Analysis, Design, and Experimental Validation of a High-Isolation, Low-Cross-Polarization Antenna Array Demonstrator for Software-Defined-Radar Applications

Abstract

In a software-defined radar (SDR) system, most of the signal processing usually implemented in hardware is implemented by software, thus allowing for higher flexibility and modularity compared to conventional radar systems. However, the majority of SDR demonstrators and proofs of concept reported in the open literature so far have been based on simple antenna systems. As a result, the full potentialities of an SDR approach have not been completely exploited yet. In this work, we propose a flexible antenna module to be integrated into an active electronically scanning array (AESA) with controlled sidelobe level over a wide angular range, exhibiting polarization reconfigurability with a low cross-polarization level and high isolation. For this purpose, analytical and numerically efficient techniques for the synthesis of the aperture distribution and the correct evaluation of the radiating features (e.g., beamwidth, pointing angle, sidelobe levels, etc.) are presented in order to grant real-time control of the digital beamforming network. A sub-array module demonstrator is fabricated and measured to corroborate the concept.

1. Introduction

In the last few years, the attention on software-defined radar (SDR) has grown exponentially. As a matter of fact, this approach offers a series of advantages that goes from digital beamforming to hardware flexibility [1]. The possibility to drive an active electronically scanning array (AESA) with an SDR platform [2,3,4,5] would pave the way for a fully digital radio-frequency (RF) front-end with unprecedented capabilities in terms of reconfigurability, re-use of hardware, and modularity.

Recently, SDR platforms have been configured and tested to drive passive bistatic radar (PBR) systems based on linear arrays to perform detection and tracking functions. These works [6,7,8,9] were mainly focused on the software implementation of the calibration network and of the digital signal processing (DSP) architecture. However, in order to fully exploit the potentialities of the SDR approach, design flexibility has to be ensured at the hardware level. Antenna elements that allow for dynamic polarization and beam steering are thus required, especially when used in large planar AESA, to reach the high gain and high coverage requirements of radar tracking systems.

In order to respond to the needs of design flexibility and high performance, inspired by the recent works on multifunction phased array radar (MPAR) [10,11,12], we propose here an attractive unit cell fed with two ports that can be independently controlled in amplitude and phase, and that is capable of providing low cross-polarization and high isolation levels without using additional techniques, such as decoupling networks. In order to corroborate the antenna performance, a 4 × 8 array demonstrator is fabricated whereas the gain and reflection coefficient of a 4 × 6 array are measured, showing good agreement with the expected theoretical results for the single radiating element in a periodic environment.

This preliminary demonstration paves the way for targeting the development of larger AESA panels, thanks to the modularity of the proposed sub-array architecture. Specifically, we will also discuss here the system requirements of a 40 × 40 planar AESA capable of scanning a very directive beam over a wide angular range with controlled sidelobes in C band. Efficient analytical and numerical methods for the realization of a Tseng–Cheng beamforming network [13] and the evaluation of the main radiating properties are proposed to easily interface the AESA with an SDR platform. This latter investigation reveals that the proposed demonstrator looks suitable for achieving the ambitious system requirements of the 40 × 40 AESA.

The paper is organized as follows. In Section 2, we describe the targeted antenna performance of the AESA and present analytical methods for both the design of the digital beamformer and the evaluation of the AESA radiating features. In Section 3, the antenna element is described and designed to ensure polarization flexibility, low cross-polarization, and high isolation. In Section 4, the fabrication of a 4 × 8 prototype and experimental validation of a $4\times 6$ demonstrator is shown. Finally, conclusions are drawn in Section 5.

2. Array Design

In radar applications, some antenna requirements are crucial. In particular, the half-power beamwidth (HPBW) is related to the spatial resolution of the radar, the sidelobe level (SLL) is related to false target detection, and the scan range defines the region of space where the antenna can properly detect a target without a beam-shape degradation. Even the radiating element polarization is crucial, and dual-polarized elements are interesting for multifunctional and reconfigurable systems.

