Impulse-Noise Cancellation using
the Common Mode Signal
Oana Graur
Electrical Engineering and Computer Science
Jacobs University
Campus Ring 7
28759 Bremen
Germany
Type: Guided Research Thesis
Date: June 2, 2009
Supervisor: Prof. Dr.-Ing. W. Henkel
A BSTRACT
Impulse noise has long been known for causing transmission errors. The current
thesis presents a method of mitigating impulse noise using a Normalized LMS
canceler which takes as reference input the common mode signal and subtracts
the adapted version from the differential mode signal. A simple detection method
for the impulse noise was devised. The effect of the step size on the stability and
convergence of the algorithm was investigated. Multiple Matlab simulations were
conducted for the training phase of the canceler, using sets of impulses generated by various sources. In the end, the simplifications this model involves were
explored and suggestions for improvement were given.
Contents
1 Introduction
3
2 Impulse noise
4
3 Adaptive Filtering
5
4 Impulse noise detection
7
5 Structure of the canceler
7
6 Simulation Results
8
6.1 NLMS Cancellation . . . . . . . . . . . . . . . . . . . . . . . . .
9
6.2 Training phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
7 NEXT and FEXT
15
8 Conclusions
18
2
1 Introduction
Transmission quality is often degraded by bursts of high amplitude referred to
as impulse noise [1]. Noise can appear in a transmission system due to several
reasons such as electromagnetic coupling of radio signals, inductive coupling of
magnetic fields, and capacitive coupling of electric fields.
Normally, transmission over copper cables is carried out using differential-mode
(DM) signals. These signals propagate through a pair of wires. One wire carries
the signal as we perceive it and the other one carries the opposite phase signal.
Unlike single-ended signals, which are dependent on an external reference level,
the differential signals are only referenced to the pair, which brings a few advantages. Since the receiver measures the difference between the two wires, the net
result is twice as large and leads to improved signal to noise (SNR) ratio.
xDM (t) = x1 (t) − x2 (t)
(1)
A common mode (CM) signal appears on both lines of a 2-wire cable, in-phase
and with equal amplitudes. Technically, the CM signal is the arithmetic mean of
the signals x1 (t) and x2 (t) and can be measured at the receiver from the center tap
of a balun [3].
x1 (t) + x2 (t)
xCM (t) =
(2)
2
A 50 Ω termination resistor was included in the upper branch and a 1 MΩ in the
lower CM branch. The reason to use a high-ohmic input impedance for the CM
line was that the characteristic impedance for common-ground is undefined and
an arbitrary value can have a detrimental effect on the common mode signal when
the baluns are not perfectly symmetric.
3
Figure 1: CM and DM signals
2 Impulse noise
Considerable investigations have been carried out in analyzing the performance of
a communications channel under the influence of Gaussian noise. Unfortunately,
those results were not always consistent with error rates of practical communication channels. More indepth investigations revealed that in the majority of the
cases, the impulse noise was essentially non-Gaussian. The reason why the Gaussian distribution does not accurately estimate impulse noise is because impulse
noise exhibits large amplitudes known as outliers that occur too frequently to fit
to a Gaussian model. Impulse noise has a high peak to rms ratio and most of the
energy contained below 200 kHz. It has also been discussed that impulse noise is
comprised of bursts of impulses rather than of individual pulses spaced at various
intervals. Some of the causes responsible for generating impulse noise are power
line arcing, ac wiring, lightning strikes, electrical engines, welding and switching
of home appliances [2]. For Internet users, high amounts of impulse noise can
cause the service to slow down and result in errors in downloaded data. With
IPTV, even low levels of impulse noise can result in video pixelation or breaks in
the audio content.
Measurements of impulse noise have been taken, both in DM and CM with a
sampling rate higher than the Nyquist rate. Figure 2 presents an impulse measured both in DM and in CM. Please note the different amplitudes of the waveforms. Before attempting cancellation, the impulses were received and saved into
nonoverlapping blocks of length N for convenience.
