In monopulse radar antennas, the synthesizing process of the sum and difference patterns must be fast enough to achieve good tracking of the targets. At the same time, the feed networks of such antennas must be as simple as possible for efficient implementation. To achieve these two goals, an iterative fast Fourier transform (FFT) algorithm is used to synthesize sum and difference patterns with the main focus on obtaining a maximum allowable sharing percentage in the element excitations. The synthesizing process involves iterative calculations of FFT and its inverse transformations; that is, starting from an initial excitation, the successive improved radiation pattern and its corresponding modified element excitations can be found repeatedly until the required radiation pattern is reached. Here, the constraints are incorporated in both the array factor domain and the element excitation domain. By enforcing some constraints on the element excitations during the synthesizing process, the described method provides a significant reduction in the complexity of the feeding network while achieving the required sum and difference patterns. Unlike the standard optimization approaches such as genetic algorithm (GA), the described algorithm performs repeatedly deterministic transformations on the initial field until the prescribed requirements are satisfied. This property makes the proposed synthesizing method converge much faster than GA.
Conventional approaches for synthesizing sum and difference patterns require the use of two separate element excitations for one monopulse radar antenna, for example, Taylor excitation [
Currently, this drawback of the conventional approaches can be overcome by using subarrays [
In [
This paper introduces a simple and fast algorithm for synthesizing sum and difference patterns with a number of common element excitations for the purpose of reducing the feeding network. For active antenna arrays, the element excitations can be implemented by using a number of digital attenuators and phase shifters. More number of elements results in a complex feeding network. Thus, higher sharing percentage in the element excitations results in lower complexity and cost. By using an iterative fast Fourier transform (FFT) with a specific constraints on the required element excitations and their corresponding radiation patterns, it is possible to reach a good compromise among the sum/difference patterns quality and the complexity of the required feeding networks. Unlike the current optimization approaches, the proposed method may be regarded as a deterministic method. Therefore, it delivers results much faster as compared to standard optimization approaches.
Consider an array of an even number of isotropic elements
It can be seen from these two equations that the element excitations
As we mentioned earlier, for linear arrays with equally spaced elements, the radiation pattern and its corresponding element excitation are related together by an inverse Fourier transformation. The algorithm starts by updating the sum and difference patterns of the corresponding initial excitations in an iterative manner using 4096point inverse FFT. During each update, only the sidelobe values that exceed the prescribed sidelobe are modified, and the other sidelobe values are left unchanged. After this modification, a direct 4096point FFT is performed on the adapted sum and difference patterns to get two new sets of the excitation coefficients
The constraints on the element excitations are performed by replacing the values of a number of the element excitations of the synthesized sum pattern to be equal to that of the synthesized difference pattern.
It is worth mentioning that the constraints are applied to a certain number of element excitations of the corresponding sum pattern (this means that the rest of the element excitations of the synthesized difference pattern remains unchanged). Finally and after applying these element excitation constraints, the fast Fourier transform is performed to get new sum and difference patterns.
To validate the effectiveness and the convergence speed of the described method, a number of numerical experiments have been performed on a 2.4 GHz Laptop equipped with a 4 GB of RAM. In the following examples, the synthesis of equally spaced linear arrays composed of
As a first example, the iterative FFT algorithm is used to synthesize sum and difference patterns to reach a prescribed sidelobe requirement. In this case, the required sidelobe level of both patterns is chosen to be −24 dB. Note that, in this example, none of the constraints have been applied to the element excitation, that is,
Sum and difference patterns (a) and the corresponding amplitude excitations (b) for the case of no common excitation and
In the second example, both radiation patterns are resynthesized subject to twelve common amplitude excitations (i.e., 60% of the overall number of array elements is shared for synthesizing sum and difference patterns). The resulting element excitations and the corresponding array patterns are shown in Figure
Sum and difference patterns (a) and the corresponding amplitude excitation (b) for the case of fixed difference pattern, 60% common excitation, and
The sidelobe level (SLL), beam width (BW), taper efficiency, and runtime for these aforementioned two cases (i.e., no common excitations and 60% common excitations) and for other cases that use different percentage of sharing excitations are reported in Table
Patterns features when
Difference pattern  Sum pattern  Sum pattern with common excitations constraints  