In this work, we aim at investigating a planar $40\times 40$ dual-polarization AESA capable of producing a pencil beam (about $3^{\circ }$ of half-power beamwidth at broadside) over an angular range of ${60}^{\circ }$ on the elevation plane and a full coverage on the azimuthal plane, with a sidelobe level of $-30$ dB to be maintained for the whole scanning range and over the design frequency range, also exhibiting polarization reconfigurability with a low cross-polarization level and high isolation. C-band patch antennas are considered for the design, as shown in Section 3, with a fractional bandwidth of $10\%$ and equally spaced over a rectangular lattice by $d_x=d_y={\lambda }_0/2$. The main frequencies in the design band will be called $f_{min}$ for the lower frequency, $f_{max}$ for the higher frequency, and $f_0$ for the center frequency. In particular, $f_{min}=0.95f_0$ and $f_{max}=1.05f_0$. The latter properties can be demanded by the element factor and will be discussed in Section 3. In the next subsections, we focus on the design of the array factor (AF) with analytical techniques to meet the constraints of having a pencil beam with controlled SLL over the entire scanning range.

2.1. Analytical Techniques

2.1.1. Sidelobe Level

We first examine the abovementioned specifications to find the most suitable technique for the realization of an efficient digital beamformer. The requirement of an invariant SLL considerably restricts the available beamforming techniques. As a matter of fact, apart from optimization algorithms (see, e.g., the excellent reviews in [14,15] and refs. therein), only the Tseng–Cheng technique [13] ensures a constant SLL at arbitrary azimuthal cuts and for any scan angle. Moreover, the Tseng–Cheng technique has the following advantages compared to other techniques: (i) it is fully analytical; (ii) it is computed once, without requiring the amplitude coefficients to be refreshed as the scan angle changes; and (iii) being based on the Dolph–Chebyshev technique, it provides the minimum beamwidth for a given SLL [13]. All these aspects make the Tseng–Cheng technique particularly attractive for its implementation in an SDR-driven beamforming network.

In order to implement the Tseng–Cheng technique, we assume an even number of elements along the two main directions, i.e., $N_x=N_y=M=2N$, and quadrantal symmetry of the amplitude coefficients, i.e., $a_{mn}=a_{-m,n}=a_{m,-n}=a_{-m,-n}$ (the extension to rectangular lattices has been proposed in [16]). Under this hypothesis, the AF can conveniently be recast as [13,17]

$$\mathrm{AF}(\theta ,\varphi )=4\sum _{m=1}^N\sum _{n=1}^N{a}_{mn}cos\left[(2m-1)u\right]cos\left[(2n-1)v\right]$$

(1)

where $u=0.5k{d}_x(sin\theta cos\varphi -sin{\theta }_{0}cos{\varphi }_0)$ and $v=0.5k{d}_y(sin\theta sin\varphi -sin{\theta }_{0}sin{\varphi }_0)$; $d_x\left(d_y\right)$ is the spacing along the $x\left(y\right)$-axis, $\theta$ and $\varphi$ are the elevation and azimuthal angles of the spherical coordinate system, and $k=2\pi /\lambda$ is the free-space wavenumber, with $\lambda$ being the free-space wavelength. As shown in [13], an invariant SLL over the entire angular range can be achieved if the amplitude coefficients are set equal to

$$\begin{array}{cc}\hfill a_{mn}& =\frac{1}{N^2}\sum _{p=1}^N\sum _{q=1}^N{T}_{M-1}\left[w_{0}cos\frac{\pi (p-\frac{1}{2})}{2N}cos\frac{\pi (q-\frac{1}{2})}{2N}\right]\hfill \\ \hfill & cos\left(\frac{\pi (m-\frac{1}{2})(p-\frac{1}{2})}{N}\right)cos\left(\frac{\pi (n-\frac{1}{2})(q-\frac{1}{2})}{N}\right)\hfill \end{array}$$

