Grating-Lobe Mitigation Using Parasitic Scatterers and Principal Component Analysis

This communication presents a method for suppressing grating lobes that occur in sparse antenna arrays. The method is based on parasitic antenna elements that are terminated to reactive loads. The loads are computed using a computationally efficient algorithm that is based on principal component analysis (PCA) and strives to maximize realized gain toward grating-lobe-free (GL-free) scan directions. The developed method is demonstrated at 5 GHz by both electromagnetic simulations and prototype measurements using a $3\times3$ element antenna array with one-wavelength interelement distances. The example shows that the grating-lobe level is suppressed by 8.4 dB, and the main beam gain is increased by 3.9 dB compared to an array without parasitic scatterers.


I. INTRODUCTION
Wireless systems have already adopted millimeter-wave frequencies, and even terahertz frequencies are studied for future.Thus, beam steering becomes more important and the number of antenna elements in an antenna array increases.Due to the losses, the integrated circuit implementing the beam forming should be tightly integrated with the antenna array at these frequencies.When the size of the integrated circuit becomes larger than the space reserved for the antenna array, the antenna elements are forced further away from each other, and the array becomes sparse [1].
Antenna arrays with inter-element distance larger than half wavelength are more susceptible to generate grating lobes.Grating lobes can be seen as spatial aliasing artifacts since sparse arrays do not fulfill the generalized Whittaker-Kotelnikov-Shannon criterion [2], [3].Thus, there exists a fundamental barrier that prevents scanning the full angular space with a sparse array.Nevertheless, by limiting the scan area to a grating-lobe-free (GL-free) window, the beam can be scanned inside the window without suffering from grating lobes.
One method to suppress the grating lobes is to divide a large array into smaller subarrays and rotating or displacing them [4], [5], [6].Thus, the grating lobes are divided into multiple smaller lobes to reduce their magnitude.If the full array is formed from dense subarrays, the subarrays can be phased so that they form nulls toward the grating lobe directions [7].However, this method works only with certain scan directions assuming that the subarrays have The authors are with the Department of Electronics and Nanoengineering, Aalto University, 00076 Espoo, Finland (e-mail: albert.salmi@aalto.fi;jan.bergman@aalto.fi).
Color versions of one or more figures in this communication are available at https://doi.org/10.1109/TAP.2023.3333569.
Digital Object Identifier 10.1109/TAP.2023.3333569fixed phasings.Additionally, a phase-center-displacement method can suppress the grating lobes if the phase-center locations of the array elements are movable [8].The methods presented in earlier studies typically lead to great suppression of grating lobes with single beam-steering direction.However, the beam-steering performance is often decreased.The fundamental limit for spatial sampling restricts the GL-free scan window, and by taking this restriction into account already in the antenna element design stage, the GL-free beam-steering performance could be achieved better than in previous studies.
Parasitic elements have been recently utilized in many antenna design problems.In [9], the parasitic elements reduce the active voltage standing wave ratio.In [10], parasitic waveguide elements limit the radiation to the GL-free window, similarly as we do in this work.These methods require optimization of parasitic terminations in a multidimensional variable space, which is a computationally demanding task when the number of parasitic elements is large.
The grating-lobe-mitigation method presented in this communication utilizes parasitic elements for limiting the antenna array scan area into the GL-free window.The driven elements in the array couple to the parasitic elements which then scatter the coupled waves.The superposition of the scattered fields and the fields of the driven elements form embedded element patterns that cover the GL-free window and minimize the radiation toward the other directions.Thus, the grating lobes that occur outside of the GL-free window are mitigated while the antenna array's main beam can be steered within the GL-free window.
In this communication, we avoid heavy optimization by using principal component analysis (PCA).It reduces the multidimensional optimization problem into a computationally lightweight line-search problem.The feasibility of PCA for dimensionality reduction of optimization is explored in [11].Furthermore, the method is simple from the antenna design perspective as well.Any type of dense antenna array can be selected as the starting point, followed by converting the antenna elements that cannot be actively driven to parasitic ones.
The applicability of the method is demonstrated at a single frequency point using a nine-element planar patch-antenna array with 40 parasitic elements.The parasitic elements are terminated to transmission lines, which implement the passive reactive loads.The proposed test antenna is compared to a sparse array without any parasitic elements as well as an array that has nonterminated patches between the driven elements.The results are validated using manufactured prototypes.

