any frequency
for all frequencies
Unitary determinant
From
it also follows that
(7)
From Property 3), it follows that the 2PN model of a cable is
a reciprocal 2PN (i.e., a 2PN that satisfies the Reciprocity Theorem). The Reciprocity Theorem states that any transfer function with the dimension of impedance or admittance remains
unchanged if the points of excitation and response are interchanged. From Property 4), it also follows that a cable is a symmetrical channel, in the sense that it exhibits the same behavior
, the transfer
if driven from either side. In fact, since
function (5) of a PL cable is the same from either side, provided
that source and load impedances are interchanged. Interestingly,
for the more general case of two or more different cables spliced
(Section IV).
together, we have that
C. Special ABCD Matrices
(3)
In (3), we have changed the signs of and according to
the usual convention that considers currents that flow from the
source to the load as positive. Usually, the transmission matrices
and backward
in (1) and (3) are referred to as forward
transmission matrices, respectively.
Given the 2PN in Fig. 1, the ABCD parameters allow us to
calculate the following useful quantities:
(4)
1For ease of notation, hereinafter, the explicit dependency on the frequency
of A; B; C; D; Z ; ; Z ; Z ; V , and I (x = 1; 2) has been omitted
Particular attention must be given to bridged taps or, in general, to series and shunt impedances along the line. In fact, in
these cases, the ABCD matrix takes a particular form that differs from that of (6). As a general result, any linear line discontinuity can be embedded in the model in the form of 2PN.
1) Shunt Impedances Along the Line: A bridged tap occurs
whenever a branching connection is spliced onto the cable,
a situation that is very common both to indoor and outdoor
wiring. The end of the branch can be either open or terminated.
A bridged tap can be viewed as a three-port section, but one of
the ports appears as a load impedance to the line between the
two sections on each side of the bridged tap. Such a situation
can still be modeled as a cascade of 2PNs since the bridged tap
can be modeled as a shunt impedance across the line with the
impedance equal to the input impedance of the bridged tap. In
this case, the forward transmission matrix of a bridged tap is
(8)
GALLI AND BANWELL: A NOVEL APPROACH TO THE MODELING OF THE INDOOR POWER LINE CHANNEL—PART II
1871
where the input impedance looking into the bridged tap
can be calculated as in (4). If the branching connection is unterminated (i.e., infinite load), the input impedance in (4) becomes
(9)
so that we have the following particular case:
(10)
2) Series Impedances Along the Line: The same approach
can be used to model simple series impedances along the line. In
this case, the forward transmission matrix of a series impedance
along one of the cables of the two-conductor TL becomes
(11)
3) Transformers: As shown in [1], the differential and the
companion circuits are tied by a transformer. The forward transmission matrix of a transformer with turns ratio is
Fig. 2.
Final circuit model of the power-line link in Fig. 7 of [1].
the chain rule to (11) and (12), we obtain the following forward
transmission matrix:
(12)
(15)
D. Chain Rule
In general, a TL is made of several sections and each
section may consist of different cables of different lengths.
An important property of transmission matrices is that they
easily allow us to handle tandem connections of 2PNs. For a
given network configuration, the overall ABCD matrix of the
end-to-end circuit is obtained by exploiting the chain rule (i.e.,
multiplying the ABCD matrices of the single portions of the
is the forward transmission matrix
network). Therefore, if
of the th section, the overall forward transmission matrix
of the end-to-end circuit consisting of
sections is given by
the following relationship:
(13)
Similarly, the overall backward transmission matrix of the
end-to-end circuit is given by the following relationship:
Each forward transmission matrix
is of the kind in (6)
and, therefore, satisfies all of the properties in (7). In particular,
from Property 4) in (7) we can write
(14)
As an example of the application of the chain rule, we can
now compute the forward transmission matrix of a transformer
on the second port. This is a
followed by a series resistor
practical case since it is the case of the modal transformer folas shown in the proposed model of [1]. Applying
lowed by
III. ANALYTICAL COMPUTATION OF THE TRANSFER FUNCTION
In this section, we will consider the network topology in
Fig. 7 of [1] and its corresponding circuit model shown in Fig. 2.