(2)

where $T_{M-1}[·]$ is the Chebyshev polynomial of the first kind and order $M-1$, $w_0$ is related to the SLL through the expression $w_0=cosh[1/(M-1)\mathrm{arccosh}\left(\mathrm{PSR}\right)]$, where PSR is the peak-to-sidelobe ratio, i.e., $\mathrm{PSR}=\sqrt{1/\mathrm{SLL}}$ (assuming all quantities in linear scale). In order to avoid grating lobes, the usual condition for untapered distributions is modified as follows:

$$\frac{d_{x\left(y\right)}}{\lambda }<\frac{1-[arccos(1/w_0)]/\pi }{1+sin{\theta }_{\max }}$$

(3)

where ${\theta }_{\max }$ is the maximum scan angle. As is clear from expression (2), its straight implementation would require four nested loops to evaluate all the coefficients (viz., two for the m and n elements indices, and two for the p and q summation indices), thus resulting in time-consuming computations. To take advantage of the MATLAB potentialities in handling matrix calculations, it is therefore important to recast (2) to avoid as much as possible the use of nested loops. Indeed, the two summations in (2) can conveniently be expressed in a quadratic form:

$$a_{mn}=N^{-2}{\underline{t}}_p\left(m\right){\underline{\underline{T}}}_{pq}{\underline{t}}_q\left(n\right)$$

(4)

where ${\underline{t}}_p\left(m\right)=cos\left[\pi (m-0.5)(\underline{p}-0.5)\right]/N$, ${\underline{t}}_q\left(n\right)=cos\left[\pi (n-0.5)(\underline{q}-0.5)\right]/N$, and ${\underline{\underline{T}}}_{pq}=T_{2N-1}\{w_{0}cos[\pi ((\underline{p}-0.5)/2N]cos\left[\pi ((\underline{q}-0.5)/2N]\right\}$, with $\underline{p}=1,2\cdots N$ and $\underline{q}=1,2\cdots N$ being row and column vectors of N elements, respectively.

This calculation for the Tseng–Cheng tapering coefficients is very efficient, considering that the MATLAB implementation of (4) considerably reduces the execution time from 18 min to fractions of a second (on a laptop with 32 GB RAM and a 3.5 GHz quad-core processor) for the previous case study.

2.1.2. Beamwidth

As is known, the HPBW (${\Theta }_{x\left(y\right)0}$) at broadside along the principal planes of a uniform planar array is well approximated by the formula [18]:

$${\Theta }_{x\left(y\right)0}=\pi -2arccos\left(\frac{0.886\lambda }{2N_{x\left(y\right)}{d}_{x\left(y\right)}}\right)\simeq \frac{0.886\lambda }{N_{x\left(y\right)}{d}_{x\left(y\right)}}$$

(5)

where the last approximation is obtained in the limit of large arrays (we recall that $arccos\left(x\right)\simeq \pi /2-x$ for $x\to 0$).

As the beam is scanned at a generic angle (${\theta }_0$, ${\varphi }_0$), the beamwidth over the two principal planes changes according to the following equations [18]:

$$\begin{array}{c}\hfill {\Theta }_{\mathrm{h}}={\left(\sqrt{{cos}^2{\theta }_0[{\Theta }_{x0}^{-2}{cos}^2{\varphi }_0+{\Theta }_{y0}^{-2}{sin}^2{\varphi }_0,{\varphi }_0]}\right)}^{-1}\end{array}$$

(6)

$$\begin{array}{c}\hfill {\Psi }_{\mathrm{h}}={\left(\sqrt{{\Theta }_{x0}^{-2}{sin}^2{\varphi }_0+{\Theta }_{y0}^{-2}{cos}^2{\varphi }_0}\right)}^{-1}\end{array}$$

