**Specific Emitter Identification Based on Fractal Features**

Janusz Dudczyk

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67894

#### Abstract

If we take into consideration the fact that the radar signal recognition and identification process is an integral part of contemporary combat operations, the importance of the fractal analysis increases significantly. For this reason, the fractal analysis is used in the process of radar sources identification on the contemporary battlefield. Radar Signal Recognition (RSR) with the use of classical methods, that is based on statistical analysis of basic measurable parameters of a radar signal, such as Radio Frequency (RF), Amplitude (A), Pulse Width (PW) or Pulse Repetition Interval (PRI) is not enough to carry out the distinction process of particular copies of the same radar type. Only by this approach, the identification process of particular copies in a set of the same type emitters can be carried out. As a result, it is possible to maximize Correct Identification Coefficient (CIC) in the final stage of the recognition process, which is realized in Electronic Warfare (EW) systems. One of the most important elements of the whole recognition and identification process, which is realized in ELectronic INTelligence (ELINT) battlefield system, is building a measurement data vector, then a radar's metrics and the same database. This approach is called Specific Emitter Identification (SEI).

Keywords: fractal feature, pattern of radar, Specific Emitter Identification (SEI), radar vector parameters, ELINT system, Radar Signal Recognition (RSR), Correct Identification Coefficient (CIC)

## 1. Introduction

Developing an innovative method for generating distinctive features extracted from radar signals in order to achieve explicit identification is a main goal in the process of Specific Emitter Identification (SEI). As a result, it is possible to maximize Correct Identification Coefficient (CIC) and identify particular copies of radars of the same type in ELectronic INTelligence (ELINT) system on the contemporary battlefield. The presented achievements in this

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chapter concern methods and techniques adjusted to electromagnetic emitter source recognition and identification with particular reference to the systematic approach, that is, from the acquisition process, through initial data transformation, main transformation, classification and identification and building a model in the DataBase (DB). Only a comprehensive approach can be coherently fitted into a harmonious whole of all the ELINT recognition systems' processes on the contemporary battlefield. As a result, this approach may contribute to an increase in effectiveness of activity by minimization of time, which is necessary for the decision process realized in Electronic Warfare systems.

Generally, it is possible to distinguish the main task, which has a significant contribution to the development of recognition and identification of radar signals, that is, developing innovative methods for generating distinctive features of radar signals [1–5]. As mentioned above, the task is connected with developing a method for setting the structure of basic measurable parameters of a radar signal in the form of time-frequency-formalized Pulse Descriptive Word (PDW) [5, 6], developing and implementing in SEI process, an advanced Hierarchical Agglomerative Clustering Algorithm (HACA) based on the 'bottom up' agglomerative formula, which makes it possible to receive dendrograms of hierarchical clustering for pulse repetition intervals and their distinctive analysis [1], applying the inter-pulse modulation analysis to extract distinctive features [7], developing an innovative histogram procedure to build PRI decomposition [2] and finally developing an innovative method for defining a transformation attractor of radar signals measurement clusters [3, 4]. This approach is called Specific Emitter Identification.

## 2. Classic recognition of radar signals

As a general rule, the systems aiming at acquisition, analysis and recognition of radar signals on the contemporary battlefield are autonomous systems, which are made of electronic recognition devices doing ELINT tasks. These systems accomplish complex procedures in the scope of emission acquisition, analysis, transformation and radar emission recognition with the range of wavelengths 0.5–18 GHz, long-term data archivization and full synthesis and fusion of information. These systems, in most cases, are made of the following subsystems and modules, that is, radar signal acquisition subsystem, radar signal processing and analysis subsystem, database management subsystem and communication module1 between particular subsystems.

Accomplishing basic tasks in the designed systems of radar signal acquisition, analysis and recognition on the contemporary battlefield is based on:

• automated searching for and detecting<sup>2</sup> electromagnetic emitters (in the range of wavelengths 0.5–18 GHz);

<sup>1</sup> Communication module between particular subsystems and components of acquisition system and radar signal analysis on the contemporary battlefield is not the subject of this chapter.

<sup>2</sup> In the aspect of tasks connected with searching for and detecting signals and measuring their parameters, the specialist equipment of the radar signal acquisition, analysis and recognition system makes it possible to accomplish the tasks above on the land, from the air or the sea and may be adjusted to a plane, helicopter, a ship or installed in a special container case.

• automated parameter measurement of the detected emitters;

chapter concern methods and techniques adjusted to electromagnetic emitter source recognition and identification with particular reference to the systematic approach, that is, from the acquisition process, through initial data transformation, main transformation, classification and identification and building a model in the DataBase (DB). Only a comprehensive approach can be coherently fitted into a harmonious whole of all the ELINT recognition systems' processes on the contemporary battlefield. As a result, this approach may contribute to an increase in effectiveness of activity by minimization of time, which is necessary for the decision

Generally, it is possible to distinguish the main task, which has a significant contribution to the development of recognition and identification of radar signals, that is, developing innovative methods for generating distinctive features of radar signals [1–5]. As mentioned above, the task is connected with developing a method for setting the structure of basic measurable parameters of a radar signal in the form of time-frequency-formalized Pulse Descriptive Word (PDW) [5, 6], developing and implementing in SEI process, an advanced Hierarchical Agglomerative Clustering Algorithm (HACA) based on the 'bottom up' agglomerative formula, which makes it possible to receive dendrograms of hierarchical clustering for pulse repetition intervals and their distinctive analysis [1], applying the inter-pulse modulation analysis to extract distinctive features [7], developing an innovative histogram procedure to build PRI decomposition [2] and finally developing an innovative method for defining a transformation attractor of radar signals

measurement clusters [3, 4]. This approach is called Specific Emitter Identification.

As a general rule, the systems aiming at acquisition, analysis and recognition of radar signals on the contemporary battlefield are autonomous systems, which are made of electronic recognition devices doing ELINT tasks. These systems accomplish complex procedures in the scope of emission acquisition, analysis, transformation and radar emission recognition with the range of wavelengths 0.5–18 GHz, long-term data archivization and full synthesis and fusion of information. These systems, in most cases, are made of the following subsystems and modules, that is, radar signal acquisition subsystem, radar signal processing and analysis subsystem, database

Accomplishing basic tasks in the designed systems of radar signal acquisition, analysis and

• automated searching for and detecting<sup>2</sup> electromagnetic emitters (in the range of wave-

Communication module between particular subsystems and components of acquisition system and radar signal analysis

In the aspect of tasks connected with searching for and detecting signals and measuring their parameters, the specialist equipment of the radar signal acquisition, analysis and recognition system makes it possible to accomplish the tasks above on the land, from the air or the sea and may be adjusted to a plane, helicopter, a ship or installed in a special container case.

management subsystem and communication module1 between particular subsystems.

process realized in Electronic Warfare systems.

