Abstract
Phase-only beamforming offers a hardware-efficient alternative to amplitude tapering, but achieving low sidelobe levels while maintaining high array efficiency remains challenging. This paper proposes a novel two-objective optimization algorithm for phase-only beamforming. The proposed method combines Genetic Algorithm with Sequential Quadratic Programming to enhance the performance. This approach utilizes a multi-objective optimization framework to minimize peak sidelobe level (PSLL) while simultaneously maximizing array efficiency. The proposed method performs better than the existing phase-only beamforming methods in terms of both PSLL and array efficiency. Computer simulation results are presented for the PSLL and array efficiency of linear arrays with 20, 32, 40, 55, 60, 80, and 100 elements to validate these findings, with results compared to other phase-only methods. Furthermore, it is shown that the proposed method exhibits enhanced performance compared to well-known amplitude tapering windows in managing the trade-off between peak sidelobe level, beamwidth, and array efficiency. For two-dimensional planar arrays, the proposed method demonstrates improved performance compared to the existing phase-only beamforming methods in both peak sidelobe level and array efficiency criteria. A comparison has been performed for circular planar arrays with 32 and 172 elements, demonstrating that the proposed method achieves an improvement of 8.8% to 21.8% in array efficiency while simultaneously improving the PSLL by approximately 1.5 dB. For square 15 by 15 planar arrays, a considerable PSLL enhancement of at least 6.8 dB is obtained compared to the existing phase-only approach while achieving narrower beamwidth.
Supplementary Information
The online version contains supplementary material available at 10.1038/s41598-026-46940-w.
Keywords: Phased array antennas, Genetic algorithms, Quadratic programming, Multi-objective optimization method, Antenna radiation pattern synthesis
Subject terms: Engineering, Mathematics and computing
Introduction
The common approach for reducing PSLL of an array antenna is the implementation of an amplitude tapering window1. However, this method requires the use of non-uniform amplitude excitations for array elements, which results in an increase in antenna beamwidth and leads to hardware complexity. Moreover, the practical application of such non-uniform amplitude excitations often needs the incorporation of power attenuators, which can cause significant loss of power.
Phase-only pattern synthesis methods offer an effective way of minimizing the PSLL through optimization algorithms that obtain optimal phase values for array elements, while applying uniform amplitude excitations2–6. This leads to a simpler implementation process compared to the amplitude method. In these methods, antenna beamwidth is often significantly smaller than those associated with amplitude windowing procedures. Furthermore, the use of uniform amplitudes eliminates the need for power attenuators and decreases the gain reduction (array efficiency loss) compared to amplitude windowing approaches. However, it is important to note that the array efficiency loss is not zero in phase-only methods, as explained in Section “Optimization problem”. Another option in phase-only synthesis methods is interference nulling while maintaining low sidelobe level7–10 in many applied systems including FDA radars11,12. This is achieved by using suitable objective function and/or imposing an extra constraint to suppress the pattern in the interference directions. Beamwidth widening is another application of phase-only beamforming so that the antenna can radiate to a wider region without reducing its size and radiated power13,14.
Current optimization methods for phase-only pattern synthesis can be categorized into two main groups. The first category includes convex optimization algorithms, including gradient search approaches. These methods are usually caught in a local minimum and can only access locally optimal solutions. To obtain more effective solutions, they often require multiple initial points or may need to be combined with other methods. Examples of convex optimization approaches for phase-only pattern synthesis include2 and15, which are based on steepest descent and gradient descent methods, respectively. Another convex approach is presented in3, which utilizes the CVX convex optimization model. This model employs techniques such as linear programming, semi-definite linear programming, second-order programming and quadratic programming16.
The second category consists of evolutionary and statistical optimization algorithms. Some examples of this category are genetic algorithm (GA)17–20, particle swarm optimization (PSO)4,20, differential evolution algorithm9,21, ant colony algorithm22, simulated annealing5, mayfly algorithm23 and electromagnetism-like approach24. While these algorithms may not essentially lead to locally optimal solutions, they are not caught in local minima and can usually obtain solutions better than those of convex optimization methods. Moreover, these methods can operate from arbitrary initial points.
GA is a flexible optimization algorithm inspired by natural selection and the principles of genetics. GA was first introduced by John Holland during the 1960s and 1970s25 and has since gained popularity as a method for solving complicated problems across numerous areas, such as engineering and machine learning.
Sequential Quadratic Programming (SQP) method6,26–29 uses both first and second-order derivatives, resulting in faster convergence than gradient search algorithms. In addition, to impose problem constraints, the Lagrange multiplier method can be used to find efficient solutions (near optimal solutions that achieve optimality when all constraints can be expressed as equality constraints). It has been proved that the solutions obtained by Lagrange multiplier method are efficient, even for nonconvex problems30. Nevertheless, for severely nonconvex problems, the efficient methods’ performance may degrade, and to achieve the desired results, appropriate initial points should be used.
In phase-only pattern synthesis methods, as with other sidelobe level reduction approaches, we often experience lower array efficiency compared to conventional methods without tapering. This subject is not addressed in the majority of existing works which have only studied the PSLL. When considering the actual pattern, the reduction in array efficiency can be observed in the all existing methods. This issue can be significant in practice and should be taken into account when evaluating power consumption.
In31, GA and SQP were used with GA employed only to find an initial solution. Subsequently, SQP was applied in a separate stage. In fact, GA and SQP were not combined, and GA’s capabilities were not fully utilized.
In this paper, a phase-only optimization algorithm is presented that combines GA and SQP. A modified GA is used that incorporates SQP in its reproduction step to perform the local search, ultimately leading to a near-optimal solution. In other words, SQP is incorporated into GA allowing the full utilization of the capabilities of both methods. Also, Lagrange multiplier approach is utilized for imposing the problem constraints. In the proposed combined optimization algorithm, a two-objective cost function is used to simultaneously minimize the PSLL and maximize array efficiency. Furthermore, a constraint on array efficiency or PSLL is implemented to guarantee the desired array efficiency or PSLL.
The proposed two-objective optimization method is capable of establishing a tradeoff between PSLL, array efficiency and beamwidth so that we can achieve the desired PSLL value with minimal reduction of array efficiency and minimal increase in beamwidth. Similarly, one can achieve the desired array efficiency and beamwidth with minimum PSLL value.
Commonly, controlling the trade-off between PSLL and beamwidth has been performed using amplitude windows like Chebyshev, Taylor and Kaiser. The simulation results show that while the phase-only beamforming benefits from uniform amplitude excitations, it demonstrates better performance compared to amplitude windowing approaches for controlling the trade-off between PSLL, array efficiency and beamwidth. Although this advantage can be observed in some existing phase-only methods, they have not included array efficiency in the optimization problem and cannot control the trade-off efficiently. The proposed phase-only approach offers a systematic way for implementing such a trade-off with adjustable PSLL or array efficiency.
Optimization problem
Problem formulation
In phase-only pattern synthesis, while uniform amplitude excitations are applied for all elements, the phases of elements are adjusted so that desired pattern is achieved. The phase of each element can be considered as two terms, the conventional beamforming term and the added phase from phase-only method.
For an arbitrary M-element array with the elements position vectors 
, the antenna pattern is obtained from1,32,33
 |
1 |
where 
denotes the inner product of two vectors, 
is a unit vector in an arbitrary direction, 
is the unit vector in the desired mainlobe direction, 
is the phase applied to the 
’th array element due to phase-only approach and 
. The PSLL is then defined as
 |
where 
is the set of sidelobe angles.
