Topology estimation of a digital subscriber line

International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN
0976 – 6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 4, July-August (2013), © IAEME
101
TOPOLOGY ESTIMATION OF A DIGITAL SUBSCRIBER LINE
M.Bharathi#1
, S.Ravishankar#2
, Vijay Singh#3
#1, #2
Department of Electronics and Communication Engineering, R V College of Engineering
Bangalore, India,
#3
DRDO, Bangalore
ABSTRACT
Topology estimation of a Digital subscriber line (DSL) is critical for an operator to commit a
quality of service (QoS) requirement. Single Ended Loop Testing (SELT) is the most preferred and
economical way for estimating the copper loop topology. A new method employing a combination of
complementary code based Correlation Time Domain Reflectometry (CTDR) and Frequency
Domain Reflectometry (FDR) for loop topology estimation is developed. Use of existing modem
without any additional hardware in the measurement phase is the unique advantage of this method.
Since the measurement is done online the effect of cross talk and AWGN is also considered. In this
proposed SELT method approximate loop estimation is first obtained from CTDR measurements. An
optimization algorithm based on FDR is then used to predict a more accurate loop topology.
Employing FDR measured data and the FDR data of the approximate CTDR predicted topology, an
objective function is defined. The objective function is then minimized using Nelder-Mead
multivariable optimization method to get an accurate loop estimate. Tests carried out on typical
ANSI loops shows good prediction capability of the proposed method. No prior knowledge of the
network topology is required in this process. For the estimated loop topology capacity in terms of
data rate is calculated and is compared with the capacity of the actual loop.
Keywords: Digital subscriber line (DSL), Frequency domain Reflectometry (FDR), Correlation time
domain Reflectometry (CTDR), and Loop qualification.
I. INTRODUCTION
It is important for a service operator to evaluate the Quality of Service (QoS) afforded over a
subscriber loop under realistic circumstances. Apart from the data rate performance specified for
typical services, a QOS also prescribes the delay in the transmission (in ms), the packet loss and
BER. Subscriber line conditions include the transfer function of the line which is a function of the
line topology and noise Power Spectral Density (PSD). The line topology is unravelled first, and then
INTERNATIONAL JOURNAL OF ELECTRONICS AND
COMMUNICATION ENGINEERING & TECHNOLOGY (IJECET)
ISSN 0976 – 6464(Print)
ISSN 0976 – 6472(Online)
Volume 4, Issue 4, July-August, 2013, pp. 101-118
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102
performance is computed for the topology considering AWGN and crosstalk noise. A double ended
loop measurement allow easy estimation of loop impulse response and the noise PSD, but needs a
test device at the far end of the loop and is not economical prior to a service commencement. An
economical method would require a reuse of the network operator’s central office (CO) side DSL
modem resources to perform measurements.
The physical loop consists of gauge changes, bridge taps and loop discontinuities that result
in a change of characteristic impedance. The generated echo from these discontinuities when a signal
is injected into the physical loop is analysed to extract details of location and the type of
discontinuity. S. Galli et al [1-4] have employed pulse TDR based techniques to characterize the
loop. A pulse is considered as a probe signal and is transmitted through the loop and the reflections
produced by each discontinuity are observed in time. The time domain reflection which contains the
signature of the loop is then analysed to predict the loop topology. Clustering of the TDR trace [2-3]
and the use of statistical data [4] are included to reduce the time and to increase the accuracy
respectively. These techniques provide a good estimation of the loop but are computationally
intensive and cannot be easily implemented in current DSL modems. A more practical method
described by Carine Neus et al [6] uses one port scattering parameter S11 in time domain and
estimates the loop topology. The S11 measurement is however done off line with a vector network
analyser over the entire band width [5]. David E. Dodds [7, 8] has proposed FDR for identifying the
loop impairments. In the measurement phase a signal generator is used to probe the line up to 1.3
MHz in steps of 500 Hz and the reflections are coherently detected. However if there are multiple
discontinuities close to each other (<100m), detecting all discontinuities in a single step may not be
possible. If the discontinuities are far from each other the order of variation of the reflection makes it
difficult to predict all the discontinuities in a single step.
SELT Estimation is performed in three phases. The measurement phase during which CTDR
and FDR measurements are captured; termed as SELT – PMD function in G.SELT [20] and a second
phase called as interpretation phase when an analysis is done for topology estimation; termed as
SELT-P function in G.SELT. In a third phase the data rate is calculated (i.e again a SELT-P function)
for the estimated topology. No separate test equipments are required and the measurement is done
online in the bundle without a need to access copper. In this paper the logical interfaces required by
G.SELT [20] are retained. However the interpretation and analysis of the data is vendor specific and
this paper describes a novel method.
The measurement phase of the proposed method reuses the blocks of the current DSL modem
and hence only a small code is needed that can be easily compiled into any modem. In this step the
line is sounded sequentially once by employing CTDR and next by employing FDR. In the
interpretation phase, the first step consists in analysing CTDR results to obtain an approximate
estimation of the distance and the type of the discontinuities [9]. The topology learning from the
CTDR application is used to generate an FDR data for the estimated loop. In the second step of the
interpretation phase the generated FDR data is compared with a target (measured) FDR data in a
mean squared sense to arrive at an exact estimate of the loop topology. The analysis of measured
data may be performed in the modem to a limited extent or offline where more computing resources
are available. In the third phase the capacity is calculated from the accurately predicted topology
obtained in the previous loop estimation phase. Good predictability has been observed for a variety
of ANSI loops with different reach and with multiple bridge taps [11].
Section II of this paper details the hybrid method for loop topology estimation. In section III
measurement and interpretation phase of topology estimation are dealt along with the results for
ANSI loop topology. The bit rate of the estimated channel is calculated considering AWGN noise of
-140 dBm/Hz and cross talk.