116 Fractal Analysis - Applications in Physics, Engineering and Technology

2. Classic recognition of radar signals

recognition on the contemporary battlefield is based on:

on the contemporary battlefield is not the subject of this chapter.

lengths 0.5–18 GHz);

1

2


The analysis of radioelectronic situation on the contemporary battlefield and long-term radar signal measurements makes it possible to admit that during the process of signal recognition and classification, there is a phenomenon of penetrating ranges of radar parameters, many ranges of particular parameters for single radars, different types of emissions (constant, pulse, interrupting), complexity of sounding pulses and specific work properties (signal polarization depending on the weather), decrease in frequency of repetition and top pulse as the beams go above the horizon or the change of top power in case of e.g. weather condition changes. The general block diagram of radar signal acquisition, analysis and recognition is presented in Figure 1.

The radar recognition system (see Figure 1) is only able to recognize and classify particular types of radars. A definitely more advanced one is the recognition process understood in a 'narrow sense'. Its aim is to identify these signals, thus their emission sources as well. I deal with the recognition understood in a 'narrow sense', which concerns identification of particular radar copies of the same type depending on the detail level.

The process of distinguishing the radar emission source even 'a single copy' is the exact identification of radar signal source in the aspect of SEI. Thus, applying in the acquisition system, the

Figure 1. The process of information processing in the radar recognition system.

analysis and recognition of radar signals, inter-pulse analysis of signal parameters (defining the type of inter-pulse modulation and estimation of parameters' modulation), intra-pulse analysis (defining the type and parameters of intra-pulse modulation on the basis of a single pulse), applying innovative methods for generating distinctive features, using fast-decision identification algorithm and advanced DataBases prepared as a result of modelling entity relationships and using AI3 devices is an immanent specificity of ELINT systems and makes it possible to identify particular copies of radar emitters with the use of the above dedicated methods for generating distinctive features' signals in Electronic Warfare (EW) systems. As concerns contemporary used ELINT class systems, data classification and recognition techniques are currently developed fields of science, it is not possible to formulate optimal model of their structure and maximize the recognition and classification function as well as the identification function.

In Figure 2, a block diagram of the acquisition, analysis and identification system, including the subsystems that implement modern and advanced methods for generating distinctive fractal features, is presented.

<sup>3</sup> AI (Artificial Intelligence), a term of artificial intelligence, which in BD systems is realized on the basis of using artificial neural networks (so-called AI bionic trend) and expert systems, based on predicate calculus (so-called AI pragmatic trend).

Figure 2. The process of distinctive fractal features generation in the RSR system.

analysis and recognition of radar signals, inter-pulse analysis of signal parameters (defining the type of inter-pulse modulation and estimation of parameters' modulation), intra-pulse analysis (defining the type and parameters of intra-pulse modulation on the basis of a single pulse), applying innovative methods for generating distinctive features, using fast-decision identification algorithm and advanced DataBases prepared as a result of modelling entity relationships and using AI3 devices is an immanent specificity of ELINT systems and makes it possible to identify particular copies of radar emitters with the use of the above dedicated methods for generating distinctive features' signals in Electronic Warfare (EW) systems. As concerns contemporary used ELINT class systems, data classification and recognition techniques are currently developed fields of science, it is not possible to formulate optimal model of their structure and maximize the

In Figure 2, a block diagram of the acquisition, analysis and identification system, including the subsystems that implement modern and advanced methods for generating distinctive

AI (Artificial Intelligence), a term of artificial intelligence, which in BD systems is realized on the basis of using artificial neural networks (so-called AI bionic trend) and expert systems, based on predicate calculus (so-called AI pragmatic

recognition and classification function as well as the identification function.

Figure 1. The process of information processing in the radar recognition system.

118 Fractal Analysis - Applications in Physics, Engineering and Technology

fractal features, is presented.

3

trend).

## 3. Innovative method for generating distinctive features based on fractal analysis

RSR with the use of classic techniques, which are based on the statistical analysis of basic measurable parameters, such as radio frequency, signal amplitude, pulse width, and pulse repetition interval, is completely not sufficient for SEI process to carry out the process of distinguishing particular radar copies of the same type.

For this reason, on the stage of initial data processing, a method for defining the structure of basic measurable parameters of a radar signal in the form of formalized time-frequency Pulse Description Word was developed. These vectors are input data into the further process of generating distinctive features, in the main processing stage [3, 4]. As mentioned above, the PDW vector is a formalized data structure of record type, where particular fields consist of frequency parameters and time parameters of radar signal according to Eq. (1), where Nr(k) is the number kth of the pulse, tp(k) is the time of appearing kth pulse in [µs], A(k) is the amplitude of kth pulse, PW(k) is the width of kth pulse in [µs], PRI(k) is the Pulse Repetition Interval of kth pulse in [µs], RF(k) is the Radio Frequency of kth pulse in [MHz], n is the number of pulses in the record of these which are qualified to the analysis while k is the number of pulses in the measured sample.

$$\mathbf{PDW} = \begin{bmatrix} \text{Nr}(1) & t\_p(1) & A(1) & \text{PW}(1) & \text{PRI}(1) & \text{RF}(1) \\ \text{Nr}(2) & t\_p(2) & A(2) & \text{PW}(2) & \text{PRI}(2) & \text{RF}(2) \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \text{Nr}(k) & t\_p(k) & A(k) & \text{PW}(k) & \text{PRI}(k) & \text{RF}(k) \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \text{Nr}(n) & t\_p(n) & A(2) & \text{PW}(n) & \text{PRI}(n) & \text{RF}(n) \end{bmatrix} \tag{1}$$

The effect of the further transformation of PDW vector is Basic Signal Vector VB, whose fields are of particular signal frequency and time parameters, according to Eqs. (2)–(4). The time parameters of the vector VPRI <sup>B</sup> are as follows: minimum pulse repetition interval value PRImin, average pulse repetition interval value PRIEV, maximum pulse repetition interval value PRImax, the number of values of pulse repetition interval nPRI, the number of values of average pulse repetition interval nPRIEV, the minimum value of pulse width PWmin, the average value of pulse width PWEV and the maximum value of pulse width PWmax.

$$\mathbf{V\_{B}^{PRI}} = \left[ \mathbf{PRI}\_{\text{min}}, \mathbf{PRI}\_{\text{EV}}, \mathbf{PRI}\_{\text{max}}, \mathbf{n} \mathbf{PRI}, \mathbf{n} \mathbf{PRI}\_{\text{EV}}, \mathbf{PW}\_{\text{min}}, \mathbf{PW}\_{\text{EV}}, \mathbf{PW}\_{\text{max}} \right] \tag{2}$$

