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Evidence of chaos and fractals in ECM efffectiveness simulation: theory may help in the development of electronic-countermeasures techniques against radar-homing missile seekers. (Technology).

The purpose of this article is to introduce new evidence that chaos theory has application to problems facing the EW community of researchers, scientists, engineers, operators, and administrators. At least one paper has already appeared in this publication, speculating on the importance of chaos theory to the interpretation of results of EW trials and simulation studies, as well as to the fundamental structuring of such activities (1). The application of chaos theory involves the broad accommodation of the emergent properties of nonlinear systems and processes. The opportunity to apply new mathematical approaches is a byproduct of the plummeting cost of computing power and the affordability of previously unimagined processor speeds. This has allowed researchers and engineers to develop and use high-fidelity software models to solve fundamental EW problems, problems that without exception involve nonlinear processes. This in turn implies the potential for chaotic behavior, which is a natural, normal, and often vexing attribute of nonlinear processes.

This article presents direct evidence that chaos theory is important to solving EW-related problems. As a point of reference, the term "chaos theory" is well defined in (2, p. 194): "...whereas chaos is a condition, chaos theory is the amalgam of methods used to scrutinize nonlinear, dissipative, deterministic problems that have randomness embedded in them." Specifically, the evidence we wish to advance is related to the use of software simulation for the purpose of developing electronic-countermeasures (ECM) techniques against the seeker of a terminal-phase-radar homing missile. Although the choice of this software application is somewhat arbitrary, it is a good one, because the interaction of a homing missile and an active angle-deception waveform involves numerous and deeply nested instances of nonlinear feedback. As a result, the behavior of the simulation model can become chaotic. This leads to the hypothesis that chaos theory may be applicable to the design and effectiveness assessment of ECM techniques . It is important to distinguish between chaotic behavior and noise. While chaotic behavior is unpredictable in a specific instance, it is different from randomness, because there may be underlying patterns that can be exploited. The technical challenge is in finding these patterns, given that they may be difficult or impossible to recognize using conventional analysis techniques.

The work reported here involved research into the effects of cross-polarization (crosspol) jamming (3) against active radio-frequency (RF) missile seekers. Specifically, this article describes a duality between calculation of the Mandelbrot set and calculation of the effects of crosspol jamming on a generic terminal-phase seeker. The research showed that effects of crosspol jamming can be summarized as a graphical object embedded in the countermeasure space, and that this object appears to have characteristics of a fractal. Previously unsuspected patterns of seeker behavior are deduced from this object. This work was conducted in 1994-1995 in collaboration with the Defence Research Establishment Ottawa (DREO).

A Fragmented Artform

The word "fractal" was first coined by Benoit Mandelbrot (4) and comes from the Latin adjective "fractus," meaning irregular or fragmented. The term fractal refers to a set of points, called collectively a "fractal object," which has certain properties. Loosely speaking, these properties may be stated as follows:

(i) the set has self-similar features,

(ii) the set has structure at all scales, and

(iii) the dimension of the set is not an integer. (4,5)

The Sierpinski gasket (15), an example of a fractal object, is shown in Figure A. It is defined by the recursive application of a law: a triangular hole is bored in triangle, such that the sides of the hole are one half of the length of the side of the triangle. Triangular holes with the same relative dimension are then bored in each side of the resulting sub-triangles. The law that produces the Sierpinski gasket is recursive, because each application of the law operates on the result of the previous application.

The Mandelbrot set is another example of a fractal object. The study of the Mandelbrot set and of other fractal objects is relevant to EW problems because the nature of these objects arises from nonlinear recursive processes similar to those embedded in EW and ECM problems. The Mandelbrot set is defined by the following recursive equation, where Z and C are complex numbers:


The equation is recursive, because Z appears on both the left- and right-hand sides of the equation. C is a point in the complex plane tested for set membership. The criterion for set membership is that the magnitude of the complex number Z must remain bounded after n iterations of the above equation. The algorithm to generate the Mandelbrot set comprises the following steps:

1. Choose a test point C in the complex plane.

2. Let Z = 0 initially.

3. Repeatedly apply the above equation each time calculating a new value of Z. Sufficient iteration are required to determine if the magnitude of Z converges or diverges. In the former case, the point C is said to belong to the Mandelbrot set. In the latter case, the point is said to be outside the Mandelbrot set.

4. The complex plane is probed by testing all points of interest.

Because Z and C are complex and because the above equation involves a nonlinear operation (squaring), it is generally impossible to predict the outcome of the above equation for a given value of C, if there are a large number of iterations.