II. METHOD FOR GRATING LOBE MITIGATION
We study a sparse antenna array where the distance between driven elements is larger than λ /2.The idea of the method is to embed additional elements between the driven ones as shown in Fig. 1(a).Since the antenna array is sparse, there is space for interleaving these parasitic elements.Part of the power fed to the driven elements couples to the parasitic elements and the coupled power then scatters back from the terminations of the parasitic elements.Thus, the passively terminated parasitic scatterers also affect the radiation properties of the array.
The antenna array with its parasitic elements is described as an N -port system.The problem setting is shown in Fig. 1(b).The model consists of N D driven elements and N P parasitic elements.The voltage waves a D n and b D n are incident and coupled at driven elements, respectively.At parasitic ports, a P n and b P n are used to describe those waves.The ratios between the waves at the ports are described using scattering parameters as shown in Fig. 1(b).The superscripts D and P refer to driven and parasitic ports, respectively.
To obtain the scattering matrix S of size N D + N P , and the embedded element patterns of each port, the full antenna array is simulated so that also the parasitic ports are driven actively.The terminations of the parasitic scatterers are computed based on this simulation data.Using the N -port system model, we formulate the algorithm for computing appropriate terminations.

A. Ideal Terminations
First, we determine the ideal terminations which depend on the beam-steering direction.Also, the ideal terminations may have negative resistance, which would require active loading.In Section II-B, we use these ideal terminations, possibly nonrealizable, for computing the scan-direction-independent and passive ones.
We denote the beam-steering directions (θ l , ϕ l ) using the index l ∈ [1, L].The elements are named by index n.When a parasitic element n is terminated to a load, it introduces a reflection coefficient r n that can be any complex number at this point.
For both driven and parasitic elements, we calculate the optimal, scan-direction dependent, incident voltage waves, i.e., feeding coefficients a D nl and a P nl that focus the main beam of the antenna array to the desired direction denoted by index l.The feeding coefficients of antenna elements can be calculated using progressive phasing, Dolph-Chebychev synthesis [12], or other synthesis methods.In this work, we use feeding coefficients that maximize the realized gain toward the main beam direction and have unit amplitudes.
As discussed in [13], [14], and [15], the feeding coefficients for maximizing the realized gain toward direction l can be computed as where operator eig λmax : C N ×N → C N , returns the eigenvector that corresponds to the largest eigenvalue of the argument matrix.The radiation matrix D l is obtained from the embedded element patterns where each element of the array is excited separately while others are terminated to generator impedances.The entry of matrix D l at row i and column j is Here e α nl is the α-polarized component of the electric field radiated by element n toward the direction denoted by index l.e α nl is obtained when the element n is driven as a part of the full array while other elements are terminated to generator impedances, i.e., it refers to the embedded element patterns.In this work, we use a 50generator impedance, although it could be chosen arbitrarily.(•) * denotes complex conjugation.
For practical reasons, we restrict the magnitudes of the feeding coefficients to be equal to one in each port.Therefore, the feeding coefficients are retracted to the complex unit circle using operator ret : where | • | gives the amplitude of its argument.Thus, the feeding coefficients are computed finally as We terminate the parasitic elements so that the fields scattering from the elements are the same as the fields radiated by those elements if they were actively driven.Thus, after terminating the elements, the incident waves a P nl should be the same as was obtained using (4).
Let a wave coupled to a parasitic element n be b P nl when the beam is focused to direction l.The coupled wave reflects back, which gives the incident wave where r nl is reflection coefficient of the element.For now, the reflection coefficient depends also on the scan direction.The coupled waves of the parasitic elements are calculated as where s PD n j and s PP n j are elements of the scattering matrices S PD and S PP , respectively, at row n and column j (see Fig. 1).
By applying (6) to (5), and solving r nl , we obtain With these reflection coefficients, the main beam is focused to the direction l similarly as if the parasitic elements were actively driven.
If the parasitic elements were terminated to impedance loads, the terminations could be calculated as where z 0n is the real reference impedance of port n.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