The frequency parameters of the signal vector VRF <sup>B</sup> are defined according to Eq. (3) and are as follows: the minimum value of the signal radio frequency RFmin, the average value of the radio frequency for the periodRFEV, the maximum value of the radio frequency in the period of changes RFmax, the number of values of the radio frequency nRF and the number of average radio frequencies in the cycle of changes nRFEV.

$$\mathbf{V\_{B}^{RF}} = \left[RF\_{\text{min}} \, RF\_{\text{EV}} \, RF\_{\text{max}} \, nRF\_{\prime} \, nRF\_{\text{EV}}\right] \tag{3}$$

The vector VB of the final structure presented according to Eq. (4) consists of parameters concerning information about the accuracy measurements of: radio frequency sigRF, pulse repetition interval sigPRI and pulse width sigPW.

$$\mathbf{V\_{B}} = [\mathbf{V\_{B}^{\text{PRI}}}, \mathbf{V\_{B}^{\text{RF}}}, \text{sigRF}, \text{sigPRI}, \text{sigPW}] \tag{4}$$

These parameters are the base for defining the brackets of acceptable changes of radar signal, that is, RF, PRI and PW are used in the estimation process of effectiveness of the Fast decision Identification Algorithm (FdIA), described in [6]. The process of developing the signal vector VB also undergoes the process of implementation and automation in the stage of initial data

Figure 3. PRI histogram for six copies of the same type of radars marked by colours.

Figure 4. 3D graphic depicting of PW, PRI and RF for six copies of the same type of radars.

The effect of the further transformation of PDW vector is Basic Signal Vector VB, whose fields are of particular signal frequency and time parameters, according to Eqs. (2)–(4). The time

average pulse repetition interval value PRIEV, maximum pulse repetition interval value PRImax, the number of values of pulse repetition interval nPRI, the number of values of average pulse repetition interval nPRIEV, the minimum value of pulse width PWmin, the average value

follows: the minimum value of the signal radio frequency RFmin, the average value of the radio frequency for the periodRFEV, the maximum value of the radio frequency in the period of changes RFmax, the number of values of the radio frequency nRF and the number of average radio frequen-

The vector VB of the final structure presented according to Eq. (4) consists of parameters concerning information about the accuracy measurements of: radio frequency sigRF, pulse repeti-

These parameters are the base for defining the brackets of acceptable changes of radar signal, that is, RF, PRI and PW are used in the estimation process of effectiveness of the Fast decision Identification Algorithm (FdIA), described in [6]. The process of developing the signal vector VB also undergoes the process of implementation and automation in the stage of initial data

of pulse width PWEV and the maximum value of pulse width PWmax.

<sup>B</sup> are as follows: minimum pulse repetition interval value PRImin,

<sup>B</sup> ¼ ½RFmin, RFEV, RFmax, nRF, nRFEV� ð3Þ

<sup>B</sup> , sigRF, sigPRI, sigPW� ð4Þ

<sup>B</sup> are defined according to Eq. (3) and are as

<sup>B</sup> ¼ ½PRImin,PRIEV,PRImax,nPRI,nPRIEV,PWmin,PWEV,PWmax� ð2Þ

parameters of the vector VPRI

VPRI

cies in the cycle of changes nRFEV.

tion interval sigPRI and pulse width sigPW.

The frequency parameters of the signal vector VRF

120 Fractal Analysis - Applications in Physics, Engineering and Technology

VRF

VB ¼ ½VPRI

Figure 3. PRI histogram for six copies of the same type of radars marked by colours.

<sup>B</sup> , VRF

processing and the main data processing by ELINT system (see Figure 2). During the analysis, in total hundreds of radar samples coming are carried out. The received record collections (e.g. six copies of the same type of radars) with basic measurable parameters of PDW are presented in the form of a graph with basic measurable parameters, that is, RF, PW and PRI in Figures 3 and 4. Figure 3 presents the PRI histogram of six tested radar copies in an overall depiction. Figure 4 presents a 3D graph of RF, PRI and PW parameters from six copies, in an overall depiction as well.

Figures 5 and 6 present 3D depicting of radio frequency and pulse width for three selected copies of the same type of radars received with the use of 'mesh' function in the MatLab software.

Figure 5. 3D graphic depicting of PW for three selected copies of the same type of radars.

Figure 6. 3D graphic depicting of RF for three selected copies of the same type of radars.

On the basis of the recordings and initial analysis to further process of identification, only these copies were admitted whose basic measurable parameters, that is, RF, PW and PRI were much the same—see Figures 5–8. Figures 5–8 present the biggest similarity of the radar signal parameters which those sources generated.

Figure 7. 3D graphic depicting of RF, PRI and PW for three selected copies of the same type of radars marked by three shades of gray.

Figure 8. 3D graphic depicting of PW, PRI and RF for three selected copies of the same type of radars marked by three shades of gray.

## 4. Defining the transformation attractor and distinctive features extraction

One of the ways to increase the number of details of definition is specific identification of electromagnetic emitter sources SEI which extracts distinctive features in the process of signal transformation. The distinctive features may be a result of the received transformations of measurement data sets. New data sets will have fractal features which will make it possible to define clearly the source of radar emission. The fractal features and the theory of fractals is adopted by researchers, especially in the field of SAR (Synthetic Aperture Radar) image transformation [8, 9], acoustic signal transformation and the analysis of radar signals. New possibilities of Digital Signal Processing (DSP) in Frequency Modulated Continuous Wave (FMCW) radar and fractal image compression is a promising brand new compression method [10, 11]. It should be noted that the identification of emitter sources based on classical methods of the analysis of basic parameters is currently inefficient. The methods of SEI [12, 13] should be used in order to identify, more precisely, a radar copy of the same type.

#### 4.1. An attractor of transformation

On the basis of the recordings and initial analysis to further process of identification, only these copies were admitted whose basic measurable parameters, that is, RF, PW and PRI were much the same—see Figures 5–8. Figures 5–8 present the biggest similarity of the radar signal

Figure 7. 3D graphic depicting of RF, PRI and PW for three selected copies of the same type of radars marked by three

Figure 6. 3D graphic depicting of RF for three selected copies of the same type of radars.

parameters which those sources generated.

122 Fractal Analysis - Applications in Physics, Engineering and Technology

shades of gray.

