1 Introduction

Cables are present in almost all systems in many fields and their maintenance is often complicated. The increase of the complexity of modern systems has come with the increase of wire lengths and giving rise to complex networks. So, systems offering fast and accurate diagnosis are required.

When cables are part of a critical system, some faults can have severe consequences. It is thus necessary to set up a monitoring function for the wired network state, permanently and in real time. This involves the integration of the diagnostic system into the cable network to help preventing the damage that can be caused by cable defects.

In terms of wire diagnosis [1], the most widely used approaches for faults detection in cables are time-domain reflectometry (TDR) [2] techniques.

Even if TDR has proved its efficiency in detecting and locating faults in simple wired networks, for complex network, the use of a single injection point (only one reflectometer) is not sufficient to cover the entire network. Indeed, this result alone leads to ambiguities about the exact location of the fault. Also, this may be explained by the signal attenuation due to the traveled distance and multiple junctions.

For that, the solution is to perform distributed TDR, which consists of measuring at several points of the network under test, by implementing several sensors at different extremities of the network in order to maximize the diagnosis coverage and quality.

Each sensor performs a diagnosis and provides its own interpretation as to the location of the defect. The communication between these sensors must allow the fusion of data to make decision about fault location. The methods of distributed diagnosis such as the M-sequence method [3] and the method of selective averages [4] are based on the analysis of different reflected signals and do not allow sensors to communicate with each other.

The proposed technique is based on phase modulation of the test signal of MCTDR reflectometry [5]. The idea is to analyze not only reflected part, but also the transmitted part of the diagnostic signal. Indeed, the transmitted signals provide additional information on the fault characteristics and make it possible to integrate of the communication between the network distributed reflectometers. Besides, the proposed method enables monitoring and communication without interfering with existing signals (of the monitored network) and without interference between different reflectometers (sensors) signals.

The second part of this paper explains the MCTDR reflectometry principle. Section 3 introduces mathematical models of developed signals and cable response. Results about method robustness are presented in Sect. 4.

2 MCTDR Reflectometry

Reflectometry is a well-known method used for defects detection in electrical cables. It takes the principle of a radar by injecting a wave into a cable, and analyzing the reflected signals. This makes it possible to deduce information on the discontinuities encountered in the cable (Fig. 1).

Fig. 1.
figure 1

Reflectometry principle.

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The properties of reflected signals permit to get information about the impedance discontinuity. The reflectogram, given by correlation between the incident MCTDR signal and reflected one, allows identifying the type of defect and its location. In fact, by measuring the delay 2τ between the incident wave and the reflected one and knowing the propagation velocity vg of the cable, it is possible to calculate the location (relative to the injection point) ΔL of the defect, by the following equation:

$$ \Delta {\text{L}} = 2\uptau{\text{vg}}/2 $$

When the wave injected into a transmission line (with a characteristic impedance ZC1) encounters an impedance discontinuity ZC2 during propagation, a portion of its energy will be returned to the injection point.

Each discontinuity is associated with a reflection coefficient which provides information on the polarity and the amount of energy returned to the incidence point. The reflection coefficient is given by the following equation:

$$ \Gamma _{1} = \frac{{Z_{c2} - Z_{c1} }}{{Z_{c2} + Z_{c1} }} $$

Modern reflectometry-based methods use complex diagnosis signals with specific mathematical properties. The reflectogram is obtained by the correlation of the measured signal and the injected signal.

Among the TDR methods, we use the MCTDR. The MCTDR signal is designed as the sum of a finite number of sinusoids at a given set of frequencies, chosen outside of the useful system signals:

$$ s_{n} = \frac{2}{\sqrt N }\sum\limits_{k = 0}^{N/2} {c_{k} { \cos }\left( {\frac{2\pi k}{N}n + \theta_{k} } \right)} $$

MCTDR method enables an accurate control of the injected signals spectrum in order to adapt it to system constraints, and supports a distributed diagnosis architecture that enables unambiguous and full diagnosis coverage for complex wire grid topologies. Moreover, it has good autocorrelation properties allowing a good accuracy of defects localization (Fig. 2).

Fig. 2.
figure 2

MCTDR signals: signal spectrum and autocorrelation.