B. Realizable Terminations Determined by Using Principal Component Analysis
In practice, we terminate the parasitic elements to fixed loads since we cannot adjust them separately for different scan angles due to the complexity of a tunable system.Naturally, we also restrict the magnitude of the reflection at parasitic elements to be below one in order to implement them passively.It is typically favorable to terminate the elements to purely reactive loads to minimize losses.Therefore, we retract the reflection coefficients to the unit circle.The goal is to find the vector of reflection coefficients r ∈ {r ∈ C N P : |r n | = 1} that maximizes the resulting realized gain toward the desired scan directions.
The fixed reflection coefficients are determined from the scandirection-dependent reflection coefficients using PCA.In PCA, the different scan directions are samples of the data, and the reflection coefficients are the variables.
Let R ∈ C N P ×L be the data matrix constructed from reflection coefficients and let p 1 ∈ C N P be the first principal component of the data.Along p 1 , the variance of R is maximized [16].Consequently, a data point from the complex plane {r = c p 1 : c ∈ C} is close to all the points of data R.A reasonable conclusion is that the suitable vector of fixed reflection coefficients lies also on that particular complex plane.
The use of PCA drastically simplifies the optimization task.Without PCA, we would have an optimization problem with N P complex variables.Now, the number of complex variables is reduced to one, since we assume that the vector of suitable variables is on the plane spanned by the first principal component.
The first principal component p 1 is computed using MATLAB's pca() function [17].We use the eigenvalue decomposition algorithm together with centralization of the data and take all rows of the data into consideration.
Next, we force the constant modulus constraints for the reflection coefficients by using the retraction operator (3).Consequently, we have a one-variable optimization problem since only the phase of spanning factor c should be found.The objective in this line search is to maximize the realized gain to all desired scan directions.That is, we should find the most appropriate phase scaling factor φ ∈ [0, 2π] by solving where g l (r) is the realized gain toward the direction (θ l , ϕ l ) when the parasitic elements are terminated.It can be computed as where η is the wave impedance.The total system input power P in is fixed because the input signals have equal amplitudes.In (10), the feeding coefficients of driven ports a D nl are the same as computed earlier from (4).However, the incident voltage waves of parasitic ports a P nl (r) depend now on the reflection coefficients.The incident voltage waves of parasitic ports are entries of the vector where diag : C N → C N ×N gives the diagonal matrix of its argument vector.Equation ( 11) is derived from ( 5) to (6).
The line search ( 9) is computed by using MATLAB's fminbnd() function.Finally, the fixed, realizable, and scan-independent

C. Scan Direction Grid at GL-Free Window
We consider a half-space where spherical coordinates θ and ϕ are restricted to intervals [0, (π/2)] and [0, 2π], respectively.Using the presented method the gain is striven to be maximized to the directions {(θ l , ϕ l )} L l=1 .Choosing these target directions is a critical step in the method.Different choices of the scan direction grid may emphasize certain directions too much over the other scan directions.For instance, choosing a fixed angular spacing would emphasize the polar direction, in this case, broadside of the array, over other directions.
In this work, the scan grid has L u × L v equally spaced grid points in rectangular area in uv-coordinate plane.The coordinates in the grid denoted by index l are where mod is the modulo operator and ⌈•⌉ the ceiling operator.The uv-coordinates can be transformed into spherical coordinates as The limits u min , v min , u max and v max are chosen so that grating lobes do not emerge into the scan area.Let d x and d y be the interelement distances between the driven elements in Cartesian xand y-directions.The inter-element distances are given in wavelengths.Then, the grating lobes do not appear in the scan area if −(1/2d x ) < u < (1/2d x ) and −(1/2d y ) < v < (1/2d y ), which defines the limits.
Finally, the presented algorithm is summarized in the flowchart presented in Fig. 2.

