An optimal design of a cylindrical polarimetric phased array radar (CPPAR) for weather sensing is presented. A recently introduced invasive weed optimization (IWO) technique is employed to obtain the desired radiation pattern of the CPPAR. Instead of optimizing each element excitation in a large array (with expensive calculation costs), the modified Bernstein polynomial distribution, defined by seven parameters, is used to optimize the current distribution for the CPPAR. The simulation results show that the desired sidelobe levels (SLLs) and beam width are achieved in a computationally effective manner. Furthermore, the imaged feed arrangement is used to suppress the cross-polarization level. Both co-polar and cross-polar radiation patterns for broadside and off-broadside directions are presented to show the performance of the optimized CPPAR.
 Phased array radar (PAR) technology has recently been introduced to the weather community. The first phased array radar dedicated to weather observation, the National Weather Radar Testbed (NWRT) was developed in Norman, Oklahoma [Zrnic et al., 2007]. Operating at a wavelength of 9.38 cm, the NWRT is able to make reliable weather measurements. Compared to conventional reflector antennas with mechanically steered beams, the NWRT takes advantage of electronic beam steering, resulting in shorter surveillance times and faster data updates. In addition, the NWRT has the capability to steer the beam mechanically in the azimuth direction, allowing for multiple measurements of the same meteorological volume [Yu et al., 2007; Heinselman et al., 2008; Zhang et al., 2011a; Le et al., 2009; Zhang and Doviak, 2007, 2008; Yeary et al., 2010].
 While PAR is starting to receive attention in the weather community, radar polarimetry has already matured to the stage where Weather Surveillance Radar 1988 Doppler (WSR-88D) radars are being upgraded with dual-polarization capability [Doviak et al., 2000]. It is desirable to combine electronic beam steering and polarimetry capabilities. However, a planar polarimetric phased array radar (PPPAR) has some deficiencies when the beam is scanned off-broadside. The PPPAR, with multiple faces to scan the whole azimuth space, suffers from the disadvantages of increase in beam width, loss of sensitivity and coupling in dual-polarizations when the beam is pointed away from the broadside angle [Zhang et al., 2009].
 The cylindrical polarimetric phased array radar (CPPAR) has been recently proposed to overcome the deficiencies encountered with PPPAR. The CPPAR principle and potential performance was studied by Zhang et al. [2011b]as compared with the WSR-88D radar. It has the advantages of azimuth scan-invariant pattern and orthogonal polarizations. Although the CPPAR has several advantages compared to PPPAR, there is a concern as how to achieve desired performance for a given antenna size, including low sidelobe level (SLL) (<−27 dB), narrow beam width (<1.0 degree) and low cross-polarization. To achieve high performance, it is desirable to use a pattern synthesis method to optimally determine weight.
 Generally, the numerical analysis of large conformal array antennas is very expensive and requires an extremely large computational domain. On the other hand, the common mathematical and computational simplifications of planar array cannot be applied directly to conformal arrays. Direct array pattern synthesis techniques including Fourier methods [Taylor, 1952; Josefsson and Persson, 2006], aperture projection methods [Chiba et al., 1989; Schuman, 1994], adaptive array methods [Zhou and Ingram, 1999; Sureau and Keeping, 1982], alternative projection methods [Steyskal, 2002; Vescovo, 1995] and least mean square methods [Vaskelainen, 1997; Dinnichert, 2000] have been used for pattern synthesis of conformal array antennas. Although these techniques are fast, the effect of coupling between elements is not usually taken into account. Optimization techniques including genetic algorithm (GA), particle swarm optimization (PSO), and simulated annealing (SA) are also applied to the conformal array antennas synthesis problem [Yang et al., 2009; Sun et al., 2010; Li et al., 2010; Boeringer and Werner, 2005; Ferreira and Ares, 1997]. Although optimization techniques are more efficient in obtaining the desired radiation pattern, they are very expensive for large array antennas, especially when an accurate analysis is considered. On the other hand, a huge number of evaluations is needed to obtain the optimized values for all the variables. Therefore, applying any optimization method to a CPPAR with a few thousand elements will be very expensive.
 In this paper, an optimal design of the cylindrical polarimetric phased array radar (CPPAR) for weather measurements is presented. A recently introduced optimization technique, Invasive Weed Optimization (IWO) [Karimkashi and Kishk, 2009, 2010] is employed to optimize the elements excitation current of the CPPAR to obtain the desired SLLs and beam width. A two dimensional (2D) modified Bernstein polynomial [Boeringer and Werner, 2005] is used to define the amplitude distribution of the CPPAR to improve computational efficiency. Optimizing the modified Bernstein polynomial distribution (defined by seven parameters) instead of each element's amplitude substantially reduces the computational burden. By using such a smooth and unimodal amplitude distribution, the active element pattern technique is used where the element pattern of each element is calculated in the presence of its neighboring elements.
