eTDR Fault Localization Accuracy - The effect of Multiple Impedance mismatches

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eTDR Fault Localization Accuracy
The effect of multiple Impendence mismatch
By:
Francisco J. Gastiazoro
Product Manager – Broadband Business Unit
Intraway Corporation
Campillo 2541 – Buenos Aires – Argentina
+54 (911) 2184-6020
francisco.gastiazoro@intraway.com

All contents are Copyright © 2015 Intraway Corporation. All rights reserved. Important Notices and Privacy Statement.
Contents_____________________________________________________________________________ 2
Introduction (Executive Summary)________________________________________________________ 3
Micro-reflections in Transmission Lines____________________________________________________ 3
Standing Waves_____________________________________________________________________________ 3
Impedance Mismatch ________________________________________________________________________ 4
Return Loss ________________________________________________________________________________ 5
Reflectometers and Micro-reflections ___________________________________________________________ 6
DOCSIS Adaptive Equalization and Pre-equalization__________________________________________ 9
Fault Localization_____________________________________________________________________ 13
Using Time Domain Reflectometer (TDR) _______________________________________________________ 13
Parabolic Interpolation Algorithm _____________________________________________________________ 14
Multiple Impedance mismatches. _____________________________________________________________ 16
Using Micro-reflection Signatures and Geo-reference _____________________________________________ 18
Conclusions _________________________________________________________________________ 21
Bibliography_________________________________________________________________________ 21

Impedance mismatches are present in many transmission lines, including coaxial cables in HFC outside plants that
create echo, known as micro-reflection. From DOCSIS 1.1, the standard provides means to undermine this effect
by applying equalization techniques in the transmitter and the receptor.
Adaptive equalization filters located at the Cable Modem and the CMTS are the response to most linear
impairments such as micro-reflections. This solution improves typical monitoring indicators such as the RxMER.
Monitoring tools that only focus on degradation of such data loses an important source of proactive maintenance.
This paper introduces detailed information about this effect in order to provide a theoretical and technical
background which helps understand how to interpret pre-equalization data that helps in fault localization.
The obtained conclusions will allow cable operators to effectively use TDR and micro-reflection signatures along
with Cable Modem geo-reference to narrow down the possible locations of the impairment and respond swiftly to
proactive indicators before it affects the customer’s satisfaction.
Standing Waves
When a wave that travels through a medium faces another one with the same frequency and amplitude doing the
same in the opposite direction, we are in the presence of a standing wave. This phenomenon is encountered in
many scenarios in real life such as in sound propagation in guitar’s strings and, of course, in what is the subject of
this paper, electric signals.
What you can see in the line is a series of nodes (zero displacement) and anti-nodes (maximum displacement) at
fixed points. Figure 1 depicts this effect throughout a transmission line at two different moments.
Figure 1 – Standing Waves
When the amplitude of the second wave, the one travelling in the opposite direction, is not equal to the original one,
we are in the presence of a partial standing wave. Due to losses in the transmission lines and other effects later
studied, this is the scenario we are finding in real life applications. This real life standing wave is presented in the
next figure and shows a scalloped sinusoidal form that varies in function of the Reflection Coefficient.

Γ = 1 Γ = 0.5 Γ = 0.1
The degree to which the resulting wave is a pure standing wave or just a travelling wave is measured by the
Standing Wave Ratio (SWR) also called Voltage Standing Wave Ratio (VSWR) when referring to electric signals.
VSWR is defined as the ratio of the partial standing wave’s amplitude at an antinode (maximum) to the amplitude at
a node (minimum) along the line. As the purpose of the paper is to analyze the effects on transmission signals, we
are going to use VSWR. This indicator is of much importance for further analysis and it is defined as:
(1)
Impedance Mismatch
The most basic model of a transmission system is comprised of a Source, a lossless Transmission Line and the
Load (the receiver). Each element has its own impedance, known as the characteristic impedance Z. The condition
that is assumed when analyzing the system’s behavior is that all impedances are the same. In other words Zs = ZT
= ZL. This implies that all the signal energy is absorbed by the load.
Figure 2 – Transmission Line Model
That model is especially useful to understand how the system will behave under ideal/theoretical conditions (where
there is no loss in any form).
When the impedances of the transmission line and the load are not the same we are in front of impedance
mismatch. Impedance mismatches behave like mirrors, all or some of the energy delivered to the load is reflected
back to the line, causing standing waves. A reflectometer is any element in the line that creates reflected waves,
therefore, impedance mismatches act as reflectometers.
In real life, there is no perfect match between any two elements in the transmission line; every connector, amplifier,
node, splitter, coupler, power inserter, feeder tap, terminator, and even the cable itself represent an impedance
mismatch of some sort. For instance, we have the manufacturing errors (there is always an error margin),
components damaged by the weather/environment conditions, no perfect fit between a coax connector and the
cable itself, etc.