The propagation times on the various parts of the network are
shown together with the respective characteristic impedances.
It is worth noting that both characteristic impedances and
propagation times for the differential mode and the pair mode
are different.
The equivalent cascaded 2PN representation of the circuit in
Fig. 2 will now be derived. Among the several topologies that
can be chosen, we will consider the case of
, and
. Since the circuit model in Fig. 2 allows
us to consider the three-conductor PL link as if it consisted of
two two-conductor TLs coupled through a transformer, we can
represent the PL link in Fig. 2 as the one shown in Fig. 3.
As described in Section II(c), bridged taps can be viewed as
a three-port network, where one of the ports appears as a load
impedance to the line between the two sections on each side of
the bridged tap. The transformer that “ties” together the differential model and its companion model describing the pair-mode
propagation is inserted onto the “hot” and “return” cables as
if it were a bridged tap. This means that it appears as a shunt
impedance across the “hot” and “return”, and this impedance is
equal to the input impedance seen looking into the transformer.
We can then represent the PL link in Fig. 7 of [1] as cascaded
2PNs as shown in Fig. 4.
With respect to Fig. 4, we have the following definitions:
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Fig. 3.
Circuit equivalent to the one in Fig. 2.
Fig. 4.
Equivalent cascaded 2PN representation of the circuit in Fig. 3.
1)
2)
forward matrix of a
cable feet long;
forward matrix of a
cable feet long
placed as an unterminated bridged tap onto the main cable
(hot-return);
3)
forward matrix of a
cable 60 ft long placed
as a shorted bridged tap onto the main cable (hot-return);
forward matrix representing the breaker (return4)
ground wires shorted) (i.e., the companion circuit model
and its modal transformer);
input impedance of the companion circuit (i.e.,
5)
the impedance seen looking into the companion circuit
through the modal transformer).
is the one
The structure of matrices
given in (8). The computation of matrices
is
trivial and is skipped here. As far as the computation of matrix
is concerned, it is useful to represent the companion model
.
in Fig. 2 as cascaded 2PNs and compute its input impedance
Matrix
has, then, the following structure:
(16)
This is shown in Fig. 4, where the 2PN of the 2:1 modal trans.
former has been computed for the case of
The overall ABCD matrix from (X) to (Y) can be found by
exploiting the chain rule (see Section II(d)) that states that the
overall ABCD matrix of the end-to-end circuit is obtained by
multiplying the ABCD matrices of the single portions of the
network. Finally, we may represent the PL link in Fig. 7 of [1], as
GALLI AND BANWELL: A NOVEL APPROACH TO THE MODELING OF THE INDOOR POWER LINE CHANNEL—PART II
the simple 2PN circuit shown in Fig. 1 where the overall forward
transmission matrix form of X to Y is given by
(17)
Once the overall ABCD matrix of the end-to-end circuit is
obtained, the transfer function form is given by
(18)
Describing the behavior of a PL link using ABCD modeling
leads to the possibility of unveiling an interesting property of
the transfer function never pointed out in the previous literature:
symmetry. In fact, as will be shown in the following section, it is
possible to prove mathematically that the PL channel, regardless
of its topology, is a symmetric channel (i.e., exhibits the same
transfer function from either side, provided that the source and
load impedances are the same).
for an elegant proof of this statement). Moreover, the symmetry
property is satisfied if and only if both Properties 1) and 3) in
(7) are satisfied. In the case of a network made of several sections, Property 1) in (7) is not satisfied anymore by the overall
transmission matrix. In fact, we have
(21)
confirming that the product of two matrices of the kind of (6) is
no longer of the kind of (6).