A pictorial representation of the Mandelbrot set can be generated by representing the complex plane as a two-axis graph: the x-axis represents the real part of a complex number C = a +jb, and the y-axis represent the imaginary part, as shown in Figure B. Values of C found to lie within the Mandelbrot set are represented by colored dots while those outside of the set are shown in black. It is not necessary to carry out an infinite number of iterations of the defining equation to test a particular point for set membership; instead, the magnitude of Z is examined after some large number of iterations. If the magnitude of Z has exceeded some predetermined threshold value--say. 2.0--then the point C is said to be outside the set. If the magnitude of Z is at or below the threshold value, then the point is said to be inside the set.

The Mandelbrot set is a true fractal object, comprising self-similar features and detail on all scales. The complex axes are shown in the figure, to give the reader an appreciation that the Mandelbrot set lies in the complex plane, and where it lies in the plane.

The boundary of the Mandelbrot set is of special interest in understanding its fractal nature. The boundary contour is lace-like and scale-dependent; it depends on the magnification used. As illustrated by the insets of Figure B, the closer one examines the boundary, the more detail is seen.

Points outside the Mandelbrot set are of interest as well; although all diverge after n iterations, some fail the threshold test (diverge) sooner than others. It is possible to classify by color those which diverge after, say, 10 iterations, those that diverge at between 10 and 20 iterations, between 20 and 30 iterations, and so on, as illustrated in Figure C. Given that the test point is the "initial condition" applied to the recursive equation defining the set, the set is sensitive to (i.e., is a strong function of) initial conditions near the boundary. This insight is important to solving many EW-and ECM-related problems for which the outcome is sensitive to (i.e., is a strong function of) the initial conditions of the weapon-versus-EW engagement.

In order to develop the hypothesized connection between ECM effectiveness and the calculation of fractal objects and chaos theory, it is useful to select an example ECM problem on which to base calculations. We chose crosspol jamming as a departure point, because the effect of this technique produces angle errors in the radar and depends on the relative pointing direction of the radar antenna and radome, if present. The recursive and nonlinear nature of the process of angle tracking the jammer is intuitively evident in this problem. Other jamming techniques could as well have been chosen, either against tracking radars, surveillance radars, targeting or synthetic-aperture radars, or others.

Crosspol jamming involves the transmission of a jamming signal at or near orthogonal to the principal plane of polarization of the victim radar. The intent is to exploit the unintentional (and from the radar's perspective, undesirable) response inherent in many radars to cross-polarized returns. A specific implementation, swept crosspol jamming, involves sweeping the transmitted polarization about the orthogonal position. For illustrative purposes, the crosspol waveform is specified to be triangular in time and defined by two parameters: polarization sweep rate and polarization sweep limits, where the limits are chosen to be symmetrical relative to perfect cross polarization of the radar. The effect of crosspol jamming on radar angle tracking can be calculated by using a high-fidelity computer model to simulate the radar's response to various jamming waveforms, each defined by an unique combination of polarization sweep rates and limits.

From an ECM perspective, it is desirable to introduce angle-tracking errors in the servo loops of the radar. For this reason, the following heuristic measure of effectiveness (MOE) is adopted: Crosspol waveforms "inside" the set of effective waveforms must cause an angle error greater than a threshold value of 1.5 radar beamwidths. This is strongly analogous to the methodology described for calculation of the Mandelbrot set, except instead of testing points in the complex plane, we explore ECM effectiveness by candidate testing jamming waveforms in a plane defined by sweep rates and limits. Since the behavior of a simple, nonlinear, recursive mathematical equation is characterized by a fractal object, there is reason to believe that the behavior of the complex nonlinear recursive process of radar angle tracking in the presence of ECM can also be characterized by a fractal object.

Representing Reality

A series of approximately 15,000 simulation runs were performed to calculate a pictorial representation of the behavior of a simulated static-geometry engagement between a homing seeker tracking in azimuth and elevation planes, and a crosspol jammer. In each run, the simulation began with the seeker pointed at the jammer. The jamming was then applied for t seconds, and the MOE was evaluated against the recorded seeker antenna pointing position -- i.e., the time at which the angle deflection exceeded 1.5 antenna beamwidths was determined for the given crosspol waveform. A different crosspol waveform was used in each simulation. There were no noise sources in the simulation runs.