103
II. HYBRID METHOD FOR LOOP TOPOLOGY ESTIMATION
CTDR method needs no prior knowledge of the loop for topology estimation but the accuracy
of prediction is limited due to the variation in propagation velocity with the frequency and the gauge
of the copper medium. On the other hand, FDR method can predict the topology with higher
accuracy but requires a reasonable initial knowledge of the topology. A hybrid method is developed
to overcome these limitations by combining the two methods. Approximate loop topology is
obtained with CTDR method and is used as the initial guess for FDR based optimization method for
accurate prediction of the topology. The details of the developed CTDR and FDR optimization
method are outlined below.
A. Complementary CTDR method
Spread spectrum (SS) techniques afford a possibility of providing measurements with
improved SNR without sacrificing response resolution. Proposed CTDR method uses the DMT
modem with its bit loading algorithms [11] for measurement. The PN sequence is distributed over all
the tones in every DMT symbol. The carriers employ Quadrature amplitude modulation (QAM)
constellation. The frequency domain signal generated by the constellation encoder is converted to a
time domain signal )(tp using inverse discrete Fourier transform (IDFT).This time domain signal
)(tp is transmitted through a loop with an echo transfer function )(th and correlated with its echo
signal )(tv at the receiver to obtain the correlated signal )(tW that is expressed as,
)1())(*)(*()()()()( thtpktptvtptW ⊗=⊗=
Where, k is the proportionality constant. If the autocorrelation of the probe signal can be
approximated as delta function, then the correlated signal is
)2(})(*))(({)( thtLktW δ=
Here, L is the number of elements in the code, operator ⊗ represents correlation operation and
operator * represents convolution operation.
In our implementation complementary codes are used as a probe signal to increase the range
of predictability. Complementary codes are set of codes whose out of phase autocorrelation sums to
zero. So the sum of the auto correlation of the two member sequence is a delta function [10].
)3(2 kkkkk LBBAA δ=⊗+⊗
Where, kδ is the delta function and kk BA , are the complementary code pairs of length L.
Ideally the auto correlation of the individual sequences ( kk BA , ) has side lobes but gets cancelled
when added together. The peak of added signal at zero shift will be 2L.
A complementary code of K
L 2= is generated with K=10. Tone numbers 0-511 are loaded
with 2 bits per tone with this L element code pair. Unipolar version of each of the complementary
codes ( uniuni BA , ) [10] and its one’s complementary form ( uniuni BA ',' ) are generated and these 4
codes are used to probe the line.
B. Application of Complementary codes for loop topology estimation
The steps involved in using the complementary codes for the loop topology estimation is
shown in Fig.1.
1. Generate complementary codes kA and kB .
)(tW

104
2. Generate the unipolar version and its one’s complemented form for kA and kB .
3. For uniA , simulate the reflected signal ( )(thuniA ∗ ) where, )(th is the impulse response of the
channel.
4. For uniA' , simulate the reflected signal ( )(' thuniA ∗ ).
5. Obtain the received signal for the sequence kA )(*')(* thuniAthiunAAr −=
6. Obtain the correlated signal kAArAW ⊗=
7. Repeat steps 3-6 for the second Golay sequence to obtain BW .
8. Sum BWAWW += .
uniA
uniA
uniB
uniB
A
W
B
W
W
+
+
−
−
+
+
A
r
B
r
)(th
Correlation
Correlation
)(th
)(th
)(th
kA
kB
uniA
uniA
uniB
uniB
A
W
B
W
W
+
+
−
−
+
+
A
r
B
r
)(th
Correlation
Correlation
)(th
)(th
)(th
uniA
uniA
uniB
uniB
A
W
B
W
W
+
+
−
−
+
+
A
r
B
r
)(th
Correlation
Correlation
)(th
)(th
)(th
kA
kB
Fig.1. Functional diagram of Complementary CTDR for loop testing
The position of the peak in the correlated signal (W) used to estimate the location of
discontinuity )(d is given by
)4(
2
max.tv
d =
Where, v is the velocity of propagation in the twisted pair and maxt is the peak position.
The peak amplitude of the signal depends on the length of the loop section and the type of
discontinuity (reflection coefficient). Using the first peak amplitude of the correlated signal the
reflection coefficient of the first discontinuity can be estimated. The peak amplitude for an open
circuit discontinuity (reflection coefficient = 1) for different lengths is plotted in Fig.2. Using Fig.2
as reference, for a CTDR estimated first segment length and amplitude, the reflection coefficient can
be calculated.
)5(
1ρwithlengthestimatedtheforAmplitude
lengthestimatedtheforAmplitude
)()1(
=
=fρ
0 2 4 6 8 10 12
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Length of the line (Kft)
Peakamplitude
26 AWG
24 AWG
7 8 9 10 11 12
0
1
2
x 10
-3
0 2 4 6 8 10 12
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Peakamplitude
26 AWG
24 AWG
7 8 9 10 11 12
0
1
2
x 10
-3
Fig.2. Peak amplitude variation of the reflected signal with line length

