Phase-Only Beam Broadening of Contiguous Uniform Subarrayed Arrays Utilizing Three Metaheuristic Global Optimization Techniques

Continuous coverage of wide areas is desirable in many application realms including radar and cellular communications. Radars are challenged with finding efficient means to scan wide angular extents to provide detection, tracking, characterization and measurement of objects within their coverage area. Similarly, the cellular communication industry needs to efficiently provide voice and data communications over very broad coverage areas [10]. This coverage problem can be addressed by steering the main beam across the coverage area using mechanical or electronic means, but beam broadening has the advantage of continuous coverage of a wider angular extent instead of using a time-multiplexing scheme as utilized by beam steering.

In a phased array radar, broadening of the main beam can be accomplished by either adjusting the amplitude or phase of the individual elements of the array or by adjusting both the amplitude and phase. In today’s modern phased arrays that utilize solid-state transmit amplifiers which operate most efficiently in saturation, it is often desired to utilize phase-only approaches to beam broadening as presented in [10, 4, 3] to optimize power efficiency. However, these published methods of beam broadening assume that the phase of each element in the array can be independently adjusted which is not true of contiguous uniform subarrayed arrays.

In a contiguous uniform subarrayed array architecture the elements are grouped together in a manner that only allows the amplitude and phase of each subarray, not each element, to be adjusted. Contiguous uniform subarrayed arrays are commonly used in large array designs to reduce cost by limiting the numbers of control elements. However, the subarrayed architecture also makes it more difficult to form a low-ripple broadened main beam and to reduce the sidelobes. There is currently a gap in the available literature regarding the efficiency of global optimization methods to synthesize antenna patterns with a broadened main beam for subarrayed architectures when the synthesis is constrained to be phase-only, utilize contiguous uniform subarrays, and produce low sidelobes. [5, 13, 12] address synthesis with subarrayed architectures but they concentrate on optimization of combined sum and difference patterns, not broadening the main beam. [2] proposes a synthesis method for subarrayed arrays but this method is amplitude only, utilizes subarrays of unequal sizes, and does not address beam broadening. [11, 6, 9, 7] approach our desired domain space but utilize hybrid subarrays where the amplitude is dependent on the subarray amplitude but the phase of each element in the subarray can still be adjusted independently. The closest published material that this author has found is [10] which presents a beam broadening approach for subarrayed architectures where the synthesis is constrained to be phase-only and utilize contiguous uniform subarrays, but this method constrains the widened full array beam widths to only those beam widths that are multiples of the subarray beam width. This method also produces unacceptable sidelobe levels.

Metaheuristic global optimization methods such as Simulated Annealing (SA) [18], Genetic Algorithms (GA) [19, 1], and Particle Swarm Optimization (PSO) [20, 17] have been used to find solutions for several radar array architectures utilzing both amplitude or phase adjustment. These optimization problems cannot be optimized efficiently utilizing brute force techniques due to computational limits. These global optimization algorithms are capable of searching complicated solution spaces with many local optima to find the global optimum. Due to the dimensionality and computational difficulty of radar beam broadening problems, these techniques are well-suited for synthesis of antenna patterns using various cost evaluation functions.

This paper compares three of the most popular global optimization algorithms and their ability to generate broadened main beam patterns in subarrayed radar architectures. It compares the efficiency, the rate at which a sufficient solution is found, and effectiveness, the relative quality of the solutions, of these techniques in terms of cost function evaluation of the pattern generated from a given phase setting for 40x40 element arrays and 80x80 element arrays. Simulation results of pattern optimization for these subarrayed arrays are given to demonstrate the validity of these techniques and their comparability.

The mathematical model of the optimization algorithms utilized in this paper are presented in Section II which is followed by Section III which presents the cost function used. Section IV shows the simulated results of the various global optimization algorithms. Finally, Section V presents the conclusions that are drawn from the results.

In this paper a cost function determines how close the generated array factor, $AF_{\mathrm{gen}}$, is to the desired array factor, $AF_{\mathrm{des}}$. The desired array factor is defined as a normalized flat-top function with a main beam and sidelobes. The main beam consists of a desired angular width with an amplitude of 0 dB. The sidelobes are the points in $AF_{\mathrm{des}}$ outside this main beam with a desired sidelobe level defined in negative dB. $AF_{\mathrm{gen}}$ is computed by taking the IFFT of the element excitations and is normalized before comparing it to $AF_{\mathrm{des}}$. The cost function, $C$, is defined as:

where ${E}_{\mathrm{mb}}$ is the error calculated for the points that are in the main beam and below 0 dB, $\theta_{\mathrm{mb}}$, and is defined as:

and ${E}_{\mathrm{sl}}$ is the error calculated for the points that are outside the main beam and above the desired sidelobe level, $\theta_{\mathrm{sl}}$, and is defined as:

.

The logarithm is used in (3) to reduce the values returned to a more condensed range so that the convergence rate of the cost function can be more easily visualized. For a more tangible performance measure, (3) can be converted to a percentage pattern effectiveness metric as

where $\beta$ represents the number of angular points used to evaluate $AF_{\mathrm{gen}}$ vs. $AF_{\mathrm{des}}$.

The results presented in this paper were generated using a computer simulation that synthesized an approximation of the array pattern where each element was assumed to be an isotropic radiator. The array pattern was generated with an FFT-based approach for computational efficiency. This simulation contained adjustable parameters that allowed arbitrary arrays to be modeled based on the number of elements in each subarray, the number of subarrays in the array, the spacing between elements, and the number of bits for each phase shifter. The desired antenna pattern was described using parameters for width (12 degrees) of the broadened main beam and maximum sidelobe level (-13 dB).