On the basis of the previous considerations, the forward and
backward transmission matrices of a PL link are indeed different. However, it can be proved that the forward and backward
transfer functions of a PL link are the same provided that source
and load impedances are equal. As previously mentioned, this
cannot be proven by means of the Reciprocity Theorem but has
to be proved in some other way.
and given in
Let us consider the general expressions of
(13) and (14)
IV. SYMMETRY OF THE FREQUENCY-TRANSFER FUNCTION
Matrix multiplication does not usually satisfy the commutain (13) is different from
in
tive property and, therefore,
(14). The multiplication of two matrices is commutative if and
only if the two matrices share all common eigenvectors. However, this is not the case because the eigenvectors of a matrix of
the kind in (6) are
(19)
These eigenvectors depend on the characteristic impedance of
the cable and, therefore, the transmission matrices pertaining to
two different kinds of cables (i.e., with two different characteristic impedances) have different eigenvectors. Although it may
seem counterintuitive, this means that even a simple connection
consisting of two different cables spliced together has different
overall forward and backward matrices. However, even if the
overall forward and backward transmission matrices of a PL netand
work are different, the reciprocity Property 3) in (7) for
is still maintained. In fact, we have
(20)
and
satisfy all of the properties in (7),
since matrices
for any . Therefore, the cascade of reciprocal 2PNs is still a reciprocal 2PN. However, the fact that the whole network may be
modeled as a reciprocal 2PN does not mean that the PL channel
is a symmetric channel (i.e., has the same frequency response
in both directions) as it is sometimes stated (see, for example,
Section 3.8.1 in [2]). In fact, the Reciprocity Theorem holds for
transfer functions with the dimension of a transimpedance or
transadmittance and does not hold for dimensionless transfer
functions, such as the voltage gain (5) (see Section 1.8 of [3]
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(22)
It is easy to prove that the forward and backward chain matrices of a link coincide with the forward matrices of the original
link and the one that exhibits a symmetric topology with respect
to the middle point of the link, respectively. The matrices in (22)
are the chain matrices of the whole connection and they satisfy
only Property 3) in (7) (i.e., they have unitary determinant; on
satisfies all of the properties in
the other hand, each matrix
(7), for any . On the basis of matrix multiplication properties
and of induction, it is possible to show that the following relationships always hold:
(23)
and
Now, the forward and backward transfer functions
of the whole link can be calculated on the basis of the
chain matrices in (22)
The difference
between the reciprocals of the previous
quantities is shown in the equation at the bottom of the following
page.
Exploiting the relationships in (23), we finally obtain
(24)
Now, if
, we can write
(25)
Since
and
are always nonzero quantities and are always different for different cable configurations, we can state
and
are equal, the forward and backthat if and only if
ward transfer functions are the same. See [10] for the case of the
twisted-pair channel.
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!1
Fig. 5. Comparison of the transmission response without bonding (R
, top trace) and with bonding (R = 0, lower trace). The network model in Fig. 7
of [1] was simplified to isolate the effect of ground bonding by disconnecting the simulated feed line, disconnecting the 15-ft stub, and differentially terminating
the far outlet with a 147-
resistor.
V. EXPERIMENTAL RESULTS: ISOLATION OF RESONANT MODES
AND SYMMETRY OF THE TRANSFER FUNCTION
We are building a networked “smart” home in one of our
labs. This home will be equipped with several wired and wireless networks, such as Ethernet, IEEE 1394, IEEE 802.11, and
HomeRF. It will also be a EIA/CEA 851 VHN Home Network
compliant test bed, where VHN is a home network standards
effort under the auspices of the Electronic Industries Alliance
(EIA)/Consumer Electronics Association (CEA). Power cables
have been deployed in such a way that several different topologies can be investigated. For our experiments, we chose an indoor wiring model having two branches with five outlets and a
cabreaker box as shown in Fig. 7 of [1]. Type NM-B
bles were used in the branch circuits. The U.S. NEC wiring
and grounding practices were also followed. Because of storage
and logistical constraints, a 50- RJ-58 cable was used to emulate 60 ft of
cable in the mains feed. Transmission measurements were performed with a network analyzer using balun
transformer coupling at the locations X and Y indicated in Fig. 7
of [1]. These transformers suppress unwanted pair mode crosstalk that afflicted the TDR measurements. In some cases, measurements were also made in the reverse direction with X and Y
exchanged.