105
When the discontinuities are closely spaced it is difficult to distinguish the cross correlation
peaks. Step by step ML principle [4] is used in identifying the discontinuity type at the calculated
location. Data de-embedding technique is used to mask the reflections of known discontinuities
from the measured echo signal to unravel the signatures of the unknown discontinuities. Thus by
incorporating the de-embedding process, overall predictability of the CDTR method is enhanced.
After identifying each discontinuity (i) in a successive manner, an auxiliary topology ( )(i
Aux )
is formed which consists of all the previously identified discontinuities followed by an infinite loop
section. The reflection ( ir ) due to this auxiliary topology is generated and is removed from the total
reflection )(tv to get a de-embedded TDR trace iD .
)6()( ii rtvD −=
The trace Di consists of echoes from the rest of discontinuities in the line and is correlated
with the input signal )(tp to arrive )(tWi . )(tWi is the correlated signal after removal of echoes from
the known discontinuities (first i discontinuities) and hence brings out the next peak (i+1) and
discontinuity. This process is continued until there is no identifiable peak in the resultant signal. In
this way after identifying each discontinuity the reflection due to the identified discontinuity is
removed from the total reflection to enhance the predictability of the following discontinuities.
The effect of AWGN (-140dBm/Hz) and cross talk is added in the simulation as the measurement
is done online. Cross talk is a slowly varying signal across the symbols and so gets cancelled due to
the subtraction of the reflected signal (step 5 &7) shown in Fig.1. To mitigate the effect of AWGN
noise, averaging over number of symbols is carried out. This averaging improves the signal to noise
ratio (SNR) and hence increases the dynamic range.
The predicted line topology (Ф) from CTDR contains length and gauge of all the line sections.
The prediction accuracy is improved using the proposed FDR based optimization method. This
optimization method works by comparing FDR signal of predicted and actual loops. The FDR
received signal for the predicted topology Ф is simulated using the mathematical model described in
the next section.
C. Model for the FDR received signal
In FDR method the PN sequence is sent sequentially (one tone in each DMT symbol). The
received echo signal is a function of the reflection (ρ) and transmission (τ) coefficients at each
discontinuity.
The reflection coefficient (ρ) [16] is
)7()(
ZbZa
ZbZa
f
+
−
=ρ
Where, Za and Zb are the frequency dependent characteristic impedance before and after the
discontinuity. Similarly, τ is given by [16]
)8(
2
)(
ZbZa
Za
f
+
=τ
In the above equations, ρ and τ varies with frequency as the characteristic impedance is a
function of frequency which is given by [18],
)9(
CjG
LjR
Z
ω
ω
+
+
=
The frequency dependant RLCG parameters in the above equation are obtained empirically as
described in [18] and used in our computation for the transfer function of the 24 AWG and 26 AWG
UTP lines. In the equations that follow we assume that the transmitted signal is a Discrete Multitone

106
signal with ‘N’ tones conforming to the tone spacing and bandwidths as detailed in the DSL
standards [11, 12].
The observed reflected signal along with the effect of noise, when the nth
tone is sounded is
given by
)10(
1
)()()()( ∑
=




 +=
M
i
fNonfiRnfR n
Here )()(
n
i
fR is the received signal from the ith
echo path when the nth
bin is sounded. Here
)( nfNo is the noise in the nth
tone and M is the number of echo paths in the loop. Further,
)11()()()()()(
nn fiHechofSnfiR =
Where )( nfS is the spectrum of the transmitted data and the )()(
nfiHecho is the transfer function of
the ith
echo path and is given by
)12()()()()())1(,...)2(,)1(()()(
nnn fifiHiFfiHecho ρτττ −=
Here ))1(,...)2(,)1(( −iF τττ is a frequency dependant function that includes the transmission
coefficients of all the discontinuities preceding the ith
discontinuity and )()(
nfiρ is the reflection
coefficient of the ith
discontinuity. )()(
nfiH is the transfer function of the round trip path. The total
received signal is sum of received signal of over all the tones.
∑=
n
fnRfR )13()()(
D. FDR Optimization
An iterative optimization process based on FDR method is developed. Nelder-Mead
algorithm is chosen for this optimization as it can solve the multidimensional unconstrained
optimization problems by minimizing the objective function. Tone numbers 6-110 is sounded
individually with two bits as lower frequency tone offer lower attenuation and hence better range.
The steps involved in this algorithm is
1. Simulate FDR received signal for the guess topology ),( fnR Φ .
2. Obtain an FDR measurement ( ))(ˆ fnR .
3. Calculate the objective function (MSE)
)14()(ˆ),(
1
2








−Φ= ∑
=
N
n
fnRfnROE
4. Obtain the accurate line topology by minimize OE using Nelder-Mead simplex
optimization algorithm.
Nelder-Mead optimization algorithm iteratively improves Ф in terms of line segment lengths
until the best solution (close match) is found. This algorithm works with constructing vectors with
updating each variable (Each line segment lengths) of Ф, one at a time by increasing 5%. Initial
simplex consists of the newly created ‘n’ vectors along with Ф. The algorithm updates the simplex
repeatedly until the best solution is found. Nelder-Mead algorithm has a limitation that it can
converge to local minima. To overcome this local minima problem optimization is performed with a
different initial guess whenever the objective function value is greater than 1e-4.