 This paper is organized as follows. Section 2 presents the modeling and design principles of the CPPAR. Application of the IWO to the CPPAR is presented in section 3. The antenna optimization results are shown in section 4. Finally the conclusion is drawn in section 5.
2. CPPAR Modeling and Design
 The configuration of the CPPAR consisting of M × N dual polarized microstrip patch radiating elements is shown in Figure 1. Although, in principle, any number of beams can be formed, it has been found that a four-beam system is appropriate in either the planar or cylindrical configuration [Zhang et al., 2011b]. Each active sector of the antenna, a quarter of the cylinder, generates a beam with the broadside direction along the bisector of that sector. The azimuth beam steering is achieved by commutation, and the elevation beam steering is electronic and ranges from 0 degree to 30 degrees.
 Each microstrip patch antenna working at the frequency of 2.8 GHz is designed on an RT/Duriod 5880 substrate of thickness of 3.175 mm. The square patch antenna is excited by using 50 Ω probe feeds (Figure 2).
 In order to account for the mutual coupling between elements, a 3 × 3 array is modeled when the central element is excited and the other elements are terminated to matched loads. The simulated results are shown in Figure 3. These results indicate that the coupling between elements mostly affects the cross-polar radiation pattern. Including the neighboring elements in calculating the element's active pattern is adequate. It should be mentioned that the cross polarization level is below −40 dB at the vertical plane (E-plane).
 After obtaining the element pattern, the total radiated field of the CPPAR can be obtained by:
where EPm,n(θ, ϕ) are the active element pattern, A(m, n) are the element excitation complex current, k is the free space wave number, R is the radius of the cylinder and d is the distance between elements. φm and znare the location of each element on the cylinder coordinate system. It is desirable for CPPAR to have a comparable performance to WSR-88D, which has a reflector antenna with a diameter of 8.54 m. To have the same effective size of the WSR-88D reflector for each active sector, the CPPAR should have a height of 8.54 m and a radius ofR = 6.05 m.
3. CPPAR Optimization
 In this section, the optimization algorithm is applied to the CPPAR antenna to obtain the desired SLLs and beam widths for both the broadside and off-broadside radiation patterns. The IWO algorithm is briefly described and then the optimization procedure, inter-element spacing between elements and cross-polarization minimization are discussed.
3.1. IWO Algorithm
 The IWO algorithm has been introduced recently. This algorithm has attracted much attention and been applied to different problems. It has been shown that the IWO can outperform both the GA and the PSO in the convergence rate as well as the final error level. Considering the algorithm process, the key terms used to describe this algorithm should be introduced. Some of these terms are presented in Table 1. Each individual or agent, a set containing a value of each optimization variable, is called a seed. Each seed grows to a flowering plant in the colony. The meaning of a plant is one individual or agent after evaluating its fitness. Therefore, growing a seed to a plant corresponds to evaluating an agent's fitness [Karimkashi and Kishk, 2010].
Table 1. Some of the Key Terms Used in the IWO
Each individual in the colony containing a value of each optimization variable
A value representing the goodness of the solution for each seed
One agent/seed after evaluating its fitness
The entire agents or seeds
The number of plants in the colony
Maximum number of plants
The maximum number of plants allowed to produce new seeds in the colony
 1-First of all, theN optimization parameters (variables) should be chosen. For each of these variables in the N-dimensional solution space, a maximum and minimum value should be assigned (Define the solution space).
 2-Each seed takes a random position over thed dimensional problem (Initialize a population).
 3-Each initialized seed grows to a flowering plant. In other words, the fitness function returns a fitness value to be assigned to each plant, and then these plants are ranked based on their assigned fitness values (Evaluate fitness and ranking).
 4-Every plant produces seeds based on its assigned fitness or ranking. The number of seeds each plant produces depends on the ranking of that plant and increases from its minimum possible seed production (Smin) to its maximum (Smax) (Reproduction).
 5-The produced seeds in this step are being dispread over the search space by normally distributed random numbers with mean equal to the location of producing plants and varying standard deviations. The standard deviation at the present time step can be expressed by:
where itermax is the maximum number of iterations. σinitial and σfinal are defined initial and final standard deviations, respectively and n is the nonlinear modulation index (Spatial dispersion).
 6-After that all seeds have found their positions over the search area, the new seeds grow to the flowering plants and then, they are ranked together with their parents. Plants with lower ranking in the colony are eliminated to reach the maximum number of plants in the colony,Pmax (Competitive exclusion).
