Figure 1: LPMA on arbitrary surface with backfire radiation pattern

The purpose of this research is twofold:

The log-periodic impedance-modulated feed has been successfully used in traditional LPMA design, where the ground plane is assumed to be infinite [1]. This study will demonstrate that the impedance-modulated feed template can still be used to design an LPMA mounted on a realistic platform, provided that the feed is not forced to be log-periodic. Moreover, it will be shown that optimization of the feed alone is sufficient, that is, the external antenna structure itself can be scaled log-periodically and the above threefold performance degradation can be overcome by optimization of the feed alone. The feed parameter search space was found too complex and multimodal for standard local-search algorithms, such as simplex and Newton-based methods [8]-[10] to yield satisfactory results. Initial optimization attempts with these classical techniques revealed the presence of multiple local minima in vastly different areas of the search space. Genetic algorithms (GAs), although relatively slow in comparison to gradient-based optimizers, have become very popular due to their proven ability to thoroughly scan highly multimodal search spaces and the ease with which they can be implemented [11]-[15]. Unfortunately, for problems such as the one studied here, where multiple performance goals may conflict, the simple GA is still an inadequate optimizer since it involves combining multiple competing goals, which can significantly complicate the optimization problem. This study, therefore, uses a GA capable of multiple objective optimization known as a Pareto GA. In the past, such GAs have been used to solve dual-objective electromagnetic optimization problems [12], [14]. Here, a three-objective Pareto GA is applied to antenna feed design to minimize field degradation, feed reflection, and feed inefficiency of an LPMA mounted on a complex body. Additionally, the designs produced by the Pareto GA are shown to outperform those produced by the simple GA.

Standard Log-Periodic Antenna Arrays

When fed appropriately, a log-periodic antenna array can maintain consistent performance over a broad range of frequencies. This performance is achieved by scaling successive cells of the array by a constant. The external radiating structure of the standard LPMA, shown in Figure 2, can be defined using a small set of design parameters.

Figure 2: LPMA with illustration of image

The heights of, and the spacing between, neighboring antenna elements progresses geometrically by a given constant, t. A second parameter relates element length to the interelement distance between element n and n+1.

LPDAs and LPMAs naturally operate in backfire mode [2], [5]. To achieve backfire radiation, the antenna feed must be designed so that the element-to-element phase shift arising from signal propagation through a feed line section is equal to, or slightly less than, the element-to-element phase shift arising from free-space signal propagation [1]. Because LPDA feed lengths are constrained to be equal to the boom lengths, proper phasing is established through feeder transposition between adjacent dipoles [2]. Since transposition is not possible in the LPMA, excess feed lengths are added to achieve the required phase shift per cell, and an impedance-modulated feed is used to prevent the occurrence of structural stop-bands [1].

A standard log-periodic impedance-modulated feed is composed of transmission-line sections of characteristic impedance Z0 or modulated characteristic impedance ZM (Figure 3). The lengths are determined as follows. The lengths of the modulated feed sections on either side of the nth monopole are identical. Both the lengths of the modulated feed and the lengths of the nonmodulated feed are scaled by t for successive cells.

Figure 3: Diagram of an impedance-modulated feeder

In theory, to achieve an infinite bandwidth the array structure would also have to be of infinite extent along its feed axis. In practice, the array is truncated on both ends, which leaves a finite operating band over which the array acts consistently. To improve the frequency independent performance of the array, truncation sections of characteristic impedance Z0 are placed at both ends of the feed. The truncation length between the source and the shortest element serves to electrically approximate the missing section of transmission line that would be present in an LPMA or LPDA with an infinite number of elements [3]. The truncation length at the load end permits more flexible matching to the real load impedance.

Design Goals and Parameters

To design an LPMA mounted on an arbitrary platform, the optimization technique presented here will minimize the field degradation D, the feed reflection G, and the feed inefficiency X.

These parameters are qualitatively described here and will be quantitatively defined in the next section where the analysis of the antenna system is detailed. The field degradation D is a measure of the discrepancy between the desired patterns and the calculated patterns for the design being considered over the operating band. The feed reflection G refers to the reflection observed at the input of the feeder. Specifically, G is defined as the maximum magnitude of the feed reflection coefficient observed over the operating band. The feed inefficiency X indicates the maximum fraction of the power transferred by the source into the feed that is consumed by the load, measured over the operating band. Since G and X both range from zero to one, the field degradation measure is constructed so that 0 < D < 1 for acceptable patterns. For many practical platforms, the standard LPMA feed (described in Section 2) cannot deliver adequate performance. Since the ground plane structure considered here is assumed to be neither log-periodic nor infinite, interelement transmission-line lengths are no longer forced to scale by t. Additionally, the modulated impedance values ZM are allowed to differ from monopole to monopole to support radiating elements with varying impedances.