For any 2-dimensional planar array with elements distributed in x–y plane, (1) can be rewritten as
 |
2 |
where 
is the direction of mainlobe in spherical coordinates and 
is the position of mth element in x–y plane.
For a uniform linear array (ULA) with distance 
between the elements, the antenna pattern can be formulated as1
 |
3 |
where 
is the mainlobe angle (the angle between the mainlobe center and the plane orthogonal to the array) and 
is an arbitrary angle.
For unambiguous ULA with 
, the PSLL value can be found from
 |
where 
and 
are the angles of first positive null and first negative null in the pattern, respectively.
The largest and default range of 
is 
. However, according to the angular interval of interest, this range can be restricted as the set 
. In this case, the attainable PSLL can be decreased and the PSLL value is obtained from
 |
The unconstrained problem of PSLL minimization can be formulated as
 |
4 |
Although the phase-only approaches have not a gain reduction like that of amplitude windowing methods, there is often an array efficiency loss compared to the conventional uniform excitation method. The array efficiency is defined as the ratio of peak power density of the under-study array to peak power density of the uniformly excited array and is obtained from2,24
 |
where 
and 
are the values of peak power density for the under-study array and the uniformly excited array, respectively. According to (1), the array efficiency for the phase-only array is obtained from
 |
To minimize the PSLL and maximize 
(or minimize 
) simultaneously, one can use the two-objective problem
 |
However, there is a trade-off between PSLL and array efficiency and if one is improved the other is degraded. To attain the desired point in this trade-off, we utilize a constraint for the minimum value of array efficiency. The problem is then written as
 |
5 |
If the lower bound 
is changed, the optimal value of one objective parameter (PSLL or 
) will be reduced and the other is raised. Therefore, 
can be adjusted so that the desired trade-off point is achieved. The constraint in (5) can be written as standard inequality form 
.
Similarly, the optimization problem can be formulated so that guarantee the desired PSLL value
 |
6 |
Trade-off between the antenna parameters
In general, the PSLL suppression leads to increasing the beamwidth and decreasing array efficiency, gain and directivity of array antennas. While this effect is well-known for amplitude tapers32, similar effect can be seen in other techniques of PSLL reduction like phase-only beamforming methods and time-modulation34–36 techniques as well.
Directivity, 
, at the mainlobe direction is defined as 
where 
is the total radiation power32. From 
we conclude 
. Therefore, directivity is explicitly proportional to the array efficiency 
.
Gain, 
, is related to directivity by 
where 
is the conductivity and dielectric efficiency32. Thus, 
and the gain is also proportional to array efficiency.
Beamwidth has a reverse relation with directivity which is approximately given by32:
 |
where 
and 
are azimuth and elevation 3 dB beamwidths, respectively.
Thus, directivity and gain are linearly proportional to array efficiency and beamwidth is inversely proportional to array efficiency and if we control the trade-off between PSLL and array efficiency, it is equivalent to controlling the trade-off between PSLL and directivity, gain and beamwith as well.
Genetic algorithm
The use of GA for phase-only pattern synthesis is a relatively recent development. One of the key advantages of using GA for phase-only pattern synthesis is its ability to handle nonconvex optimization problems. Traditional convex optimization algorithms, like gradient descent, can get caught in locally optimal points and fail to obtain the globally optimal values. GA addresses this problem by exploring multiple points in the search space simultaneously, and by keeping a various population of candidate solutions.
The flexibility is another important advantage of GA. It can easily handle multiple objectives, such as minimizing the sidelobe level and maximizing the array efficiency. It can also include nonlinear constraints, such as imposing a minimum value for array efficiency.
GA is usually consisted of the following steps:
1. Initialization
At the beginning, the initial population is generated. Each individual has a sequence of genes called chromosome. In an optimization problem, each variable can be interpreted as a gene.
2. Fitness Evaluation
The fitness function is a criterion for the ability of each individual to solve the problem. In many references, the fitness function is defined equal to the cost function that should be minimized. We will define the fitness function of the suggested method in Sect. “Suggested algorithm” .
3. Selection
Some individuals of the current population are chosen as parents of the next generation. The individuals with greater fitness value have greater probability to be selected.
4. Reproduction
The selected parents generate the offspring by means of the genetic operators. The main genetic operators are:
Crossover: Creates new offspring by combining parents’ genes.
Mutation: Makes random changes in offspring genes to increase the variety of population.
5. Replacement
A part of individuals of the current population are replaced by the new offspring.
6. Termination
The algorithm is terminated when a criterion such as achieving the desired fitness value, or the maximum number of iterations is satisfied.
Sequential quadratic programming
The SQP method involves approximating the cost function with a quadratic function in each iteration6,24,26,27. The resultant quadratic programming (QP) problem can then effectively be solved using the numerical techniques like quasi-Newton approach.
In general, the SQP can be used to solve the nonlinear problem
 |
7 |
where 
is the problem variables vector, 
is the vector of nonlinear inequality constraint functions and 
is the vector of nonlinear equality constraint functions.
At the 
th iteration of SQP where 
, the change vector 
is found by solving the QP problem
 |
8 |
where 
is the Lagrangian function, 
and 
are Lagrange multipliers for non-equality constraints and equality constraints, respectively.
The quadratic programming problem (8) can be solved using the quasi-Newton method5. After finding 
, the variables vector at the subsequent iteration is obtained as 
. The iterative process is repeated until the absolute value of difference between the cost function values at the last two iterations becomes smaller than the given cost function tolerance 
.
The above SQP method is also capable of solving multi-objective optimization problems via minimax approach in which the cost function of (7) is set to 
, where 
is a vector containing objective functions. In this case, the cost function of Lagrange dual problem is
 |
In general, in minimax approach, it is recommended that the objective functions (
and 
in our problem) to be divided by their desired values (
and 
) for normalization and use 
and 
as objective functions. However, in constrained minimax optimization problems which are solved using the Lagrange multiplier methods, the Lagrange multiplier,
, acts as an automatic scaling coefficient that keeps balance between the objectives. For example, in our approach when we use the normalization, the problem (5) will become
 |
and the Lagrange dual problem cost will be
 |
 |
Similarly, for the problem (6)
 |
 |
It can be seen that in the all cases, the effective weight applied to the objective functions depends on 
and the Lagrange multiplier acts as an automatic scaling coefficient mechanism. Thus, the optimization method is not sensitive to objective scales 
and 
. However, an approximate normalization will be useful for faster convergence.
From the computational complexity aspect, SQP is more efficient compared to traditional second-order nonlinear optimization algorithms. This is achieved by using quadratic approximation and exploiting the quasi-Newton approach that reduces the computational complexity by applying suitable approximation of Hessian matrix and its inverse at each iteration37.
Suggested algorithm
In the suggested method a new operator of self-reproduction based on SQP is added in the reproduction step of GA in which 
individuals with the best fitness values generate 
new children using SQP as a local search method. Each of the selected individuals are given to SQP algorithm as the initial point and the output will be a new offspring, i.e. SQP(× 0, SqpParams) is calculated with ‘ × 0 = selected individual’ as the initial point of SQP algorithm. In the replacement step, to guarantee the convergence, the best 
individuals must be kept in the next generation and not be replaced. The fitness function of the suggested algorithm is defined as
 |
where 
and 
are desired values of 
and 
, respectively. Our goal is to minimize the fitness function and obtain a solution that satisfies 
and 
(termination condition). The constraints are similar to those defined in (5) and (6).