 7-After this process carried out for all of the plants, the process is repeated at step 3 until either the maximum number of iteration is reached or the fitness criterion is met (Repeat).
3.2. Optimization Procedure
 The IWO with restricted boundary condition is applied to the problem of synthesizing the far field radiation pattern of the CPPAR antenna. The phase shifts are chosen to make a phase front in the direction of the chosen scan angle. Therefore, only the amplitude weights of elements are optimized to achieve the desired sidelobe levels. The objective is to obtain sidelobe levels less than a tapered sidelobe mask decreasing linearly from −30 dB to −40 dB in the range of 1° ≤ |θ| ≤ 10°, 1° ≤ |φ| ≤ 10° and less than −40 dB for |θ| > 10°, |φ| > 10°. A “do not exceed criterion” is utilized in the objective function. That is, an error will be reported if the obtained radiation pattern exceeds the desired sidelobe level. The optimization process for a few thousand radiating elements is computationally very expensive. In order to avoid expensive computations in finding optimal weights for each element, a modified Bernstein polynomial [Boeringer and Werner, 2005] is utilized to define the current distribution of the array antenna. In other words, instead of optimizing the excitation coefficients of array elements, a two dimensional function defined by seven variables specifies a smooth and uni-modal current distribution on the array elements. A one dimensional modified Bernstein polynomial is defined as [Boeringer and Werner, 2005]:
where A, C0, C1, N0, and N1 specify the shift of the excitation maximum, the left and right endpoint values, and the left and right sharpness of the peak of f(u). For the cylindrical array antenna, a two dimensional modified Bernstein polynomial is generated by multiplication of two 1D functions in the ϕ and z directions.
 Since the desired envelope is symmetrical in the azimuth plane, we exploit the symmetry of current distribution in this plane. Therefore, f(ϕm) is defined with two parameters since N0 = N1, C0 = C1, and A = 0.5. After some simple manipulation f(ϕm) can be expressed as:
 Using the new expression, only seven parameters, instead of ten, are optimized to obtain the desired current distributions. Having fewer variables makes the optimization process simpler and faster.
 It should be noticed that the current distribution can be defined as the summation of some orthogonal functions like Bessel and Cosine; however, it might not be realizable in practice due to the coupling effects that tend to smooth out the effective amplitude distribution. Furthermore, using the modified Bernstein polynomial causes a gradual decrease of sidelobe levels by moving the observation angle away from the main beam. In addition, the simulation results show that the maximum sidelobe levels occur in the principle planes. Thus, defining the sampling points at the principle planes and around the main beam guarantees obtaining the desired sidelobe levels in all the directions.
3.3. Inter-element Spacing
 In order to achieve a cost-efficient design for the CPPAR, the number of elements should be minimized. The advantage of minimizing the element number is clear from the point of view of T/R modules. Minimizing the number of elements for periodic array antennas means increasing element spacing, causing the appearance of grating lobes. In order to compromise between the number of elements and appearance of grating lobes, a case study is done. Simulated radiation patterns of the CPPAR for different element spacing in both the vertical and horizontal planes are shown inFigure 5. It should be noted that the radiation patterns in the vertical plane are computed for the case where the main beam is pointed to its maximum scan angle (θ = 30 degrees). Therefore, element spacing in the vertical and horizontal planes is chosen to be 0.65λ and 0.6λ, respectively. Such element spacing yields 122 elements in each vertical column and 592 elements in each ring around the cylinder.
3.4. Cross-Polarization Minimization
 In order to suppress the level of cross-polarization, the mirrored feed arrangement is considered [Woelders and Granholm, 1997; Granholm and Woelders, 2001; Rahmat-Samii et al., 2006]. For the sake of illustration, an array consisting of identical 2 × 2 subarrays is considered. The feed arrangements for the conventional (baseline) and mirrored feed arrangements are shown in Figures 6a and 6b, respectively. The ports with a negative sign ‘-’ mark are fed 180° out of phase compared to the port marked with a positive sign ‘+’. It will be seen that using the image feed arrangements substantially suppresses the cross-polarization level.
4. Optimization Results
 In this section, the simulated radiation patterns of the optimized CPPAR for different scan directions are presented. The optimization is applied to the co-polar pattern in each scan direction and the cross-polar pattern is computed for the two arrangements shown insection 3.3. It should be mentioned that only the vertically polarized radiation pattern are shown for brevity. In this case, Eθ is copular and Eφis cross-polar component, respectively. The parameters used for the IWO are summarized inTable 2.
