We apply Pareto genetic algorithm (GA) optimization to the design of antenna elements for use in the Long Wavelength Array (LWA), a large, low-frequency radio telescope currently under development. By manipulating antenna geometry, the Pareto GA simultaneously optimizes the received Galactic background or “sky” noise level and radiation patterns of the antenna over all frequencies. Geometrical constraints are handled explicitly in the GA in order to guarantee the realizability, and to impart control over the monetary cost of the generated designs. The antenna elements considered are broadband planar dipoles arranged horizontally over the ground. It is demonstrated that the Pareto GA approach generates a set of designs, which exhibit a wide range of trade-offs between the two design objectives, and satisfy all constraints. Multiple GA executions are performed to determine how antenna performance trade-offs are affected by different geometrical constraint values, feed impedance values, radiating element shapes and orientations, and ground conditions. Two different planar dipole antenna designs are constructed, and antenna input impedance and sky noise drift scan measurements are performed to validate the results of the GA.
 The Long Wavelength Array (LWA) is a synthesis imaging radio telescope currently under development to provide extremely high sensitivity and resolution in the frequency range of 20 MHz to 80 MHz. The LWA will be used in cosmological studies involving the high-redshift universe and the epoch of reionization, other astrophysical studies involving acceleration of relativistic particles, solar and space weather research, and ionospheric research [Ellingson et al., 2009]. It is currently planned that the LWA will consist of 52 stations spread over a diameter of roughly 400 km. Each station is a phased array of 256 antenna elements placed over a diameter of approximately 100 m.
 In order to maximize sensitivity, the antenna elements are required to receive the Galactic background or “sky” noise at a much higher level than the “self” noise generated by system electronics so that the receiver is sky noise dominated. The sky-to-system noise dominance is primarily influenced by mismatch loss between the antenna and the input feed and absorption of incident energy by the ground beneath the antenna. A wide field of view is required to perform beamforming over a wide range of incident angles. This implies that a low directivity or wide beam width antenna element is desired. To simplify calibration of the instrument, the element patterns should be smoothly varying and have minimal ripple. The antenna elements must also provide circular polarization to avoid distortion induced by the ionosphere. It is also desired to minimize the monetary cost of the antenna design due to high number of elements in the system.
 Dipole antennas have been used successfully in other low-frequency radio telescopes. In these systems, the dipoles are generally placed close to the ground to maintain wide beam width radiation patterns. Additionally, two dipoles are typically crossed, which permits reception of circularly polarized waves by proper combination of the dipole outputs. For example, a thin wire “inverted-V” dipole antenna is used in the LOFAR low-frequency demonstrator [Gunst et al., 2006]. While this design is simple and low cost, it does not provide wideband impedance matching to a low impedance (nominally 100 Ω) input feed, which is planned for LWA. To improve performance with a low impedance feed, dipoles consisting of broadband “drooped” planar elements were developed for the Long Wavelength Demonstrator Array (LWDA), a predecessor to the LWA. This design exhibits sky noise dominated operation, and wide beam width and low axial ratio radiation patterns over the 60 MHz to 80 MHz band [Kerkhoff and Ellingson, 2005]. These properties make planar dipoles attractive for use in the LWA. The much wider bandwidth required for the LWA, however, makes it difficult to maintain high sky noise dominance and desirable radiation patterns over all frequencies, so that a trade-off must be made between these two objectives.
 Pareto genetic algorithms (GA) [Goldberg, 1989; Srinivas and Deb, 1995] have become popular for complex antenna design problems involving multiple objectives. Pareto GA has been used to optimize, for example, electrically small wire antennas [Choo et al., 2005] and Yagi antennas [Kuwahara, 2005]. There are few results reported, however, involving Pareto GA optimization of wideband antennas, and no known results involving optimization of such antennas for radio astronomy. In this paper, a Pareto GA is used to optimize planar dipole antennas in terms of both sky noise reception and radiation pattern quality for use in the LWA. Through this approach, the range of available performance trade-offs between the two design objectives will be determined for a number of different design variations. These include different geometrical constraint values, feed impedance values, antenna element shapes and orientations, and ground conditions.
 The paper is organized as follows: The Pareto GA optimizer is first described. Results from GA optimization and measurements of selected antenna designs are then presented. Finally, conclusions are given.