The easiest way to make fractals is by using a set of affine transformations, which are contractions or narrowing transformations. In this case, the set of affine transformations is Iterated Function System (IFS). A recording of radar signal was made. Further frequency values, for which the recording was made, correspond to particular measurement points. By transforming the sets of measurement points in the scope of their symmetry or left-side/right-side asymmetry, what was received was the attractor of transformation which can be a fractal in a special case. As a result, the attractor of generalized measurement function appeared, which was the result of the procedure of SEI described here. While doing the analytical procedure of defining the attractor of measurement function, right-side measurement vectors p<sup>r</sup> and left-side ones p<sup>l</sup> with the beginning in the particular point of reference f0, so that <sup>p</sup><sup>r</sup> ¼ ½p<sup>r</sup> <sup>1</sup>, p<sup>r</sup> 2,…,p<sup>r</sup> N� <sup>T</sup> and <sup>p</sup><sup>l</sup> ¼ ½p<sup>l</sup> <sup>1</sup>, p<sup>l</sup> 2,…,p<sup>l</sup> M� <sup>T</sup> were assigned. In order to define the desirable selective features, the <sup>T</sup> : <sup>p</sup><sup>r</sup> ! <sup>t</sup> transformation was done. In this transformation, <sup>t</sup> is the image of the <sup>p</sup><sup>r</sup> vector in the form of a vector with coordinates corresponding to the p<sup>l</sup> vector. For the transparent record of the transformation above with the use of vectors p<sup>r</sup> and p<sup>l</sup> , the mapping was written in the Euclidean plane, that is, <sup>T</sup> : <sup>E</sup><sup>1</sup> ! <sup>E</sup><sup>2</sup> . In the issue, which is considered here, these transformations are linear mappings, so they can be written in the matrix form as <sup>t</sup> <sup>¼</sup> <sup>T</sup> <sup>ð</sup>p<sup>p</sup>, <sup>A</sup>Þ, in which A is the matrix of a given transformation. Depending on the received symmetry or asymmetry (right/left-hand) of measurement points, they will create different dispersion graphs. An example of right-side and left-side asymmetry dispersion graph is presented in Figures 9 and 10 and precisely described in the work of Dudczyk [13]. The number of measurement points is chosen empirically and is a double value of the maximum filter's width of the IF frequency from the superheterodyne receiver, which is used in the measurement procedure, that is, 40 MHz. As a result of this assumption, the critical original number of measurement points is as follows N = M = 80.

Depending on the received symmetry or asymmetry (right/left-side) of measurement points, it is possible to create dispersion graphs. Measurement points presented in Figure 11, transformed and depicted together, form the so-called measurement function Kðf <sup>n</sup>Þ. Figure 11 shows the

Figure 9. Depiction of transformation of measurement points in a 2D Euclidean space for N > M, that is, the range of right-hand asymmetry.

Figure 10. Depiction of transformation of measurement points in a 2D Euclidean space for N < M, that is, the range of lefthand asymmetry.

Figure 11. A graph of measurement points dispersion after transformation—attractor of transformation.

<sup>T</sup> : <sup>p</sup><sup>r</sup> ! <sup>t</sup> transformation was done. In this transformation, <sup>t</sup> is the image of the <sup>p</sup><sup>r</sup> vector in the form of a vector with coordinates corresponding to the p<sup>l</sup> vector. For the transparent record of

tions are linear mappings, so they can be written in the matrix form as <sup>t</sup> <sup>¼</sup> <sup>T</sup> <sup>ð</sup>p<sup>p</sup>, <sup>A</sup>Þ, in which A is the matrix of a given transformation. Depending on the received symmetry or asymmetry (right/left-hand) of measurement points, they will create different dispersion graphs. An example of right-side and left-side asymmetry dispersion graph is presented in Figures 9 and 10 and precisely described in the work of Dudczyk [13]. The number of measurement points is chosen empirically and is a double value of the maximum filter's width of the IF frequency from the superheterodyne receiver, which is used in the measurement procedure, that is, 40 MHz. As a result of this assumption, the critical original number of measurement points is

Depending on the received symmetry or asymmetry (right/left-side) of measurement points, it is possible to create dispersion graphs. Measurement points presented in Figure 11, transformed and depicted together, form the so-called measurement function Kðf <sup>n</sup>Þ. Figure 11 shows the

Figure 9. Depiction of transformation of measurement points in a 2D Euclidean space for N > M, that is, the range of

Figure 10. Depiction of transformation of measurement points in a 2D Euclidean space for N < M, that is, the range of left-

, the mapping was written in the

. In the issue, which is considered here, these transforma-

the transformation above with the use of vectors p<sup>r</sup> and p<sup>l</sup>

124 Fractal Analysis - Applications in Physics, Engineering and Technology

Euclidean plane, that is, <sup>T</sup> : <sup>E</sup><sup>1</sup> ! <sup>E</sup><sup>2</sup>

as follows N = M = 80.

right-hand asymmetry.

hand asymmetry.

coordinate plane, where an abscissa (the value of x) is marked as a f x <sup>n</sup> and an ordinate (the value of y) is marked as a f y n.

On the basis of distinctive streaks that were formed, such hypothesis can be proposed: functions gAðf <sup>n</sup>Þ, gBðf <sup>n</sup>Þ, gCðf <sup>n</sup>Þ and gDðf <sup>n</sup>Þbelong to the class of linear functions, in whichgAðf <sup>n</sup>Þ, gBðf <sup>n</sup>Þ, gCðf <sup>n</sup>Þand gDðf <sup>n</sup>Þ will be the regression lines for the streaks formed through the measurement points [14]. Linear equation of regression for the presented case is defined with the following equation gðf <sup>n</sup>Þ ¼ α � f <sup>n</sup> þ β, in which α can be expressed as a vector ½αA,αB,αC,αD� T and β can be expressed as a vector ½βA, βB, βC,βD� <sup>T</sup> and <sup>g</sup>ð<sup>f</sup> <sup>n</sup><sup>Þ</sup> can be expressed as a vector <sup>½</sup>gAð<sup>f</sup> <sup>n</sup>Þ,gBð<sup>f</sup> <sup>n</sup>Þ,gCð<sup>f</sup> <sup>n</sup>Þ, gDð<sup>f</sup> <sup>n</sup>Þ�<sup>T</sup>. To define the value of <sup>α</sup> and <sup>β</sup>, Eq. (5) should be minimalized.

