Specific Emitter Identification Based on Fractal Features

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) battle- field system, is building a measurement data vector, then a radar's metrics and the same database. This approach is called Specific Emitter Identification (SEI).


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 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][2][3][4][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.

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 module 1 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 2 electromagnetic emitters (in the range of wavelengths 0.5-18 GHz); 1 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. 2 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; • analysis of measured signal parameters in a thick electromagnetic environment (a few thousand or more pulses) and initial measurement data processing (initial parametric selection and/or signal reduction); • main signal processing (extraction of basic features and estimation of basic measurable radar signal parameters i.e. signal radio frequency, amplitude, pulse width, pulse repetition interval) on the basis of statistical functions concerning estimating e.g. average values of parameters, class models and hypothesis verification; • radar signal archivization in files with measurement data in the DataBase and updating procedure as well as DataBase structure modification; • radar types recognition and classification on the basis of radar signals received from them by comparing signal parameters with the model in BD in the shortest time possible and using expert's knowledge in the process of emitter sources recognition, classification and location; • visualization of signals, measurement data clusters, results of recognition and other data and distinctive information presented in a tabular and graphic form; • depict results of recognition on a digital map and automatically adjusting a unit to the area; • possibilities of import and analysis of measurement data from electronic recognition, which is received from other sources and data measurement and recognition information fusion; • creating a simulation software to generate warfare scenarios, test correctness of emission sources classification, recognition and location procedures and to estimate the effectiveness of the system and trainings for operators.
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 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 AI 3 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. Figure 1. The process of information processing in the radar recognition system. 3 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).

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.  The effect of the further transformation of PDW vector is Basic Signal Vector V B , whose fields are of particular signal frequency and time parameters, according to Eqs. (2)-(4). The time parameters of the vector V PRI B are as follows: minimum pulse repetition interval value PRI min , average pulse repetition interval value PRI EV , maximum pulse repetition interval value PRI max , the number of values of pulse repetition interval nPRI, the number of values of average pulse repetition interval nPRI EV , the minimum value of pulse width PW min , the average value of pulse width PW EV and the maximum value of pulse width PW max .
The frequency parameters of the signal vector V RF B are defined according to Eq. (3) and are as follows: the minimum value of the signal radio frequency RF min , the average value of the radio frequency for the period RF EV , the maximum value of the radio frequency in the period of changes RF max , the number of values of the radio frequency nRF and the number of average radio frequencies in the cycle of changes nRF EV .
The vector V B 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.
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 V B also undergoes the process of implementation and automation in the stage of initial data 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.    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

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.

An attractor of transformation
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 r and left-side ones p l with the beginning in the particular point of reference f 0 , so that p r ¼ ½p r 1 , p r 2 , …, p r N T and p l ¼ ½p l 1 , p l 2 , …, p l M T were assigned. In order to define the desirable selective features, the  T : p r ! t transformation was done. In this transformation, t is the image of the p r vector in the form of a vector with coordinates corresponding to the p l vector. For the transparent record of the transformation above with the use of vectors p r and p l , the mapping was written in the Euclidean plane, that is, T : E 1 ! E 2 . In the issue, which is considered here, these transformations are linear mappings, so they can be written in the matrix form as t ¼ T ðp p , AÞ, 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 n Þ. Figure 11 shows the  On the basis of distinctive streaks that were formed, such hypothesis can be proposed: functions g A ðf n Þ, g B ðf n Þ, g C ðf n Þ and g D ðf n Þbelong to the class of linear functions, in whichg A ðf n Þ, g B ðf n Þ, g C ðf n Þand g D ðf n 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 n Þ ¼ α Á f n þ β, in which α can be expressed as a vector ½α A , α B , α C , α D T and β can be expressed as a vector ½β A , β B , β C , β D T and gðf n Þ can be expressed as a vector ½g A ðf n Þ, g B ðf n Þ, g C ðf n Þ, g D ðf n Þ T . To define the value of α and β, Eq. (5) should be minimalized.
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: in which m 10 and m 01 are sample 1st moments, m 20 is sample 2nd moment and m 11 is mixed sample 1st moment. After further transformations, the regression equation is as follows: where and μ 11 means mixed 2nd central moment and μ 20 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 (P AB , P BC , P CD , P AD ), were formed. Then, with the use of characteristic points of coordinates, the measurement function Kðf n Þ was formed.

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 n 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 a k , a kÀ1 , …, a 0 are characteristic parameters of measurement function Kðf n Þ, as shown in Figure 12.
The formalized notation of the measurement function Kðf n 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 n Þ in the bracket 〈f min n , f max n 〉, respecting Eq. (12).
Simultaneously, the arc length of the generalized measurement function Kðf n Þ as the second distinction feature of the radar emission source will be represented through the arc length L of the function Kðf n Þ in the brackets 〈f min n , f max n 〉, respecting Eq. (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 V B 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.

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 functionKðf n Þ going through all particular characteristic points P n , in which n = 0,1,…,k gr . 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 functionKðf n Þ preserves the character of not decreasing function in a particular bracket 〈f min n , f max n 〉 and out of definite character, it shows prediction features. Simultaneously,Kðf n Þ is located in an area that is mapped by the lower limiting straight F L ðf n Þ and upper limiting straight F U ðf n Þ and is symmetric relatively to the symmetrical limiting straight F S ðf n Þ according to Eqs. (14)- (16), in which f x 2 , f x 3 , f x gr are the abscissae of characteristic points and f y 2 , f y 3 , f y 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.
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.

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 Figure 13. The fractal character of generalized measurement function. 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 n BÀP is the number of correct comparisons of basic features' vectors V B (presented according to Eq. (4)), to extended vectors V EXT (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.
The number of n BÀP correct comparisons is set according to Eq.
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.   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         [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,  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.

Conclusion
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.