To choose the values of 
and 
we recommend the following process. At first, select the objective with stricter and more defined requirement as the constraint variable and use the related problem. Then, set the desired value of that objective equal to the constraint boundary. For the other objective select an approximate desired value using the unconstrained problem or existing known solutions (the accurate value is not required for desired value since Lagrange method finds its optimal value by automatic scaling as explained before). For example, if we have the requirement 
we can use the problem (5) and set 
. Then, set 
slightly higher than the optimal solution of unconstrained problem (4).
The parameters setting of GA are shown in Table 1.
Table 1.
Genetic algorithm parameters in the proposed optimization method. Some of the parameters are chosen similar to their default value in MATLAB software38. The recommended value intervals of the GA parameters for which the algorithm can be executed without significant increase in computational load or degrading the performance in the most computers are shown in third column.
| Parameter | Value/Description used in experiments of this manuscript | Recommended value interval |
|---|---|---|
| Constraint Tolerance | 0.001 | 10−5–0.002 |
| Creation Function | Generates a random population of uniform distribution | – |
| Crossover Function | Creates a random binary vector indicating the genes taken from each parent | – |
| Crossover Fraction | 0.75 | 0.6–0.08 |
| Function Tolerance | 10−6 | 10−8–10−5 |
| Maximum Generations | 5 × (Number of variables) | 4 × (Number of variables)−20 × (Number of variables) |
| Mutation Function | Adds random values with distribution  to the parent vectors |
– |
| Nonlinear Constraint Algorithm | Augmented Lagrangian Algorithm | – |
| Population Size | 200 | 150–300 |
| Population Type | Double vector | – |
| Selection Function | In the used selection function, a line is found that each parent is associated with a section of the line of length proportional to its scaled value. The algorithm moves along the line in steps of equal size. (the function selectionstochunif in MATLAB) | – |
In selecting the GA parameters, there is a trade-off between the performance and the speed of algorithm. To improve the performance, one should increase population size and maximum generations and decrease constraint tolerance, function tolerance and crossover fraction. However, increasing / decreasing these parameters too much aimed at improving the performance, imposes large computational load without significant improvement in the performance. The values in experiments of this manuscript (second column of Table 1) are selected based on default values in MATLAB38 but optimized so that the algorithm can be executed in reasonable time on a personal computer (our computer specifications: core i7 @ 2.8 GHz, 16 GB of RAM) and without degrading the performance.
The recommended value intervals of the GA parameters for which the algorithm can be executed without significant increase in computational load or degrading the performance in the most computers are shown in third column in Table 1.
The proposed algorithm has a faster convergence compared to the conventional GA and consequently, the maximum number of generations is set to 5 × (Number of variables) which is 20 times lower than that of MATLAB GA default value. Since the total computational complexity of GA is approximately proportional to the number of generations, this can reduce the computational complexity of GA about 20 times. However, it is important to consider the computational complexity of the SQP as well.
The summary of suggested algorithm is illustrated in Flowchart 1.
Flowchart 1.
Flowchart of the suggested algorithm.
It can be seen in Flowchart 1 that SQP is incorporated into the internal loop of GA to form a unified algorithm.
The full description of suggested algorithm with more details is brought in Algorithm 1.
Algorithm 1.
Suggested optimization algorithm for solving problems (4), (5) and (6).
In Algorithm 1, 
denotes the elements position matrix, 
is the vector of inequality constraints, GaParams is the GA parameters, SqpParams is the SQP parameters and 
denotes the result of SQP with initial point 
and other SQP parameters in SqpParams and 
(
, 
) denotes the 
individuals with minimum fitness in 
. The number of SQP iterations per generation should be given as SqpParams.Iterations.
Simulation results
Simulation setting for the proposed method are as follows. GA parameters (GaParams in Algorithm 1) are the same as those values in Table 1, columns 1 and 2. The recommended value of the objective function tolerance for SQP is 10−6 and the recommended quasi-Newton solver is the Brayton method26. We have adjusted 
, 
, and the number of SQP iterations per generation (SqpParams.Iterations) equal to 4.
Uniform linear arrays
Unconstrained problem
First, the results of PSLL minimizing problem (4) are investigated. Table 2 compares the PSLL values of the suggested phase-only algorithm with the case of uniform excitations and also with GA and PSO approaches. The parameters used for GA are the same as those in Table 1 and the parameters adjusted for PSO are the default values in38 except for maximum stall iterations that was set to 20 × (Number of variables) in our experiment (instead of 20) to improve the performance. The best solution in 10 independent experiments of GA and PSO is reported in Table 2.
Table 2.
Comparison of PSLL values in the proposed unconstrained phase-only method with the case of uniform excitations for a ULA with 20 to 100 elements.
| PSLL(dB) | ||||
|---|---|---|---|---|
| Number of elements | Proposed phase-only method | Uniform excitations | Genetic algorithm | Particle swarm optimization algorithm |
| 20 | − 16.4 | − 13.2 | − 15.3 | − 15.6 |
| 32 | − 17.6 | − 13.2 | − 16.7 | − 16.6 |
| 40 | − 18.4 | − 13.2 | − 16.8 | − 17 |
| 55 | − 19.6 | − 13.3 | − 17.5 | − 17.5 |
| 80 | − 20.5 | − 13.3 | − 18.6 | − 18.7 |
| 100 | − 20.8 | − 13.3 | − 18.8 | − 19.1 |
The optimal values of phases in the proposed unconstrained phase-only method are shown in Table 7 in the appendix.
Table 7.
Optimal phases in the proposed unconstrained method for ULA.