2. Pareto GA Optimization
 Genetic algorithm (GA) optimization, is a stochastic search technique based on the principles of natural selection and evolution [Goldberg, 1989]. A GA operates in an iterative fashion, evolving a population of candidate solutions by means of genetic operators such as crossover and mutation, and selection of the most fit based upon a predefined cost function. Pareto GA is a variant of this technique well suited to multi-objective optimization. In a popular Pareto-based approach, called nondominated sorting GA (NSGA) [Srinivas and Deb, 1995], a separate cost function is defined for each objective, and the GA assigns ranks to each member of the population based upon how well its cost function values compare with other members in the population. Any member who has at least one superior cost function value in each and every pairwise comparison with the remainder of the population is considered to be rank-1. All rank-1 individuals form a so-called Pareto “front”, which when fully evolved, indicates the full range of available trade-offs that exist between the different objectives.
 An NSGA is used to perform multi-objective optimization of planar dipole antennas for use in the LWA. The original antenna geometry considered is depicted in Figure 1. The eight design parameters included in GA optimization are shown in Figure 1. In the GA, sharing is implemented by generating niche counts based upon Euclidean distances between designs in objective space. The niche counts are used to scale the nominal fitness values generated by Pareto sorting. As in a traditional GA, selection, crossover, mutation, and fitness evaluation are performed in each generation. An elitism operator is implemented in the Pareto GA by performing a second selection step at the end of each generation. In this step, selection is performed on the combined population consisting of both parent and child individuals resulting from crossover and mutation in order to return the population to its original size [Cui et al., 2005]. Roulette wheel selection is used in the initial selection step, while fitness ranking is used in the second selection step. A binary coded chromosome is used to represent antenna designs in the GA. Constraints upon antenna geometry are handled explicitly in the GA, and may be placed on individual dimensions of the antenna. Additionally, geometrical checks are performed to verify that an element design does not intersect its counterpart element in a crossed dipole antenna, and that no portion of the element extends below the ground plane. If an element design violates any of the constraints placed on individual dimensions or fails any of the geometrical checks, its fitness is set to zero, effectively removing it from the population.
 A simulation code based upon the method of moments (MOM) formulation described by Trintinalia and Ling  is used to estimate antenna performance in the GA. In this formulation, first-order triangular patch basis functions are used to model the current distribution over a planar surface. A Gaussian delta gap feed model is used to excite the antenna. An infinite PEC ground beneath the antenna is achieved by imaging the basis functions of the antenna across the ground plane. An infinite lossy ground is approximated using a reflection coefficient-based model similar to the one described by Burke and Poggio . EasyMesh version 1.4 (B. Niceno, Information about EasyMesh v1.4, available at http://www-dinma.univ.trieste.it/nirftc/research/easymesh/, 2002) is used to generate triangular mesh discretizations of antenna designs for use in MOM simulation.
 Note that although a geometrical check is performed to verify that crossed dipoles do not intersect, only a single dipole out of a crossed dipole pair is considered in MOM simulation in order to minimize GA run time. This is an acceptable simplification to make for the planar dipole elements considered here since coupling between crossed planar dipoles of similar design has been shown to be low [Erickson et al., 2006]. Note that this may not be true for other types of antennas. For instance, planar dipoles with much wider radiating elements than those considered here exhibit significant coupling when operated in a crossed configuration [Suh et al., 2004]. In such a case, it may be necessary to include both dipoles in GA simulation and properly account for the coupling between them in the GA cost function definitions.
 The GA simultaneously optimizes antenna performance in terms of sky noise frequency response and radiation pattern quality over the 20 MHz to 80 MHz operating band. The cost function used in the GA for sky noise frequency response is
and Nf is the number of frequencies evaluated. Equation (1) is the rms error over all frequencies, f, between the equivalent noise temperature at the terminals of an antenna due to sky noise, TA, and a reference temperature profile, Tref. As implied by the equation, it is desired to achieve an antenna temperature at least as high as the reference temperature profile over all frequencies.
 An approximation for the sky noise temperature received by a lossless, low-directivity antenna is [Ellingson, 2005]
where k is Boltzmann's constant, c is the speed of light, and If is intensity. In equation (3), the intensity is assumed to be that of the Galactic polar region [Cane, 1979], which is well approximated by
where fMHz denotes frequency in MHz, Ig = 2.48 × 10−20 and Ieg = 1.06 × 10−20. The sky noise temperature as seen at the antenna terminals when losses are considered is given by
where (1 − ∣Γ∣2) represents mismatch loss efficiency between the antenna and feed line, and er is radiation efficiency. er is dominated by the loss due to absorption by the ground beneath the antenna; conductive loss in the antenna elements is generally small and is ignored here.
