# Detection and Localization of Interference and Useful Signal Extreme Points in Closely Coupled Multiconductor Transmission Line Networks

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

_{V}and S

_{I}are the matrices of modal voltages and currents; E0 and ED are the diagonal matrices E0 = diag(exp(–${\gamma}_{1}$x), …, exp(–${\gamma}_{{N}_{k}}$x)) and ED = diag(exp(–${\gamma}_{1}$∙(l–x)), …, exp(–${\gamma}_{{N}_{k}}$ (l–x))); ${\gamma}_{{N}_{k}}$ is the propagation constant for the k-th MCTL section; N

_{k}is a number of conductors in the k-th MCTL section; l is the length of MCTL section; and x is the coordinate along the MCTL section (the number of x is defined by n

_{TLS}parameter). The calculation of S

_{V}, S

_{I}, and

**E**(x) is described in [36]. C1 and C2 are constant vectors calculated as:

_{k}}, $j\in ${1, …, 2N

_{k}} with one nonzero value in each column, is the selector matrix that maps the terminal currents of the k-th MCTL section; ${Y}_{k}$ is the conductance matrix of the k-th MCTL section; V is the vector of the node voltage waveforms; and E is a constant vector with the entries determined by the independent voltage and current sources.

## 3. Proposed Approach for Signal Extreme Points Detection and Localization

**Preparation stage**Formulating a problem to be solved, the aim, and expected results. Designing cross sections and a circuit diagram of REE (taking into account all galvanic connections between circuit diagram components), adding all parameters of active and reactive components (for a transmission line section, the per unit length coefficient matrices of electromagnetic (L) and electrostatic (C) induction, conductivities (G), and resistances (R); for the excitation signal source, the type of a signal and its parameters).

**Calculation stage**Setting up the start and end points of the pulse propagation, the number of a segment along each MCTL section, and the speed of displaying. Calculating signal waveforms. Detecting and localizing signal extreme points. Performing optimization to obtain an optimal solution.

**Final stage**Analyzing the obtained results, i.e., signal waveforms, whose propagation along each conductor of each MCTL is displayed in dynamics; detected signal extreme points; localization of these points in the circuit diagram.

## 4. Developed Algorithms used in the Approach

#### 4.1. Visualization Algorithm of Signal Propagation along the Circuit

#### 4.2. Algorithm for Defining the Allowed Ways of Signal Propagation along the Circuit

- (1)
- The index of node A appears in the system (when a user chooses it). A = 2.
- (2)
- The variable of the list type called “list” is created (empty at first). The start node index is assigned to a current node index a (a = A = 2).
- (3)
- The circuit components connected to node a are numbered.
- (4)
- The check of other unprocessed components is performed. If there is an unprocessed component, then go to step 5, otherwise, go to step 7. (Numbers of the steps are shown in Figure 3.)
- (5)
- The processing of each numbered component is performed in succession. The component end connected to the node a is defined as input and the opposite one is defined as output.
- (6)
- The index of the node connected to the output end is assigned to node b (b = 3).
- (7)
- The node a index is added to the end of the “list” (list contains 2).
- (8)
- The check of another node with b index is performed. If there is a node with b index, then go to step 9, otherwise, go to step 11.
- (9)
- There is a node with b index. The index of node b is translated to the node a index (a = 3).
- (10)
- The current numbered component (trl
_{1}) becomes completed (shown by green ticks in Figure 5). Other elements connected to node a are numbered again. Therefore, we go to step 3 of the algorithm. - (11)
- The algorithm will work until nodes b will run out.

#### 4.3. Algorithm for Detecting and Localizing Signal Extreme Points

_{max}is the maximum voltage value, U

_{min}is the minimum voltage value, U

_{n}is the current matrix element (during the 1st algorithm run n = 1 in the blocks 3 and 9); and n is the number of matrix elements. After the algorithm run, the signal extreme point is displayed. According to the fact that the response is calculated in each segment of each MCTL section, the numbers of the conductor segment and the section (where the signal is localized) are known. The development of this approach resulted in creating an additional program module for TALGAT software [37]. Using this module, ordinary coupled lines were, first, studied and, then, compared with the results of CST MWS software. Then the real PCB bus of a spacecraft ANS was investigated under different conditions.

^{2}). The complexity of the proposed approach is O(n

^{3}).