A. Isolation of Resonant Modes
Referring to the wiring model shown in Fig. 7 of [1], transmission experiments were conducted between nodes (X) and (Y).
Swept frequency transmission measurements were performed
balun transformers were used with
from 0.3 to 30 MHz. 1:
resistor matching networks to interface the 50- network analyzer ports to the test points. Experiments were conducted with
, and matched sending
various values chosen for
.
and receiving impedance
The practical significance of ground bonding and the necessity of the modeling presented in Part I of this work was demonstrated in an early experiment. The network model was simplified to isolate the effect of ground bonding by disconnecting the
emulated feed line, disconnecting the 15-ft stub, and differentially terminating the far outlet with a 140- resistor.
Fig. 5 compares the transmission response without bonding
, top trace) and with bonding (
, lower
(
trace). The diagrams to the right of the graphs are meant
to illustrate the two network topologies: the dashed line in
the lower diagram signifies pair-mode excitation. The upper
trace in Fig. 5 shows relatively benign 3-dB attenuation over
0.3–30 MHz in the absence of bonding. Addition of bonding
creates significant resonant attenuation at 3.27 and 9.95 MHz,
with less pronounced attenuation at 16.89 and 23.27 MHz. The
decreased attenuation at 16.89 and 23.27 MHz suggests that the
coupled mode is lossy. This experiment clearly demonstrates
that bonding creates significant frequency dips that are not
described by a model that omits the effects of mode coupling.
By selectively disconnecting the bonding shunt, 15-ft, 25-ft,
and 60-ft branches, 12 distinct topology variations can be obtained. Fig. 6 summarizes the observable notches for all 12 distinct topologies. The box labeled “B” indicated the presence of
bonding and the diamond indicates the breaker box where the
GALLI AND BANWELL: A NOVEL APPROACH TO THE MODELING OF THE INDOOR POWER LINE CHANNEL—PART II
Fig. 6.
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Table summarizing resonant dips for all 12 considered topologies. The first topology (first row) is the compete topology shown in Fig. 7 of [1], with
R = 0 = 140 , and
= 2 . The 12th topology (last row) is the topology shown in the upper part of Fig. 5, but with a terminating impedance
= 140 on the last bridged tap. This particular topology exhibited a benign attenuation profile without notches.
;R
R
R
mains feed is (eventually) connected; the mains feed is always
shorted, except for the eighth topology where it is open. The
second bridged tap may be unterminated or terminated in resis.
tance
The first topology (first row) is the complete topology shown
. The
in Fig. 7 of [1], with
eleventh (twelfth) topology is the topology whose transfer function from X to Y is shown in the lower (upper) part of Fig. 5.
By examining Fig. 6, we notice that the ground bonding shunt
(eleventh topology) creates dips at 3.3, 9.9, 16.7, and 23.3 MHz;
(tenth topology) creates dips at 7.0 and 21 MHz, the
15-ft branch (ninth topology) creates a 11.4-MHz dip, and the
(seventh topology) creates dips at 4.8,
mains feed with
9.8, 14.9, 19.8, and 24.8 MHz. SPICE simulations based on the
equivalent circuit in Fig. 2 are consistent with the measured results for all 12 topology variations investigated.
Looking at Fig. 6, it is interesting to note that if the link
contains the unterminated 15-ft branch (the first bridged tap
in Fig. 7 of [1]), the transfer function always contains a notch
at 11.4 MHz. Similarly, in all of the topologies containing the
, notches at 4.8, 9.8, 14.9, 19.8,
60-ft. mains feed with
and 24.8 MHz are always present. Also, in all of the cases where
, dips at 7.0
the second bridged tap is unterminated
and 21 MHz are always present. On the basis of these experiments, we can say that if a certain feature (e.g., a bridged tap),
the bonding, the mains feed, etc., is always present in the link,
then corresponding notches will be always present in the transfer
function. This behavior can be exploited by PL modems as described in Section VI. Another interesting aspect of these results
is the possibility of actually exploiting the observation of those
resonant modes to infer the actual topology, or at least the main
features of it. The possibility of identifying (partially) the network on the basis of its resonant modes is currently under investigation.