| Number of elements | PSLL (dB) | Angular range | Optimal Phases (degrees) |
|---|---|---|---|
| 20 | − 16.4 | [− 90, 90] | 4.068157213 − 12.41694032 − 10.82564028 8.289391343 3.480327521 |
| 9.55090597 4.661984848 0.147716139 − 1.898630958 − 0.038820569 | |||
| − 0.008240592 0.764667168 − 0.227276096 2.942705855 12.4945091 | |||
| − 6.378391818 39.64530901 − 110.6345034 17.79248622 − 2.761810343 | |||
| 32 | − 17.6 | [− 90, 90] | − 1.842604446 21.00886312 48.16088481 0.624388657 13.7522379 |
| 25.3178212 11.77044635 16.57406303 18.5073033 4.80703729 | |||
| 10.63261097 15.72458716 13.95713054 15.96063518 19.60833263 | |||
| 15.88212214 14.43591419 16.25408225 19.84305964 13.67050794 | |||
| 15.82651858 19.68070067 − 0.982795888 − 8.167151956 25.73414371 | |||
| 53.59867606 − 23.71997428 − 172.0126583 − 3.712392148 92.51472152 | |||
| − 2.849919192 − 1.461599878 | |||
| 40 | − 18.4 | [− 90, 90] | 25.63581094 35.64398538 − 28.07402188 5.093549285 60.54268711 |
| 33.57747574 6.169844158 12.61256398 21.49838973 13.780154 | |||
| − 4.192089542 15.59986845 22.68374829 9.16268718 7.506271963 | |||
| 7.952861942 12.9454576 12.32573094 11.52111971 14.63548728 | |||
| 11.71625347 13.69604102 13.3718974 14.22252281 6.333858702 | |||
| 15.7679245 9.886512954 9.461583582 1.09358455 23.66716878 | |||
| 17.34417017 − 66.37422055 − 7.222230469 108.0873544 − 6.565929527 | |||
| 162.1907623 7.529597051 − 44.27100639 28.92516639 26.18207474 | |||
| 55 | − 19.6 | [− 90, 90] | 13.47373981 25.37156358 − 27.86249272 − 33.24491426 − 116.9365654 |
| − 42.62757745 73.37981182 − 23.71569977 43.52994046 − 68.54343764 | |||
| 17.15291978 8.382785397 28.53192012 − 20.78282085 5.798083137 | |||
| − 7.092077572 20.69143793 − 10.57599002 − 6.390001538 − 1.237339547 | |||
| 5.684512502 − 7.362831981 − 6.940319253 − 5.265771599 − 3.766213299 | |||
| − 5.466248232 − 4.505077479 − 7.1201221 − 6.264887617 − 4.246887075 | |||
| − 5.578791736 − 6.292144162 − 5.55601749 − 1.277821834 − 5.10365166 | |||
| − 4.432946273 − 2.064630938 3.458880584 6.608299078 − 6.309898745 | |||
| − 16.30681003 41.40361821 12.83170069 − 52.31784651 12.50815346 | |||
| 60.30066073 − 10.87626183 − 67.00711752 − 258.3791716 − 36.74434746 | |||
| − 77.29369092 − 3.416156838 − 44.20352786 16.46483268 18.49754066 | |||
| 55 | − 21.3 | [− 60, 60] | 0.611390551 17.82083922 − 0.85096287 − 35.20373154 0.854921748 |
| 111.0133003 − 5.652667131 − 16.08190731 − 8.326395775 43.19977705 | |||
| − 19.55671615 36.56244466 − 39.44117772 20.70456434 − 11.22726173 | |||
| 8.658297956 − 17.9867957 7.041068545 6.657435823 − 12.7247264 | |||
| 7.06427425 − 5.049984397 2.997246916 2.149265536 − 6.664419069 | |||
| 6.45669126 − 2.843331158 − 2.581511686 − 0.785989724 1.165482176 | |||
| − 2.688879596 − 0.73555562 8.434856414 − 17.12821507 16.73996637 | |||
| − 20.46055327 21.81809465 − 29.3880815 39.50616355 − 54.77837865 | |||
| 40.07137758 − 59.13258795 − 8.754781133 50.84318455 − 56.48665484 | |||
| 75.89841703 − 79.4945226 59.70113262 − 75.31604839 77.8863861 | |||
| − 65.61856529 66.39925041 − 65.13256737 56.64346151 − 29.02750399 | |||
| 80 | − 20.5 | [− 90, 90] | 1.05392988 − 4.551358924 − 25.47802455 − 24.52340647 − 9.579796945 |
| − 222.7349455 85.85628534 − 10.78129645 33.71524106 − 25.59412111 | |||
| − 119.8001641 − 17.39520297 − 22.59667688 45.74034472 − 7.073825032 | |||
| − 89.1048239 29.94116322 3.812856304 − 15.37118472 − 83.25362781 | |||
| − 7.760213283 − 45.83308133 0.464569199 − 34.56058317 − 15.52891348 | |||
| − 32.67262433 − 11.5087284 − 14.46785945 − 13.88083759 − 25.69335581 | |||
| − 20.17849965 − 4.348818739 − 18.57737336 − 23.65839791 − 12.05289592 | |||
| − 14.18035104 − 20.67432149 − 10.20782977 − 10.28388916 − 18.51205738 | |||
| − 17.56908456 − 10.75866091 − 17.7198365 − 22.2308663 − 8.230716241 | |||
| − 15.62264404 − 24.28645603 − 13.5594 − 15.86791127 − 14.57739031 | |||
| − 17.47877803 − 18.28509094 − 21.35542502 − 21.32929095 − 34.13915505 | |||
| − 25.32414111 − 27.45730431 − 21.30446037 − 43.62002337 − 0.839933711 | |||
| − 40.12208393 − 42.60743928 25.86806792 − 18.47550179 − 51.63345296 | |||
| − 43.73027692 48.41926841 − 80.17417596 10.38185863 171.1270055 | |||
| − 61.85443813 9.729980551 3.593329391 0.332703644 92.33269759 | |||
| − 15.43358597 − 24.18364933 − 61.2362075 − 2.310938485 3.194492338 | |||
| 100 | − 20.8 | [− 90, 90] | 7.532531593 − 23.75471824 − 11.37709679 − 89.60343311 − 24.19060939 |
| 28.51590561 6.700953177 − 21.22650666 62.51342935 − 115.084554 | |||
| 5.188938929 − 72.17756806 31.83665394 127.6356055 28.29116504 | |||
| − 13.77337511 − 71.74453378 − 61.55344428 4.079498842 43.67977283 | |||
| − 34.87277746 − 1.883340505 − 33.60266724 − 67.26127633 7.260419459 | |||
| − 13.12341086 − 10.38271533 − 4.213919954 − 25.50603228 9.580270285 | |||
| − 29.06377728 − 3.653984069 − 2.589130655 − 12.7210291 − 13.45021381 | |||
| − 7.353457872 − 14.0837818 − 15.78780791 − 9.071391282 − 16.50578341 | |||
| − 13.45462837 − 12.78958781 − 11.27742615 − 7.507377181 − 14.38589594 | |||
| − 10.88344623 − 12.25673929 − 14.47523642 − 6.44207812 − 10.49097793 | |||
| − 10.70265096 − 12.31806427 − 7.908202622 − 10.24656986 − 14.17704085 | |||
| − 9.37591865 − 16.52618341 − 11.93774438 − 9.933226261 − 16.43983943 | |||
| − 8.711453516 − 15.64120001 − 14.33016453 − 8.686566517 − 15.40144324 | |||
| − 12.42027781 − 15.68152618 − 13.15189819 − 16.11872917 − 12.8364393 | |||
| − 1.259547673 − 6.459443928 − 28.06844464 16.34402019 − 57.17393062 | |||
| 34.15288356 − 24.96878559 − 41.48238757 29.24345182 − 54.8144667 | |||
| − 68.19314059 42.79310101 32.24357892 − 129.6594264 − 3.708026709 | |||
| − 22.97204604 31.91353535 − 20.03167325 − 61.09182038 − 21.1980556 | |||
| 35.20311613 103.7301475 222.9560639 75.4994084 − 11.87158688 | |||
| − 44.43086612 − 14.72800519 − 18.23352474 − 19.76406237 − 26.4568903 |
According to the results in Table 2, for 20 to 100 element ULAs, the PSLL of the proposed algorithm ranges from -16.4 dB to -20.8 dB. These results demonstrate the significant improvement of the proposed method compared to the conventional method with the PSLL of about -13.3 dB. According to Table 2, the proposed algorithm has also better performance compared to GA and PSO methods.
Also, the proposed algorithm performs better than other phase-only pattern synthesis approaches for the unconstrained problem. For example, the PSLLs of 20, 32, 40 and 55 element arrays obtained by the EM-Like phase-only method24 are -16.1 dB, -17.3 dB, -17.9 dB, and -18.3 dB respectively. This is while in the proposed algorithm, the PSLL values are lower and it has better performance.
If the angular range of interest is limited, a lower PSLL will be attainable. However, outside that angular range, the PSLL will be higher. In Table 2, no angular limit has been imposed; therefore, the total angular range of − 90◦ ≤ θ ≤ 90◦ is used resulting in a PSLL of -19.56 dB for a 55-element array. However, for example, if the angular range of interest is limited to − 60◦ ≤ θ ≤ 60◦, the PSLL of -21.28 dB is accessible for the same array. The patterns of the mentioned 55-element array obtained from the proposed method with angular ranges of − 90◦ ≤ θ ≤ 90◦ and − 60◦ ≤ θ ≤ 60◦ are illustrated in Figs. 1and Fig. 2, respectively.