## 5. Investigation of Coupled Transmission Lines

#### 5.1. Meander Line with Two Turns

_{r}) is 4.5, and the separation (s) between the conductors is 0.217 mm. The simulation was carried out without losses. Each conductor of the meander line was divided into 50 segments. Therefore, the voltage waveforms were calculated in 200 points along the whole length of the meander line (the MCTL section includes four conductors). The voltage maximum that is 1.14 times higher than the steady-state level was detected and located in segment 46 of the second conductor (shown by the arrow in Figure 9). The waveforms of the localized maximum in comparison with the ones found using CST WMS are presented in Figure 10. A good agreement of the obtained results is observed. The simulation in the CST MWS took 1080 s. while the TALGAT software took 1 s.

#### 5.2. Meander Line with Two Turns

_{r}) is 4.5, and the dielectric thickness (h) is 0.3 mm. The aim of the investigation was to calculate signal waveforms with extreme points at different separations (s) (from 0.4 mm down to 1 μm) between the conductors. The different separations were made in order to increase the mutual coupling between the line conductors. The most interesting results of this investigation are presented in Figure 12.

_{max}) was localized with s = 8 μm (Figure 12b). It is twice as high as the voltage amplitude at the input (U

_{2}). The comparison of the obtained result with CST MWS is shown in Figure 13. A good agreement of the obtained results is also observed. The simulation in the CST MWS took 840 s while the TALGAT software took 0.5 s.

## 6. Investigation of Spacecraft ANS PCB Bus

#### 6.1. Influence of Ultrashort Pulse Duration on Localization of its Extreme Points in PCB Bus

_{1}) has the rise, top, and fall times of 1 ns, the second (U

_{2}) – 100 ps, and the third (U

_{3}) – 10 ps, and therefore the whole durations are 3, 0.3, and 0.03 ns. Such choice of excitation parameters is determined by the fact that in such way not only useful signals, but interference ones, are considered.

_{b}) and the end (U

_{e}) of the conductor, and the waveforms with the voltage maximum (U

_{max}) and minimum (U

_{min}) values, appearing under each excitation, are presented.

_{1}, and minimums under excitations U

_{1}and U

_{2}. By this reason the waveforms of these signals and their locations are not shown in Figure 17 and Figure 18. Table 1 contains the values of the voltage peaks and the numbers of segments with their locations.

_{1}). The extreme points of the waveforms presented in Figure 17a and their location coincide with the waveforms at the conductor’s ends. Let us consider the U

_{2}and U

_{3}excitations which can be relegated to high-speed or interfering signals because their durations are shorter. Under the U

_{2}excitation, voltage maximum is 0.59 V (Figure 17b) which is 18% higher than the steady-state level of 0.5 V. The maximum is localized in segment 10 (Table I) in one of the transmission line sections with five conductors that are located on another layer (Figure 18a). Under the U

_{3}excitation, not only the voltage maximum of 0.552 V that exceeds the 0.5 V level by 10% is detected, but also a minimum of minus 0.18 V (Figure 17c) or minus 36% of 0.5 V, which is lower than the level of zero. Moreover, it is shown in Table 1 and Figure 18b,c that the extreme points are localized in absolutely different places.

_{3}excitation. Besides the appearance of peak values, it was found that there appeared multiple reflections and the signal amplitude at the output of the line decreased. It was 0.4 V in Figure 17c, which is 20% lower than the 0.5 V level. The investigation shows specific aspects of detecting and localizing extreme points of the ultrashort pulse with various durations. For example, the highest maximum value (about 0.6 V) is detected under the U

_{2}excitation as we can see from Table 1. The extreme points are located in different transmission line sections and segments along these conductors (i.e., in different places of the PCB).

#### 6.2. Influence of Ultrashort Pulse Duration on Localization of Crosstalk Extreme Points in PCB Bus

_{1}). The signal waveforms calculated for conductor 3 are presented in Figure 20a, but their extremes and location are similar to the waveforms calculated at the ends of the conductor. Let us consider the excitations U

_{2}and U

_{3}which can be regarded as interference signals. These excitations have the shorter durations than the useful ones. Some voltage maximums are detected, but they are located in the diagram nodes. The voltage minimum calculated under the excitation U

_{2}is minus 0.126 V (25.2% of 0.5 V). The location of this extreme point is in the segment 5. But the voltage minimum calculated under the excitation U

_{3}is minus 0.199 V (39.8% of 0.5 V) and is located in segment 8.