(7)

where ${\Theta }_{\mathrm{h}}$ and ${\Psi }_{\mathrm{h}}$ represent the HPBW on the elevation plane, and its orthogonal plane, respectively. When the array has the same number of elements and the same spacing over the x and y axes (as in the case shown discussed here), ${\Theta }_{x0}={\Theta }_{y0}={\Theta }_{0\mathrm{h}}$, thus ${\Psi }_{\mathrm{h}}={\Theta }_{0\mathrm{h}}$ is constant, whereas ${\Theta }_{\mathrm{h}}={\Theta }_{0\mathrm{h}}sec{\theta }_0$, giving rise to an unavoidable pattern distortion in the scanning towards endfire (viz., for ${\theta }_0={60}^{\circ }$, $sec{\theta }_0=2$, hence ${\Theta }_{\mathrm{h}}$ is doubled). While the numerical evaluation of the beamwidth along the plane $\varphi ={\varphi }_0$ is quite straightforward, a few words should be spent on the evaluation along the orthogonal plane. Indeed, the AF is represented as a function of $\theta$ and $\varphi$, hence the HPBW over the ${\varphi }_0$-plane is simply given by the distance between the $-3$ dB points along $\theta$, while the HPBW over the orthogonal plane needs an explicit definition of the angle $\psi$. This definition can be recovered from simple geometrical considerations and leads to the following expression:

$$\psi =arcsin[sin\tilde{\theta }sin(\tilde{\varphi }-{\varphi }_0)]$$

(8)

where $\tilde{\theta }$ and $\tilde{\varphi }$ are the zeros of the equation $tan\theta cot{\theta }_0=sec(\varphi -{\varphi }_0)$, that is, the equation of the plane orthogonal to the unit vector $\widehat{\theta }$ in (${\theta }_0$, ${\varphi }_0$), i.e., the plane defined by $\widehat{r}·\widehat{\theta }{|}_{{\theta }_0,{\varphi }_0}=0$, where $\widehat{r}$ is the position unit vector. We should note that $\tilde{\theta }$ and $\tilde{\varphi }$ should be sought by means of a root-finding algorithm. Once all the $\tilde{\theta },\tilde{\varphi }$ pairs are found, the AF calculated at ($\tilde{\theta },\tilde{\varphi }$) must be rearranged according to the ordering given by $-\pi /2\le \psi (\tilde{\theta },\tilde{\varphi })\le \pi /2$. Then, the $-3$ dB points along $\psi$ can finally be found.

However, when tapering techniques for SLL reduction are taken, the HPBWs are notably affected and accounted for by the spreading factor $s_{\mathrm{f}}$. In particular, if one applies a Dolph–Chebyshev technique (as that of Tseng–Cheng) to achieve an SLL of $-30$ dB, the $s_{\mathrm{f}}$ is given by [18]

$$s_{\mathrm{f}}=1+0.636{\left\{\frac{2}{\mathrm{PSR}}cosh\left[\sqrt{{\left(\mathrm{arccosh}\mathrm{PSR}\right)}^2-{\pi }^2}\right]\right\}}^2.$$

(9)

Therefore, ${\Theta }_{x0}$ and ${\Theta }_{y0}$, as they appear in (5), have to be multiplied by $s_{\mathrm{f}}$, as given by (9).

2.1.3. Bandwidth

As is well known from array theory [17,19], the array bandwidth is severely limited by the phase shifters unless one resorts to true time-delay devices. The latter usually require expensive and complex architectures that can only be justified when the array needs to ensures an excellent performance over a considerably large bandwidth. Conversely, for relatively small fractional bandwiths as those considered here, phase shifters are usually preferred. As a result, the beam is exactly steered at a point in space $({\theta }_0,{\varphi }_0)$ only at the center frequency $f_0$, i.e., when the u and v terms appearing in (1) are evaluated at $k_0=k\left(f_0\right)$. As a result, for $f\ne f_0$, the beam is squinted by an angular deviation $\Delta \theta$ from the theoretical pointing angle ${\theta }_0$, given by the formula

$$\Delta \theta =|arcsin(sin{\theta }_0{f}_0/f)-{\theta }_0|.$$

(10)