 In general, the measured sky noise temperature deviates somewhat from equation (5) since equation (3) assumes a constant antenna collecting area over all angles (which is not true for a real antenna) and equation (4) is technically only valid at the Galactic pole transit time. Nonetheless, equation (5) can be used to assess the relative minimum sky noise values received by different antenna designs. A sky map-based approach will be described in section 3.2, which more accurately simulates absolute sky noise temperatures. However, this method is computationally intensive and not suitable for use in the GA optimizer.
 Also not considered in equation (5) is the increase in antenna temperature due to ground noise. As suggested by Ellingson , a worst case approximation for this contribution, which assumes ground noise is received through the lower half of an isotropic radiation pattern, is 145 K. Ground conductivities are typically such, however, that the ground acts like a reflector. This reduces the antenna backlobe so that the received ground noise should be much lower than worst case. Since received sky noise values in the LWA band are relatively high (many hundreds to thousands of K), and again, due to the focus in this study on relative comparisons of different antenna designs, ground noise is ignored.
 The cost function for radiation pattern quality is
which is the maximum over all frequencies of the rms error over all observation angles between the gain pattern of an antenna, Gant, and a reference gain pattern, Gref; Nθ and Nϕ are the number of elevation and azimuth angles, respectively, over which patterns are evaluated. In the calculation of C2, Gant is normalized at each frequency to the maximum over all elevation and azimuth angles. It is desired to minimize both C1 and C2.
3.1. Pareto GA Designs
 The Pareto GA is first used to optimize the original planar dipole geometry. The dipole is assumed to operate directly over an “average” lossy ground with a relative permittivity of 13 and a conductivity of 0.005 S/m. The GA evaluates each antenna design in 10 MHz increments between 20 MHz to 80 MHz. In evaluating C1, the reference temperature profile, Tref, is chosen so that a sky noise dominance, D = TA/TS where TS is the noise generated by system electronics, of 10 dB is achieved. Assuming TS = 250 K, the nominal reference temperature is Tref,nom = 2500 K. At lower frequencies where Tsky ≥ Tref,nom, Tref is set to Tref,nom. At higher frequencies where this is not the case due to the rapid decrease in sky noise temperature with frequency, Tref is set to Tsky, since it is not possible to achieve performance any better than Tsky. A feed impedance of 100 Ω is assumed in calculating mismatch loss. Gref = sinθ is used to calculate C2. In addition to the constraints which forbid crossed elements from intersecting and elements extending beneath the ground, an additional constraint is imposed that the total width of the dipole element, wT = w1 + 2w2 (referring to Figure 1), be less than 42 cm. This constraint is intended to limit the material costs of generated antenna designs.
 The eight design parameters are encoded in the GA using a 44 bit chromosome. A population size of Np = 200 is used and the GA runs for 300 generations before terminating. A crossover rate of 0.5 and mutation rate of 0.1 are used. The niche radius used in the Pareto sharing scheme is rniche = (1/Np) = 0.005.
 The GA took 119 hours to execute using six 1.0 GHz AMD Athlon processors. The Pareto fronts generated by the GA at generations 1, 50, and 300 are shown in Figure 2. After the first generation, there are very few designs on the front. The front is greatly improved and contains many more designs after generation 50. By the final generation, the front is improved further and contains nearly the entire population (177 out of 200 designs.) The final front is relatively smooth and evenly sampled over a wide range of values for both objectives.
 Three designs from the final Pareto front, labeled 1, 2, and 3 in Figure 2, are selected for detailed analysis. The sky noise frequency responses calculated using equation (5) for these three designs are compared in Figure 3. Also included in Figure 3 for comparison are the sky noise received by a lossless antenna (Tsky), and the reference sky noise profiles, as defined earlier, for both Tref,nom = 2500 K (D = 10 dB) and 1000 K (D = 6 dB). Design 1, which has the highest (worst) C1 value of the three designs, exhibits the lowest sky noise temperatures over most frequencies. The bandwidths over which design 1 performs better than the D = 10 dB and D = 6 dB reference profiles are relatively narrow. Sky noise temperatures consistently increase between designs 1 and 2 and between designs 2 and 3, particularly at the low and high ends of the operating band. The bandwidths over which designs 2 and 3 satisfy the reference profiles are also increased. Though none of the designs satisfy the D = 10 dB profile above 43 MHz for the assumed system noise, design 3 exceeds the D = 6 dB profile over the entire 20 MHz to 80 MHz band.