Table 2. IWO Parameter Values for the Array Optimization
4.1. Broadside Pattern
Figure 7shows the simulated radiation patterns of the CPPAR in both the vertical and horizontal planes. It can be seen that the desired sidelobe levels and beam widths are achieved. Moreover, the effect of mirrored feed arrangement on the suppression of the cross-polar pattern is observed. The simulated co-polar and cross-polar radiation pattern images of the CPPAR are shown inFigure 8. The higher sidelobe levels in the principle planes confirm the affectivity of the observation points defined on the principle planes. Figure 8bshows that higher cross-polarization levels occur at the locations far beyond the main beam. The optimized current distribution of the array is shown inFigure 8c. It is seen that a very smooth tapering is achieved.
 In another effort, the optimized radiation pattern of the CPPAR is compared to the case where WSR-88D tapering is applied to the CPPAR. The comparison between the simulated radiation pattern of these two are shown inFigures 9a and 9b. The mainlobe of the optimized CPPAR pattern is very close to that of WSR-88D pattern, but the sidelobe level is much lower. A comparison between the current distribution of these arrays in the azimuth plane are shown inFigure 9c. Although the modification in the current distribution is minor, it causes great improvement in the radiation pattern of the array. In other words, the sidelobes levels are very sensitive to the small deviations of the current distribution, indicating the importance of the optimization.
4.2. Off Broadside Patterns
 The simulated radiation pattern of the array for scan directions of 10 degrees, 20 degrees and 30 degrees are shown in Figures 10–12. It is seen that desired SLLs and beam widths are obtained for all the scan directions. It is also seen that the mirrored feed arrangement suppressed the level of cross-polarization in all scan directions.Table 3shows the maximum cross-polarization Level for all the scan directions. It can be observed that by increasing the scan direction away from the broadside angle, the cross-polarization level is increased. It should be noticed that the maximum cross-polarization occurs at angles far from the main beam.
Table 3. The Maximum Cross-Polarization Level for Different Scan Angles
θ = 90°
θ = 80°
θ = 70°
θ = 60°
Max cross-polarization level (dB)
4.3. CPPAR Sensitivity
 In Figure 13, the minimum detectable reflectivity factor of CPPAR versus the distance from the antenna for different scan directions and different maximum power is compared to that of the WSR-88D [Doviak and Zrnic, 1993]. The reflectivity factor is with D as the drop diameter and N(D) as the drop size distribution, which is the normalized radar reflectivity of the backscattering cross section per unit volume for Rayleigh scattering. The minimal detectable reflectivity is defined such that the signal-to-noise (SNR) is zero dB. The calculations were made usingDoviak and Zrnic [1993, equation (4.35)] by assuming 1 μs pulse and ignoring the transmission losses. It is shown that increasing the maximum allowed power from each single patch from 10 W to 100 W makes the CPPAR compatible with the WSR-88D having a 475 kW peak power. It is also seen that by increasing scan direction (θ) the minimal detectable reflectivity is increased, meaning a reduced sensitivity. Fortunately, such sensitivity reduction is within 3 dB and the reduced sensitivity is acceptable because the radar does not need to make measurements at long range at high elevation. It is noted that the sensitivity calculation applies to a passive CPPAR without taking processing gain into account. Had an active CPPAR been used with N T/R modules, there would be a sensitivity gain of 10log(N) dB.
 The IWO algorithm was applied to optimally design the cylindrical polarimetric phased array radar for weather sensing applications. A Modified Bernstein polynomial defined by seven variables was optimized to assign the current amplitude to the microstrip patch elements of the CPPAR. Using the modified Bernstein polynomial, not only is the computational domain reduced, but a very smooth current distribution is assigned to the CPPAR antenna, increasing the antenna efficiency and minimizing the effect of coupling between elements. After minimizing the number of elements of the CPPAR, the amplitude weights of elements are optimized while the phase weights are chosen to make a phase front in the direction of the chosen scan angle. In addition, the imaged feed arrangement was used to suppress the cross-polarization level. The simulation results show that the desired sidelobe levels and beam widths are achieved for the broadside and off broadside angles. However, by increasing the scan angle in elevation, the beam width was slightly increased. Moreover, the cross-polarization level has been suppressed by using the imaged feed technique. For the broadside beam pattern a very low cross-polarization level is obtained around the main beam. However, when the beam is scanned off broadside, the cross-polarization level increased. Considering the fact that an alternative transmission CPPAR system has much less stringent requirements, even the increased cross-polarization level is acceptable. Using the CPPAR for simultaneous transmission requires heavier suppression of cross-polarization and should be subject to more studies.
 The work is supported by NOAA grant NA08OAR4320904 and NSF grant AGS-1046171.