The color-coded results of these simulation runs are presented in Figure D, comprising a knee-shaped colored object, embedded in a black background. The color key is shown in the figure; waveforms for which the angle threshold was exceed between 0 sec and 0.92 sec after simulation start are coded yellow; those for which the threshold was exceeded between 0.92 sec and 1.95 sec are coded orange, etc. The calculated object appears to be fractal, in that (a) there are selfsimilar features (lobe structures), and (b) there is lace-like detail at increasing magnification (see inset detailed views). Based on a more comprehensive analysis beyond the scope of this paper, the features of this object -- knee shape, presence of lobe structures, and lace-like edges -- appear to be robust to various features and parameters of the radar model. Timeouts for the simulations were 12 sec, 30 sec, and 60 sec, as indicated.

Because the software model used in this investigation is completely deterministic, the structure of the object in Figure D is causal. A noticeable feature of this object is the "slant" color contours within the lobes. The boundaries of these contours were found to correspond to contours of constant waveform period, suggesting that, for this model, the time required for the ECM to cause an angle disturbance that exceeds the threshold value is related to the ECM waveform period. Normalizing to half the waveform period can highlight the relationship between this time and the period of the ECM waveform. This is done because the waveform is fundamentally triangular in time, so that the second half of the waveform is the mirror image of the first half. There are, thus, two passes of the jammer polarization through the radar's crosspolarization position in each ECM cycle, which means there are two opportunities to cause angle-disturbance events in each waveform period.

An additional analysis was conducted using a radar model with slightly different antenna characteristics. Approximately 80,000 simulation runs were conducted to generate the fractal objects shown in Figure E. The data were reprocessed to normalize the time that the angle threshold was exceeded to half the waveform period. Figure E(a) represents the unprocessed data for this second radar model. Figure E(b) represents crosspol waveforms in which the angle threshold was exceeded after an odd number of half waveform periods -- i.e., waveforms in which the angle threshold was exceeded during the positive slope (clockwise rotation) of the triangular waveform, and Figure E(c) represents waveforms for which the threshold was exceeded after an even number of half cycles (counterclockwise rotation). In both cases, the lobes represent "harmonics," loosely speaking, of ECM half cycles associated with catastrophic angle disturbance. By comparing Figures E(b) and E(c), it is apparent that the seeker's angle-tracking loop in this simulation is more susceptible to the clockwise sweep direction than to the counterclockwise direction. Such susceptibility biases have been observed in experimental trials, although elaboration of these results is outside the scop e of this article.

Information derived from a fractal object will have a number of practical applications. For example, the fractal object presented here suggests that the seeker may be more susceptible to clockwise sweep rather than counterclockwise sweep. If a judiciously planned series of measurements reveals this to be true of an actual missile seeker, then it is theoretically possible to "parse" a missile engagement accordingly. If the range of the missile during the engagement is known or can be confidently deduced, and if the missile seeker is known to be most susceptible to clockwise half of a triangular crosspol waveform, then it is possible to define sub-ranges in the missile flight corresponding to clockwise polarization sweeps. By choosing the waveform start polarization and/or waveform period, it may be possible to tailor the jamming to maximize the probability of a radical trajectory maneuver at a desired range or within a subset of desired ranges. Alternatively, this fractal object, or a similar analysis based on calculation of fractal objects, may indicate that a positive-slope saw-tooth waveform is preferable to triangular or negative-slope saw-tooth waveforms. As another example of insight and information derived from the object, consider the effect of sweep rate in Figure E(a) for sweep limits between +/-9 deg/s and +/-11 deg/s. At low sweep rates, the MOE is met; the angle error induced in the seeker exceeds the threshold. As the sweep rate increases, the jamming appears to become ineffective, then becomes effective again at high sweep rates. A simplistic, conceptual understanding of crosspol jamming based on linear processes and dwell time in the window of susceptible polarizations predicts a monotonic decrease in the magnitude of angle disturbances as sweep rate increases. In the context of the fractal object, the non-monotonic change in ECM effectiveness is seen as a consequence of the overall pattern of behavior, rather than as an anomaly indicating a software bug in the simulation or erroneous experimental measurements.

We are aware of one other instance of an apparently fractal object being calculated using ECM effectiveness assessment software. Following the principles of the methodology described in this paper, Dr. David King examined the possibility that a simulation of cross-eye jamming against a terminal-phase active-radar-homing missile is chaotic (7). He confirmed that the simulation results can become chaotic, and he prepared diagrams showing the fine structure of the missile miss-distance surface as a function of ECM parameters.