Figure 3 - Incident, Transmitted and Reflected Signals in a Transmission Line
How severe are those impedances mismatches? To respond to this we use VSWR.
The ratio of reflected to incident wave (or Voltage) is known as the reflection coefficient, Γ, which is mathematically
defined as Γ = E–
/E
+
where E–
is the reflected wave’s voltage and E
+
is the incident wave’s voltage. The same
expression can be re-written as:
(2)
ZL and ZT are not usually purely resistive impedances, so Γ has to be considered a complex number. Using phasor
notation we will refer the reflection coefficient as:
(3)
The magnitude of Γ vary from 0 to 1, where 0 indicates all power is absorbed by the load (no reflection), and 1
indicates 100% reflection (short circuit, open circuit or pure reactance).
VSWR relates to the reflection coefficient through the following mathematical expression:
(4)
The VSWR ranges from 1 (No standing wave) to ∞ (a perfect standing wave).
Return Loss
Cable industry generally uses return loss rather than VSWR to characterize impedance mismatches. In its simplest
form, return loss is defined as the difference, in decibels, between the amplitude of the incident wave and the
reflected wave.
R(dB) = 10 𝑙𝑜𝑔
𝐼𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑃𝑜𝑤𝑒𝑟
𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑 𝑃𝑜𝑤𝑒𝑟
So Return loss can also be seen as.
R = 10 𝑙𝑜𝑔 (
𝑉+2
𝑉−2)

R = 20 𝑙𝑜𝑔 (
𝑉+
𝑉−
)
R = 20 𝑙𝑜𝑔 (
1
|Γ|
)
The Reflection coefficient Γ is:
|Γ| =
𝑉𝑆𝑊𝑅 − 1
𝑉𝑆𝑊𝑅 + 1
So
R = 20 𝑙𝑜𝑔 (
𝑉𝑆𝑊𝑅 + 1
𝑉𝑆𝑊𝑅 − 1
)
For example, assume that an incident wave’s amplitude is +30 dBmV, and an impedance mismatch causes a
reflection whose amplitude is +14 dBmV. Here the return loss is 30 dBmV – 14 dBmV = 16 dB, a fairly typical value
for passive devices such as taps and splitters
The incident and reflected voltages derived from the +30 dBmV (31.62 mV) incident signal and +14 dBmV (5.01
mV) reflected signal. From these, the reflection coefficient is
Γ =
𝑉−
𝑉+
Γ =
5.01 𝑚𝑉
31.62 𝑚𝑉
Γ = 0.1584
Next, let’s calculate the voltage standing wave ratio
𝑉𝑆𝑊𝑅 =
1 + |Γ|
1 − |Γ|
𝑉𝑆𝑊𝑅 =
1 + 0.1584
1 − 0.1584
𝑉𝑆𝑊𝑅 = 1.38
Reflectometers and Micro-reflections
An echo, like the sound, is a variation of the original signal that arrives at the receptor or transmitter a time after the
original signal has, due to the effect of reflections. For an echo to reach the receptor, at least two reflectometers
have to be present in the transmission line. In other words when multiple impedance mismatches exist, there will be
multiple reflections, ending up, therefore, in a situation where the echo is encountered in the receptor side.
If only one reflection is present, we still encounter a standing wave, however that wave has no effect on the signal
in the receptor side. The only effects visible are the ones related to the changes introduced solely by the
malfunctional element in the line.