A. Prototype
To demonstrate the feasibility of the presented method, three different arrays of patch antenna elements are designed.Each array has a total of nine driven elements in a 3 × 3 planar configuration Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.with one-wavelength separations.Two of the arrays have additional parasitic elements between the driven elements, effectively resulting in 7 × 7 element arrays with half-wavelength element separation.The third array consists of only the 3 × 3 configuration of driven elements and represents the reference case of a sparse array.
The patch antenna is selected for its simple and well-known design process as well as to achieve a planar array with the antenna elements on the opposite side of the printed circuit board (PCB) from the feed lines.The feed lines of the parasitic elements are used as the passive terminations, and this configuration allows changing the lengths of the transmission line terminations without affecting the radiation pattern of the parasitic elements.
The three different antenna arrays are all based on a fullydriven 7 × 7 element array optimized for 5 GHz.The array with parasitically terminated elements, shown in Fig. 3(a), is created by realizing the desired active feed of the element with the best passive parasitic termination at the port.The two reference cases, shown in Fig. 3(b) and (c), are created by removing the feed lines completely, and by removing both the feed lines and antenna elements from the parasitic element locations, respectively.Therefore, the nine driven elements are identical in every array.
To realize the parasitically terminated array, all of the elements are initially simulated with a discrete port in CST as shown in Fig. 4(a).Then, the elements that will remain driven are chosen and the rest are converted to parasitic elements.The excitations of the parasitic elements are replaced with passive terminations, in this case, shorted transmission lines with specific lengths as shown in Fig. 4(b).Therefore, the proposed design process of a parasitically terminated array is almost as straightforward as designing a typical, fully excited antenna array.
The feed lines, as well as the parasitic terminations, in the proposed design, are 50-coplanar waveguides (CPW) in a curved layout.This peculiar shape is chosen to fit a sufficiently long transmission line within one 0.5λ × 0.5λ cell together with the antenna element, making the design infinitely scalable.Furthermore, the shape of the curve prevents the CPW from passing behind the patch element, resulting in uniform transmission line characteristics for the full length of the line.In an actual system with radio frequency integrated circuit (RFIC) chips driving the elements, a multilayer PCB could be used to isolate the parasitic terminations from any possible effects from the chips.However, a two-layer PCB is used in this work for simplicity.
The difference in the length of the grounded CPW in the parasitic element terminations and the initial discrete excitation location defines the reflection coefficient of that element.In order to realize the reflection coefficient r n in element n at the desired frequency  point of 5 GHz, the length of the line should be calculated as where β and z tl are phase constant and line impedance of the transmission line at 5 GHz, respectively.Those parameters are obtained through simulations.Negative l n indicates that the CPW line should be shortened from its initial length.We use MATLAB's atan() function for computing tan −1 () [17].Any imaginary part of l n is omitted.The fixed-length transmission lines result in narrowband terminations, which suits the point-frequency analysis of this work.
In order to illustrate the computation method for determining the parasitic terminations, the ideal and realized reflection coefficients of two parasitic elements are plotted in Fig. 5.The blue diamonds are the scan-direction dependent ideal reflection coefficients.That is, they are the entries of matrix R at rows corresponding to the element numbers, here 10 and 34.The red circles are the final realized reflection coefficients that are computed based on the ideal ones using the presented PCA algorithm, i.e., the entries 10 and 34 of the vector computed with (12).The corresponding termination transmission-line lengths are l 10 = −3.5 mm and l 34 = 2.6 mm.
In our case, the number of grid points in ( 13) and ( 14) is L u = L v = 16.The elapsed time in PCA evaluation is 11 ms, and the line search (9) converges in 10 ms.
The prototype arrays are manufactured on a 1.52-mm thick Rogers RO4350B substrate.The manufactured parasitically terminated array is shown with its key dimensions in Fig. 6.The operating frequency was verified by measuring the S-parameters using a vector network analyzer, and the radiation patterns were measured using an MVG StarLab 6-GHz system.