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The idea is to use the phases (θk) to encode messages. At the injection, a technique of digital modulation is used to allow the transmission of information via the diagnosis MCTDR signal. The modulated phase of each carrier of the MCTDR signal represents a symbol of digital information sent by the reflectometer.

We have verified that the use of the signal phase does not modify MCTDR main characteristics to have the good property of autocorrelation.

3 Analytic Models

3.1 Selectives Averages

Since several reflectometry modules are injecting test signals simultaneously, specific signal processing methods are needed to remove interferences between concurrent modules (Fig. 3).

Fig. 3.
figure 3

Example of wired network.

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The idea of the method of selective averages [4] is to insert orthogonal weighting coefficients into the transmitted signal and at the reception during the calculation of the average. This method makes the filtering independent of the test signal. Each sensor p connected to the network is then identified by its coefficients sequence b(p):

$$ b^{(p)} = (\beta_{1}^{(p)} , \ldots ,\beta_{M}^{(p)} )^{T} $$

where p is the index of the sensor.

At the reception, with the averaging step, the calculation of the result is done with M acquisitions. Each acquisition is multiplied with one coefficient βm.

In fact, each sensor receives all the signals from the other sensors. To estimate the signal \( y^{(j)} \) coming from the sensor j, it must multiply the M acquisitions \( \left( {a_{m} } \right) \) by the coefficients b(j):

$$ y^{\left( j \right)} \frac{{\sum\limits_{m = 1}^{M} {\beta_{m}^{\left( j \right)} a_{m} } }}{{\sum\limits_{m = 1}^{M} {\beta_{m}^{{\left( j \right)^{2} }} } }} $$

The selective averaging method makes it possible to obtain negligible interference noise level. Where there is a synchronization between the reflectometers, the interference noise can be canceled by the use of the Walsh-Hadamard sequences. Besides, this method enhances the SNR (Signal to Noise Ratio).

3.2 Cable Response

Cable response is given by cable losses and all impedance discontinuities (Fig. 4). It gives information about the propagation behavior of the signal along the cable and the amount of energy returned to the source.

Fig. 4.
figure 4

Reflection and transmission coefficients related to cable faults.

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Each cable fault (or impedance discontinuity) is associated with a reflection coefficient. Soft faults [6,7,8] do not cut the communication but are necessary to be detected. But, hard defects (open or short circuit), interrupt completely the communication and the network working and require a rapid intervention.

We are interested in the transmitted part of the test signal after passing through the wired network, to ensure communication between two reflectometers. The cable transfer function H, is given by:

$$ H\left( f \right) = \prod\nolimits_{i} {\left( {1 - \varGamma } \right)e^{ - \gamma \left( f \right)l} } $$

(Γj)j>0 represent impedance discontinuities reflection coefficients, l is the cable length and γ is the propagation parameter, depending on frequency f, given by:

$$ \upgamma\left( {\text{f}} \right) = \alpha \left( f \right) + j\beta \left( f \right) $$

The real part α represents the attenuation of the wave amplitude during its propagation, and the complex part β represents the wave phase rotation.

After passing through the wired network, the injected signal X(f) becomes:

$$ Y\left( f \right) = H\left( f \right)X\left( f \right) $$

3.3 Frequency Analysis of MCTDR Signal

With a digital system operating at the sampling frequency fe = 1/Te, the Fourier transform of the MCTDR signal (at DAC output) is written as follows:

$$ X\left( f \right) = sinc\left( {\pi fT_{e} } \right)\sum\limits_{n = 0}^{M - 1} {\sum\limits_{k = 0}^{N - 1} {c_{k} e^{{j\theta_{k} }}\updelta\left( {f - \left( {\frac{k}{N} + n} \right)f_{e} } \right)} } $$