The main consequence of the beam squint is that the $-3$ dB fractional bandwidth $\Delta f/f_0$ is affected. In particular, for a linear array of length L it is determined by the expression [19]

$$\frac{\Delta f}{f_0}=0.886s_{\mathrm{f}}\frac{{\lambda }_0}{Lsin{\theta }_0}\simeq s_{\mathrm{f}}{\Theta }_{0\mathrm{h}}csc{\theta }_0$$

(11)

with ${\lambda }_0$ being the center wavelength. Again, the last expression is obtained in the limit of large arrays using the right-hand side of (5).

2.2. Numerical Results

The analytical techniques presented in the previous paragraphs are exploited here in a proof of concept to test their effectiveness and accuracy in the array synthesis and its performance evaluation.

The proof of concept is chosen to fit the constraints mentioned at the beginning of Section 2, namely, $\mathrm{SLL}<-30$ dB, ${\theta }_{0,max}={60}^{\circ }$, ${\Theta }_{0\mathrm{h}}=3^{\circ }$ over the $f_{min}$–$f_{max}$ range. In order to fulfill the SLL requirement, we have to accept a spreading factor of about $1.3$, but only a slight modification to the condition for avoiding grating lobes ($w_0\simeq 1.0057$). We thus set $d_x=d_y={\lambda }_0/2$, which mostly satisfies (3). With these numbers at hand, it is found from (5) that $N_x=N_y=40$ provides for ${\Theta }_{0\mathrm{h}}$ around $2.5^{\circ }$ that, after taking into account the spreading factor (viz., $s_{\mathrm{f}}\simeq 1.3$), yields $3.3^{\circ }$. Under this condition, the $-3$ dB fractional bandwidth at the maximum scan angle would be around $6.6\%$. Using (11), it is found that the targeted bandwidth performance would be reached by either limiting the maximum scan angle to ${40}^{\circ }$, or broadening the broadside beamwidth to about $4.5^{\circ }$. In the following, we consider the original setting, thus accepting a reduction in the operating bandwidth.

Therefore, we consider a $40\times 40$ planar array with $d/\lambda =0.5$ spacing among the elements and apply the Tseng–Cheng technique to obtain an invariant SLL of $-30$ dB. The distribution of the amplitude coefficients is reported in Figure 1. As shown, the coefficients at the four corners are almost zero, thus the corresponding elements may not be fed, with minor impact on the radiating performance. In particular, in Figure 2a–d a comparison is shown among the normalized power radiation patterns produced with an ideal (i.e., keeping all the coefficients) Tseng–Cheng distribution (see Figure 2a), and those produced by setting to 0 the coefficients below $0.01$ (see Figure 2b), $0.05$ (see Figure 2c), and $0.1$ (see Figure 2d), when the array is scanned at $({\theta }_0={60}^{\circ },{\varphi }_0={45}^{\circ })$. For a threshold of $0.01$ no appreciable difference is seen with the ideal case, but in the former case one would achieve a reduced filling factor of $83\%$. As the threshold raises, the filling factor reduces further (viz., $62\%$ and $43\%$ for thresholds of $0.05$ and $0.1$, respectively), but the SLL increases as well (viz., peaks of $-26$ dB and $-21$ dB for thresholds of $0.05$ and $0.01$, respectively). As a result, further reductions in the number of elements should not be considered as this would require optimization techniques for sparse arrays, an aspect that goes beyond the scope of this work.

The effect of the element reduction is even more appreciable looking at the 1-D normalized power patterns along the two principal planes (solid black and gray dashed lines in Figure 2e–h). It is noted that the patterns along the elevation plane are more affected than those along the orthogonal plane in all cases. Interestingly, the beamwidths are not affected by the element reduction, and their numerical evaluations agree well (a percent error less than $10\%$) with the formulas provided in (5)–(7).