Fig. 1.
Pattern of the array with 55 element obtained from the proposed algorithm for 
.
Fig. 2.
Pattern of the array with 55 element obtained from the proposed algorithm for − 60◦ ≤ θ ≤ 60◦.
It is worth noting that in the patterns of Figs. 1 and 2 the maximum value of pattern is set to 0 dB. To measure the array efficiency, the actual pattern without normalization should be investigated. The actual patterns of the array with 55 elements in the proposed algorithm and uniform excitation method are compared in Fig. 3. It can be observed that at the peak of the pattern, there is a 1.6 dB loss of array efficiency in the proposed method.
Fig. 3.
Actual patterns of the array with 55 elements obtained from the proposed algorithm and uniform excitation method.
Constrained problems
Now, we consider the two-objective problems of minimizing PSLL and maximizing array efficiency, focusing on the trade-off between PSLL and array efficiency (problems (5) and (6)). Controlling the trade-off between array efficiency and PSLL using an optimization algorithm has not been studied in the earlier phase-only pattern synthesis methods. Nonetheless, the array efficiency values in terms of PSLL are provided in some references.
A computer simulation has been performed to evaluate the ability of proposed method for solving two objective constrained problems. The fitness function curve of the proposed method versus the number of iterations in 5 independent experiments for a 55-element array along with the average ofaverage fitness function (over 5 experiments) are shown in Fig. 4.
Fig.4.
The fitness function curve of the proposed method versus the number of iterations in 5 independent experiments for a 55-element array along with the average of 5 experiments.
It can be seen in Fig. 4 that the optimal fitness value obtained form all 5 experiments are very close together and in all experiments the proposed method has been converged to the desired fitness of 1 (equivalent to desired PSLL of -18.5 dB and desired efficiency of %76.5).
Table 3 compares PSLL and array efficiency in the proposed and existing phase only methods for the arrays with 20, 32, 40, 55, 60, 80 and 100 elements.
Table 3.
Comparison of PSLL and array efficiency in the proposed and existing phase-only methods in2,24,31 and3 for linear arrays.
| Number of elements | PSLL(dB) | Array efficiency (%) | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Proposed algorithm | 2 | 24 | 31 | 3 | Proposed algorithm | 2 | 24 | 31 | 3 | |
| 20 | − 16.4 | − 15.8 | − 16.2 | − 16.3 | − 16.1 | 83.7 | 81.8 | 83.5 | 83.2 | 83.6 |
| 32 | − 17.5 | N/A | − 17.3 | − 17.4 | N/A | 79.6 | N/A | 78 | 75 | N/A |
| 40 | − 18, − 18.2* | − 17.9 | − 17.8 | − 18.1 | − 18.1 | 78.4, 76.8* | 76.6 | 78 | 65.9 | 74.7 |
| 55 | − 18.5 | N/A | − 18.3 | − 18.4 | N/A | 76.5 | N/A | 75 | 76.5 | N/A |
| 60 | − 19.1 | N/A | − 18.8 | − 18.8 | N/A | 73.9 | N/A | 71.5 | 71.8 | N/A |
| 80 | − 20.1, − 20.4* | − 20.1 | − 19.8 | − 19.8 | − 20.3 | 70.3, 68.5* | 69.6 | 70 | 70 | 67.3 |
| 100 | − 20.5 | N/A | − 20.1 | − 20.2 | N/A | 69.3 | N/A | 69 | 69.1 | N/A |
In the recent work of3, utilizing a phase-only method for 20, 40 and 80 element arrays yielded PSLLs of − 16.1 dB, − 18.1 dB and − 20.3 dB and array efficiencies of 83.6%, 74.7% and 67.3%, respectively. In the all cases the proposed method has better performance in terms of PSLL and array efficiency, as shown in Table 3.
In Fig. 5, the running time of the proposed method for different numbers of elements in the linear array case is compared to that of the method in31. The results show that while the proposed method performs better in terms of array efficiency and PSLL, it is approximately 3 times faster than the method in31. The optimal values of phases in the proposed phase-only constrained method are shown in Table 8 in the appendix.
Fig. 5.
The running time of the proposed method for different number of elements in linear array case is compared to that of31.
Table 8.
Optimal phases in the proposed constrained method for ULA.
| Number of elements | PSLL (dB) | Array Efficiency (%) | Optimal Phases (degrees) |
|---|---|---|---|
| 20 | − 16.4 | 83.7 | − 24.8588, − 6.6228, − 7.4335, − 31.9314, − 24.0413, − 26.1811, − 22.3884, − 20.4725, − 19.7522, − 20.5586, − 19.8178, − 23.0303, − 22.7946, − 22.9210, − 28.9167, − 15.3558, − 66.0040, 83.1013, − 41.8581, − 20.3574 |
| 32 | − 17.5 | 79.6 | 13.6724 11.2012 51.5200 − 5.5295 9.9549 25.7103 16.2577 |
| 18.1008 13.7233 5.4027 15.0562 14.9322 15.4615 13.6107 | |||
| 15.1285 14.5149 13.7029 12.9306 11.4028 14.8793 10.4501 | |||
| 13.8682 4.2677 − 13.7997 20.0500 65.1685 − 23.6228 − 153.2438 3.9359 | |||
| 64.2857 − 9.4404 1.4927 | |||
| 40 | − 18 | 78.4 | − 4.8670 − 7.7456 22.0424 − 17.0480 125.1168 − 20.7738 22.6067 − 95.5491 |
| 9.5040 − 2.6403 − 12.3532 − 3.6888 0.3553 − 1.1929 − 1.3913 | |||
| − 2.0619 − 2.0915 − 1.5963 − 1.7100 − 1.5993 − 1.1579 − 0.8120 − 1.1047 | |||
| 0.4566 − 0.6153 − 1.6809 − 2.4540 − 2.3546 − 1.6942 − 6.1314 − 12.9774 | |||
| 24.5633 − 36.1617 − 24.4359 − 6.2592 90.1781 24.6139 − 27.2979 | |||
| 3.9369 1.1417 | |||
| 40 | − 18.2 | 76.8 | 27.8023 27.1795 − 23.7205 4.7897 66.6950 20.8970 4.0606 |
| 1.6591 19.1046 10.9602 7.8088 14.8660 20.2161 8.1876 12.0292 | |||
| 12.6041 13.0776 15.1184 14.2623 14.8830 12.4215 11.7655 | |||
| 11.6204 12.2253 9.6600 12.1702 11.4945 4.4463 − 1.8040 | |||
| 35.5443 28.8056 − 55.3921 − 8.6782 110.0844 − 12.2587 158.7702 10.9382 | |||
| − 31.7800 22.5453 16.0898 | |||