B. Symmetry of the Transfer Function
Fig. 7 shows the measured transfer function of the link in
Fig. 7 of [1] from (X) to (Y) and from (Y) to (X), with
and
. The response is nearly identical in
both directions with matched sending and receiving impedance
of 140 , thus confirming the results found in Section IV on the
symmetry of the PL channel.
Fig. 7 can be compared with Fig. 8 that shows the simulated
transfer function obtained using the equivalent cascaded 2PN
representation derived in Section III.
Although the similarity of the two figures is striking and,
more important, all of the frequency notches are accounted for,
there are some differences in the two transfer functions. These
differences are essentially due to the fact that our software simulator could only compute ABCD matrices of telephone cables
(i.e., American Wire Gauges 19, 22, 24, 26). Therefore, the
ABCD calculation in our computer analysis was made by using
19 AWG (the thickest telephone cable) and modifying its characteristic impedance appropriately, but not perfectly since characteristic impedances and propagation constants are complex
functions of frequency. Essentially, we have modified only the
asymptotic value of the characteristic impedance of the 19 AWG
cable in order to match the characteristic impedance of the PL
cable under test (140 for the differential mode, and 58 for
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Fig. 7.
IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 3, JULY 2005
Plot of the measured transfer functions (from X to Y, and from Y to X) for the topology in Fig. 7 of [1].
the pair mode). Similarly, in order to take into consideration different propagation times, we have slightly modified the length
of the cables in the simulations. Since the velocity propagation
of an AWG 19 cable is approximately 70% of the speed of light
in vacuum, these length modifications were of the order of a few
tenths of a foot.
Let us consider the transfer function that we would have if the
mains feed and the pair-mode excitation due to bonding at the
breaker box were neglected, as in previously reported models.
The transfer function in this case exhibits three notches at 7,
11.4, and 21 MHz as also shown in the table in Fig. 6 for the
third topology; the notches of the third topology are the combination of those of the ninth and tenth, that contain the first unterminated bridged tap and the second unterminated bridged tap,
respectively. On the other hand, if we consider the full model
here proposed, many frequency notches not accounted for appear as clearly shown in the actual transfer function in Figs. 7
and 8.
VI. EXPLOITATION OF THE DETERMINISM PRESENT IN THE
POWER-LINE CHANNEL
As experimentally verified, it is possible to isolate reflections
and resonant modes on the basis of specific topologies. Moreover, it was also mathematically proven and experimentally verified that the PL channel is a symmetric channel regardless of its
topology as long as source and load impedances are the same.
In the following subsections, some possible ways of exploiting
these deterministic aspects of the PL channel will be mentioned.
A. Mapping the Network
Although some appliances in the home may be switched on
or off several times during the day, some appliances are usually
always on and their input impedance varies slowly during the
Fig. 8. Simulated transfer function obtained on the basis of the equivalent
2PN representation obtained in Section III. Compare this plot to the measured
transfer function shown in Fig. 7.
day. Some other features of the network can be considered constant during the day: the presence of the mains feed, the bonding,
some unused plugs in certain rooms, etc. Therefore, it is not
unreasonable to assume that the transfer function between two
PL modems located at two home network nodes may always
exhibit notches at certain frequencies. This suggests the possibility that PL modems could effectively try to map particular
features of the entire PL home network. This mapping could be
accomplished adaptively by continuously transmitting training
sequences or by embedding into the modems some a priori information on the topology of the in-home PL wiring.