_{3}as it shown in Table 1 (the full duration of the pulse was 0.03 ns). The highest minimum value (minus 0.199 V, 39.8% of 0.5 V) is also under the excitation U

_{3}. Considering that the maximum acceptable crosstalk level in the ANS PCB bus should be less than 10% of signal amplitude in the active conductor, it follows that all detected extreme points (excluding those under the excitation U

_{1}) dissatisfy this condition.

#### 6.3. Simulation of ESD Effects on PCB Bus

## 7. Optimization by Genetic Algorithms

#### 7.1. Optimization of Ultrashort Pulse Duration by Criteria of Peak Voltage Maximization in PCB Bus

_{P}). This is caused by the necessity to check the convergence of the fitness function results. The diagram of convergence of the U

_{max}values with a different number (n) of fitness function (product of the number of chromosomes and the number of populations) calculations is shown in Figure 24. A total of 20 voltage waveforms were calculated in each segment along each conductor of each MCTL section from Figure 15 with the obtained results for the highest fitness function value (run 2 from Table 3, when the number of populations was 75), but only the waveforms at the beginning (U

_{b}) and end (U

_{e}) of the conductor and also the waveforms with the voltage maximum (U

_{max}) values are presented. The results are presented only for one active and one passive conductor with the highest crosstalk amplitude.

#### 7.2. Optimization of Ultrashort Pulse Duration by Criteria of Peak Voltage Maximization in PCB Bus

_{m}) ranged from 0.01 up to 0.08, and, in the second part, the crossover coefficient (k

_{c}) ranged from 0.1 up to 0.8. In the first part, k

_{c}= 0.5 and,, in the second part k

_{m}= 0.1. Three parameters, t

_{r}, t

_{d}, and t

_{f}, were optimized in the range from 1 ns down to 0.01 ns when the chromosome number was five and the number of populations was 26 (so the total GA calculation number was 130). The sum of peak voltages at the ends of the PCB bus conductors 1, 3, and 5 was maximized. The aim of the optimization was to define the rise, flat top, and fall duration values of the ultrashort pulse, at which the sum of voltages (U

_{SUM}) in the preset points will be maximal. It is important to notice that it is necessary to choose a greater number of GA calculations to provide a more complete investigation. The simulation of excitation by several sources is also useful, but due to the fact that the work presents the preliminary stage of the investigation only, it was decided to choose a small number of calculations and one source only. The values of the sum of maximum voltages in the preset points for the first part of the investigation are presented in Table 4. The presented results are obtained with k

_{c}= 0.1 and different k

_{m}for ten GA runs.

_{C}is the calculation number and N

_{R}is the run number. The presented results are obtained when k

_{m}= 0.03 because with such k

_{m}the U

_{max}in Table 4 is the highest. Convergence diagrams of the arithmetic average of ten runs with different k

_{m}are shown in Figure 27b. The GA run results of the second part of the investigation (the sum of maximum voltages at I, II, and III points when k

_{m}= 0.1 and k

_{c}is different) are presented in Table 4.

_{c table}= 0.8 because with such k

_{c}the U

_{max}in Table 3 is the highest. The convergence diagrams of the arithmetic average of ten runs with different k

_{c}are shown in Figure 28b. As regards the best fitness function result, all obtained values are similar and differ in the third decimal place. Additionally, the highest result (0.55704 V) is obtained with k

_{m}= 0.03 (Table 4). Let us consider the results of the second part of the investigation. As regards the best fitness function result, the situation is the same as for the first part of the investigation, all obtained results are similar and differ in the third decimal place only. Meanwhile, the highest result is 0.55448 V (obtained with k

_{c}= 0.8).

_{c}= 0.8 it is at the 70th calculation). However, these coefficient variations hardly influence the detected peak voltages of the ultrashort pulse and crosstalk. The obtained peak voltages have the same amplitudes.

#### 7.3. Optimization of PCB Bus Loads by Criteria of Peak Voltage Minimization

_{min}, the minimum value of the sum of peak voltages at points I, II, and III) with different runs and the number of the fitness function calculations are presented in Table 6, where N

_{R}is a number of a run, and n is the number of fitness function calculations. The GA was launched ten times for each combination of the chromosome number and the population number. It was made in order to check the convergence of the fitness function results. The convergence diagram of the U

_{min}values with a different n is shown in Figure 29. The dependences of the minimum voltage values on the n are shown in Figure 30.