 The normalized principal plane copolarized gain patterns for the three GA designs at 30 MHz and 80 MHz are given in Figure 4. Also included for comparison in each plot is the reference gain pattern, sinθ. All three designs exhibit relatively wide beam width and smoothly varying radiation patterns at 30 MHz, though designs 2 and 3 exhibit reduced E plane beam widths as compared with design 1. At 80 MHz, all three designs exhibit some pattern ripple, though the patterns of design 1 follow the reference pattern relatively closely. Design 2, on the other hand, exhibits reduced E plane beam width and increased ripple in the H plane, while design 3 exhibits low gain near the zenith and high sidelobes, which is undesirable.
 Since the LWA requires a circularly polarized antenna, crossed dipole antennas consisting of GA dipole designs are considered. If the two elements of a crossed dipole antenna radiate no cross-polarized fields and are fed in quadrature, the axial ratio of the antenna is given by [Balanis, 1997]
Since the GA dipole designs exhibit low cross-polarized fields (< −20 dB relative to copolarized fields over all frequencies), the axial ratio in the principal planes of the crossed dipole antenna can be approximated by substituting the copolarized E and H plane fields of the single dipole (given in Figure 4) for Eθ and Eϕ in equation (7), respectively. The axial ratios for designs 1, 2, and 3 at 80 MHz are compared in Figure 5. As can be seen, crossed dipole antennas with well-behaved single element patterns (such as design 1) tend to exhibit lower axial ratio values, and thus, better reception of circularly polarized waves.
 The dimensions of the GA designs are provided in Table 1. The total length of the dipole elements, lT = l1 + l2, increases between designs 1 and 2 and between designs 2 and 3. The increased length leads to better impedance matching, and thus improved sky noise reception at lower frequencies. Additionally, the element lengths of designs 2 and 3 are such that the 3λ/2 resonance is moved closer to 80 MHz, which improves impedance matching and sky noise reception at higher frequencies. As mentioned by Kerkhoff and Ellingson , impedance matching is improved and ground loss is reduced by moving the element away from the ground. This can be accomplished by (1) increasing the distance between the feed and the bend point, lb, (2) decreasing the droop angle, α, or (3) increasing the height of the feed point above the ground, H. Feature 1 is found in design 2, while features 2 and 3 are made use of in design 3. These features, however, degrade the radiation pattern performance, as pointed out by Kerkhoff and Ellingson . Also, increasing the length of the element will tend to degrade high-frequency radiation patterns by introducing sidelobes.
Table 1. Dimensions of Selected Pareto GA Designs
 The results presented in this section clearly demonstrate that a trade-off must be made in the antenna design between radiation pattern quality and sky noise reception. The results also show that, despite the simplicity of the cost function definitions given by equations (1) and (6), they are effective in evaluating this trade-off over a wide operating frequency range.
3.2. Validation of GA Results
 Two antenna designs were constructed and measurements performed to verify the accuracy of MOM simulation, and thus, the Pareto GA results. A design considered to be the “baseline” for LWA [Erickson, 2006] was selected for evaluation since its performance was already well understood. The dimensions of this design are included in Table 1, and its cost function values are shown in Figure 2 where they are slightly behind the Pareto front. Pareto GA design 2, described in section 3.1, was also evaluated. Pictures of the constructed antennas are provided in Figure 6.
 Input impedance measurements were performed on the antennas. A procedure based upon scattering parameter analysis was used to separate the desired antenna response from the undesired response of the 180°-phased power combiner connecting the antenna to the measurement instrument [Kerkhoff, 2008]. Since the measurements were performed without a ground screen, the permittivity and conductivity of the soil beneath the antenna were measured using the ground probe-based technique described by Smith and Nordgard . The measured ground parameters were averaged over 20 MHz to 80 MHz, and the resulting values were used in the simulation results that follow.
Figure 7 compares the measured and simulated input impedances, which agree well for both antenna designs. In order to make a proper comparison with simulation results, the measurements were performed after having removed the center supporting mast depicted in Figure 6, using the struts near the ends of the elements to temporarily maintain the antenna geometry. This was necessary due to a shift in the full-wave resonance frequency caused by the dielectric properties of the mast, which cannot be modeled in the MOM code used.