107
The flowchart of the complete proposed method is shown in the Fig.3.
Correlate
with input
Estimate
discontinuity
distance
Estimate
discontinuities by
maximum likelihood
and successive
decomposition
R(Ф,fn)
R^ (fn)Error
function
Correlation TDR for approximate topology
FDR based optimization
Measured echo
signal in time
domain
Measured echo
signal in freq
domain
Approximate
topology(Ф)
Error minimization using
Nedler- Mead algorithm
Final predicted
topology
Correlate
with input
Estimate
discontinuity
distance
Estimate
discontinuities by
maximum likelihood
and successive
decomposition
R(Ф,fn)
R^ (fn)Error
function
Correlation TDR for approximate topology
FDR based optimization
Measured echo
signal in time
domain
Measured echo
signal in freq
domain
Approximate
topology(Ф)
Error minimization using
Nedler- Mead algorithm
Final predicted
topology
Fig.3. Flow chart of the proposed method
III. SIMULATION RESULTS AND DISCUSSION
Test loops are defined to emulate possible scenarios as per ITU recommendation 996.1[11]
shown in Fig.4, that include a variety of reach, gauge change and bridge taps. The applicability of the
method is tested in the presence of AWGN (-140 dBm/Hz ) and the cross talk defined in [11].
Test loop4
12 Kft
26AWG
Test loop3
Test loop2
Test loop1
9 Kft 4 Kft
26 AWG 24AWG
3 Kft
26AWG
6 Kft
26AWG
0.5 Kft
26 AWG
9 Kft 2 Kft 0.5 Kft 0.5 Kft
26 AWG 24 AWG 24 AWG 24 AWG
1.5 Kft
26 AWG
Fig.4. Test loops
Test loop1: Correlation results in amplitude versus time lag and is converted to the desired units of
amplitude versus distance using equation 4. For test loop 1 the variation of correlation amplitude
with distance is shown in Fig. 5. The positive peak indicates the discontinuity as an open end of loop
as the other possible discontinuities (bridge tap and gauge change) have negative reflection
coefficient [1]. The possible gauge types with 12.71 Kft length and open end termination are listed in
Table 1 along with the mean square error between the simulated TDR of the possible topology and
the target TDR. For the predicted line (12.71 Kft, 26 AWG), error of 5% in the length is observed
which is due to the use of average VoP in the distance calculation. The CTDR predicted topology is
used as an initial guess for the FDR based optimization method.

108
0 10 20 30 40
-6
-4
-2
0
2
4
6
8
x 10
-6
X: 12.71
Y: 5.318e-006
Distance(Kft)
Crosscorrelationamplitude
Fig.5. Distance Vs correlation amplitude for test loop1
TABLE 1
POSSIBLE TOPOLOGIES FOR TEST LOOP1
Sl. No Hypothesized discontinuity Possible topology MSE
1 End of loop(open) 12.7 Kft line,26 AWG 8.70e-5
2 End of loop(open) 12.7 Kft line, 24 AWG 5.83e-4
The FDR signal for test loop 1 is shown in Fig.6. It is observed that the signal amplitude is
low in the order of 1e-3 and the rate of decay is steep. Fig.7 shows the convergence plot using the
Nelder-Mead algorithm with CTDR predicted loop as an initial guess. With 16 iterations MSE is
converged to 8.05e-6 and the predicted line is 12.0001 Kft 26 AWG.
0 1 2 3 4
x 10
5
-2
0
2
4
6
8
x 10
-4
ReflectedSignal(Volts)
Frequency (Hz)
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6
1
2
3
x 10
-5
ReceivedechoSignal(Volts)
0 1 2 3 4
x 10
5
-2
0
2
4
6
8
x 10
-4
Frequency (Hz)
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6
1
2
3
x 10
-5
0 1 2 3 4
x 10
5
-2
0
2
4
6
8
x 10
-4
Frequency (Hz)
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6
1
2
3
x 10
-5
Fig.6. FDR signal for test loop1
0 2 4 6 8 10 12 14 16
10
-6
10
-5
10
-4
10
-3
Iteration
MSE
Current Function Value: 8.0545e-006
Fig.7. Convergence plot for test loop1