System Discretization

To reduce the time required per evaluation, the complete analysis of the antenna system is divided into two parts: the radiating structure analysis (an external problem), and the feed-network analysis (an internal problem). The external problem, analyzed by an integral-equation-based method of moments (MoM) code, is computationally expensive, whereas the solution of the internal problem is computationally trivial. The fixed external structure (i.e., the platform and monopoles) can be adequately characterized through a single MoM analysis at each frequency k of interest to completely characterize its behavior in the presence of an arbitrary feed network. The performance of different feeds connected to the fixed external structure may then be analyzed with the method described below. To describe the feed-element interactions at each frequency k, the complete antenna admittance matrix needs to be extracted for the system of radiating elements using the MoM. Specifically, by successively placing a unit voltage source at the base of each of the N monopoles in turn, and setting the excitation of the other monopoles to zero, the currents developed at the base of all array elements by each excitation can be determined.

Genetic Algorithms and Pareto Optimization

The principles underlying GA-based optimizers are uncomplicated and elegant. As their appellation implies, GAs are based on the principles that drive the genetic reproduction cycle and the process of natural selection described by Charles Darwin in The Origin of Species and later expanded upon in The Descent of Man [16], [17]. Tersely described, a GA takes a population of candidate designs and encourages evolution toward the design goals, within the constraints of the population's established genetic structure. GA-based optimization has many advantages over more traditional techniques. As mentioned earlier, GAs are very well suited for optimizing multimodal functions. Of additional interest to the current study, GAs possess an intrinsic ability to optimize both discrete and continuous parameters simultaneously and do not require derivative information. For the problem considered here, the first capacity allows the GA to select continuous lengths of cable with impedances chosen from a small discrete set of values, which are readily available from commercial sources (a critical feature for the design of practical feeds). Unfortunately, GAs may be relatively slow, as they generally require more objective function evaluations than classical optimization techniques. This weakness is compensated by their ability to find strong local or global optima and GAs are generally more efficient than other stochastic optimizers such as simulated annealing or Monte Carlo methods. Moreover, the Pareto GA, which is described below, efficiently uses all of the objective function evaluations by locating a subset of the population representing optimal tradeoffs between the design parameters.

Numerical Results

The specific grounding structure used to demonstrate the design technique introduced in this paper is an aircraft wing. The compressed wide-band LPMA is mounted on the airfoil's trailing edge. This configuration has potential for use in synthetic aperture radar systems. A model of a prospective target platform for the antenna is shown in Figure 4.

Figure 4: Example target platform for LPMA design with rudimentary nacelles and winglets

Goals 2 and 3 are self-explanatory, while goal 1 requires further discussion. Two pattern cuts are analyzed. The E-plane characterizes the radiated power over a conical cut with its vertex at the origin and a base that intersects the unit sphere at -30° elevation. The H-plane characterizes the radiated power measured around a circle that is concentric with the fuselage and aligned with the trailing edge of the wing. The ground is assumed to be in the range of -10° to -50° from the horizon. Within the context of this problem, front is defined as directly outboard of the wing, at a given elevation angle below the horizon. Accordingly, back is defined as directly inboard at that same angle of elevation. The front-to-back ratio must be maintained over that entire elevation range to ensure reliable reception at different observation altitudes. Figure 5 illustrates the H- and E-planes as defined above, including arcs where the front-to-back ratio must be maintained, and illustrates the patterns supplied as ideal. Each pattern cut was evaluated at 180 equally spaced points.

Figure 5a: Ideal pattern definition for E- and H-planes relative to a single wing model

The number of bits used for each feed parameter is given in Figure 6. The remaining Pareto-GA parameters were Np = 10 000, pcross = 0.85, pmut = 0.005, r = 0.08, E = 1.75 and Nfreq = 8. The Nfreq = 8 frequencies used for optimization were 25.0, 30.0, 35.0, 45.0, 55.0, 65.0, 75.0, and 88.0 MHz. The low end of the band contains an extra frequency point, 30.0 MHz, since that portion of the frequency band is more problematic and benefits from additional weighting in the optimization process. Limits on the continuous valued parameters and the databases for the discrete parameters are also given Figure 6. Note that the allowable ranges of the feed lengths scale by t, which encourages log-periodicity while not strictly enforcing it.