 To validate simulated antenna sky noise response and radiation patterns, a series of drift scan measurements were performed. Drift scans involve measuring the time evolution of the total received power due to sky noise. The test setup used consists of an active balun, which presents a 100 Ω impedance and a 250 K noise temperature to the antenna, analog filters and amplifiers, and a spectrum analyzer. The total power collected by the antenna was measured in 10 kHz intervals between 20 MHz and 80 MHz every five minutes for a period of two days. Iterative outlier removal and averaging were applied to the data to reduce noise, primarily due to radio frequency interference (RFI). A Y factor calibration [Pozar, 1990] was performed on the test setup using a calibrated noise diode to remove the electronics noise from the sky noise measurements. Measurements were performed at a rural site roughly 35 miles west of Austin, TX.
 The drift scan response is predicted by convolving simulated antenna radiation patterns with an all-sky map of radio continuum. In particular, the sky noise temperature for a lossless antenna at time t is given by [Rogers et al., 2004]
where M denotes the all-sky map, i and j are the map indices for galactic latitude and longitude, respectively, U is the normalized antenna power pattern, b is the galactic latitude, and θ and ϕ are the elevation and azimuth, respectively, of a given source. The noise temperature received by the antenna when losses are taken into account, TA, is given by equation (5). The Haslam 408 MHz all-sky map [Haslam et al., 1982] is used for M. The map is scaled in frequency to the LWA band using the procedure described by Kerkhoff . From equation (8), it is evident that the only antenna characteristic which affects the shape of the drift scan curve with time is its radiation patterns. This provides an indirect means to validate simulated radiation patterns.
 Simulated and measured drift scan data for the two antenna designs, given in terms of sky noise temperature at the antenna terminals, are compared as a function of frequency in Figure 8. Except between 54 and 60 MHz, the measurements above 50 MHz are mostly free of RFI, and agree well with simulation. The measurements indicate that GA design 2 exhibits improved sky noise reception as compared with the baseline design in this frequency range. A comparison is more difficult to make at lower frequencies due to the presence of RFI. The RFI is reduced at some frequencies, though, such as 26.5 and 38 MHz for the baseline design, and 25 and 43 MHz for GA Design 2, where the results appear to agree well. An offset between simulation and measurement is evident, however, at 22.5 MHz for both designs. These offsets are believed to be due, in part, to approximations inherent to the ground model used in the MOM code. Some error in the simulation may also be due to not explicitly accounting for HII absorption near the Galactic center, which attenuates the measured sky noise below 50 MHz [Polisensky, 2007]. Despite this disagreement, the simulations correctly predict the significant increase in low-frequency sky noise level provided by GA design 2 over the baseline design, which is evident in the measurements.
 The drift scan data for GA design 2 are presented as a function of time at four different frequencies in Figure 9. A scaling factor (which is noted in each plot legend) has been applied to the simulated results to simplify the comparison of temporal trends. Although reasonable agreement between measurement and simulation is achieved over most times and frequencies, interference in the measurements causes some disagreement. After 3700 LST, however, the interference is generally reduced, and the drift scan curve shapes agree well at all frequencies. Similar agreement is also achieved with the baseline design, as documented by Kerkhoff . The agreement in drift scan curve shapes indicates that: (1) the measurements at frequencies where strong RFI is not evident in Figure 8 are indeed dominated by sky noise, thus validating conclusions made about the relative receive performance of the two antennas, and (2) the simulated radiation patterns are accurate.
3.3. Pareto GA Study of Design Variations
 The Pareto GA optimizer is now used to study how performance trade-offs are affected by changing different aspects of the antenna design problem presented in section 3.1. The effect of changing the element width constraint value, wT,max, is first considered assuming the original antenna geometry shown in Figure 1, average ground conditions, and a 100 Ω feed impedance. The GA is rerun for wT,max values of 28 cm and 60 cm and the resulting Pareto fronts are compared with the front for wT,max = 42 cm in Figure 10a. While there is little improvement for C2 < 0.11, significant improvement in performance is achieved for higher values of C2 by increasing wT,max. In comparing designs from each front, which satisfy C2 = 0.20, it is found that the antenna input impedance varies less over the operating band as the element width is increased. This improves the broadband match to a low impedance feed line, which is evident in Figure 11. By improving impedance matching, sky noise reception is also improved. The disadvantage of a wider element, however, is its larger size and higher monetary cost.