Even though this may look trivial, it is not. This assumption is based on the fact that if it happens in real life, we are
not in the presence of an impedance mismatch between the cable modem and the coax cable, or the mismatch is
so little that the second reflection is below the noise level already present in the line. It is important to take this
conclusion into account in fault localization as we will see later.
In a high-speed data scenario a micro-reflection is reflection with a short time delay. That is, a close-in reflection
whose time delay relative to the incident signals ranges from less than a symbol period to perhaps several symbol
periods. From now on, we will be focusing on micro-reflections.
Let us see the effect of two reflectometers on a lossless line to see the effect of an echo in the receptor’s side on
the received signal.
Figure 4 – Echo Generated by Two Reflectometers
The incident wave at the second reflectometer is represented as:
The first reflector is represented as the phasor . Conversely, the second reflector is represented as
.
The reflected wave will take a Δt delay to reach the second reflector, this Δt is the round trip delay that takes from
bouncing at the second reflector, travels towards the first reflector, bounces again and finally, travels back to the
second reflector to be summed with the incident signal. That time delay has an effect in the signal’s phase and it’s
represented mathematically as the phasor .
Adding up all the effect to the incident signal, the reflected signal is represented as:
(5)
So, the resultant signal that continues through the line until it reaches the receptor is the sum of both the incident
and the reflected signals.
The time delay introduces an interesting effect now, depending on which frequency it is transmitted, the resulting
wave has different effects over the amplitude and the phase. This effect is known as a linear distortion.
The next step is to look further on the frequency response of this system. To do so we are going to use phasor
analysis. Figure 5 shows the possible value the resulting signal can have when summing both phasors at different
frequencies.

Figure 5 – Phasor Representation of the Sum of Reflected and Incident Wave
The amplitude of the signal changes with the frequency reaching its maximum when both, the incident and the
reflected signals, are in phase, and reaching the minimum when both are out of phase. Mathematically this is:
and (6)
As any other periodic signal, each value is repeated each 2π radians.
So the shape will repeat periodically every as shown in Figure 6.
Figure 6 – Amplitude Ripple in the Frequency Response
The first conclusion is that micro-reflection creates a linear distortion that affects the frequency response of the
system as a whole. This effect is known as amplitude ripple. The amplitude ripple is characterized by its amplitude
swing.
(7)
If we replace the amplitude ripple can be calculated in the same way as the VSWR in (4)
using this equivalent reflection coefficient. VSWR ≃ Amplitude Ripple.
This leads to a second conclusion: we can know the magnitude of the reflected signal by measuring the amplitude
ripple in the frequency domain.

It is worth noticing that the reflections do not stop after the second reflection, they continue to do so until they reach
the noise floor. All the conclusions reached in this section are still valid when we extend the analysis to more than
one echo at the receptor.
If we replace a lossless line with a real one, the amplitude of the ripple will be lower because the reflected wave
loses energy when travelling between the reflectors, but the effect remains the same. This loss is included in the
equivalent reflection coefficient.
An adaptive filter is one whose coefficients (parameters that define its behaviour) change over time “adapting” itself
to the input signal or the context where the signal lives. DOCSIS 1.1, 2.0 and later are capable of equalizing using
adaptive filters in the transmitter, a process which is called pre-equalization, and in the receiver, just called plain
equalization.
Those filters are designed to cancel, or at least to mitigate, the effect of the channel impairments. This is been done
in two ways:
1. Cancelling the impairment effects once the signal has traveled the channel (transmission line), that is, at
the receiver.
2. Pre-distorting the signal before it leaves the source so that when it has finished traveling the transmission
line, the impairment effects are compensated.
Mathematically, in terms of the frequency response, both scenarios can be represented as
and respectively. Where X(z) is the
transmitted signal and H(z) is the transfer function of both the channel and the filter. It is easy to see that if the
system is able to reach the condition where , the received signal is equal to the
transmitted signal.
It is beyond the scope of this paper, but it can be demonstrated that, to successfully cancel the linear impairments,
such as micro-reflections at the receiver, an Infinite Impulse Response Filter (IIR) is needed. However, to
implement this in the transmitter, the same effect can be achieved with a Finite Impulse Response Filter (FIR).
In practice, a FIR filter are also implemented at the receiver, which do not compensate entirely the channel
impairments, but it attenuates the effects greatly, reaching even to a steep drop below the noise floor already
present in the signal. FIR filters require more computational capacity to do the same task than the IIR, but, in the
case of the latter, implemented with adaptive coefficients, it is difficult to assure its stability as they are more prone
to have their poles outside the region of convergence due to a series of effects not relevant to this paper.
Mathematically, the transfer function of a FIR filter can be described as: (8)
Most FIR filters are implemented using a Lattice scheme and their coefficients are calculated based on a Minimum
Square Error strategy. The following diagram of a four tap filter exemplifies this:

Figure 7 - Four Tap FIR Filter with Equalization
Now, let us see how to mitigate the micro-reflection effect with equalization at the receiver and then with pre-
equalization at the transmitter.
To keep it simple, the transmission line (the channel) will introduce a single micro-reflection, of half the value of
transmitted signal, inverted and with separation between reflectors of one tap symbol. Graphically this can be seen
as:
Figure 8 - Micro-Reflection Representation
When the equalizer is implemented at the receiver (CMTS), with a 3 stage tap FIR Filter, the transfer function that
best compensate the channel effect is represented with the followings coefficients:
(9).
Figure 9 depicts what happens to the signal with a micro-reflection at different stages to finally show the output
signal.

Figure 9 - A Signal with Echo through an Equalizer at the Receptor.
The micro-reflection was not completely cancelled by the filter, but it is reduced eightfold and delayed 4 symbols
resulting in a lower amplitude ripple.
On the other hand, when the filter is implemented in the transmitter and the signal is pre-distorted, the transfer
function is the following:
(10)
Figure 10 depicts what happens to the signal at different stages to finally show what happens when the output
signal encounters the micro-reflection.

Figure 10 - A Signal with Echo through an Equalizer at the Transmitter
As seen in the picture, the micro-reflection is summed to the pre-distortion added by the filter resulting in elimination
of the former.
Looking at pre-equalization tabs coefficients we can see how separate the micro-reflections from the original source
are. This is the base for what comes next, fault localization.
In DOCSIS 1.1 cable modem support 8-tap upstream pre-equalization and DOCSIS 2.0 and later versions support
24-tap upstream pre-equalization. CMTS also implements equalization filter at the end of the upstream path.
A Cable Modem has no way of knowing what the upstream channel response is. Due to this, ranging and station
maintenance bursts transmitted by the modem are evaluated by the CMTS. The CMTS upstream receiver has its
own adaptive equalizer, from which equalizer coefficients that are transmitted to each modem derive. The modems,
in turn, use those equalizer coefficients to configure their internal adaptive pre-equalizers.
Currently available CMTS burst receivers, which incorporate DOCSIS 2.0 and later 24-tap adaptive equalization,
support main tap positions of 2 through 10. Adaptive equalizer tap position 8 is generally the default setting.
How does that affect the maximum time delay the filter can correct when dealing with micro-reflections?
 Assuming that adaptive equalizer tap #8 is the main tap, that results in a usable span is: (24 – 8) x
0.1953125 μs = 3.13 μs.
 If the adaptive equalizer main tap is changed to #10, the usable span becomes (24-10) x 0.1953125 =
2.73 μs, and
 if the adaptive equalizer main tap is changed to #2 the usable span becomes (24-2) x 0.1953125 = 4.3 μs.
Another way of seeing the value of coefficients is to think of them in terms of energy. Figure 11 shows a real Cable
Modem pre-equalizer filter expressed as tap energy.