$$E[f\_n^Y - \mathfrak{a} \cdot f\_n - \mathfrak{B}]^2 = \min \tag{5}$$

$$\begin{cases} \frac{\partial}{\partial \alpha} E[f\_n^Y - \mathfrak{a} \cdot f\_n - \mathfrak{B}]^2 = -2E[(f\_n^Y - \mathfrak{a} \cdot f\_n - \mathfrak{B})f\_n] \\ \qquad \frac{\partial}{\partial \beta} E[f\_n^Y - \mathfrak{a} \cdot f\_n - \mathfrak{B}]^2 = -2E[(f\_n^Y - \mathfrak{a} \cdot f\_n - \mathfrak{B})] \end{cases} \tag{6}$$

After comparing the calculated derivatives Eq. (6) to zero, appears the system of normal equations in which after replacing the expected values with particular moments of equation systems, the following equation can be written:

$$\begin{cases} \mathfrak{a} \cdot m\_{20} + \mathfrak{B} \cdot m\_{10} &=& m\_{11} \\ \mathfrak{a} \cdot m\_{10} + \mathfrak{B} &=& m\_{01} \end{cases} \tag{7}$$

in which m<sup>10</sup> and m<sup>01</sup> are sample 1st moments, m<sup>20</sup> is sample 2nd moment and m<sup>11</sup> is mixed sample 1st moment. After further transformations, the regression equation is as follows:

$$\log(f\_n) = \frac{\mu\_{11}}{\mu\_{20}} \cdot f\_n + \left(m\_{01} - \frac{\mu\_{11}}{\mu\_{20}} m\_{10}\right) = \mathfrak{a}\_{21} f\_n + \mathfrak{B} \tag{8}$$

where

$$\mathbf{a}\_{21} = \begin{bmatrix} \frac{\mu\_{11}^A}{\mu\_{20}^A}, \frac{\mu\_{11}^B}{\mu\_{20}^B}, \frac{\mu\_{11}^C}{\mu\_{20}^C}, \frac{\mu\_{11}^D}{\mu\_{20}^D} \end{bmatrix}^T = \begin{bmatrix} \alpha\_{A\prime} \ \alpha\_{B\prime} \ \alpha\_{C\prime} \ \alpha\_D \end{bmatrix}^T \tag{9}$$

$$\mathcal{B} = \begin{bmatrix} m\_{01}^{A} - \frac{\mu\_{11}^{A}}{\mu\_{20}^{A}} m\_{10^{\circ}}^{A} \ m\_{01}^{B} - \frac{\mu\_{11}^{B}}{\mu\_{20}^{B}} m\_{10^{\circ}}^{B} \ m\_{01}^{C} - \frac{\mu\_{11}^{C}}{\mu\_{20}^{C}} m\_{10^{\circ}}^{C} \ m\_{01}^{D} - \frac{\mu\_{11}^{D}}{\mu\_{20}^{D}} m\_{10^{\circ}}^{D} \end{bmatrix}^{T} \\ = \begin{bmatrix} \boldsymbol{\beta}\_{A'} \ \boldsymbol{\beta}\_{B'} \ \boldsymbol{\beta}\_{C'} \ \boldsymbol{\beta}\_{D} \ \boldsymbol{\end}^{T} \ (10)$$

and μ<sup>11</sup> means mixed 2nd central moment and μ<sup>20</sup> means 2nd central moment. As a result of further transformations, four linear regression equations were given. The particular equation system given by the regression equation allows to calculate characteristic points of coordinates. Examples of four characteristic points presented in Figure 11 in the form of black points, such as (PAB, PBC, PCD, PAD), were formed. Then, with the use of characteristic points of coordinates, the measurement function Kðf <sup>n</sup>Þ was formed.

#### 4.2. Distinctive fractal features extraction

As a result of further transformations, four equations of linear regression were received. Then, it is possible to draw a measurement function Kðf <sup>n</sup>Þ in the form of a product k degree given k + 1 characteristic points, defined by the Lagrange's polynomial formula in accordance with Eq. (11), where ak, ak�<sup>1</sup>, …, a<sup>0</sup> are characteristic parameters of measurement function Kðf <sup>n</sup>Þ, as shown in Figure 12.

Figure 12. The image of measured function according to Lagrange polynomial and fractal features extraction.

$$K(f\_n) = a\_k f\_n^k + a\_{k-1} f\_n^{k-1} + a\_{k-2} f\_n^{k-2} + \dots + a\_0 \tag{11}$$

The formalized notation of the measurement function Kðf <sup>n</sup>Þ allows to extract distinctive features through defining the space area under the measurement function and the arc length of the function, which appeared for the SEI process. The feature S is the value of the space area of a closed surface expanding from the generalized measurement function Kðf <sup>n</sup>Þ in the bracket 〈f min <sup>n</sup> , f max <sup>n</sup> 〉, respecting Eq. (12).

<sup>g</sup>ð<sup>f</sup> <sup>n</sup>Þ ¼ <sup>μ</sup><sup>11</sup> μ<sup>20</sup>

126 Fractal Analysis - Applications in Physics, Engineering and Technology

<sup>α</sup><sup>21</sup> <sup>¼</sup> <sup>μ</sup><sup>A</sup> 11 μA 20 , μB 11 μB 20

<sup>01</sup> � <sup>μ</sup><sup>B</sup> 11 μB 20 m<sup>B</sup> <sup>10</sup>, m<sup>C</sup>

coordinates, the measurement function Kðf <sup>n</sup>Þ was formed.

4.2. Distinctive fractal features extraction

where

<sup>β</sup> <sup>¼</sup> <sup>m</sup><sup>A</sup>

Figure 12.

<sup>01</sup> � <sup>μ</sup><sup>A</sup> 11 μA 20 m<sup>A</sup> <sup>10</sup>, m<sup>B</sup> � <sup>f</sup> <sup>n</sup> <sup>þ</sup> <sup>m</sup><sup>01</sup> � <sup>μ</sup><sup>11</sup>

, <sup>μ</sup><sup>D</sup> 11 μ<sup>D</sup> 20

and μ<sup>11</sup> means mixed 2nd central moment and μ<sup>20</sup> means 2nd central moment. As a result of further transformations, four linear regression equations were given. The particular equation system given by the regression equation allows to calculate characteristic points of coordinates. Examples of four characteristic points presented in Figure 11 in the form of black points, such as (PAB, PBC, PCD, PAD), were formed. Then, with the use of characteristic points of

As a result of further transformations, four equations of linear regression were received. Then, it is possible to draw a measurement function Kðf <sup>n</sup>Þ in the form of a product k degree given k + 1 characteristic points, defined by the Lagrange's polynomial formula in accordance with Eq. (11), where ak, ak�<sup>1</sup>, …, a<sup>0</sup> are characteristic parameters of measurement function Kðf <sup>n</sup>Þ, as shown in

Figure 12. The image of measured function according to Lagrange polynomial and fractal features extraction.