## 8. Use of Evolution Strategies

- (1)
- The initialization of a population ${P}_{\mathsf{\mu}}=\{{a}_{1},$ …,${a}_{\mathsf{\mu}}\}$ with the use of μ parent chromosomes.
- (2)
- The generation of λ offspring $\widehat{a}$ forming the offspring population ${\widehat{P}}_{\lambda}=\{{\widehat{a}}_{1},$ …,${\widehat{a}}_{\lambda}\}$ where each offspring $\widehat{a}$ is generated by the following steps:
- (3)
- Select (randomly) ρ parents from P
_{μ}(if ρ = μ take all parental individuals instead). - (4)
- Recombine the ρ selected parents to form a recombinant individual r.
- (5)
- Mutate the strategy parameter set
**s**of the recombinant r. - (6)
- Mutate the objective parameter set, y, of the recombinant, r, using the mutated strategy parameter set to control the statistical properties of the object parameter mutation.
- (7)
- The selection of a new parent population (using deterministic truncation selection) from either the offspring population ${\widehat{P}}_{\lambda}$ (this is referred to as comma-selection, usually denoted as “(μ, λ) selection”), or the offspring ${\widehat{P}}_{\lambda}$ and parent P
_{μ}population (this is referred to as plus-selection, usually denoted as “(μ + λ) selection”). - (8)
- Go to 2 until the termination criterion is fulfilled.

_{max}) in the preset point (for ten ES runs) and the best solutions with different ISs of the ES are shown in Table 7. (Cells with the highest U

_{max}for each IS are colored in yellow.) The signal waveforms calculated with IS = 3, 0.3, 0.03 ns for the highest U

_{max}are shown in Figure 34.

_{max}arithmetic averages (of 10 runs) for IS = 3, 0.3, 0.03 ns in their dependence on the ES iteration number (N

_{I}) are shown in Figure 35, Figure 36 and Figure 37, respectively. It is necessary to check the appearing of signal peaks along the whole conductor with the use of excitation parameters which have been obtained as a result of the optimization. The voltage waveforms along the active conductor are shown in Figure 38a, where the waveforms are the following: U

_{b}, at the input; U

_{e}, at the end; and U

_{max}, with the highest peak voltage. The voltage maximum localization is shown in Figure 38b. The voltage waveforms calculated along the passive (nearest to active) conductor with the highest amplitude of the crosstalk and its maximum localization are presented in Figure 39.

_{max}values are very similar and differ in the fourth decimal place. The same situation is observed with the best solutions obtained after each ES cycle – the differences are near 5 ps. The highest U

_{max}is obtained in the first ES run for all ISs. The small differences in the obtained results hardly change the voltage waveforms calculated in the V34 node. As we can see from Figure 34, the signal waveforms coincide.

_{max}arithmetic average for each IS. The strongest change of U

_{max}is observed when IS = 300 ps, starting from 0.5 V (Figure 36). However, after 30th calculation, it becomes almost the highest and other changes are within the bounds of 30 mV. Before the 30th calculation, the U

_{max}change has strong spikes, which are possibly caused by a strong mutation of an offspring. When IS = 3 ns (Figure 35), the U

_{max}arithmetic average has a smooth rising character without strong spikes (in the range up to 30 mV) and starting at 0.52 V it reaches the maximum value in the 69th calculation, and when IS = 30 ps (Figure 37), the U

_{max}changes the least of all and starting at 0.55 mV it reaches the maximum in the 39th calculation.

## 9. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

x | A coordinate along an MCTL section |

V(x) | The vector describing the voltage in x |

I(x) | The vector describing the current in x |

S_{V} | The matrix of modal voltages |

S_{I} | The matrix of modal voltages |

E0, ED | The propagation matrices |

${\gamma}_{{N}_{k}}$ | The propagation constant |

N_{k} | A number of conductors in the k-th MCTL section |

l | A length of an MCTL section |

C1, C2 | The constant vectors describing the mode amounts |

n_{TLS} | A parameter describing a number of x |

n | A number of MCTL sections |

V | The vector describing the voltages calculated in each segment of each MCTL section along the whole conductor |

W | The matrices describing the lumped memory elements of the network |

H | The matrices describing the lumped memoryless elements of the network |

D_{k} | The selector matrix that maps the terminal currents of the k-th MCTL section |

Y_{k} | The conductance matrix of the k-th MCTL section |

E | The constant vector with the entries determined by the independent voltage and current sources |

L | The matrix of electromagnetic induction |

C | The matrix of electrostatic induction |

R | The matrix of conductivities |

G | The matrices of resistances |

trl_{k} | A k-th MCTL section |

U_{max} | A maximum voltage value (in the block diagrams)A waveform with the maximum voltage value (in the graphics) |

U_{min} | A minimum voltage value (in the block diagrams)A waveform with the minimum voltage value (in the graphics) |

U_{b} | A voltage waveform at the beginning of a conductor |

U_{e} | A voltage waveform at the end of a conductor |

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**Figure 3.**Block diagram of the algorithm for defining the allowed ways of signal propagation along the circuit.