4
Differential Mode Signal
0.01
0.005
0
−0.005
−0.01
0
1000
2000
3000
4000
5000
6000
7000
8000
6000
7000
8000
Common Mode Signal
0.2
0.1
0
−0.1
−0.2
−0.3
0
1000
2000
3000
4000
5000
Figure 2: Impulse measured in DM and CM
3 Adaptive Filtering
The concept of adaptive filtering has found a lot of applications over the past three
decades, ranging from channel equalization and echo cancellation to mitigation of
narrowband interference in wideband signals [6]. The Least Mean Squares Algorithm is typically used due to its low computational complexity and ease of implementation [7]. LMS does not require measurements of the correlation functions,
nor does it require matrix inversions. LMS is a linear adaptive filtering algorithm
that consists of two basic processes: a filtering process which involves computing the output of a transversal filter and generating an estimation of the error by
comparing this output to a desired signal. The second process involves automatic
adjustment of the coefficients of the filter in accordance with the estimation of the
error. Since the LMS algorithm involves the feedback of the error in its operation,
the issue of stability is of critical importance. For this reason, we define J(n) to
be the mean-squared error produced at time n. A convergent algorithm satisfies
the following condition
J(n) → J(∞)
n→∞
5
(3)
The output of a filter of size M is given by
y(n) = hT (n)xM (n)
(4)
xM (n) = [x(n) x(n − 1) . . . x(n − M + 1)]T
(5)
h(n) = [h1 (n) h2 (n) . . . hM (n)]T
(6)
where xM (n) holds the M current samples within the filter taps at time n, h(n)
holds the current weighting coefficients of the filter and hT denotes the Hermitian
transpose of h(n). The LMS algorithm assumes zero mean of the input signals.
The LMS update algorithm is described by the following equations
h(n + 1) = h(n) + µe∗ (n)xCM (n) ,
(7)
e(n) = xDM − hH (n) · xCM (n) ,
(8)
where n is the time index, the boldface characters represent vectors, h(n) holds
the current adaptive weights, e(n) is the current error, xDM (n) and xCM (n) are
the current blocks of inputs of size M × 1 within the tapped delay line of the filter. M is the length of the FIR filter and µ is the step size factor. This algorithm
is sensitive to values of the input vector x(n) and choosing a step size for which
the algorithm would be stable and converge can prove to be problematic. For this
reason, an adaptation of the algorithm has been proposed in [6], which normalizes the power of the input. The Normalized LMS (NLMS) is described by the
following set of equations
h(n + 1) = h(n) +
µe∗ (n)xCM (n)
xH
CM xCM
e(n) = xDM − hH (n) ∗ xCM (n)
(9)
(10)
The step size µ directly affects how quickly the filter will converge. A small value
of µ will lead to a small change in coefficients and a slow convergence. A larger
value of the step size will decrease the time required for convergence. In case this
value is too large, the coefficients will change too fast and the filter will not be
stable anymore. It has been shown [6] that a tradeoff between convergence time
and stability is given by the following optimal learning rate
µopt =
E[|y(n) − ŷ(n)|2 ]
,
E[|e(n)|2 ]
(11)
where y(n) is our DM signal and ŷ(n) is the CM signal already filtered. However,
this formula cannot be used in practice as it requires a-priori knowledge about the
6
signal. Instead, experimentation is used. It can be shown that the value of the error
will eventually converge to a minimum, given an optimum choice of the step size.
This error, however will not be zero since the system includes outside noise and
interference. It can be noticed from the formulas above that the LMS algorithm
requires 2M + 1 multiplications and 2M additions per iteration. In other words,
the computational complexity of the LMS is O(M ).