 Also included in Figure 10a is the final front from a GA run in which the feed impedance is increased to 150 Ω for wT,max = 42 cm. It has been shown for individual antenna designs that sky noise reception can be enhanced by properly adjusting the feed impedance [Ellingson, 2005]. The GA results agree with this conclusion, and indicate that significant improvement can be achieved over the full range of pattern cost function values. As is evident in Figure 11, a better broadband match is achieved by increasing ZL, which, in turn, improves sky noise reception. The Pareto front for this case is comparable to that for wT,max = 60 cm and ZL = 100 Ω, which suggests that using a higher feed impedance to improve antenna performance may be more cost effective than increasing antenna size regardless of radiation pattern quality.
 Next, the effects of changing the radiating element shape and orientation are studied. Four new antenna geometries are considered, which are similar to the original geometry, but have the following differences: (1) An additional outward taper is added at the bottom (feed) end of the element; (2) an additional downward element bend (for a total of two) is allowed; (3) an additional inward taper is added at the far end of the element; (4) the bend is removed, and the element is oriented perpendicularly(or vertically) relative to the ground; the element is allowed to tilt downward from the feed point. The new antenna geometries are shown in Figure 12.
 The GA is rerun for each of these geometries assuming an average ground and wT,max = 42 cm. The final fronts for geometries 1 and 2 are nearly identical to the front for the original geometry, and therefore, these two geometries are not considered further. The final fronts for the other antenna geometries are compared in Figure 10b. There is some improvement in performance with geometry 3 as compared with the original geometry for lower values of C1. The inward taper added to the far end of the element in this geometry appears to improve high-frequency radiation patterns somewhat. Geometry 4 performs much worse than the other geometries, particularly for intermediate values of C1. Comparing GA designs from each front for a common value of C1, the element height above the ground is much higher for geometry 4 than for the other geometries (by up to 46%). This is necessary in order for the vertically oriented element to clear the ground. For this range of antenna heights, the radiation patterns degrade more rapidly by increasing the height than impedance matching and ground losses improve. This leads to a degraded performance trade-off.
 Finally, the effects of different ground conditions are considered. A ground screen may be desired to buffer the response of the antenna against changes in ground permittivity and conductivity. The effects of a ground screen are approximated in MOM simulation by an infinite PEC ground. The final GA fronts for PEC and average grounds assuming the original antenna geometry and wT,max = 42 cm are compared in Figure 10c. For C2 < 0.17, a better performance trade-off is achieved with a lossy ground than a PEC ground. This can be understood by considering Figure 13, which compares the 80 MHz radiation patterns of designs from each front for C1 = 1360. The antenna image due to PEC ground perturbs the dipole radiation patterns causing a poor match to the sin(θ) reference pattern at lower elevation angles. The effect of the antenna image is not as pronounced over lossy ground. For C2 > 0.17, a significantly better trade-off can be achieved over PEC ground since ground absorption is eliminated. It is noticed that for a given value of C1, the designs from the PEC front have consistently shorter element lengths than those from the lossy ground front (by up to 25%.) This length reduction would reduce the materials cost of the antenna, which would offset the additional cost of the ground screen somewhat.
 Pareto GA optimization has been applied to the design of broadband planar dipole antennas for use in the LWA radio telescope. It was shown that the GA generates a Pareto optimal set of antenna designs providing a wide range of trade-offs between sky noise reception and radiation pattern quality over the 20 MHz to 80 MHz operating band. An analysis of GA-generated designs showed how different design parameters control the performance trade-off between objectives. Two planar dipole designs were constructed, and input impedance and sky noise drift scan measurements were performed on each. In general, good agreement was achieved between measurement and MOM simulation, validating the results of the GA.
 The Pareto GA was used to study the effect of different design variations on antenna performance. By imposing a constraint on antenna width, a trade-off between antenna performance and monetary cost may be made. Through proper selection of the feed impedance presented to the antenna, improved performance can be achieved without increasing antenna size, and regardless of radiation pattern quality. For the assumed constraints, the radiating element shape has a relatively small effect on performance. The orientation of the radiating elements, however, has a more significant effect, with a horizontal orientation performing better than a vertical one. Finally, it was found that the ground beneath the antenna has a significant effect with a perfectly conducting ground being superior in most, but not all cases to a lossy ground.
 This work was supported in part by ARL:UT IR&D, the National Science Foundation, under grant ECCS-0725729, and the Texas Higher Education Coordinating Board.