109
Test loop2: Fig. 8 shows the correlation amplitude variation with distance for test loop2. The
amplitude of the peak at 9.03Kft is -3.54e-6 and indicates a negative reflection coefficient at first
discontinuity. All the possible topologies with this observation and their mean square error with
actual echo signal are listed in Table 2
0 10 20 30 40
-5
-4
-3
-2
-1
0
1
2
x 10
-6
X: 9.039
Y: -3.54e-006
Distance(Kft)
Fig.8. Distance Vs correlation amplitude for test loop2
TABLE 2
POSSIBLE TOPOLOGIES AT FIRST DISCONTINUITY (TEST LOOP2)
Hypothesized type Possible topology (dotted
line indicates infinite
length)
MSE
Gauge change
26 AWG
9.03 Kft
24AWG26 AWG
9.03 Kft
26 AWG
9.03 Kft
24AWG
1.05e-6
Bridge tap
(Taps with Open
end) 26 AWG
9.03 Kft
26AWG
26AWG
26 AWG
9.03 Kft
26 AWG
9.03 Kft
26AWG
26AWG 1.11e-5
Bridge tap
(Taps with Open
end) 26 AWG
9.03 Kft
26AWG
24AWG
26 AWG
9.03 Kft
26 AWG
9.03 Kft
26AWG
24AWG 1.25e-5
Bridge tap
(Taps with Open
end) 24 AWG
9.03 Kft
24AWG
24AWG
24 AWG
9.03 Kft
24 AWG
9.03 Kft
24AWG
24AWG 1.11e-4
Bridge tap
(Taps with Open
end) 24 AWG
9.03 Kft
24AWG
26AWG
24 AWG
9.03 Kft
24 AWG
9.03 Kft
24AWG
26AWG 1.03e-4
From the above table, topology with gauge change (first row in Table 2) is identified as the
correct topology till first discontinuity. The correlated signal after the removal of the echo due to first
discontinuity ( )(1 tW ) is given in Fig.9 which indicates the next discontinuity at 13.56Kft (segment
length = 13.56Kft–9.03Kft). The type of discontinuity is concluded as end of loop as the reflection
coefficient of this discontinuity is positive and there is no bridge tap identified previously in the loop.
Thus, the CTDR predicted loop topology is 9.03 Kft of 26 AWG followed by 4.53 Kft of 24 AWG.

110
0 10 20 30 40
-3
-2
-1
0
1
2
3
4
x 10
-6
X: 13.56
Y: 2.915e-006
Distance(Kft)
Fig.9. De-embedded signal ( )(1 tW )for test loop2
The FDR reflection for this test loop is shown in Fig.10. Fig.11 shows the MSE variation for
a range of line sections. This figure indicates the variation of MSE with the prediction accuracy of
the individual line segments. Nelder-Mead optimization algorithm with the CTDR predicted loop as
an initial guess estimates the line as 9.00 Kft in series with 3.99 Kft and the MSE is 6.62e-06. The
convergence plot using Nelder-Mead algorithm is shown in Fig.12.
0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 10
5
-2
-1
0
1
2
3
4
5
x 10
-4
Frequency (Hz)
2.5 3 3.5 4 4.5
0
2
4
6
8
x 10
-6
0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 10
5
-2
-1
0
1
2
3
4
5
x 10
-4
Frequency (Hz)
2.5 3 3.5 4 4.5
0
2
4
6
8
x 10
-6
0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 10
5
-2
-1
0
1
2
3
4
5
x 10
-4
Frequency (Hz)
2.5 3 3.5 4 4.5
0
2
4
6
8
x 10
-6
Fig.10. FDR signal for test loop2
8
8.5
9
9.5
10
3
3.5
4
4.5
5
0
2
4
6
8
x 10
-3
1s t line length2nd line length
MSE
Fig.11. Error surface for test loop 2

111
0 5 10 15 20 25 30 35
10
-6
10
-5
10
-4
10
-3
Iteration
MSE
Predicted Topology
9Kft
26AWG
3.99Kft
24AWG
0 5 10 15 20 25 30 35
10
-6
10
-5
10
-4
10
-3
Iteration
MSE
Predicted Topology
9Kft
26AWG
3.99Kft
24AWG
Predicted Topology
9Kft
26AWG
3.99Kft
24AWG
Fig.12.Convergence with final predicted topology for test loop2
Test loop 3: Correlation amplitude versus distance plot for test loop 3 is shown in Fig.13. A bridge
tap has two reflections: one from the location of bridge tap with reflection coefficient of ~ (-0.3) and
the other from the open end of the bridge tap with reflection coefficient equal to 1. Hence a negative
peak followed by positive peak within a very short distance is expected. In Fig.13, the amplitude
value of the first peak at 3.107 Kft is -0.001976 which corresponds to negative reflection coefficient.
All the possible topologies are listed in Table 3 along with their computed mean square error with
actual signal.
5 10 15 20 25 30 35 40
-2
-1.5
-1
-0.5
0
0.5
1
1.5
x 10
-3
X: 3.107
Y: -0.001976
Distance(Kft)
Fig.13. Distance Vs Correlation amplitude for test loop 3
From the table, topology with bridge tap (Second row in Table 3) of 26 AWG lines is
identified as the Auxiliary topology ( )1(
Aux ) till first discontinuity. The correlated signal after the
removal of the echo due to this first identified topology ( )(1 tW ) is given in Fig.14. Further de-
embedding using the identified tap length of 0.57Kft (3.67Kft –3.10 Kft) results in )(2 tW (Fig. 15)
which helps in predicting the next segment length as 6.22Kft (9.32Kft – 3.10Kft) and the identified
discontinuity is end of loop with open termination.