**Figure 5.**Display of each step of the algorithm before the complete processing of the first circuit component.

**Figure 26.**Voltage waveforms along the passive conductor (

**a**) and the crosstalk maximum location (

**b**).

**Figure 27.**Convergence diagrams of the fitness function values for each run with k

_{m}= 0.03 (

**a**) and the arithmetic average of runs with different k

_{m}(

**b**).

**Figure 28.**Convergence diagrams of the fitness function values for each run with k

_{c}= 0.8(

**a**) and the arithmetic average of runs with different k

_{c}(

**b**).

**Figure 39.**Voltage waveforms along the passive conductor (

**a**) and the localization of its maximum (

**b**).

Excitation | Figure | U_{max} | U_{min} | ||
---|---|---|---|---|---|

Voltage, V | Segment, Figure | Voltage, V | Segment, Figure | ||

U_{1} | 17a | 0.530 | 1 | −0.05 | 1 |

U_{2} | 17b | 0.597 | 10 (18a) | −0.11 | 20 |

U_{3} | 17c | 0.552 | 5 (18b) | −0.18 | 3 (18c) |

Excitation | Figure | U_{max} | U_{min} | ||
---|---|---|---|---|---|

Voltage, V | Segment, Figure | Voltage, V | Segment, Figure | ||

U_{1} | 20a | 0.031 | 1 | −0.031 | 1 |

U_{2} | 20b | 0.139 | 1 | −0.126 | 5 (21a) |

U_{3} | 20c | 0.292 | 1 | −0.199 | 8 (21b) |

N_{P} | Number of a Run | t, s | t_{r}, ns | t_{d}, ns | t_{f}, ns | U_{max}, V |
---|---|---|---|---|---|---|