, μC 11 μC 20

<sup>01</sup> � <sup>μ</sup><sup>C</sup> 11 μC 20 m<sup>C</sup> <sup>10</sup>, m<sup>D</sup>

" #<sup>T</sup>

" #<sup>T</sup>

μ<sup>20</sup> m<sup>10</sup>

> <sup>01</sup> � <sup>μ</sup><sup>D</sup> 11 μ<sup>D</sup> 20 m<sup>D</sup> 10,

¼ α21f <sup>n</sup> þ β ð8Þ

¼ ½βA, <sup>β</sup>B, <sup>β</sup>C, <sup>β</sup><sup>D</sup> � <sup>T</sup> <sup>ð</sup>10<sup>Þ</sup>

¼ ½αA, <sup>α</sup>B, <sup>α</sup>C, <sup>α</sup>D� <sup>T</sup> <sup>ð</sup>9<sup>Þ</sup>

� �

$$S = \int\_{f\_n^{\min}}^{f\_n^{\max}} K(f\_n) df\_n = \int\_{f\_n^{\min}}^{f\_n^{\max}} (a\_k f\_n^k + a\_{k-1} f\_n^{k-1} + a\_{k-2} f\_n^{k-2} + \dots + a\_0) df\_n \tag{12}$$

Simultaneously, the arc length of the generalized measurement function Kðf <sup>n</sup>Þ as the second distinction feature of the radar emission source will be represented through the arc length L of the function Kðf <sup>n</sup>Þ in the brackets 〈f min <sup>n</sup> , f max <sup>n</sup> 〉, respecting Eq. (13).

$$L = \int\_{f\_{\text{n}}^{\text{min}}}^{f\_{\text{n}}^{\text{max}}} \left[ 1 + \left( \frac{\partial \mathcal{K}(f\_{\text{n}})}{\partial f\_{\text{n}}} \right)^{2} \right]^{\frac{1}{2}} df\_{\text{n}} = \int\_{f\_{\text{n}}^{\text{min}}}^{f\_{\text{n}}^{\text{max}}} \left[ 1 + \left( ka\_{\text{f}} f\_{\text{n}}^{k-1} + (k-1)a\_{k-1} f^{k-2} + \dots + a\_{1} \right)^{2} \right]^{\frac{1}{2}} df\_{\text{n}} \quad (13)$$

According to Eqs. (12) and (13), it is possible to extract two additional distinctive features, that is, the length of measurement function and the value of area which is included under this function. The presented method of features extraction makes it possible to estimate numerical surface areas under the measurement functions (feature S) and the distance of arc of these functions (feature L). Then the vector of basic measurable parameters of radar signal was extended with two additional features.

Given in that way, two additional features expand the VB vector of the basic features of radar signal measurable parameters, such as PW, PRI and RF. And these features are a good separation measure in the SEI process. The way of defining these two additional features and using them in the process of identification of the radar copies of the same type was presented in further part of this chapter.

#### 5. Fractal of generalized measurement function

Generalization of the method of radar signal identification on the basis of the transformation fractal is defining the generalized measurement function <sup>K</sup>^ð<sup>f</sup> <sup>n</sup><sup>Þ</sup> going through all particular characteristic points Pn, in which n = 0,1,…,kgr . Figure 13 presents the fractal character of the measurement function received as a result of the transformation of the set of measurement points.

The generalized measurement function <sup>K</sup>^ <sup>ð</sup><sup>f</sup> <sup>n</sup><sup>Þ</sup> preserves the character of not decreasing function in a particular bracket 〈f min <sup>n</sup> , f max <sup>n</sup> 〉 and out of definite character, it shows prediction

Figure 13. The fractal character of generalized measurement function.

features. Simultaneously, <sup>K</sup>^ <sup>ð</sup><sup>f</sup> <sup>n</sup><sup>Þ</sup> is located in an area that is mapped by the lower limiting straight FLðf <sup>n</sup>Þ and upper limiting straight FUðf <sup>n</sup>Þ and is symmetric relatively to the symmetrical limiting straight FSðf <sup>n</sup>Þ according to Eqs. (14)–(16), in which f x <sup>2</sup>, f <sup>x</sup> <sup>3</sup>, f <sup>x</sup> gr are the abscissae of characteristic points and f y <sup>2</sup>, f <sup>y</sup> <sup>3</sup>, f <sup>y</sup> gr are the ordinates of characteristic points. Figure 13 presents a fractal character of the generalized measurement function which has the form of a contraction mapping.

$$F\_L(f\_n) = \begin{bmatrix} f\_{g^r}^y - f\_2^y \\ f\_{g^r}^x - f\_2^x \end{bmatrix} \cdot (f\_n - f\_2^x) + f\_2^y \tag{14}$$

$$F\_{\rm II}(f\_n) = \begin{bmatrix} f\_{\rm gr}^y - f\_3^y \\ f\_{\rm gr}^x - f\_3^x \end{bmatrix} \cdot (f\_n - f\_3^x) + f\_3^y \tag{15}$$

$$F\_S(f\_n) = \begin{bmatrix} f\_{\mathcal{S}^r}^y \\ f\_{\mathcal{S}^r}^x \end{bmatrix} \cdot f\_n \tag{16}$$

It should be mentioned that the received shape of the measurement function (according to Figures 11 and 12) is an individual model of a radar emission source. 'An individual model' means 'lines on the fingers' of the radar which make a clear identification possible.

#### 6. Results of analysis

To compare the received results with other RSR methods based on, for example, Fast decision Identification Algorithm of emission source pattern described in Ref. [6], or out-of-band radiation of radar devices described in Ref. [13], or inter-pulse modulation described in Ref. [7] and intra-pulse analysis of a radar signal shown in Ref. [15] or method based on data modelling presented in Ref. [16], Correct Identification Coefficient (CIC) is set according to Eq. (17), where nB�<sup>P</sup> is the number of correct comparisons of basic features' vectors VB (presented according to Eq. (4)), to extended vectors VEXT (with two additional features L and S) in a particular class, where N is the number of all comparisons divided by the number of test collections.

$$\text{CIC} = \frac{n\_{B-P}}{N} \tag{17}$$

The number of nB�<sup>P</sup> correct comparisons is set according to Eq. (18), where <sup>γ</sup><sup>j</sup> <sup>i</sup> function assigns to a pair of vectors <sup>ð</sup>V<sup>i</sup> <sup>B</sup>, <sup>V</sup><sup>j</sup> EXTÞ the value which equals '1' if i = j, or the value which equals '0' if i 6¼ j. The example of CIC received values are presented in the following part of this chapter.

$$m\_{B-P} = \sum\_{i=1}^{I} \sum\_{j=1}^{J} \gamma^{i\cdot j} \tag{18}$$

The process of identification was made on the basis of length measurement and the decision about the criterion of minimal distance classification. A correctness estimation of tests with particular class were Mahalanobis, Euclidean and Hamming distances (metrics) [17, 18]. The criterion of classification was the criterion of 'the nearest neighbour', which was used as one of the basic threshold criteria [19]. In order to assess the quality of the classification/identification process, the Correct Identification Coefficient was defined.