No common excitations 0%  40%  50%  60%  
SLL [dB]  −24.0  −24.0  −18.0  −16.32  −15.32 
BW [deg]  22.8301  14.8328  13.261  13.014  12.920 
Taper efficiency  0.5716  0.9288  0.9580  0.9572  0.9587 
Runtime [second]  0.012129  0.008816  0.1209  0.1127  0.1212 

0.8573  —  —  —  — 
It is worth mentioning that the taper efficiency is defined as follows:
From this table, it can be remarked that the greater the percentage of sharing excitations, the poorer the SLLs of the corresponding sum patterns. Nevertheless, the taper efficiency is slightly increased, and the beam width is significantly decreased when increasing the percentage of sharing excitations.
As mentioned in the previous example, the excitations of the difference pattern were fixed, whereas the excitations of the sum pattern are constraints to 60% of common excitations. Accordingly, the SLL in the resulting sum pattern was increased. However, this pattern is crucial in the radar applications. Thus, it is desirable to keep its peak sidelobe at low level. By using the proposed method, the excitations of the sum pattern can be easily fixed while those for the difference pattern can be constraints. As with the previous example, 60% of common excitations are used. The resulting element excitations and the corresponding array patterns are shown in Figure
Sum and difference patterns (a) and the corresponding amplitude excitation (b) for the case of fixed sum pattern, 60% common excitation, and
In the next example, the convergence speed of the iterative FFT algorithm is compared to the standard GA. For fair comparison, it is assumed that the initial excitations or populations for both algorithms are uniform. In addition, the updating processes of these two algorithms are terminated immediately after reaching the allowed number of iterations or prescribed SLL value. In Figure
Variation of the maximum sidelobe level versus the number of iterations for the sum pattern.
In the last example, the synthesis of sum and difference patterns with required
Sum and difference patterns (a) and the corresponding amplitude excitation (b) for the case of no common excitation and
Sum and difference patterns (a) and the corresponding amplitude excitation (b) for the case of fixed sum pattern, 50% common excitation, and
The feeding network complexity and cost are the main challenging issues in practical implementation of the tracking array antennas. Since the amplitudeonly control of the element excitation is considered in this study, therefore the complexity and cost in terms of required numbers of amplifiers or attenuators are considered.
The required number of attenuators needed in implementing the feeding network of the sum and difference patterns under different percentage of sharing excitations is reported in Table
Complexity of the feeding network for different cases.
Array patterns  Feeding network complexity  

Separate excitation without sharing, 0%  Sharing percentage, 20%  Sharing percentage, 50%  Sharing percentage, 80%  
Number of attenuators  Peak SLL (dB)  Number of attenuators  Peak SLL (dB)  Number of attenuators  Peak SLL (dB)  Number of attenuators  Peak SLL (dB)  
Sum pattern 

−24 

−22 

−16 

−15 
Difference pattern 

−24 

−24 

−24 

−24 
Total number of attenuators  2 
—  2 
—  2 
—  2 
— 
It has been shown that it is possible to generate both sum and difference patterns with a certain number of common excitations. The complexity of the feeding network can be significantly reduced with respect to that case of separate (no sharing) excitations. This reduction comes at the cost of changing the peak SLL in the sum pattern or in the difference pattern. The results show that the greater the percentage of sharing excitations, the poorer the SLLs of the corresponding beam patterns. Thus, the designer needs to trade off between the required sidelobe level and the feed network’s complexity.
More importantly, the proposed method converges 100 times faster than the standard GA because the core calculations were based on direct and inverse fast Fourier transforms.
The method can be easily extended to synthesize other patterns such as flattop beam and pencil beam with common element excitations. Moreover, the principle of the proposed approach may be also extended to the planar arrays having different shapes such as rectangular and circular with uniform element spacing.
The author declares that they have no conflicts of interest.