4 Impulse noise detection
Although many other detection methods for impulse noise have been successfully
described in the literature [9], the current thesis presents a simple method which
uses linear interpolation. The CM input vector is split into nonoverlapping blocks
of size L and out of each block, one maximum value is chosen. That is, after k
distinct blocks, k − 1 local maxima can be linearly interpolated and from the corresponding (k − 1)L samples, the ones above a certain threshold can be flagged
as being corrupted by impulse noise. This is necessary because, within an impulse, there are samples with amplitude values below the threshold for which the
filter would not adapt otherwise. Once flagged, the CM signal goes through the
adaptive FIR filter which updates the coefficients at every new flagged sample
that goes in. Under ideal assumptions, the resulting output would contain the DM
signal undistorted by the impulse noise except for a minimum residual error. The
disadvantage of this interpolation method is that it introduces delay due to the
need of a buffer of size L. The choice of the threshold and of the window size L
have a critical effect on the output and have to be treated accordingly. The second
subplot of Fig. 5 shows a CM impulse after interpolation.
5 Structure of the canceler
The typical Least Mean Squares canceler is illustrated in Fig. 3. An adaptive
canceler requires two inputs: a primary input that consists of the signal that needs
to be corrected and a reference input which is correlated with the undesired signal,
in our case, the impulse noise. The reference signal is processed by an adaptive
FIR filter whose coefficients are adjusted at every new incoming input sample in
order to minimize the error. In this case the DM signal represents the signal to be
corrected and the CM represents the reference signal. The FIR filter is presented
in Fig. 4.
7
Figure 3: Least Mean Squares Canceler
The idea of using the CM signal as a reference signal to detect and cancel noise
and RFI interference has previously been investigated in [4] [5]. Due to the electromagnetic coupling, the CM signal and the DM signal are strongly correlated
within a transmission system. The CM signal consists of three components: independent noise, noise correlated with the noise in DM and a component correlated
with the desired signal from DM [3].
For this simulation, the canceler was designed to adapt the weights while a detected impulse passed through the delay taps of the FIR. Once the impulse was
outside of the transversal taps, the filter retained the last set of coefficients and
filtered the rest of the data using them.
Figure 4: FIR filter
6 Simulation Results
For the first part of the simulation, sets of measurements were provided containing
only the impulses and some background noise with no additional desirable signals.
A more complex model that includes the transfer functions, NEXT, FEXT, and the
8
coupling from DM to CM is described in Section 7. The peak amplitude of the
impulses measured was 5 mV in DM and 0.5 V in CM. Simulations with different
step factors have been run. Also, measurements were provided containing impulse
samples generated by a welding machine with CM amplitudes ranging between
10V and 40V.
6.1 NLMS Cancellation
Applying the above algorithm to an impulse yields the result shown in Fig. 5. The
first subplot depicts the DM impulse before attempting cancellation, the second
subplot shows the CM envelope of the impulse after interpolation and the third
one superimposes the input – in red – and the output – in blue – of the NLMS
canceler.
0.01
Voltage [V]
Differential Mode
0.005
0
−0.005
−0.01
0
1000
2000
3000
4000
5000
Sample [n]
6000
7000
8000
Voltage [V]
0.5
Common Mode
Signal Envelope
0
−0.5
0
1000
2000
3000
4000
5000
Sample [n]
6000
7000
8000
Voltage [V]
0.01
Differential Mode
LMS Output
0.005
0
−0.005
−0.01
0
1000
2000
3000
4000
5000
Sample [n]
6000
7000
8000
Figure 5: Impulse noise in DM, impulse noise in CM and envelope, filtered output
Figure 6 presents two other examples of canceled impulse noise. For testing purposes, the canceler was trained using multiple sets of impulses and then data was
9
Voltage [V]
0.01
0.005
0
−0.005
−0.01
0
1000
2000
3000
4000
5000
Sample [n]
6000
7000
8000
0
1000
2000
3000
4000
5000
Sample [n]
6000
7000
8000
Voltage [V]
0.01
0.005
0
−0.005
−0.01
Figure 6: Canceled impulses, step size = 0.05
blindly filtered in order to study the behaviour of the output. In practice, the training phase can be initialized using a higher step size and then eventually, the step
size can be decreased. Training can be continued with a smaller step size to ensure adaptivity in case the impulse coupling changes. The following subsection
provides an overview of multiple Matlab simulations which took into account different parameters.