With respect to the bandwidth performance, the beam squint $\Delta \theta$ and the gain loss are evaluated as a function of the frequency in Figure 3a,b. It is seen that the numerical evaluation of the beam squint (solid black line in Figure 3a) compares well with the theoretical prediction of (10) (dashed gray line in Figure 3a), and reveals a beam squint of about $3^{\circ }$ within the $-3$ dB bandwidth shown in Figure 3b. The beamwidths along the principal planes and as a function of frequency are also shown in Figure 4, where their numerical evaluation is compared against the analytical expressions provided in Equations (6) and (7), accounting for the spreading factor. As commented before, the agreement is good, showing a percent error lower than $10\%$, which remains rather constant with respect to the frequency. Interestingly, it is noted that ${\Psi }_{\mathrm{h}}$ is considerably less affected by the frequency with respect to ${\Theta }_{\mathrm{h}}$.

3. Element Design

3.1. Antenna Structure

In Section 2, an array was designed to obtain a radiation pattern featuring a pencil beam with an HPBW of about $3^{\circ }$ and an SLL lower than $-30$ dB to be scanned over an angular range of ${60}^{\circ }$ on the elevation plane and ${360}^{\circ }$ on the azimuthal plane. At this stage, it is necessary to design the antenna element to meet the requirements of low cross-polarization and isolation over the $f_{min}$–$f_{max}$ band, and ensure polarization reconfigurability. In this regard, the low cross-polarization requirement calls for elements that are symmetric along the principal planes, such as square patch antennas [18]. The latter, however, rarely maintain such performance over fractional bandwidths as large as $10\%$, unless stacked configurations (typically with the top substrate thicker than the bottom one) are considered [20,21]. As per the polarization reconfigurability, two ports are considered and fed with aperture-coupled techniques to ensures the required isolation [10,22], achievable even with electromagnetic band-gap (EBG) structures [23]. These aspects led us to consider the antenna architecture proposed in [10] as a reference structure to be suitably modified and optimized to meet our requirements in the $f_{min}$–$f_{max}$ band.

3.2. Full-Wave Results

The aforementioned antenna, that is shown in Figure 5a, consists of a dual-polarized aperture-coupled stacked patch made of three dielectric substrates. On the bottom of the lower substrate (Rogers Kappa438) two printed microstrip lines are used to excite two slots on the top of the same substrate. The plane where the slots are located is also used as a ground plane. On the lower part of the upper substrate there is a parasite patch, slightly smaller than the main patch, printed on the top of the same layer. Both upper and middle dielectric layers are realized with a Taconic TLX-8 laminate.

The radiating element was simulated in CST [24] in a 2-D periodic environment, in order to take into account the coupling effects that occur in a 2-D array. Coaxial feeders, excited in their fundamental TEM mode, were simulated to properly account for their effect in the antenna design. Two wave ports are necessary to properly excite both horizontal and vertical polarization, referred to, respectively, as the H port and V port from now on. The S-parameters results obtained from the simulation of the optimized antenna are shown in Figure 5b. The isolation between ports is rather satisfactory, with values below $-30$ dB, as well as a reflection coefficient that is well below $-10$ dB in the frequency range of interest, showing double-resonance behavior related to the presence of the main and parasitic patches of the proposed antenna structure.

In Figure 6, the radiation patterns are shown on the principal planes, and as expected from the reflection coefficients a low cross-polarization level and a symmetric co-polar pattern are obtained for both the H- and V-port excitations. To complete the picture, full-wave 3-D radiation patterns for the co-polar and cross-polar components of both H and V port are shown in Figure 7. A detailed view of the resulting antenna is also shown in Figure 8, whereas in

Table 1. Design parameters for the radiating element.