Figure 11 – Cable Modem Pre-equalization Energy Tap Representation
Using Time Domain Reflectometer (TDR)
A time Domain Reflectometer (TDR) is an instrument which locates fault in cables by transmitting an incident signal
and measuring the time until the reflection is received. Knowing that, time interval allows the instrument to detect
the distance to the impairment by following this simple mathematical formula.
(11)
Where the Vp is the velocity of propagation in the cable, which depends on the dielectric characteristic of the line
(e.g. the coaxial cable has a velocity of propagation around 87% of the speed of light in a vacuum) and T is the
round-trip delay (the sum of the time that takes the incident signal to reach the impairment and the time it takes the
reflection to reach the source again).
Knowing which kind of cable was used in the transmission line, the only thing remaining to know is the time that it
takes a reflected signal to arrive to the source.
As presented previously in this paper, DOCSIS pre-equalization coefficients hold information about the symbol
delay and energy of the micro-reflection and other linear impairments. However, there is a trick; the time delay
information present in the equalizer measured the micro-reflections from the receiver point of view as was
expressed in the Reflectometers and Micro-reflections
.
One of the sources of micro-reflection is given by a broken feeder acting as the first reflectometer and the Cable
Modem itself (or a splitter inside the house) acting as the second one. Figure 12 describes this scenario.

Figure 12 – Cable Modem/Broken Feeder Impedance Mismatch Representation
To keep it simple, we will assume that the round-trip delay is exactly 3 symbols. So if we see the pre-equalizer filter
at the Cable Modem we will see something like this (assuming main tap location is at 8th position) in terms of
energy.
Figure 13 - Energy Tap - Cable Modem/Feeder Micro-reflection example
That is, 3 symbols away from the main tap, needs to be translated to seconds, or more accurately microseconds. If
the symbol rate is about 5.12 Msym/s when the upstream channel is set to 6.4 Mhz with a roll-off of 0.8, this gives
a time symbol period of 0.1953 µsec. The velocity of propagation for coaxial cable is around 260,819,438 meters
per second (855,706,819 feet per second). The TDR calculation with this numbers gives: 76.41 meters (250 feets).
So in this scenario, the TDR is telling us that the broken feeder is about 250 feets from the Cable Modem and that
is excellent news.
In real life it is very unlikely that the reflectors are located at distances that are multiple of the symbol period. When
this happens, we will see that there is a main echo tap (the one with the greater energy) and surrounding taps with
lower energy. This is because the pre-equalizer uses more than one tap to compensate for the micro-reflection.
In order to improve the accuracy and not to rely just on the main echo considering the energy in the surrounding
taps, an interpolation algorithm can be used to improve the TDR determination.
A good example of this kind of algorithm is the Parabolic Interpolator which is shown below.
Parabolic Interpolation Algorithm.
The figure below shows an example test case. Pre-equalizer taps 9, 10, and 11 have magnitudes 35 dB, 40 dB
and 29 dB, respectively. We are not concerned about other taps since the algorithm uses only a 3-point
interpolator, so the surrounding taps are plotted as zeros.

The algorithm inputs are the 3 taps around the (local) peak: (x0,y0), (x1,y1), (x2,y2), where the middle sample
(x1,y1) is the local peak of interest. The x value is the pre-equalizer tap number (typically in the range from 1-24)
and the y value is the tap magnitude in dB. It has been found empirically that using the dB values gives good
results; it is not necessary to convert from dB values to power ratios. Hence for our example we have the following
inputs to the algorithm:
x0 = 9
y0 = 35
x1 = 10
y1 = 40
x2 = 11
y2 = 29
The algorithm fits a parabola, shown in the figure as a dotted blue line, to the 3 taps. We assume the equation for
the parabola is y = a*x^2 + b*x + c.
The following code solves for the location of the peak of this parabola:
a = (y0 - 2*y1 + y2)/2; % Coefficient a in y = a*x^2 + b*x + c; note: a should be negative, otherwise no peak
exists
b = (y2 - y0)/2; % Coefficient b in y = a*x^2 + b*x + c
c = y1; % Coefficient c in y = a*x^2 + b*x + c
xm = (y0 - y2)/(4*a); % x-axis offset from max sample (samples)
ym = -(y0 - y2)^2/(16*a); % Magnitude (y-axis) offset from max sample
-4 -3 -2 -1 0 1 2 3 4
0
5
10
15
20
25
30
35
40
45
Parabolic Interpolator: y Magnitude vs x, Zoom View
Time (samples, 0=center sample #10 used by interpolator)
Magnitudeofy
Peak offset = -0.1875 samples