According to Eqs. (8) and (9), it was possible to extract two additional distinctive features, that is, the length of measurement function and the value of area which is included under this function. The results were presented in Figures 14–16.

Also, the received estimation results are presented in Figures 17–19. Appropriately crossed columns and lines of each V<sup>B</sup> vector and extended vectors VEXT present the degree of their similarity defined by the distance value. The less value of this distance means the bigger similarity of V<sup>B</sup> vector to the extended vectors. Also, in Figures 15–17, there are minimum values of the distance marked with a red dotted ellipse.

Figure 14. An attractor of transformation for Copy of Radar No. 1.

features. Simultaneously, <sup>K</sup>^ <sup>ð</sup><sup>f</sup> <sup>n</sup><sup>Þ</sup> is located in an area that is mapped by the lower limiting straight FLðf <sup>n</sup>Þ and upper limiting straight FUðf <sup>n</sup>Þ and is symmetric relatively to the symmet-

sents a fractal character of the generalized measurement function which has the form of a

� ðf <sup>n</sup> � f x <sup>2</sup>Þ þ f y

� ðf <sup>n</sup> � f x <sup>3</sup>Þ þ f y

y gr f x gr

" #

It should be mentioned that the received shape of the measurement function (according to Figures 11 and 12) is an individual model of a radar emission source. 'An individual model'

To compare the received results with other RSR methods based on, for example, Fast decision Identification Algorithm of emission source pattern described in Ref. [6], or out-of-band

y gr � f y 2

y gr � f y 3

f x gr � f x 3

means 'lines on the fingers' of the radar which make a clear identification possible.

" #

" #

FSð<sup>f</sup> <sup>n</sup>Þ ¼ <sup>f</sup>

f x gr � f x 2 x <sup>2</sup>, f <sup>x</sup> <sup>3</sup>, f <sup>x</sup>

gr are the ordinates of characteristic points. Figure 13 pre-

gr are the abscissae of

<sup>2</sup> ð14Þ

<sup>3</sup> ð15Þ

� f <sup>n</sup> ð16Þ

rical limiting straight FSðf <sup>n</sup>Þ according to Eqs. (14)–(16), in which f

FLð<sup>f</sup> <sup>n</sup>Þ ¼ <sup>f</sup>

FUð<sup>f</sup> <sup>n</sup>Þ ¼ <sup>f</sup>

y <sup>2</sup>, f <sup>y</sup> <sup>3</sup>, f <sup>y</sup>

Figure 13. The fractal character of generalized measurement function.

128 Fractal Analysis - Applications in Physics, Engineering and Technology

characteristic points and f

6. Results of analysis

contraction mapping.

Figure 15. An attractor of transformation for Copy of Radar No. 2.

Figure 16. An attractor of transformation for Copy of Radar No. 3.

Figure 17. The values of Mahal distances for Radar Copy No. 1.

A correctness estimation of tests with particular class were Mahalanobis, Euclidean and Hamming distances (metrics) [21]. The SEI estimation results are presented in Figures 20–22.

According to the SEI methods listed in this chapter, the received RER results are as follows: the use of out-of-band radiation described in the work of Dudczyk [13] and the CIC value for RSR is about 90%. The method based on fractal features described in the work of Dudczyk and Kawalec [3], and the CIC value is 91.6% for Mahalanobis metric and 96.7% for Euclidean and Hamming metrics. Very similar RER results are received in the work of Dudczyk and Kawalec

Figure 18. The values of Mahal distances for Radar Copy No. 2.

Figure 19. The values of Mahal distances for Radar Copy No. 3.

Figure 20. The values of CIC for Radar Copy No. 1.

A correctness estimation of tests with particular class were Mahalanobis, Euclidean and Hamming distances (metrics) [21]. The SEI estimation results are presented in Figures 20–22.

Figure 15. An attractor of transformation for Copy of Radar No. 2.

130 Fractal Analysis - Applications in Physics, Engineering and Technology

Figure 16. An attractor of transformation for Copy of Radar No. 3.

Figure 17. The values of Mahal distances for Radar Copy No. 1.

According to the SEI methods listed in this chapter, the received RER results are as follows: the use of out-of-band radiation described in the work of Dudczyk [13] and the CIC value for RSR is about 90%. The method based on fractal features described in the work of Dudczyk and Kawalec [3], and the CIC value is 91.6% for Mahalanobis metric and 96.7% for Euclidean and Hamming metrics. Very similar RER results are received in the work of Dudczyk and Kawalec

Figure 21. The values of CIC for Radar Copy No. 2.

Figure 22. The values of CIC for Radar Copy No. 3.

[4], where RSR is also based on the analysis of fractal features. The method based on interpulse analysis described in the work of Dudczyk et al. [7] increases the CIC coefficient up to 70%, and the method based on intra-pulse analysis described in the work of Kawalec and Owczarek [15] makes it possible to receive RSR results reaching 90% level. Data modelling applied to RSR and identification is presented in the work of Kawalec and Owczarek [16]. In this work, the value of CIC equals 98%. In the work of Dudczyk and Kawalec [6], the Fast Identification Algorithm for RER is presented. This algorithm is parameterized in three stages by implementation of three different ways to define the similarity degree of the signal vector to the pattern in the database. Based on this algorithm, the CIC value is 63%. In order to depict it, in Figure 23, there have been presented CIC values.

The presented method of features extraction makes it possible to estimate numerical surface areas under the measurement functions (feature S) and the distance of arc of these functions (feature L). Then, the vector of basic measurable parameters of radar signal was extended with two additional features. Given in that way, two additional features expand the vector of the basic features of radar signal measurable parameters, such as PW, PRI and RF, are a good separation measure in the SEI process. The way of defining these two additional features and using them in the process of identification of the radar copies of the same type was presented. The features S and L are distinctive information for good separating measure in the SEI process. Simultaneously, as a result of transformations in collections of measurement points,

Figure 23. Graphic illustration of CIC values for other RER methods.

the transformation attractor of the generalized measurement function is received. The received attractor is used later on to optimize the SEI process.