| 55 | − 18.5 | 76.5 | 10.1315 11.7492 15.5628 15.1275 12.5517 154.0263 85.7302 |
| 7.9954 8.2542 − 24.6620 − 56.4763 58.7223 31.1263 − 17.3263 12.8676 | |||
| 17.3289 15.4518 14.2995 14.7094 15.3076 15.6490 17.3099 | |||
| 16.9949 17.1046 17.7305 16.7863 18.1744 17.9513 17.8790 | |||
| 17.8864 17.0367 17.0949 17.3512 17.5364 16.8852 17.9215 | |||
| 16.7386 16.6786 19.2125 16.1936 20.1939 12.7387 − 8.9135 | |||
| 15.3504 − 9.0352 75.9608 − 76.8760 17.6529 56.2164 48.6290 | |||
| 86.8427 − 27.0787 15.9515 19.6171 21.1601 | |||
| 60 | − 19.1 | 73.9 | − 9.5847 − 0.6877 12.8907 4.8500 54.2949 − 92.3572 − 1.0863 2.5030 |
| − 43.8763 10.2809 1.2472 15.0932 30.4699 − 21.3438 23.3408 | |||
| − 4.1404 6.0298 4.2552 − 1.4919 1.5443 2.8866 0.0391 | |||
| 3.4144 0.3555 1.8684 − 0.9560 − 0.3506 1.5563 1.3763 − 0.0005 | |||
| − 0.2403 0.6925 1.8765 − 0.1513 0.3704 0.3616 2.4835 | |||
| − 0.4953 0.4785 6.3903 − 1.7521 − 2.5482 − 1.2371 5.2942 − 6.3559 | |||
| 57.0342 − 50.2530 35.0496 5.2522 122.7090 − 75.2419 11.3040 | |||
| − 29.0700 − 80.1062 140.7179 33.4696 5.8805 − 11.7413 7.1084 | |||
| − 13.7928 | |||
| 80 | − 20.1 | 70.3 | 9.4366 1.9554 16.6336 − 23.9351 37.1323 − 185.3632 70.0856 − 15.6030 |
| 88.9039 0.6034 − 68.9567 − 3.2517 5.6274 76.3804 1.1938 | |||
| − 76.1708 66.6611 45.9889 − 5.2578 − 58.8772 − 2.1198 − 12.1059 | |||
| 9.1228 − 8.0767 1.5468 9.4855 7.1655 3.6626 − 0.1431 − 1.0658 | |||
| 5.6335 5.4520 4.3912 4.5418 5.0033 6.4204 2.6368 6.9261 | |||
| 4.9601 6.2299 4.6433 6.3080 5.0867 5.2866 7.0264 | |||
| 3.5808 3.0005 3.2879 4.5448 5.3959 4.7461 4.3396 − 5.7601 | |||
| − 0.3672 − 12.0448 − 6.1589 − 2.2059 − 5.3141 − 20.2521 20.8119 | |||
| − 10.5994 − 1.2253 28.5970 − 13.2249 − 16.0968 − 20.1421 38.2742 − 90.9967 | |||
| 41.2051 142.8076 − 45.9743 2.2793 12.0838 6.8881 84.9760 | |||
| 24.8125 0.2398 − 46.0964 1.4514 14.7970 | |||
| 80 | − 20.4 | 68.5 | 2.8051 − 12.5093 − 22.7639 − 29.2026 7.6882 − 216.3533 77.3574 − 15.5567 |
| 41.2220 − 27.5219− 107.2904 − 19.4108 − 21.7650 48.3829 − 10.8906 − 91.1938 | |||
| 35.5770 3.2201 − 17.2623 − 79.2204 − 16.3037 − 43.8914 − 0.6375 | |||
| − 35.0562 − 17.3244 − 31.7968 − 12.3097 − 10.6548 − 16.9128 − 26.7804 − 21.6513 | |||
| − 6.3224 − 17.7784 − 22.3498 − 11.7392 − 15.8523 − 20.4484 − 11.6260 | |||
| − 11.6536 − 21.8748 − 17.1412 − 12.1090 − 19.1501 − 22.2460 − 10.6431 − 18.0303 | |||
| − 24.0057 − 8.3545 − 17.5141 − 17.1744 − 18.2583 − 14.1863 − 25.3583 | |||
| − 21.1249 − 39.1708 -23.9032 − 29.2023 − 16.3615 − 46.1865 − 4.9810 − 32.4408 | |||
| − 42.2153 20.4199 − 20.7260 − 49.2116 − 49.0740 42.6055 − 86.5479 | |||
| 10.5483 157.9415 − 71.0916 3.0281 4.6102 − 6.5496 89.4796 − 2.7212 | |||
| − 19.5146 − 64.9704 − 7.9257 6.2364 | |||
| 100 | − 20.5 | 69.3 | 1.8961 − 19.0905 − 11.7781 − 69.7113 − 29.9559 31.9873 0.0873 − 25.0065 |
| 67.2924 − 120.8877 1.1656 − 65.4002 31.3235 129.7566 30.8583 | |||
| − 10.9999 − 72.3425 − 64.5341 − 2.4003 51.1364 − 32.4865 − 2.0318 − 35.5397 | |||
| − 61.3743 19.1353 − 21.6075 − 3.8278 3.8755 − 20.2514 8.9703 | |||
| − 22.7312 − 4.6675 − 10.5985 − 15.3683 − 10.4813 − 13.4286 − 11.4882 − 16.4866 | |||
| − 12.1547 − 12.9931 − 10.7867 − 14.1365 − 13.9770 − 8.4729 − 12.0589 | |||
| − 9.3851 − 12.6762 − 12.9640 − 7.9320 − 10.0751 − 11.9950 − 11.9977 − 9.4670 | |||
| − 13.6854 − 12.4293 − 12.4875 − 14.7931 − 9.6206 − 12.1037 − 14.6904 − 10.8928 | |||
| − 11.4533 − 12.4743 − 7.2255 − 13.5450 − 11.2667 − 12.2336 − 14.8820 − 14.8811 | |||
| − 10.4536 0.5370 − 14.9129 − 22.3904 15.9118 − 49.0969 28.3081 | |||
| − 14.7698 − 39.6365 30.9081 − 55.0809 − 69.8388 39.5173 28.3357 − 125.2897 | |||
| − 11.2007 − 12.8505 30.3703 − 12.1135 − 62.6021 − 16.7346 15.0764 | |||
| 108.3472 222.1814 74.1363 − 13.4635 − 43.5210 − 18.9845 − 16.9008 | |||
| − 18.0694 − 14.9894 |
It should be noted that in the proposed algorithm, while an individual or its self-reproduction child obtained from SQP is the best solution, this individual or its child remains in the population and produces a new child via SQP. We will analyze the complexity based on the total number of SQP iterations executed for an individual and its offspring until reaching the final optimal solution.
The computational complexity of GA is O(
) where 
is the number of generations, 
is the population size and 
is the fitness evaluation complexity17–19. Assume that 
is the average number of locally convex regions with a local minimum that should be explored using SQP to find the desired solution and 
is the number of iterations required for finding the best local minimum (desired solution) by exploring its locally convex region using SQP. Let 
where 
is the total number of iterations of running SQP in the whole algorithm. Some parameters have different values in the proposed algorithm compared to the algorithm in31. We will show these parameters in the proposed algorithm and that of31 by indices 1 and 2, respectively. The complexity of proposed algorithm will be O(
) = O(
) where 
is the complexity of SQP per iteration. The second term in this complexity is not multiplicative to first term since in 
the total iterations of SQP in all generations are taken into account. The complexity effect of 
in the proposed algorithm is included in 
. Similarly, the complexity of algorithm of31 is O(
) = O(
). The simulation results in Fig. 5 show that the proposed algorithm has an approximately 3 times lower run time compared to that of31. The cause of this computational efficiency can be explained as follows.