Substrates/Patch/Connectors		Slots		Strips
Parameter	Value [mm]	Parameter	Value [mm]	Parameter	Value [mm]
p	26.98	$d_{x\mathrm{Hslot}}$	12.42	$d_{x\mathrm{Hstrip}}$	4.80
$h_1$	0.51	$d_{y\mathrm{Hslot}}$	10.00	$d_{y\mathrm{Hstrip}}$	13.05
$h_2$	0.79	$d_{x\mathrm{Vslot}}$	6.02	$d_{x\mathrm{Vstrip}}$	8.55
$h_3$	2.36	$d_{y\mathrm{Vslot}}$	12.56	$d_{y\mathrm{Vstrip}}$	4.80
$w_{\mathrm{p}1}$	12.9	$w_{\mathrm{Hslot}}$	1.84	$w_{\mathrm{Hstrip}}$	0.9
$h_{\mathrm{p}1}$	12.9	$h_{\mathrm{Hslot}}$	1.84	$l_{\mathrm{Hstrip}}$	8.87
$w_{\mathrm{p}2}$	14.5	$w_{\mathrm{Hslit}}$	0.26	$w_{\mathrm{Tstrip}}$	0.55
$h_{\mathrm{p}2}$	14.5	$l_{\mathrm{Hslit}}$	3.29	$l_{\mathrm{Tstrip}}$	6.00
$d_{x\mathrm{H}}$	5.50	$w_{\mathrm{Vslot}}$	1.85	$w_{\mathrm{Vstrip}}$	0.90
$d_{y\mathrm{H}}$	13.49	$h_{\mathrm{Vslot}}$	1.85	$l_{\mathrm{Vstrip}}$	11.37
$d_{x\mathrm{V}}$	9.00	$w_{\mathrm{Vslit}}$	0.15	—	—
$d_{y\mathrm{V}}$	5.50	$l_{\mathrm{Vslit}}$	2.25	—	—

a list of the various design parameters is reported.

4. Sub-Array Demonstrator

4.1. Full-Wave Results

The radiating element analyzed in the previous section is here used on a $2\times 8$ array to realize a demonstrator. This analysis is particularly useful to understand if the unit-cell simulation results (that emulate an infinite-array environment) are consistent with a finite-array simulation setup. The simulation setup is similar to the one described in Section 3, but now the finite $2\times 8$ array is simulated replacing the periodic boundary conditions with the radiation boundary conditions.

In Figure 9, the F-parameters obtained from the $2\times 8$ array simulation are shown. The F-parameters are used to calculate antenna coupling coefficient sources in the case of simultaneous excitation, when is not possible to apply the general S-parameter definition anymore, and they can therefore be considered as active scattering parameters. In particular, $F_{\mathrm{hh}\left(\mathrm{vv}\right)}$-parameters refers to the reflection coefficient of a port V(H) when all the other ports H(V) are excited, while the $F_{\mathrm{hv}\left(\mathrm{vh}\right)}$-parameters refers to the isolation between the ports H(V) when all the others V(H) are excited. In particular, in the $f_{min}$–$f_{max}$ band the $F_{\mathrm{hh}}$- and $F_{\mathrm{vv}}$-parameters are well below $-10$ dB, and have a similar behavior to the reflection coefficients obtained from the unit cell simulation for both H and V ports. The same holds for the $F_{\mathrm{hv}}$- and $F_{\mathrm{vh}}$-parameters, that are similar to the transmission coefficients values obtained from the unit-cell simulation. The good agreement between the F-parameters of the array simulation with the S-parameters of the unit-cell simulation demonstrate that the latter takes into account very well the mutual coupling effects.

Moreover, in Figure 10 and Figure 11 the 1-D and 3-D far-field radiation patterns are shown, respectively. It can be noticed that the radiation pattern of the array is squeezed along the array axis due to the array factor. On the transverse axis, the total pattern remains almost unaffected with respect to the element pattern provided by the unit-cell. In all cases, the cross-polar values remain mostly below $-20$ dB, as expected from the F-parameters.