The simulation was configured to model rectangular arrays of 40x40 and 80x80 elements where the elements were uniformly spaced a half wavelength apart. The rectangular arrays were grouped into contiguous uniform subarrays of 5x5 elements; thus, the 40x40 array had 8x8 subarrays and the 80x80 had 16x16 subarrays. Each array was also configured to utilize 6-bit phase shifters. These array configurations create solution spaces with $(2^{6})^{64}$ and $(2^{6})^{256}$ possible permutations for each respective array.

A few techniques were utilized in the simulation to shrink these solution spaces. First, array symmetry is assumed in order to simplify the problem. Symmetry is created by setting subarrays equidistant from the array center equal to the same phase. Assuming symmetry reduces the 40x40 array to only needing 9 independent phase settings instead of 64, and the 80x80 array requires 32 instead of 256. Also, since it is the relative phase of the elements and subarrays to each other that is important, the innermost subarrays were fixed to zero phase to reduce the solution space to 8 and 31 independent phase settings for each respective array. Therefore, assuming symmetry and fixing the center subarray phase to zero shrinks the solution spaces to $(2^{6})^{8}$ and $(2^{6})^{31}$ possible permutations for each respective array.

In order to effectively evaluate the stochastic algorithms, we allowed each algorithm to converge until they had the opportunity to evaluate 20,000 solutions. For each iteration of the algorithm, we recorded the best solution as convergence curves. Since the SA, GA, and PSO algorithms are not deterministic processes, they were executed 100 times with the same settings to obtain statistical significance. The convergence curves from each of the 100 executions were averaged to form a mean convergence curve for each algorithm.

For SA, constants in Table I were selected from ranges in [16]. $k$ was chosen in order to accept many solutions at the beginning but to accept fewer as the temperature neared 0. It was based on the cost obtained from $C$. The logarithm utilized in $C$ was important for ensuring that $k$ was a satisfactory value for the duration of the algorithm as the cost decreased. Since $C$ involves a 2D summation of errors, differing patterns can have magnitudes of difference in fitness, and the logarithm allows $k$ to better cover the range of costs. $\alpha$ was selected to generally cover the range of $T$ over the course of the algorithm: its value is fairly standard. In addition, the number of iterations between temperature changes was selected to allow an appropriate number of changes in the algorithm given the rough number of evaluations of $C$ desired.

Constants in Table II were empirically determined using a limited parameter search and could likely be improved with a more extensive search. This more extensive search is left as a an area for future study. Using a quarter of the population as elites in the parent selection was determined to improve convergence. Due to the use of elitism preserving genes in the GA, $r$ and $m$ were both set higher to increase the rate of testing new solutions. The selection of population size was based on the rule-of-thumb that population sizes between 20 - 50 are usually sufficient. For the smaller 40x40 array with a smaller solution space, a population of size 25 was used. However, in the case of the 80x80 radar, due to the increased solution space size, a larger population size of 50 was used.

Constants in Table III utilized in the PSO algorithm were chosen based on [15]. To allow a fair comparison between the GA and PSO algorithms, the same population sizes were used for each algorithm.

In the 40x40 radar architecture, all three algorithms in this study found the global optimum on at least some of their 100 executions. On average, PSO was slightly more effective than SA. While SA started with the steepest convergence (Figure 1), it tended to get stuck in local optima more frequently than PSO (Figure 2). Both PSO and SA outperformed genetic algorithm. However, it should be noted that each metaheuristic technique can be configured to manage the speed of convergence and the probability that the convergence stalls in a local optima instead of the global optimum. The overall poorer performance of the GA when compared to the PSO and SA is likely influenced by the GA configuration and would likely have more comparable performance if more time were spent optimizing the configuration. This increased optimization of the GA algorithm is outside the scope of this study and is an area of future work.

In the 80x80 radar architecture, the algorithms showed larger differences in their efficiency and effectiveness. Simulated annealing again showed the steepest convergence decent (Figure 6) and did find the best solution of any of the algorithms ($C=11.36$) but still suffered from stalling in local optima (Figure 5). This effect is realized by PSO showing better average effectiveness than SA even though SA’s best solution was slightly better than PSO’s best solution. Again, GA performed worse than SA and PSO, but it had less difficulty with stalling in local optima. In Figure 7 it should be noted that the ripples in the array pattern are deeper on the azimuth and elevation axes than at other orientations. While the nulls in the mainbeam are deeper than desired, actual measurements of similar arrays show that measured results typically smooth the nulls in simulated patterns so that in reality the nulls are not as deep as simulation can sometimes predict.

In this study, we explore the ability of SA, GA, and PSO to generate desired beam broadened array patterns in subarrayed arrays using phase-only modification of element excitations. We have adapted and configured each algorithm for use with the problem domain, especially the PSO algorithm where the velocity calculation needed to be modified because phase is a circular function. We find that SA and PSO are more effective and efficient than GA for the configurations shown in this study. SA is more efficient than PSO, but PSO is on average more effective than SA.

Future studies can examine these techniques for larger subarrayed arrays to better understand their effectiveness. Comparisons of these techniques utilized for both subarrayed and non-subarrayed arrays, with and without symmetry, is another potential path for research. Also, utilizing a supercomputing cluster could allow verification of the global optimum for these array configurations and better quantify the optimality of solutions found from these techniques.