In the algorithm of31
is approximately equal to 
and all the local regions are explored with the same average number of iterations. However, in the proposed method, the local regions that should be explored using SQP are around individuals which are selected based on their fitness values in GA. Meanwhile, in the next iterations many individuals (and their offspring) with worse fitness values compared to the new generation are removed from selected individuals’ group without running more SQP iterations around them and their offspring. This leads to lower number of iterations per local search and consequently lower value of 
compared to 
. Furthermore, the proposed algorithm runs SQP in each generation of GA while the algorithm of31 runs GA and SQP separately. Consequently, the proposed method needs lower number of generations to converge (i.e. 
).
Comparison to amplitude tapering windows
For classical amplitude tapering windows such as Hamming, Chebyshev, Taylor, and Kaiser, the compromise between beamwidth and PSLL is a well-known subject. Here, we will compare the performance of classical amplitude windows for establishing this trade-off with that of the proposed phase-only algorithm.
The most known windows, like Blackman, Hamming, Hann and Gaussian do not allow adjusting the PSLL and have a big beamwidth increasing coefficient (typically more than 1.5), resulting in low array efficiency, usually less than 67%. Tukey window array efficiency is also relatively low. A 55-element Tukey window with 
parameter equal to 0.77 has the PSLL of − 20 dB but its efficiency is the poor value of 36%. Triangular windows with 40 to 60 elements have the PSLL of about − 26.5 dB but also have relatively low array efficiency of about 26%.
The Chebyshev window achieves the minimum PSLL for a specific beamwidth. However, its array efficiency is relatively poor. For instance, a 60-element ULA with a PSLL of − 20dB using the Chebyshev window would have a very low array efficiency of 9.5%.
Taylor window approximates Chebyshev and is known to have better performance in terms of array efficiency. However, in many cases, it may still exhibit relatively low array efficiency. For instance, when considering a 60-element ULA with a PSLL of -15.5 dB, the Taylor window yields an array efficiency of approximately 48%, which is 25% lower than that of proposed algorithm while both have the same beamwidth and PSLL.
Kaiser window aims to concentrate maximum amount of energy in mainlobe39. It controls the PSLL through a parameter called α. Figure 6 shows a comparison of the actual pattern of a 55-element ULA using the Kaiser window and the uniform excitation case.
Fig. 6.
Pattern of 55 elements ULA for Kaiser amplitude tapering window and uniform excitation case.
Table 4 presents a comparison of the trade-off between PSLL, array efficiency, and beamwidth in the proposed algorithm and the Kaiser window for a 55-element ULA. It is important to note that both methods have the same PSLL values in each case. The results in Table 4 show that the proposed algorithm has higher array efficiency and lower beamwidth increasing factor in all cases, while both methods have the same PSLL values.
Table 4.
Comparison of the proposed algorithm and Kaiser window in PSLL, array efficiency, and beamwidth of 55-element ULA.
| PSLL (dB) | Array efficiency (%) | Beamwidth increase coefficient | ||
|---|---|---|---|---|
| Both methods | Proposed method | Kaiser window | Proposed method | Kaiser window |
| − 15.5 | 88.4 | 79.4 | 1.01 | 1.06 |
| − 16.5 | 84.6 | 73 | 1.03 | 1.09 |
| − 17.5 | 80.3 | 67.6 | 1.04 | 1.11 |
| − 18.4 | 76.7 | 63.1 | 1.06 | 1.13 |
| − 19.1 | 69.2 | 60.5 | 1.09 | 1.14 |
Two-dimensional arrays
For two-dimensional planar arrays, the proposed method demonstrates a clear improvement compared to the existing phase-only methods. For a 225 elements 15 × 15 square planar array with uniform half wavelength spacing between the elements, the proposed method achieves a PSLL of -20 dB in the whole angular range (
) with a beamwidth of 
. The array pattern in terms of 
and 
is shown in Fig. 7.
Fig. 7.
3-Dimensional radiation pattern in terms of 
and 
for two-dimensional square planar array with 15 × 15 elements.
Many of the existing phase-only beamforming references are only applicable to linear arrays and have not been extended for two-dimensional arrays. Many others did not investigate the array efficiency and even did not provide the beamwidth and only a few phase-only beamforming references like2 and40 have reported array efficiency or beamwidth data.
In the phase-only beamforming method of40, for a 15 × 15 element square array, PSLLs of − 17 dB is obtained for beamwidth of 12°. In contrast, the proposed method achieves PSLLs of − 23.8 dB for the narrower beamwidth of 8.2° (higher efficiency) as shown in Table 5. This result shows significant improvement of the proposed method compared to the IWO/CS and IWO/WDO methods in40. The optimal values of phases in the proposed method for two-dimensional square planar array are shown in Table 9 in the appendix.
Table 5.
Comparison of PSLL in the proposed algorithm and the existing phase-only algorithm in40 for 15 by 15 square planar array.
Table 9.
Optimal phases in the proposed method for square planar array.
| Number of elements | PSLL (dB) | Beamwidth (degree) | Optimal Phases (degrees) |
|---|---|---|---|
| 225 | − 23.8 | 8.2 | 339.0305 170.2628 1.8099 88.9347 298.1329 87.7712 27.1409 |
| 25.1241 99.8459 − 23.1229 84.0233 − 32.4611 − 244.1787 − 48.2095 | |||
| − 199.7066 − 244.1366 − 353.9717 157.0237 − 28.5082 115.1922 − 38.4661 | |||
| 34.4525 35.5361 − 13.3033 98.9503 − 29.7128 − 261.0775 − 39.3374 120.6714 − 15.8795 | |||
| 20.1213 − 131.5961 18.0107 111.6432 − 37.6578 78.8789 − 36.4486 | |||
| − 16.3109 82.0306 − 30.7271 116.0437 − 25.7790 90.7252 − 31.9539 | |||
| 92.3770 264.7511 62.7104 − 70.1290 15.1149 78.5877 4.4163 | |||
| 64.9132 43.0426 3.8207 71.5974 − 22.1724 74.9273 − 42.6456 | |||
| 100.1849 − 30.9853 83.0225 − 56.6150 75.5531 − 32.8004 32.4799 | |||
| 39.8395 30.8370 22.4577 28.5080 29.1342 37.4947 10.5500 | |||
| 76.2761 − 45.3844 95.7466 − 23.8324 81.3605 − 23.7077 64.7833 | |||
| 26.2704 16.6441 24.5708 19.7090 22.1758 29.9294 13.8715 | |||
| 44.5818 − 13.2544 74.4368 − 45.1068 99.5391 − 42.9025 73.8206 | |||
| 7.3405 30.7568 24.3600 24.4129 29.1918 23.2068 26.1597 | |||
| 26.8685 6.5135 70.4134 − 24.5777 81.1116 − 27.5568 74.5147 | |||
| − 25.2490 35.8769 26.5808 21.9687 21.5939 30.2137 32.6402 | |||
| 25.0548 27.5173 21.0038 46.2767 65.6191 − 27.5442 57.2127 | |||
| − 30.0514 80.1507 6.5394 11.5691 26.9612 23.1092 26.8561 | |||
| 26.3370 24.6752 32.7916 66.2616 − 24.6067 57.4120 28.9751 | |||
| − 28.8502 95.9550 − 28.3753 24.9283 23.4946 31.5107 28.6250 | |||
| 31.1377 13.2957 28.2167 19.8368 − 7.1870 103.4208 − 13.7143 | |||
| 66.2441 102.9891 − 43.9957 70.1658 − 2.0442 12.0940 31.5321 | |||
| 9.9428 35.1253 29.8210 38.3111 − 14.3477 76.5755 307.8207 | |||
| 76.5219 − 26.4180 − 31.0453 82.9306 319.5188 35.0345 127.4324 | |||
| 3.3798 36.9973 5.3960 31.0586 32.5147 − 323.9153 − 69.6100 | |||
| 70.6359 291.1382 72.4186 77.0693 − 47.1253 88.3134 250.3817 | |||
| 23.4214 24.6826 27.3608 0.1515 29.1192 − 0.0709 106.5132 | |||
| 41.3240 − 54.3267 − 324.2389 − 173.0484 295.9477 94.2269 302.1748 | |||
| 88.0963 − 41.5171 69.4337 − 24.3271 81.6770 − 31.6122 109.7867 | |||
| − 38.0105 108.6486 5.5557 − 225.3042 − 42.3637 108.2321 − 43.9919 | |||
| 88.1371 − 28.5988 64.2655 − 28.1408 97.6060 − 48.0868 105.2211 | |||
| − 31.1287 91.6636 − 49.9399 114.1035 − 25.2388 125.3260 |
Additionally, the proposed method exhibits significant improvement compared to the well-known SDM phase-only beamforming algorithm2 for circular planar arrays (see Table 6). Importantly, in the case of 32-element array, the proposed method has 8.8% higher efficiency and 1.7 dB lower PSLL. Furthermore, the efficiency of the 172-element circular array in the proposed algorithm is 21.8% higher than that of2, while its PSLL is 1.5 dB lower. The optimal values of phases in the proposed method for circular planar array are shown in Table 10 in the appendix.