Where n is the sampling index and N is carriers number. The received signal is given by:

$$ \begin{aligned} Y\left( f \right) & = sinc\left( {\pi fT_{e} } \right)\sum\limits_{n} {\sum\limits_{k = 1}^{N - 1} {c_{k} e^{{j\theta_{k} }}\updelta\left( {f - (\frac{k}{N} + n)f_{e} } \right).\prod\nolimits_{i} {\left( {1 - \varGamma_{i} } \right)e^{ - \gamma \left( f \right)l} } } } \\ & \quad = sinc\left( {\pi fT_{e} } \right)\; \cdot \;e^{ - \alpha l} \prod\nolimits_{i} {\left( {1 - \varGamma_{i} } \right)} \sum\limits_{n} {\sum\limits_{k = 1}^{N - 1} {c_{k} e^{{j\left( {\theta_{k} - \beta l} \right)}}\updelta\left( {f - (\frac{k}{N} + n)f_{e} } \right)} } \\ \end{aligned} $$

The phase of this signal is given by:

$$ phase\left( {Y\left( f \right)} \right) = - \beta \left( f \right)l + \mathop \sum \limits_{k = 0}^{N - 1} \theta_{k} \delta \left( {f - \frac{k}{N}f_{e} } \right) $$

3.4 Reception of MCTDR Signal Phases (Message)

To recover the sent message, the receiving sensor must calculate the N phases (θk) from Y(f). However, the phase of Y(f) depends on the parameters of the cable β(f) and l. Moreover we need to calculate the received signal phase on each carrier frequency band.

Therefore, we need two steps to estimate the phase sequence. We make a first measurement (Y0) by sending a zeros phases sequence: {θk} = {0}.

$$ phase\left( {Y_{0} \left( f \right)} \right) = - \beta \left( {\text{f}} \right)l $$

In a second step, we transmit message of on the sequence of the phases {θk}. Then, the receiver estimates the sent sequence:

$$ \theta_{{k_{est} }} = \left( {phase\left( {Y\left( f \right)} \right) - phase\left( {Y_{0} \left( f \right)} \right)} \right) \; \cdot \; G_{k} \left( f \right) $$

Gk is a formatting filter used to select the frequency band corresponding to the subcarrier number k:

$$ G_{k} \left( f \right) = \left\{ {\begin{array}{*{20}l} 1 \hfill & {if\,f = \left( {\frac{k}{\text{N}} + n} \right)f_{e} } \hfill \\ 0 \hfill & {elsewhere} \hfill \\ \end{array} } \right. $$

4 Method Robustness

We proved that communication quality is not affected by cable faults as signal phase don’t depend on faults reflection coefficients. Moreover, we recall that the MCTDR signal is robust to external interference thanks to good frequency spectrum management and also robust to internal interference (between sensors) thanks to the use of selective averages method.

To test the robustness of this method to high noise, a white Gaussian noise \( { \in }_{Y} \) is added to the received signal, whose signal to noise ratio varies from 0 to 30 dB:

$$ Y_{b} \left( f \right) = Y\left( f \right) + \in_{Y} $$

We show an example of simulation. We consider the example of a network consisting of two sensors placed at both ends of a cable.

A message of 256 bits is sent on the sequence of the phases of an MCTDR signal (of N = 128 sub-carriers), injected into a network by a reflectometer and estimated at a second sensor. The cable has a soft fault, with a reflection coefficient Γ = 1/3.

The 256 bits are modulated using QPSK modulation (quadrature phase-shift keying) to be inserted in the MCTDR signal as a sequence of 128 phases: {θk}k=1..128.

We notice that we find the same bits that were sent, in the case of absence of noise. Figure 5 shows the evolution of the bit error rate (BER) as a function of the signal-to-noise ratio. We tested two types of channels: stationary (the same noise is used in the phase estimation stage and the data transmission stage) and non-stationary (the noise changes).

Fig. 5.
figure 5

BER versus SNR for a stationary and non-stationary network.

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Logically the stationary cable allows having a better quality of communication. In fact, when the noise does not change between the step of channel phase estimation and the message transmission step, this allows us to estimate and compensate for a part of the noise phase. This offers a better communication quality.

We note that these results can be further improved with the use of selective averages method and also with the use an error corrector channel coding.

5 Conclusion

This paper introduce a promising communication system for network defects detection and localization. As a better fault localization offers a better security, we are designing a real-time embedded system. For future work we focus on techniques of data fusion between different sensors for an optimized distributed diagnosis, to know how to exploit them for fault localization and cable condition evaluation.

A future paper will discuss other crucial points such as synchronization, multiplexing, routing and transmission rate.