The degrees to which such fractal objects are predictive of real-world effects remains to be determined, but the correlation is expected to vary on a caseby-case basis, depending on the fidelity of the software simulation to the nonlinear processes of the real-world system of interest and on the sensitivity of the MOE as applied to real-world measurements in the presence of noise. The discrimination between chaotic behavior and the effects of system noise is key to characterizing the causal behavior of nonlinear systems, and generally requires a great deal of noise-free observation data. If this discrimination can be achieved, the advantage of applying chaos theory to ECM problems is that it potentially allows for the identification, isolation, understanding, and exploitation of causal relationships or exploitable mechanisms which might otherwise remain hidden by apparently noisy, non-causal behavior. The resulting insight may also allow the ECM engineer to ensure that EW systems are operated in non-chaotic regimes, so that their effectiveness in countering threats remains well-behaved.

Trouble in Chaos

There are, however, significant potential difficulties in the calculation and measurement of fractal objects. These are primarily related to the corruption of causal patterns by noise, the need for large quantities of noise-free data to define the fractal object, and the need for techniques of dimensional collapse. Concerning this last point, the patterns present in fractal objects useful to an ECM engineer may be embedded in a multidimensional space. If that space has more than two or three degrees of freedom, special mathematical, graphical, and software techniques will be required to make those patterns apparent and useable. For example, consider the problem of optimizing the deployment of an active off-board decoy. The ECM engineer may be required to design a system that provides optimal protection against one or several terminal-phase radar-homing missiles in an engagement. It is immediately evident that there are many degrees of freedom: decoy-launch time, decoy-launch bearing relative to the ship axis, decoy altitude, decoy-separation rate, ship aspect angle relative to the inbound missile, wind speed, range to the nearest friendly ship, and bearing of the nearest friendly ship. Sea-surface and weather conditions also figure in ECM effectiveness, as well as ship radar cross-section, scintillation characteristics based on ship size, and many other factors. Fractal objects are embedded in this multi-dimensional space.

The incorporation of stochastic elements into the calculation of fractal objects for ECM effectiveness will help determine the robustness of the objects, as well as predict the measurability of the objects. This involves replacing deterministic elements of the radar, engagement, scenario, and jammer signals with stochastic values that change from one run to the next in calculating the fractal object. Those regions of the fractal object that are not sensitive to changes in the stochastic parameters will exhibit the same patterns as are evident in non-stochastic calculations. These regions are expected to be of prime interest in exploiting the predictive power of the fractal objects in addressing ECM problems.

In conclusion, we believe that chaos theory and fractal analysis are important new tools for ECM engineers to use in gaining insight into software models and real-world systems governed by nonlinear processes. This article has presented evidence, by example, that fractals and chaos theory are important to the problem of assessing ECM effectiveness and to ECM problems in general. The chaotic behavior of software simulation involving nonlinearities has been illustrated by the calculation of a fractal object, believed to be the first of its kind in ECM applications. Based on this single though extensively examined example, the calculation of fractal objects are believed, by extension, to have practical applications to many ECM and EW problems and to the EW community at large.


(1.) T. Tucker, "Deep impact", JED, July 2001

(2.) Cambel, A. B.; Applied Chaos Theory; Academic Press, Inc., New York, 1993

(3.) Leonov, A. I.; Fomichev, K. I; Barton, W. T. translator; Monopulse Radar, Artech House, Norwood, MA, 1986

(4.) Mandelbrot, B.B.; The Fractal Geometry of Nature; W. H. Freeman, 1982

(5.) Peak, David and Frame, Michael; Chaos Under Control; W. H. Freeman and Company, New York, 1994

(6.) King, Dr. David; "Is Crosseye Chaotic?", DRA Technical Report no. DRN/SS/SSEW/TR96051/1.0, Defence Evaluation and Research Agency, Fareham, UK, November 1996

Shawn Charland is president of Sky Industries Inc., Ottawa, Ontario, Canada

Paul Pulsifer is senior scientific consultant with the ECM Section of Defence R&D Canada, an agency within the Department of National Defence that provides leading-edge science and technology services.
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Title Annotation:electronic-countermeasures
Comment:Evidence of chaos and fractals in ECM efffectiveness simulation: theory may help in the development of electronic-countermeasures techniques against radar-homing missile seekers. (Technology).(electronic-countermeasures)
Author:Charland, Shawn; Pulsifer, Paul
Publication:Journal of Electronic Defense
Geographic Code:1USA
Date:Nov 1, 2002
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