Table 6.
Comparison of PSLL and array efficiency in the proposed algorithm and the existing phase-only method in2 for circular planar arrays.
Table 10.
Optimal phases in the proposed method for circular array.
| Number of elements | PSLL (dB) | Array Efficiency (%) | Optimal Phases (degrees) |
|---|---|---|---|
| 32 | − 16 | 92.1 | − 6.3358 − 0.6673 0.5816 5.6904 − 19.3994 − 4.8496 − 0.2403 − 1.1372 |
| − 7.3567 33.8544 − 0.7581 − 0.2646 − 2.8352 0.1283 3.0819 | |||
| 0.9189 − 4.0520 − 1.5509 1.2612 1.6312 0.7655 − 2.9204 | |||
| 15.0816 − 16.7594 0.2150 − 0.3048 42.6046 − 40.2285 34.3615 | |||
| 0.7288 − 1.3196 − 40.1034 | |||
| 172 | − 19.6 | 79.8 | 18.2627 18.9534 17.4628 16.2152 15.4324 16.0562 20.8207 |
| 16.9858 19.7201 23.5777 33.3875 3.9362 32.3415 − 13.4146 | |||
| 103.7866 9.3656 13.3036 17.8497 17.6010 23.0910 − 84.0774 | |||
| − 27.1522 23.1296 17.1294 14.3780 16.6201 − 29.0333 69.2431 | |||
| 18.3428 17.9206 17.4781 13.4803 134.9294 17.3272 19.6787 | |||
| 17.4166 17.1687 17.5943 16.9712 17.2792 15.8084 − 144.1338 | |||
| 20.6061 17.2136 17.2426 8.3859 10.8018 16.0591 17.1668 | |||
| 17.6752 16.5485 17.9413 17.9575 17.7791 18.4530 12.1795 | |||
| 9.6031 17.0736 16.8365 − 47.8468 17.2162 20.2234 18.6384 17.4369 | |||
| 15.9907 17.4225 18.5302 18.1226 16.6148 17.9063 95.2656 | |||
| 17.9549 19.4484 − 12.3099 17.6338 17.2152 18.2292 17.7209 | |||
| 18.1831 17.8607 16.6522 17.7496 19.3678 19.5056 26.8273 | |||
| 17.4867 20.1503 40.1348 18.3217 17.0312 16.0078 17.4482 | |||
| 17.4780 17.6095 18.4328 17.7156 17.9994 19.0550 40.6312 | |||
| 19.3211 17.0450 34.4056 15.3217 16.4507 17.0147 17.6671 | |||
| 16.9536 17.7507 16.9917 17.8161 18.1650 22.4669 − 48.4118 | |||
| 17.4391 17.4886 10.5693 − 10.6604 19.8226 16.5583 16.3757 | |||
| 16.7031 18.0646 17.9822 17.0889 18.3889 − 20.4233 41.8196 | |||
| 18.9771 18.8858 17.5823 178.8835 7.0032 19.7831 17.3957 | |||
| 17.4932 17.3220 17.0300 17.3054 20.6278 126.8189 24.1729 | |||
| 19.4433 17.6861 14.9131− 155.2112 − 9.7491 16.7096 16.6147 16.0848 | |||
| 14.3889 84.8555 − 129.2577 18.3988 18.2219 15.7017 17.2565 | |||
| 45.3153 55.4613 6.8145 − 15.5587 60.4676 − 8.5695 16.3804 | |||
| 17.3474 18.4338 18.8267 18.1957 18.9891 19.4745 17.6300 | |||
| 17.0221 17.4274 |
The 3-dimensional pattern of proposed method for 172 element circular array in terms of 
and 
is shown in Fig. 8.
Fig. 8.
3-Dimensional radiation pattern in terms of 
and 
for two-dimensional circular 172-element planar array.
The one-dimensional actual radiation pattern of 172-element circular array in the proposed algorithm and the uniform excitation method is shown in Fig. 9. It can be deduced from Fig. 9 that in the proposed method, the PSLL is − 20.61 + 0.9788 = − 19.63 dB and the array efficiency is calculated as 10–0.09788 = 79.8%.
Fig. 9.
One-dimensional actual radiation pattern of the 172-element circular planar array in the proposed algorithm and the uniform excitation method.
The cost of increasing the performance and accuracy in the proposed method is the computational complexity. Although the computational complexity is reduced compared to the similar high performance optimization approaches, the proposed method still has more complexity compared to some fast algorithms like41 that sacrifice the performance for speed.
Conclusion
A multi-objective optimization algorithm was proposed for phase-only pattern synthesis of arbitrary array antennas. The proposed algorithm was shown to be effective for solving multi-objective problems of minimizing PSLL and maximizing array efficiency subject to an extra constraint to establish trade-off between the pattern parameters. We demonstrated that the performance of proposed algorithm performs better than the earlier optimization algorithms of phase-only pattern synthesis in terms of PSLL and array efficiency. The proposed algorithm can control the trade-off between PSLL, beamwidth and array efficiency systematically where the PSLL or array efficiency is set to the given value.
Also, in comparison to the well-known amplitude tapering windows, the proposed algorithm performs better in establishing the above-mentioned trade-off. The simulation results confirmed the advantages of the proposed phase-only pattern synthesis algorithm.
For two-dimensional planar arrays, the advantages of the proposed algorithm compared to the existing phase-only references is more obvious. For planar circular arrays, a significant improvement of 8.8% to 21.8% is achieved in terms of array efficiency while the PSLL is improved about 1.5 dB at the same time. For square planar arrays, a PSLL enhancement of at least 6.8 dB is achieved compared to the existing phase-only approach while obtaining narrower beamwidth.
Supplementary Information
Below is the link to the electronic supplementary material.
Appendix 1
Author contributions
M. Hatam, M. Mohammadpour and S. Abdi have collaborated in writing the manuscript text, analyzing the results and reviewing the text. The computer simulations are provided by M. Hatam.
Funding
There was no funding for this research.
Data availability
All data generated or analysed during this study are included in this published article, and its supplementary information files.
Declarations
Competing Interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Data Availability Statement
All data generated or analysed during this study are included in this published article, and its supplementary information files.