6.2 Training phase
The first simulation implied the training of the filter using 780 input impulses.
Different step factors were used for the study of convergence. Figures 7, 8, 9
show intermediate steps for the training phase for different step sizes.
The output errors for the step sizes used are presented in Fig. 11. As expected,
a small step size yields a slow decay and a bigger value of µ yields a very fast
decaying curve. Since the data used for this simulation contained background
noise, the error curves were expected not to be very smooth as well. On this
graph, the declining tendency is more obvious for the curves which correspond to
smaller coefficients, since the curves corresponding to larger coefficients converge
to a minimum value in a very short amount of time.
10
Input/output of system for impulse number 1
Volts [V]
0.01
Differential Mode Signal
System Output
0.005
0
−0.005
−0.01
0
1000
0
1000
0
1000
0
1000
2000
3000
4000
5000
6000
7000
Samples[n]
Input/output of system for impulse number 150
8000
2000
3000
8000
2000
3000
8000
2000
3000
Volts [V]
0.01
0.005
0
−0.005
−0.01
4000
5000
6000
7000
Samples[n]
Input/output of system for impulse number 400
Volts [V]
0.01
0.005
0
−0.005
−0.01
4000
5000
6000
7000
Samples[n]
Input/output of system for impulse number 780
Volts [V]
0.01
0.005
0
−0.005
−0.01
4000
5000
Samples[n]
6000
Figure 7: Canceler training, step size = 0.005
11
7000
8000
Input/output of system for impulse number 1
Volts [V]
0.01
Differential Mode Signal
System Output
0
−0.01
0
1000
0
1000
0
1000
0
1000
2000
3000
4000
5000
6000
7000
Samples[n]
Input/output of system for impulse number 150
8000
2000
3000
8000
2000
3000
8000
2000
3000
Volts [V]
0.01
0
−0.01
4000
5000
6000
7000
Samples[n]
Input/output of system for impulse number 400
Volts [V]
0.01
0
−0.01
4000
5000
6000
7000
Samples[n]
Input/output of system for impulse number 780
Volts [V]
0.01
0
−0.01
4000
5000
Samples[n]
6000
Figure 8: Canceler training, step size = 0.01
12
7000
8000
Input/output of system for impulse number 1
Volts [V]
0.01
Differential Mode Signal
System Output
0.005
0
−0.005
−0.01
0
1000
0
1000
0
1000
0
1000
2000
3000
4000
5000
6000
Samples[n]
Input/output of system for impulse number 150
7000
8000
2000
3000
7000
8000
2000
3000
7000
8000
2000
3000
7000
8000
Volts [V]
0.01
0.005
0
−0.005
−0.01
4000
5000
6000
Samples[n]
Input/output of system for impulse number 400
Volts [V]
0.01
0.005
0
−0.005
−0.01
4000
5000
6000
Samples[n]
Input/output of system for impulse number 780
Volts [V]
0.01
0.005
0
−0.005
−0.01
4000
5000
Samples[n]
6000
Figure 9: Canceler training, step size = 0.1
13
Input/output of system for impulse number 2
Voltage [V]
2
Differential Mode Signal
System Output
1
0
−1
−2
0
1000
0
1000
0
2000
2000
3000
4000
5000
6000
Sample [n]
Input/output of system for impulse number 64
7000
8000
2000
3000
7000
8000
4000
6000
14000
16000
Voltage [V]
2
1
0
−1
−2
4000
5000
6000
Sample [n]
Input/output of system for impulse number 112
Voltage [V]
2
1
0
−1
−2
8000
10000
Sample [n]
12000
Figure 10: Canceler training, impulses produced by a welding machine, step size
= 0.5
14
−5
6
x 10
0.00005
0.0001
0.0005
0.001
0.005
0.01
Squared error
5
4
3
2
1
0
0
1
2
3
4
Samples [n]
5
6
7
6
x 10
Figure 11: Squared error curves for different step sizes
7 NEXT and FEXT
In communications, crosstalk refers to coupling – capacitive, inductive or conductive – from one cable pair to another within the bundle. Although interference
within each pair is caused by signals within all the other pairs from the same
transmission system, this can be modelled as being produced by only one major
interferer. One way of minimizing the coupling is to twist pairs and shield them.