The network maps obtained for some particular states of the
link can be stored and appropriately utilized. Over time, it is
GALLI AND BANWELL: A NOVEL APPROACH TO THE MODELING OF THE INDOOR POWER LINE CHANNEL—PART II
1877
likely that all of the states of the channel are encountered so that
the PL modem can infer and, therefore, exploit, the actual state
of the network on the basis of the estimated channel transfer
function (e.g., with a lookup table). The obtained network mapping could be exploited in several ways. For example, by determining which are the notches always present for a particular
configuration and state of the home PL link, thus determining
in which frequency band, it is always better not to transmit information. Additionally, since some notches may be present or
not depending on the states of the channel, the availability of
a lookup table would allow the PL modem to quickly decide
which modulation/coding technique for that particular network
state is more appropriate.
is the same. As far as bit loading for the PL channel environment is concerned, it is worth pointing out that regulatory issues
may make it necessary to find bit-loading algorithms under peak
power constraints [5], and not under the usual average power
constraint.
As a final remark, it is important to note that symmetry of
the transfer function is not really equivalent to symmetry of the
channel. In fact, in the PL environment, different nodes in a
home network may be affected by different noise realizations
and, therefore, the knowledge of the state of the channel at the
transmitter is not actually “perfect.”
B. Appropriate Modulation Techniques
In this two-part work, several levels of abstraction were
crossed before obtaining a useful (from a communications
point of view) and accurate model for the indoor PL channel.
Starting from the circuit level developed in [1], we arrived at
the definition of an accurate model of the PL channel in the
frequency domain based on transmission matrices as described
in Section III. This “bottom-up” approach was deemed necessary by the authors because many aspects of signal propagation
along PL cables were still left unexplained by the most common
models reported in the literature.
The main result reported in this two-part study is the possibility of obtaining an accurate and complete model of the indoor
PL channel that takes into account both the topology of the link
and the grounding practices using an MTL approach. The circuit model obtained using this approach consists of two coupled circuits representing the propagation and interaction of the
two dominant modes: the differential and the pair modes. The
first circuit accounts for differential-mode propagation while the
second circuit, the companion model, accounts for the excitation
and propagation of the pair-mode. The circuits of the differential and companion models can also be described as a cascade
of simple two-port networks, each of which may be easily described via its transmission matrix. The possibility of describing
the PL link via transmission matrices, therefore replacing a distributed parameter circuit with a single lumped network, allowed us also to prove mathematically (and verify experimentally) that the indoor PL channel, regardless of its topology, is
isotropic (i.e., exhibits the same transfer function from either
side). The only condition necessary for such symmetry is that
the source and load impedances are the same or, equivalently,
that the output impedance of the transmitter is equal to the input
impedance of the receiver. This is a very common situation, so
that the symmetry property is indeed verified in practice. Moreover, it was also experimentally shown that it is possible to
isolate reflections and resonant modes on the basis of specific
topologies. These results allow us to affirm that the PL channel
is a more deterministic environment than commonly believed.
Current and future work will concern the exploitation of this
“deterministic” nature of the PL channel to enhance the performance of a PL modem. Some possibilities in this direction
were mentioned in Section VI. Finally, current research is also
devoted to the extension of this approach to the outdoor case,
where PL cables are used for broadband internet access over
the last mile.
The results of Section V suggest that the transfer function
of any PL link in the home is likely to contain several notches
due to the always present ground bonding and mains feed. This
is fundamentally different from the case of a home network
based on inside telephone wiring2, a case in which the frequency
transfer function will exhibit notches if and only if bridged taps
were present (see, for example, [4]).
Due to the many frequency notches present in the PL channel,
linear equalization cannot be used because it would give rise to
severe noise enhancement phenomena, a very serious drawback
in a noisy channel such as the PL [7]. Channels that exhibit
transfer functions such as the ones shown in Fig. 7 introduce
strong intersymbol interference (ISI), thus requiring powerful
nonlinear equalization techniques, combined with adaptive
modulation and precoding techniques. For example, adaptive
nonlinear equalizers such as the one proposed in [6], which
combines a recursive nonlinear symbol estimator with a channel
estimator with a low prediction order, may well cope with the
strong ISI introduced by the PL channel. Also, optimal soft
decisions may be useful in alleviating the performance degradation due to erroneous hard decisions [8]. As far as precoding
techniques, the knowledge that the channel is symmetric opens
the door to information theoretic considerations on optimal
transmission when the channel is known at the transmitter.