112
TABLE 3
POSSIBLE TOPOLOGIES AT FIRST DISCONTINUITY (TEST LOOP3)
Hypothesized
discontinuity
Possible topology
(dotted line
indicates infinite
length)
MSE
Gauge change
26 AWG
3.10 Kft
24AWG26 AWG
3.10 Kft
26 AWG
3.10 Kft
24AWG
0.0029
Bridge tap
(Taps with
Open end) 26 AWG
3.10 Kft
26AWG
26AWG
26 AWG
3.10 Kft
26 AWG
3.10 Kft
26AWG
26AWG 0.0014
Bridge tap
(Taps with
Open end) 26 AWG
3.10 Kft
26AWG
24AWG
26 AWG
3.10 Kft
26 AWG
3.10 Kft
26AWG
24AWG 0.0020
Bridge tap
(Taps with
Open end) 24 AWG
3.10 Kft
24AWG
24AWG
24 AWG
3.10 Kft
24 AWG
3.10 Kft
24AWG
24AWG 0.0020
Bridge tap
(Taps with
Open end) 24 AWG
3.10 Kft
24AWG
26AWG
24 AWG
3.10 Kft
24 AWG
3.10 Kft
24AWG
26AWG 0.0018
Fig.14. De-embedded signal ( )(1 tW )for test loop3 Fig.15. De-embedded signal ( )(2 tW )for test loop3
The CTDR predicted topology is shown in Fig. 16.
3.10 Kft
26AWG
6.22 Kft
26AWG
0.57 Kft
26 AWG
3.10 Kft
26AWG
6.22 Kft
26AWG
0.57 Kft
26 AWG
Fig. 16. CTDR predicted topology for test loop 3
0 10 20 30 40
-6
-4
-2
0
2
4
6
8
x 10
-4
X: 3.672
Y : 0.0008338
Distance(Kft)
0 10 20 30 40
-5
0
5
10
x 10
-6
X: 9.322
Y: 1.105e-005
Distance(Kft)

113
This CTDR predicted topology is used as initial guess for the FDR optimization. The FDR
signal for loop 3 is shown in Fig.17.
0 1 2 3 4 5
x 10
5
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
Frequency (Hz)
0 1 2 3 4 5
x 10
5
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
Frequency (Hz)
Fig.17. FDR received signal for test loop 3
A sensitivity study of the line segments on the variation of MSE is shown in Fig.18 and it is
inferred that the received echo signal is a strong function of the first section of the line. Optimization
algorithm predicts the line topology as 3.00 Kft parallel with 6.00 Kft with the bridge tap length of
0.5Kft. Final MSE of 7.11e-6 confirms fully converged solution.
2
2.5
3
3.5
4
5
6
7
0
0.05
0.1
0.15
0.2
First line Length2nd Line Length
MSE
Fig. 18. Error surface for test loop 3
Test loop 4: Fig. 19 shows the correlated signal amplitude with distance for test loop 4. The negative
peak with amplitude -3.19e-5 at 9.47 Kft indicates negative reflection coefficient and all the possible
topologies are analyzed and the discontinuity is identified as gauge change.
0 5 10 15 20 25 30 35
-4
-3
-2
-1
0
1
x 10
-6
X: 9.47
Y: -3.193e-006
Distance(Kft)
Fig.19.Distance Vs Correlation amplitude for test loop 4

114
)(3 tw(c):
0 10 20 30 40 50
-3
-2
-1
0
1
x 10
-6
X: 11.48
Y: -2.79e-006
Distance(Kft)
0 10 20 30 40 50 60
-1
0
1
2
3
x 10
-6
X: 13.49
Y: 1.668e-006
Distance(Kft)
0 10 20 30 40
-2
0
2
4
6
x 10
-7
X: 13.77
Y: 3.799e-007
Distance(Kft)
9.47 Kft 2.01 Kft 2.29 Kft
2.01 Kft
26AWG
26AWG 24AWG 24AWG
)(1 tw(a): )(2 tw(b):
(d): CTDR Predicted Topology
Fig.20. De-embedded signals for test loop 4
Removing the reflection from this identified segment, )(1 tW (Fig. 20(a)) helps in predicting
the next discontinuity of bridge tap at 11.48 Kft (Length of second segment is 11.48Kft-9.47Kft).
Construction of possible topologies at the second discontinuity (Table 4) helps in identifying the
gauges of all the line segments. De-embedding the next reflection based on the identified information
( )(2 tW ) helps in predicting the complete topology.
TABLE 4
POSSIBLE TOPOLOGIES AT SECOND DISCONTINUITY (TEST LOOP4)
Hypothesized discontinuity possible topology (dotted
line indicates infinite
length)
MSE
Gauge change followed by
bridge tap(Taps -Open end)
26AWG 24AWG
9.47 Kft 2.01 Kft
24AWG
26AWG
26AWG 24AWG
9.47 Kft 2.01 Kft
24AWG
26AWG 4.04e-6
Gauge change followed by
bridge tap (Taps-Open end)
26AWG 24AWG
9.47 Kft 2.01 Kft
24AWG
24AWG
26AWG 24AWG
9.47 Kft 2.01 Kft
24AWG
24AWG 4.40e-6
The CTDR estimation for test loop 4 is not complete. This is due to the very less contribution
of the far end reflection in the overall received signal. With this as the initial guess, FDR prediction
converged to MSE error of 0.0028. As the MSE error is higher than the threshold level (1e-4), further
iterations are attempted with modified initial guesses but no improvement on the MSE error is
achieved. This issue of non convergence is due to wrong specification of number of discontinuities
as the initial guess. To address these types of scenarios, the hybrid method is further improved with