Figure 6: Binary encoding of design parameters into chromosomal form

Figure 7 depicts the initial population of 10 000 randomly generated design candidates as points in the three-dimensional goal space, and shows the approximation to the Pareto front after 36 generations as a surface. Figure 8 gives a closer view of the interpolated front with the actual rank-one designs denoted by crosses. Not only does the Pareto GA provide a database of designs exhibiting optimal tradeoffs, but it also outperforms the simple GA outright in this optimization problem. Prior to using the Pareto GA, the simple GA was exhaustively applied to this problem-over 140 GA runs were performed with varying population sizes and alternate linear combinations of the three design goals. In all cases the field degradation was too large to be acceptable, even though it was naturally weighted much higher.

Figure 7: Initial population of random designs

Figure 8: Approximation to the Pareto front after fifteen generations

Figure 9 summarizes these results, with the simple GA results illustrated as circles residing over the same Pareto front shown in Figure 8. The population size of all the simple GA runs combined is 39 050. All the GAs (including the Pareto GA) were run for 65 generations to ensure convergence. The simple GA run converged (on average) after 35 generations (based on the difference between the population's average fitness and the fitness of the best member). The surface interpolated through the rank-one members of the Pareto GA population showed very little movement after the 33rd generation. Improvement in the front density was the only observable effect of continued optimization beyond that generation.

Figure 9: Pareto front compared to simple GA results

Figures 10 and 11 illustrate the performance of a single design that was chosen from the Pareto optimal database of designs represented in Figure 8. This set of figures also presents the results obtained from optimization by the simple GA and those obtained with a purely log-periodic feed with the parameters k, ZM, Z0, ZL, and ZSRC and optimized by the simplex method [8], [9].

Figures 10 and 11 show polar representation of the E- and H-plane patterns with 15-dB dynamic range sampled over the band for the Pareto GA, simple GA, and simplex designed feeds.

Figure 10: E-plane radiation patterns for (a) 25 MHz, (b) 30 MHz, (c) 35 MHz, (d) 45 MHz, (d) 55 MHz, (d) 65 MHz, (d) 75 MHz, (d) 88 MHz

Figure 11: H-plane radiation patterns for (a) 25 MHz, (b) 30 MHz, (c) 35 MHz, (d) 45 MHz, (d) 55 MHz, (d) 65 MHz, (d) 75 MHz, (d) 88 MHz

Figures 9-11 unambiguously demonstrate the power of the Pareto GA relative to the other two methods. Figure 9 shows that the entire simple GA front is dominated by the front returned by the Pareto GA. Moreover, the simplex front is so much worse than the other two it cannot even be shown on the same graph. More importantly, however, Figures 10 and 11 show that the design returned by the Pareto GA is quantitatively superior to the other two. Figures 10 and 11 demonstrate that only the Pareto GA design is able to keep a consistent pattern similar to the ideal patterns supplied in Figure 5. In the higher frequency part of the band, both the design created by the simple GA and the simplex method show significant radiation in the "back" direction. Coupling these superior patterns with the fact that the Pareto GA was the only method able to achieve the desired VSWR across the band (even though the simple GA design was chosen to minimize this very parameter) leads to the conclusion that the Pareto GA technique is indispensable for this problem. The fact that the Pareto GA returns designs that dominate the simple GA designs is very telling. The superiority of the Pareto technique may be largely due to the sharing operator, which disallows stagnation and forces the GA to search novel regions of the space. For the same reason, the sharing operator has even been found useful in single parameter problems [28]. Finally, it should be noted that the above design procedure was applied to the design of LPMAs on a variety of other platforms, including wedges, finite plates, open and closed curved surfaces, etc., with similar results (not shown here because of space limitations).

Conclusions

A method for optimizing LPMAs mounted on complex platforms was detailed. Specifically, a three-objective Pareto GA was applied to the design of a practical impedance-modulated feed for a wing-mounted LPMA. To facilitate the application of a GA, the analysis of the structure was divided into two independent procedures: an expensive MoM-based preprocessing procedure and a fast feed and antenna analysis procedure that is performed during the optimization. The use of GA-driven Pareto optimization allowed for thorough probing of the feed design parameter search space and for creation of a database of Pareto-optimal designs. For all investigated platforms, the proposed Pareto optimizer was able to deliver non-log-periodically modulated feeders that outperformed those obtained by standard GAs and the simplex method, and that restored the frequency independent behavior of the antenna in spite of the finite nature of the supporting platform.

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The above work is a collaboration between Stephen E. Fisher, Daniel S. Weile, Prof. Eric Michielssen, and William Woody from Lockheed Martin TDS. The authors would like to acknowledge a grant from Lockheed-Martin, and the AFOSR under grant F49620-96-1-0025. They also thank Paul Mayes for the benefit of his vast LPMA knowledge. Please send suggestions, comments, and inquiries to: dsw@decwa.ece.uiuc.edu.