5 | 1 | 441.407 | 0.989 | 0.0114 | 0.800 | 0.551147 |

2 | 452.437 | 0.500 | 0.0199 | 0.879 | 0.545629 | |

3 | 486.692 | 0.447 | 0.0356 | 0.582 | 0.534152 | |

4 | 486.508 | 0.707 | 0.0251 | 0.127 | 0.531801 | |

5 | 510.564 | 0.111 | 0.0550 | 0.954 | 0.524602 | |

10 | 1 | 983.870 | 0.663 | 0.0119 | 0.925 | 0.550284 |

2 | 1045.06 | 0.487 | 0.0100 | 0.772 | 0.551331 | |

3 | 1133.29 | 0.429 | 0.0102 | 0.652 | 0.549775 | |

4 | 1204.64 | 0.894 | 0.0111 | 0.923 | 0.550668 | |

5 | 1280.82 | 0.951 | 0.0184 | 0.719 | 0.547373 | |

50 | 1 | 3679.80 | 0.9160 | 0.0108 | 0.874 | 0.551185 |

2 | 3689.23 | 0.3980 | 0.0114 | 0.938 | 0.551207 | |

3 | 3686.37 | 0.7300 | 0.0103 | 0.761 | 0.551353 | |

4 | 3649.92 | 0.0135 | 0.0104 | 0.820 | 0.553568 | |

5 | 3673.96 | 0.6580 | 0.0103 | 0.789 | 0.551315 | |

75 | 1 | 5951.78 | 0.7402 | 0.01123 | 0.931 | 0.551358 |

2 | 5707.05 | 0.0123 | 0.01214 | 0.549 | 0.554590 | |

3 | 5479.93 | 0.6261 | 0.01045 | 0.785 | 0.551505 | |

4 | 5818.87 | 0.0108 | 0.01432 | 0.658 | 0.552927 | |

5 | 5880.29 | 0.8017 | 0.01031 | 0.870 | 0.551700 |

Run | k_{m} | |||
---|---|---|---|---|

0.01 | 0.03 | 0.05 | 0.08 | |

1 | 0.50819 | 0.55159 | 0.55108 | 0.54131 |

2 | 0.51607 | 0.55122 | 0.55196 | 0.55146 |

3 | 0.54913 | 0.55704 | 0.55005 | 0.54974 |

4 | 0.55094 | 0.55081 | 0.55100 | 0.55503 |

5 | 0.50799 | 0.55284 | 0.55043 | 0.55127 |

6 | 0.53153 | 0.54920 | 0.55317 | 0.55092 |

7 | 0.52685 | 0.55092 | 0.55079 | 0.54232 |

8 | 0.53983 | 0.55271 | 0.55272 | 0.55108 |

9 | 0.54009 | 0.55110 | 0.55128 | 0.54909 |

10 | 0.54309 | 0.55132 | 0.55120 | 0.55041 |

Run | k_{c} | |||
---|---|---|---|---|

0.1 | 0.3 | 0.5 | 0.8 | |

1 | 0.55444 | 0.55214 | 0.55166 | 0.55072 |

2 | 0.53083 | 0.55013 | 0.55177 | 0.55128 |

3 | 0.55169 | 0.54860 | 0.55027 | 0.55167 |

4 | 0.55041 | 0.54996 | 0.55092 | 0.52187 |

5 | 0.55279 | 0.54988 | 0.55121 | 0.55254 |

6 | 0.55102 | 0.55165 | 0.55180 | 0.55118 |

7 | 0.55150 | 0.55182 | 0.54766 | 0.55448 |

8 | 0.55120 | 0.55060 | 0.55040 | 0.55024 |

9 | 0.55191 | 0.55052 | 0.55122 | 0.55142 |

10 | 0.55058 | 0.55116 | 0.55173 | 0.55012 |

N_{R} | k_{c} | |||||||
---|---|---|---|---|---|---|---|---|

18 | 30 | 33 | 55 | 56 | 60 | 110 | 260 | |

1 | 428 | 61 | 53 | 50 | 77 | 45 | 44 | 41 |

2 | 82 | 87 | 283 | 52 | 154 | 47 | 765 | 35 |

3 | 584 | 170 | 78 | 47 | 94 | 46 | 84 | 23 |

4 | 120 | 234 | 59 | 52 | 72 | 187 | 59 | 22 |

5 | 356 | 180 | 46 | 44 | 86 | 82 | 62 | 35 |

6 | 105 | 236 | 47 | 59 | 143 | 52 | 44 | 26 |

7 | 130 | 77 | 44 | 31 | 60 | 88 | 47 | 28 |

8 | 115 | 132 | 52 | 54 | 71 | 51 | 59 | 18 |

9 | 413 | 49 | 187 | 88 | 54 | 76 | 60 | 16 |

10 | 122 | 144 | 154 | 37 | 38 | 72 | 54 | 35 |

**Table 7.**U

_{max}Values for 10 runs of evolution strategies (ES) with different initial solutions (ISs).

ES Run | IS, ns | ||
---|---|---|---|

3 | 0.3 | 0.03 | |

1 | 428 | 61 | 53 |

2 | 82 | 87 | 283 |

3 | 584 | 170 | 78 |

4 | 120 | 234 | 59 |

5 | 356 | 180 | 46 |

6 | 105 | 236 | 47 |

7 | 130 | 77 | 44 |

8 | 115 | 132 | 52 |

9 | 413 | 49 | 187 |

10 | 122 | 144 | 154 |

The best solution | 325.53ps | 330.06ps | 325.59ps |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gazizov, R.R.; Gazizov, T.T.; Gazizov, T.R.
Detection and Localization of Interference and Useful Signal Extreme Points in Closely Coupled Multiconductor Transmission Line Networks. *Symmetry* **2019**, *11*, 1209.
https://doi.org/10.3390/sym11101209

**AMA Style**

Gazizov RR, Gazizov TT, Gazizov TR.
Detection and Localization of Interference and Useful Signal Extreme Points in Closely Coupled Multiconductor Transmission Line Networks. *Symmetry*. 2019; 11(10):1209.
https://doi.org/10.3390/sym11101209

**Chicago/Turabian Style**

Gazizov, Ruslan R., Timur T. Gazizov, and Talgat R. Gazizov.
2019. "Detection and Localization of Interference and Useful Signal Extreme Points in Closely Coupled Multiconductor Transmission Line Networks" *Symmetry* 11, no. 10: 1209.
https://doi.org/10.3390/sym11101209