115
an added convergence check which adds a discontinuity to the initial guess as given in Fig.21. As the
CTDR is capable of predicting first two discontinuities and from practical understanding the
maximum number of discontinuities is not more than 4, this outer loop is set with a maximum limit
of two.
Step 1 : CTDR for initial
estimation,
Initialize COUNT
Step 2 : FDR estimation
MSE <
1e-4
Final topology
Update the initial guess,
Increment COUNT
Yes
No, COUNT < 3
Update the initial guess with
an additional bridge tap,
Initialize COUNT
No, COUNT = 3
Step 1 : CTDR for initial
estimation,
Initialize COUNT
Step 2 : FDR estimation
MSE <
1e-4
Final topology
Update the initial guess,
Increment COUNT
Yes
No, COUNT < 3
Update the initial guess with
an additional bridge tap,
Initialize COUNT
No, COUNT = 3
Fig.21. Improved Hybrid method
The converged line topology from the modified method is shown in Fig.22.
0 50 100 150 200 250
10
-5
10
-4
10
-3
Iteration
MSE
Current FunctionValue:2.2193e-005
Predicted Topology
9Kft
26AWG
0.49Kft
24AWG
1.42 Kft
26 AWG
0.75Kft
24AWG
1.47 Kft
26 AWG
1.97Kft
24AWG
0 50 100 150 200 250
10
-5
10
-4
10
-3
Iteration
MSE
Current FunctionValue:2.2193e-005
Predicted Topology
9Kft
26AWG
0.49Kft
24AWG
1.42 Kft
26 AWG
0.75Kft
24AWG
1.47 Kft
26 AWG
1.97Kft
24AWG
Predicted Topology
9Kft
26AWG
0.49Kft
24AWG
1.42 Kft
26 AWG
0.75Kft
24AWG
1.47 Kft
26 AWG
1.97Kft
24AWG
Fig.22. Convergence with the final predicted topology for test loop 4
First two line segments and the first bridge tap length of test loop4 are predicted with good
accuracy but the segment 3, 4 and tap2 length are not accurate. Fig. 23 shows the schematic
representation of the reflection from each discontinuity for test loop 4. Strength of the reflection from
each junction is calculated based on the reflection and transmission coefficients for comparison.
Signal attenuation due to the line length and gauge is not considered in this investigation as the focus
is to quantify the effect of individual reflections on the final echo. Table 5 compares the %

116
contribution of each reflection in the received signal. More than 90 % of the received signal is
contributed by the reflections R1, R2 and R3. Even though R4 reflection is considered as 5 %, due to
the higher distance of travel, attenuation will be higher and hence net overall contribution in the
received signal will be much lesser than 5%. R5 and R6 have less than 2% weightage in the received
signal even without considering the attenuation effect. This results in very feeble contribution in the
measured echo. Hence accuracy of these line segments, in the predicted topology does not influence
MSE to a significant level. This explains the reason for higher prediction error in the segments 3, 4
and tap 2 of test loop 4.
R1 R2
R3
R4
R5
R6
Fig.23. Reflections for test loop4
Line Capacity for a specified transmit energy and margin is obtained based on the SNR
profile using water filling algorithm [15]. Number of bits that can be loaded in each tone is
calculated by estimating the SNR profile at the receiver. The transfer function of the predicted loop
topology needed to compute the SNR is obtained from the Transmission matrix (ABCD) as in [14,
15]. The cross talk and the AWGN noise are also considered in this performance computation. Table
6 provides a comparison of the capacities of the estimated loops.
TABLE 5
REFLECTION STRENGTH CONTRIBUTION
Reflection
Weightage of transmission and Reflection coefficients
in the received signal (Excluding the attenuation
effect)
Sequence of approximate
reflection and transmission
coefficients
Total %
contribution
in the received
signal(Ri/∑Ri)
R1 0.03 0.03 6.8
R2 0.97*0.3*0.97 0.2823 64.5
R3 0.97*0.3*1*0.3*0.97 0.0847 19.4
R4 0.97 *0.3*0.3*0.3*0.97 0.0254 5.8
R5 0.97*0.3*0.3*1*0.3*0.3*0.97 0.0076 1.7
R6 0.97*0.3*0.3*1*0.3*0.3*0.97 0.0076 1.7
∑Ri 0.4376