Although in theory twisted pairs are perfectly balanced and noise couples onto
each conductor in a twisted pair equally, in practice this is never the case.
Far End Crosstalk (FEXT) refers to interference between two pairs in one cable as
measured at the end of the cable furthest from the transmitter. Near end crosstalk
(NEXT) is the interference between two pairs in one cable as measured at the end
of the cable nearest to the transmitter. Figure 12 presents a simplification of a
transmission system containing one major interferer. At the receiver side, in DM,
the signal is present as a sum of the desired signal after passing through the wire,
one FEXT component from the adjacent pair and one NEXT component. The
same idea is also true for the CM signal.
The cancellation algorithm described above does not take into account the signal
coupling into CM. If the desired signal couples highly into CM, the canceler will
adapt for it as well and the desirable component will be eliminated from the DM
along with the interference. Figure 13 gives a plot of the measured transfer functions both into CM and DM mode. It can be easily seen that the transfer function
for the CM is approximately –40dB for the whole range of frequencies up to 40
Mhz. Figures 14 and 15 prove the idea that both FEXT and NEXT couple higher
15
Figure 12: Transmission system including NEXT and FEXT
into DM.
0
TR DM
TR CM
-10
-20
response / dB
-30
-40
-50
-60
-70
-80
0
5e+006
1e+007
1.5e+007
2e+007
2.5e+007
frequency / Hz
3e+007
3.5e+007
Figure 13: DM and CM transfer functions
16
4e+007
-20
NEXT DM
NEXT CM
-30
response / dB
-40
-50
-60
-70
-80
-90
0
5e+006
1e+007
1.5e+007
2e+007
2.5e+007
frequency / Hz
3e+007
3.5e+007
4e+007
Figure 14: NEXT coupling into DM and CM
-30
FEXT DM
FEXT CM
-40
response / dB
-50
-60
-70
-80
-90
-100
0
5e+006
1e+007
1.5e+007
2e+007
2.5e+007
frequency / Hz
3e+007
3.5e+007
Figure 15: FEXT coupling into DM and CM
17
4e+007
8 Conclusions
Based on the idea that interference coupling is higher in common mode, a Normalized Least Mean Squares canceler was used to eliminate the impulse noise
from the differential mode signal. Matlab simulations were run using different
sets of measurements for the training phase of the canceler and the effects of near
and far-end crosstalk were discussed. For the first simulations with impulses with
small amplitudes, a drop in impulse amplitudes down to the level of background
noise was observed after training with 780 impulses and a small step size. For
the impulses generated by a welding machine which had DM peak amplitudes
of around 3V, the results were not as satisfying since there were not enough impulses available to train on and the step size chosen had to be high in order to
see a change in the output. In this case only 112 impulses were used and the
impulse noise peak amplitude dropped from 3V to 1.5V. In order to improve this
results, the canceler has to train over a longer set of impulses. For a more accurate
mathematical representation, a more suitable cancellation algorithm which does
not assume a Gaussian distribution of impulse amplitudes can be chosen. For a
practical implementation when additional desirable signals are present, the interpolation method used for impulse detection is not suitable anymore. Also, if the
NEXT and FEXT coupling into CM is high, the canceler will subtract that interference from DM even though it is not present there. All these aspects have to
be taken into account and the coupling functions have to be known beforehand to
guarantee the efficient cancellation of impulse noise using the CM signal.
18
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19