It needs to be emphasized with full conviction that referring to the works above, during the SEI procedure, the same recordings of a few hundred radar signals coming from the same type of radars are used. Only by this approach, it is possible to compare the received results. It needs to be emphasized that the RSR methods listed in this chapter differ from each other as concerns the test procedure, the compilation level, calculation time and algorithm complexity. However, the main difference is that in the process of generating distinctive features, it is possible to achieve different distinctive features from a radar signal. In that way, a quasi-optimum radar signal pattern appeared.

## 7. Conclusion

[4], where RSR is also based on the analysis of fractal features. The method based on interpulse analysis described in the work of Dudczyk et al. [7] increases the CIC coefficient up to 70%, and the method based on intra-pulse analysis described in the work of Kawalec and Owczarek [15] makes it possible to receive RSR results reaching 90% level. Data modelling applied to RSR and identification is presented in the work of Kawalec and Owczarek [16]. In this work, the value of CIC equals 98%. In the work of Dudczyk and Kawalec [6], the Fast Identification Algorithm for RER is presented. This algorithm is parameterized in three stages by implementation of three different ways to define the similarity degree of the signal vector to the pattern in the database. Based on this algorithm, the CIC value is 63%. In order to depict it,

The presented method of features extraction makes it possible to estimate numerical surface areas under the measurement functions (feature S) and the distance of arc of these functions (feature L). Then, the vector of basic measurable parameters of radar signal was extended with two additional features. Given in that way, two additional features expand the vector of the basic features of radar signal measurable parameters, such as PW, PRI and RF, are a good separation measure in the SEI process. The way of defining these two additional features and using them in the process of identification of the radar copies of the same type was presented. The features S and L are distinctive information for good separating measure in the SEI process. Simultaneously, as a result of transformations in collections of measurement points,

in Figure 23, there have been presented CIC values.

Figure 23. Graphic illustration of CIC values for other RER methods.

Figure 22. The values of CIC for Radar Copy No. 3.

132 Fractal Analysis - Applications in Physics, Engineering and Technology

Radar signal recognition with the use of classical methods, that is, based on statistical analysis of basic measurable parameters of a radar signal, such as radio frequency, amplitude, pulse width or pulse repetition interval, is not enough to carry out the distinction process of particular copies of the same radar type.

The received measurement data have a significant influence on the SEI process of radar, in which it is aimed to receive very high level of radar signal identification. Ultimately, signal source identification, which is 100%, should be characterized by the maximization of explicitness of RER procedure. It is not a trivial matter to achieve such a result. It is also known that stochastic context-free grammars (SCFG) appear promising for the recognition and threat assessment of complex radar emitters in radar systems, but the computational requirements for learning their production rule probabilities can be very onerous [20]. As shown in Ref. [21], a self-organizing map and the maximum likelihood gamma mixture model classifier and adopted Bayesian formalism are too complicated for direct analytical use in automatic radar recognition. The presented SEI method based on fractal features is realized on the basis of MatLab software package and received vectors are recorded in a dedicated database for ELINT system. The received CIC value indicates that there has been a noticeable rise in the radar signal correct identification. Comparing the received results of the identification process with other methods, it may be admitted that the presented method makes it possible to increase the value of CIC. In order to increase the CIC coefficient value, in further works on RSR use in SEI process, a common similarity matrix should be defined. This matrix should include the complexity of algorithms which are used in the RER method, estimation time, the requirements of the equipment platform and other requirements, which are significant in the process of quality estimation of a particular method. Thus, it will be possible to count automatically the similarities between vectors of basic measurable parameters for different radar copies of the same type. For ELINT systems working in real conditions, on a contemporary battlefield, the automation of RSR process and explicit identification of every single emitter in real time (with minimum time burden) are currently primary challenges for ELINT specialists

This chapter highlights the fact that the RSR process described here is a complex problem. It is also generally known that a number of aspects such as defining the DataBase, the method which creates the pattern, classification process and identification process, the criteria which are used and the method to calculate the correct identification coefficient, are currently a great challenge for researchers and, for the time being, there are no optimal solutions to them. Many solutions are still a mystery in this subject and because of the fact that they are a matter of current EW field and specific programme-device applications, they cannot be published. All attempts to implement such solutions to ELINT systems and electronic warfare should be optimized to a particular device not to overload the SEI system.

## Author details

Janusz Dudczyk

Address all correspondence to: j.dudczyk@wb.com.pl

WB Electronics S.A., R&D Department, Poznanska, Poland

## References


[8] F. Berizzi, G. Bertini, M. Martorella. Two-dimensional variation algorithm for fractal analysis of sea SAR images. IEEE Transactions on Geoscience and Remote Sensing. 2006;44(9):2361–2373. doi:10.1109/TGRS.2006.873577

which creates the pattern, classification process and identification process, the criteria which are used and the method to calculate the correct identification coefficient, are currently a great challenge for researchers and, for the time being, there are no optimal solutions to them. Many solutions are still a mystery in this subject and because of the fact that they are a matter of current EW field and specific programme-device applications, they cannot be published. All attempts to implement such solutions to ELINT systems and electronic warfare should be

[1] J. Dudczyk. Radar emission sources identification based on hierarchical agglomerative clustering for large data sets. Journal of Sensors. 2016;2016(2016):1879327. doi:10.1155/

[2] J. Dudczyk, A. Kawalec. Specific emitter identification based on graphical representation of the distribution of radar signal parameters. Bulletin of the Polish Academy of Sciences,

[3] J. Dudczyk, A. Kawalec. Fractal features of specific emitter identification. Acta Physica

[4] J. Dudczyk, A. Kawalec. Identification of emitter sources in the aspect of their fractal features. Bulletin of the Polish Academy of Sciences, Technical Sciences. 2013;61(3):623–

[5] J. Dudczyk, A. Kawalec. The use of fractal features extracted from radar signals in the process of specific identification. Przegląd Elektrotechniczny. 2014;R.90(11/2014):212–

[6] J. Dudczyk, A. Kawalec. Fast-decision identification algorithm of emission source pattern in database. Bulletin of the Polish Academy of Sciences, Technical Sciences. 2015;63

[7] J. Dudczyk, J. Matuszewski, A. Kawalec. Specific emitter identification based on an interpulses modulation of radar signals. Przegląd Elektrotechniczny. 2016;R.92(9/2016):267–

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optimized to a particular device not to overload the SEI system.

Address all correspondence to: j.dudczyk@wb.com.pl

134 Fractal Analysis - Applications in Physics, Engineering and Technology

WB Electronics S.A., R&D Department, Poznanska, Poland

Author details

Janusz Dudczyk

References

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