117
TABLE 6
ESTIMATED CAPACITY FOR THE TEST LOOPS
Test
Loop
Actual loop topology Predicted Topology Downstream
capacity of the
actual
loop(Mbps)
Downstream
capacity of the
predicted
loop(Mbps)
1 12 Kft,26 AWG 12.0Kft, 26 AWG 1.732 1.732
2 9 Kft, 26 AWG –
4 Kft 24 AWG
9 Kft, 26 AWG –
4 Kft 24 AWG
1.712 1.712
3 3 Kft, 26 AWG –
(0.5 Kft ,26 AWG)*–
6 Kft, 26 AWG
3 Kft, 26 AWG –
(0.5Kft,26AWG)*–
6 Kft, 26 AWG
2.812 2.812
4 9 Kft,26 AWG –
2Kft, 24 AWG –
(1.5 Kft,26 AWG)* –
0.5 Kft, 24 AWG –
(1.5 Kft, 26 AWG)*–
0.5 Kft, 24 AWG
9 Kft, 26 AWG –
2 Kft, 24 AWG –
(1.5Kft,26 AWG) *–
0.8 Kft, 24 AWG –
(1.1Kft,26 AWG)* –
0.7 Kft, 24 AWG
1.25 1.16
* - bridge tap line segments
IV. CONCLUSION
Performance of a DSL line is estimated with a hybrid SELT method for topology prediction
followed by data rate calculation using water filling algorithm. A two step CTDR-FDR combined
SELT method is developed to predict the twisted pair loop topology. In the first step CTDR
measurements are used to estimate the loop discontinuities as an initial guess. This estimate is further
refined using FDR based optimization method with a target (measured) FDR results.
Results of selected ANSI test loops show that this method can predict the topology with less
than 0.2 % error for single discontinuity loops. For lines with more discontinuities, the prediction
accuracy is good for the segments which contribute high for the reflected signal. Since the method is
based on matching the estimated loop reflection with the actual reflection; the prediction is not very
accurate for loops with segments that do not affect the transfer function significantly.
Maximum data rate for the predicted loops are calculated with estimated transfer function
using water filling algorithm with considering the cross talk and the AWGN noise.
V. REFERENCES
[1] Stefano Galli , David L.Waring ,”Loop Makeup Identification Via Single Ended
Testing :Beyond Mere Loop Qualification,” IEEE Journal on Selected Areas in
Communication, Vol. 20, No. 5, pp. 923-935, June 2002.
[2] Stefano Galli, Kenneth J.Kerpez, “Single-Ended Loop Make-up Identification –Part I:A
method of analyzing TDR Measurements,” IEEE Transactions on Instrumentation and
Measurement, Vol. 55, No. 2, pp. 528-537, April 2006.
[3] Stefano Galli , Kenneth J. Kerpez, “Signal Processing For Single-Ended Loop Make-Up
Identification,” in proceedings IEEE 6th Workshop on Signal Processing Advances in
WireIess Communications, pp. 368-374, 2005.
[4] Kenneth J.Kerpez, Stefano Galli, “Single-Ended Loop Make-up Identification –Part II:
Improved Algorithms and Performance Results,” IEEE Transactions on Instrumentation and
Measurement, Vol. 55, No. 2, pp. 538-548, April 2006.

118
[5] Tom Bostoen,Patrick Boets, Mohamed Zekri,Leo Van Biesen, Daan Rabijns, ”Estimation of
the Transfer function of a Subscriber Loop by means of a One-port Scattering Parameter
Measurement at the Central Office,” IEEE Journal on Selected Areas in Communciations, Vol.
20, No. 5, pp. 936-948, June 2002.
[6] Carine Neus,Patrick Boets and Leo Van Biesen, “Transfer Function Estimation of Digital
Subscriber Lines with Single Ended Line Testing,” in proceedings Instrumentation and
Measurement Technology Conference 2007.
[7] David E. Dodds, “Single Ended FDR to Locate and Specifically Identify DSL Loop
Impairments,” in proceedings IEEE ICC 2007, pp. 6413- 6418.
[8] David E. Dodds, Timothy Fretz, “Parametric Analysis of Frequency Domain Reflectometry
Measurements,” in proceedings Canadian Conference on Electrical and Computer
Engineering 2007, pp. 1034-1037, 2007.
[9] M.Bharathi, S.Ravishankar, “Loop Topology Estimation Using Correlation TDR,” in
Proceedings International Conference on Communication, computers and Devices, IIT,
Kharagpur, India, December 10-12, 2010.
[10] Moshe Nazarathy ,S.A Newton, R.P Giffard, D.S. Moberly .F. Sischka , W.R. Trutna ,S.Foster,
“Real Time Long Range Complementary Correlation Optical Time Domain Reflectometer,”
Journal of Lightwave Technology, Vol. 7, No. 1, pp. 24-38, January 1989.
[11] Test procedures for digital subscriber line (DSL) transceivers, Telecommunication
standardization sector of ITU std. G.996.1, 02/2001.
[12] Asymmetric digital subscriber line transceivers – 2 (ADSL2), Telecommunication
[13] Very high speed digital subscriber line transceivers 2 (VDSL 2), Telecommunication
[14] Dr.Walter Y.Chen , “DSL Simulation Techniques and Standards Development for Digital
Subscriber Line Systems”, Macmillan Technical Publishing.
[15] T.Starr, J.M.Cioffi, and P. J. Silverman,Eds., Understanding Digital Subscriber Line
Technology, New York: Prentice Hall,1999.
[16] John D.Ryder , Networks,Lines and Fields, Prentice Hall.
[17] Simon Haykins , Communication Systems, 4th
edition, John Wiley & Sons.
[18] Dr.Dennis J.Rauschmayer, ADSL/VDSL Principles, Macmillan Technical publishing, 1999.
[19] Gi-Hong Im, “Performance of a 51.84-Mb/s VDSL Transceiver Over the LoopWith Bridged
Taps,” IEEE Transcation on Communications, Vol. 50, No. 5, pp.711-717, May 2002.
[20] Single-ended line testing for digital subscriber lines (DSL), Telecommunication
[21] Raj Kumar Tiwari, Sachin Kumar and G R Mishra, “A Class Ab Ccii Topology Based
on Differential Pair with Modified Output Stage”, International Journal of Electrical
Engineering & Technology (IJEET), Volume 4, Issue 1, 2013, pp. 68 - 74, ISSN Print:
0976-6545, ISSN Online: 0976-6553

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