For example, treating known ISI as interference known at the
transmitter allows us to use “dirty paper” coding techniques [9].
The choice of discrete multitone (DMT) modulation may also
be an appropriate one for indoor PL channel. The considerations made in Section VI(a) and the symmetry property of the
transfer function reinforce the case of choosing DMT modulation for coping with the PL channel. In fact, DMT-based PL
modems can adaptively change the spectral efficiency in each
sub-band, depending on the particular state of the network, and
avoid loading those particular carriers located in sub-bands containing deep notches. Moreover, since a node in the network
that receives data must also estimate the transfer function of
the channel, it can perform optimal bit-loading adaptively, exploiting the knowledge that the channel in the reverse direction
2Home networking based on the exploitation of in-home telephone wiring is
today a commercially available product. The Home Phoneline Networking Alliance (HomePNA) consortium has established specifications for such modems.
HomePNA modems transmit and receive in both directions on a single wire pair,
using the two conductors in the middle of an RJ-11 or RJ-14 telephone jack.
VII. CONCLUSION
1878
IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 20, NO. 3, JULY 2005
REFERENCES
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[8] S. Galli, “Making power line communications more reliable by using
optimal soft information,” in Proc. IEEE Int. Conf. Power Line Communications Applications, Athens, Greece, Mar. 2002.
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Stefano Galli (S’95–M’98–SM’05) received the
M.S. and Ph.D. degrees in electrical engineering
from the University of Rome “La Sapienza,” Rome,
Italy, in 1994 and 1998, respectively.
After receiving the Ph.D. degree, he continued
as a Teaching Assistant in Signal Theory at the
Info-Com Department, University of Rome. In 1998,
he joined Bellcore (now Telcordia Technologies),
Piscataway, NJ, in the Broadband Networking
Research Department where he is now a Senior
Scientist. His main research efforts are devoted to
various aspects of xDSL systems, wireless/wired home networks, personal
wireless communications, power-line communications, and optical code-division multiple access (CDMA). His research interests include detection and
estimation, communications theory, and signal processing. He is a reviewer
for several IEEE journals and conferences and has published many papers.
He holds four patents and several pending ones. He also served as Co-Guest
Editor for the Feature Topic “Broadband is Power: Internet Access through
the Power Line Network” (IEEE Communications Magazine, May 2003), as
Co-Guest Editor for the Special Issue on Power Line Communications of the
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS.
Dr. Galli is serving as Chair of the IEEE Communications Society Technical
Subcommittee on power-line communications. He also served as Technical
Program Committee member of several IEEE conferences, such as the IEEE International Symposium on Power Line Communications (ISPLC’04, ISPLC’05,
ISPLC’06), the IEEE International Conference on Communications (ICC’04,
ICC’06), the IEEE Global Communications Conference (Globecon’06), the
IEEE Vehicular Technology Conference (VTC’04), and has served as the
General Co-Chair of the IEEE Workshop on Signal Processing Advances in
Wireless Communications (SPAWC’05).
Thomas Banwell (M’90) received the B.S. degree
in chemical physics from Harvey Mudd College,
Claremont, CA, in 1978, and the M.S. degree in
electrical engineering and Ph.D. degree in applied
physics from the California Institute of Technology,
Pasadena, in 1980 and 1986, respectively. He received an MD from UMDNJ-New Jersey Medical
School in 1997 and did an internship in internal
medicine.
He has been with Telcordia Technologies, Red
Bank, NJ, since 1986. He pursues problems related
to performance limitations in low-power/high-speed electronic circuits that
arise in public telecommunications network access and data processing applications and variable bit-rate optical transmission systems. His interests in
circuit theory have expanded to include modeling physiological processes such
as uterine contraction. He has authored or coauthored many technical papers
and has seven patents.