x_out = x1 + xm; % Interpolated x index (tap number)
y_out = y1 + ym; % Interpolated tap magnitude
The output is (x_out,y_out). The x_out value generally lies between integer tap numbers, giving fine time location
information. The y_out value may also be used if fine magnitude accuracy is desired. In our example we have the
following outputs:
x_out = 9.8125 (a little to the left of tap 10)
y_out = 40.2813 dB (a little higher than tap 10 magnitude).
This parabolic interpolation algorithm is courtesy of Bruce Currivan from Broadcom Corporation.
Monitoring tools, use this to elaborate an algorithm that calculates the TDR based on energies of the main echo tap
(the one which holds the greater amount of energy) and its surroundings. Figure 14 shows real time calculations of
the TDR for a series of cable modems.
Figure 14 – TDR calculations examples
Multiple Impedance mismatches.
Now we are going to analyze what happens in a different scenario. This new scenario involves an almost perfect
impedance match in the Cable Modem (the reflection caused by this mismatch is near to the noise floor), but there
are a broken feeder and a mismatched line tap both acting as reflectometers along the upstream. Figure 15 depicts
the scenario.
Figure 15 – Line Tap/Broken Feeder Impedance Mismatch Representation
Now, if the round-trip delay from the feeder to the line tap and back is exactly 3 symbols, the equalizer energy tap
will be exactly the same as Figure 13 - Energy Tap - Cable Modem/Feeder Micro-reflection example
.If we calculate the TDR in this scenario under the same conditions that the first example (same cable type and
upstream channel), it will throw the exact same distance: 76.41 meters (250 feet). But this time, the 250 feet are not
centered in the Cable Modem.

This elemental analysis can be extended to more than 2 impedance mismatches.
To add a little complexity to the analysis, we are going to show a last scenario. This involves a mismatched line tap,
a broken feeder and a Cable Modem. The round-trip delay between the line tap and the feeder is 3 symbols and the
round-trip delay between the line tap and the cable modem is 4 symbols. Figure 16 depicts this scenario.
Figure 16 – Cable Modem/Broken Feeder/Line Tap Impedance Mismatch Representation
The relationship between the reflection coefficient is: . Figure 17 shows the three
micro-reflections generated in this scenario and timing in which they arrive.
Figure 17 – Multiple Micro-Reflections Representation
The first micro-reflection to arrive is the one between the Feeder and The line tap, which does 3 symbols time later
than the original signal. The second micro-reflection to arrive is the one between the line tap and the Cable Modem,
and it arrives 4 symbols time later than the original signal. The last micro-reflection to arrive is the one between the
feeder and the Cable Modem, which arrives 7 symbols time after the original signal. It can be proven that the
relationship between amplitudes is Micro-Reflection 1 > Micro-Reflection 3 > Micro-Reflection 2. Figure 18 shows a
possible pre-equalizer energy tap for this scenario.

Figure 18 – Pre-Equalizer Energy Tap Representation in Cable Modem/Broken Feeder/Line
Tap Impedance Mismatch Representation
The main echo tap here is again the 3rd tap, and applying the TDR calculation to this information gives the exact
same distance: 76.41 meters (250 feet). And, as in the prior example, the 250 feet are not centered in the Cable
Modem.
All this leads to one simple question: how sure can we be that the main echo (the most significant one) involves the
Cable Modem and that the distance based on TDR can be centered in the Cable Modem? The answer is simple:
TDR calculation alone may not tell us the exact location because the first reflectometer is not known and it will only
be useful when the cable modem is proven to be one. Due to this we cannot based our decision by solely looking at
this parameter.
Using Micro-reflection Signatures and Geo-reference
As seen in the previous section, TDR alone do not tell us exactly where the impedance mismatches are, it only
informs the distance between the main two.
To analyze this further we need to consider the entire dataset from the same fiber node. Most of the upstream path
is shared by many cable modems, due to this, when impedance mismatches are located in common paths, all cable
modems that share that path should experience the same or almost the same pre-equalizer settings. The following
figure shows the frequency response of several Cable Modem sampled from a fiber node (for clarity only a few
were included) where there are two Cable Modem with clearly common micro-reflections.