4.2. Experimental Results

Two prototypes of this $2\times 8$ array were fabricated and used to implement a total $4\times 8$ array for the measurement setup. The prototypes were placed on a metallic plane properly shaped, and two plexiglass bars were used to secure the two prototypes on the metallic support; see Figure 12a. For that reason, two columns of radiating elements were unavailable, and the array under test was a $4\times 6$ array. The measurement setup is shown in Figure 12b, with all the measurements performed in an anechoic chamber.

The measured $S_{ij}$-parameters are shown in Figure 12c for both the V and H ports, labeled as port 1 and port 2, respectively. The measurements show a frequency shift in the resonance, that is centered to ${\tilde{f}}_0=f_0$−$0.09f_0$ instead of $f_0$. However, a fractional bandwidth of $10\%$ is achieved between ${\tilde{f}}_{min}=f_{min}$−$0.09f_0$ and ${\tilde{f}}_{max}=f_{max}$−$0.09f_0$, and the isolation between ports is lower than $-30$ dB.

The frequency shift can be attributed to different aspects:

• An important PCB curvature has been observed. This is due to the manufacturing process, and at the design frequency this issue can affect the antenna gain and the $S_{ij}$-parameters;
• Possible variations in the dielectric constant with respect to the one shown in the data sheet of the materials;
• Possible interactions between the metallic support and the radiating elements. These metallic parts are very near to the radiating board, and their effect could be not negligible.

Moreover, the expected higher-frequency resonance in the reflection coefficients for both the H and V ports is deteriorated. This effect is most likely due to the curvature of the PCB caused by the lamination process used to stack the multiple layers.

With this setup, it is possible to evaluate the total pattern from the active reflection coefficient. As is known, the total pattern can be obtained by pattern multiplication between the array factor and the active element pattern $G_{\mathrm{e},\mathrm{act}}$, which in turn is related to the active reflection coefficient ${\Gamma }_{\mathrm{act}}$ for every scan angle ${\theta }_0,{\varphi }_0$ through the relation [25]

$$G_{\mathrm{e},\mathrm{act}}(u_0,v_0)=\frac{4\pi cos{\theta }_0{p}_x{p}_y}{{\lambda }^2}(1-|{\Gamma }_{\mathrm{act}}(u_0,v_0){|}^2)$$

(12)

where $p_x$ and $p_y$ represent the periodicity of the array along the x- and y-axis, respectively, with $u_0=sin{\theta }_{0}cos{\varphi }_0$ and $v_0=sin{\theta }_{0}sin{\varphi }_0$.

The results obtained with (12) from the measurement of the scattering parameters are shown in Figure 13, whereas in Figure 14 are shown the principal planes of the active element pattern for both polarizations, and the gain plot versus frequency at broadside for both polarizations. From these results it can be seen that the active element gain has a quite flat and regular shape in the $uv$ plane and a peak value near to the theoretical expected value in the ${\tilde{f}}_{min}$–${\tilde{f}}_{max}$ band. The main reason for this behavior is due to the frequency shift reported in Figure 12b and Figure 14, where the peak gain and the minimum of the reflection coefficient are located around ${\tilde{f}}_0$.

5. Conclusions

A planar AESA capable of producing a pencil beam with invariant sidelobe level over a wide angular range and at any frequency in C band has been designed with fully analytical techniques. An efficient MATLAB implementation of the Tseng–Cheng technique is presented that looks particularly attractive for realizing a software-defined-radar (SDR) interface for real-time control of the beamforming network. The radiating performance of a case study is analyzed in detail, showing the effects of a reduction in the filling factor. A smaller $4\times 6$ demonstrator has been fabricated and measured to test the main radiating performance of the proposed antenna architecture. The flexibility of the polarization control, the low level of cross-polarization, and the good isolation between ports over a $10\%$ fractional bandwidth make this prototype attractive for future SDR applications.