Figure 19 – Common Impairments in The Frequency Response
The way of helping to determine if two or more Cable Modems have common micro-reflections, is to include in the
analysis the micro-reflection level.
The micro-reflection level can be estimated using the VSWR defined in (1) and the relationship with the ΓTotal
defined in (7).
Given the size of typical fiber nodes, the combination of echo delay and amplitude ripple of the frequency response,
known as the micro-reflection signature, is enough to determine the uniqueness of a micro-reflection. Due to this, all
Cable Modems that share the same micro-reflection signature are considered to have a common impairment.
Monitoring tools use this information combined with the TDR calculation and the Cable Modem’s geo-reference to
help isolate the location of the fault. Using the TDR we can draw a circle centered in the Cable Modem to form a
probability area. All these give us three possible scenarios:
The first scenario is given when the micro-reflection signature is not shared with other Cable Modems, such as the
one that figure 20 shows. This indicates that we can consider that there is a very high probability that the two
reflectometers are in the portion of the path which only belongs to the Cable Modem with the impairment
Figure 20 – No Common Impairment Scenario
Even more, if the TDR gives a distance that is greater than the unshared path, the impairment location is almost
certain to be within the Cable Modem premises. If it were not the case, other Cable Modems should experience the
same micro-reflections.
The second scenario is given when there is a shared micro-reflection signature, but at least one of the probability
areas do not overlap, such as the one the Figure 21 shows. In this case, only the portion of the path shared solely
by the Cable Modems with the same signature has to be taken into account as there is a high probability of a
common impairment, and the Cable Modems themselves are not acting as reflectometers.
Figure 21 – Common Impairment and not Overlapping Area

The last scenario is an extension of the last one, which is given when not only there is a shared micro-reflection
signature but all the probability areas overlap. In this case either both reflectors are in the common path or only one
is and the others are the Cable Modems. If the latter is the case, the first reflectometers should be located at the
intersection of all the areas.
Figure 22 – Common Impairment with Overlapping Area

This paper analyses the effect of an impedance mismatch on the transmission line, showing that when there are at
least two of them, a micro-reflection is received in the receptor side. This effect provokes an amplitude ripple in the
frequency response that is perceived as noise.
This theoretical background was provided to understand the need of equalization filters. A theoretical impedance
mismatch example is used to exemplify how to use pre-equalization information present in those filters to retrieve
relevant information about the micro-reflections.
It was also illustrated how time domain reflectometry uses the information provided by the main echo tap to
determine the fault’s location. By presenting simple impedance mismatched examples, the first conclusion was
presented to the audience. TDR calculation is not a reliable indicator by itself as it only provides the distance
between the two main reflectors and those two can be anywhere along the line.
Lastly and as a final conclusion, this paper makes clearly evident that with the aid of micro-reflection signatures and
geo-reference, the impairment location path can be narrowed down significantly, and that tools that take into
account that information are providing significant data that can help technicians to pinpoint the impairments faster
before the degradation affects the customer’s satisfaction.
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[2] Introduction to Signal Processing (Orfanidis, S.J), Prentice Hall, ISBN 0-1320-9172-0
[3] Proactive Network Maintenance Using Pre-equalization, CM-GL-PNMP-V02-110623, Cablelabs
[4] Carrier-Frequency Estimation of BPSK and QPSK Signals Using Spectral-Line Techniques, B. Currivan,
Proceedings of the National Science Foundation Workshop on Cyclostationary Signals, August, 1992, Yountville,
CA, USA.

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