B. Measurement Results
The measurement results showed that the best operation frequency of the prototype is 5.05 GHz instead of the design frequency of 5 GHz.This 1% frequency shift is caused by simulation and manufacturing inaccuracies.Since the antenna is a narrow band, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.the measured results are analyzed at the new operation frequency 5.05 GHz.Fig. 7 illustrates the realized gain pattern of the middle element of the array with parasitic elements.This confirms the correspondence between simulated and measured patterns.Also, we compare the results to an analytically computed pattern, which is obtained from (10) by setting a D (n=5),l = 1 and a D nl = 0 for n ̸ = 5.The analytical pattern matches almost exactly with the measured one, excluding the ripple caused by manufacturing and measurement inaccuracies.
In Fig. 8, the elements are fed in the same phase and amplitude.Since the inter-element distance is 1λ , the grating lobes occur at θ = 90 • .In the sparse array, the grating lobe level at (θ, ϕ) = (90 • , 0 • ) is −5.63 dB with respect to the main lobe, whereas in parasitically loaded array it is −14.0 dB, resulting in an 8.4 dB  suppression which is comparable with state-of-the-art [5].In this principal plane, the array with floating patches gives −16.5 dB grating-lobe level.However, in the H-plane (ϕ = 90 • ), the gratinglobe level of the array with floating patches is −16.1 dB, whereas it is −20.6 dB in the array with parasitic terminations.The main beam magnitudes at (θ, ϕ) = (0 • , 0 • ) are 12.9, 15.6, and 16.8 dB in a sparse array, array with floating patches, and array with parasitics, respectively.Thus, the main beam gain is improved by up to 3.9 dB.The proposed design with parasitics has clearly suppressed grating lobes in both planes and increased the main-beam realized gain.
The GL-free window locates at θ ∈]−30 • , 30 • [ in E-and H-planes.At window extremes, θ = ±30 • , the grating lobe is as strong as the main beam.Fig. 9 shows the realized gain patterns of the simulated parasitically-loaded array with three different scan angles inside the GL-free window, including an extreme case.The scandirection-dependent gain patterns are formed with feeding coefficients computed using (4), and the peak values of the gain patterns form the scan gain envelope as a function of the steering angle θ , also illustrated in Fig. 9. Fig. 10 shows the scan-gain envelopes for the three studied arrays.The array with parasitic elements gives improved realized main-beam gain by several decibels on the GL-free window compared to the sparse array and array with floating pads.Fig. 11 shows the total active reflection coefficients (TARCs) of the parasitically loaded array and the two reference arrays.In the parasitically loaded array, the TARC is below −6.5 dB in both principal planes.The TARC is improved by 1.4 dB at (θ 0 , ϕ 0 ) = (30 • , 0 • ), and by 2.5 dB at (θ 0 , ϕ 0 ) = (30 • , 90 • ) compared to the array with floating patches.The TARC is computed from the simulated S-parameters with excitation coefficients calculated using (4).

C. Comparison to State-of-Art
We compare the proposed method to the state-of-art in Table I.First, the table shows whether using the method in question allows the main beam to be steered without suffering from grating lobes.Also, the table shows if the referred method requires multidimensional  or a complex transition between the RFIC and the antenna.
Considering the methods that allow beam scanning, the issue in the state-of-the-art is typically a limited scalability.If the signal routing from the RFIC to the elements requires bent and overlapping feeding lines, the routing strategy raises a challenging design task, and the complexity increases when scaling up the number of elements.Also, if the method requires multidimensional and nonconvex optimization, the method introduces high computational demands when considering large arrays.
Our method is scalable and limited only by the capability of the electromagnetic simulator to solve the full-array simulation.The transmission-line-based implementation of the proposed method limits the applications to narrow-band cases, but the method itself could be extended to wide-band applications.

IV. CONCLUSION
The communication presented a novel method for tailoring the radiation patterns of antenna array elements.It is especially suitable for sparse antenna arrays where implementing parasitic scatterers is possible.The resulting patterns focused the radiation into the GL-free scan region.Consequently, grating lobes that are outside of the target scan area decreased several decibels, and realized gain increased inside the scan area.
The presented method was systematic and computationally lightweight.The method allowed designing sparse arrays starting from a dense array, and after the full-array simulation, every other element was terminated passively.The terminations were computed by utilizing PCA and line search instead of solving multidimensional optimization problem.

Manuscript received 22
June 2023; revised 19 October 2023; accepted 5 November 2023.Date of publication 22 November 2023; date of current version 15 February 2024.This work was supported by the Business Finland through ENTRY100 GHz CELTIC-NEXT Project.The work of Albert Salmi was supported by Walter Ahlström Foundation.The work of Jan Bergman was supported in part by Walter Ahlström Foundation and in part by Helsingin Puhelinyhdistys (HPY) Research Foundation.(Corresponding author: Albert Salmi.)

Fig. 1 .
Fig. 1.Model of the parasitic improved antenna array.(a) Idea of the method illustrated with three driven antenna elements, two parasitic elements, and an RFIC block.(b) N -port model of the antenna array and its ports.

Fig. 3 .
Fig. 3. Conceptual view of the three arrays (a) proposed parasitic array, (b) parasitic reference array of nonterminated floating elements, and (c) sparse reference array.Top left shows the frontal view of the arrays whereas bottom right shows the ground plane and traces with the substrate and patches hidden.Red squares show the driven element feed locations.Substrate is drawn in dark gray and copper in light gray.

Fig. 4 .
Fig. 4. Closeup of the CPWs three parasitic elements and one driven element.The bottom-left element is driven.(a) Fully excited array with discrete excitations between the CPW and ground.(b) Adjusted transmission line lengths with the end of the CPW connected to ground.

Fig. 5 .
Fig.5.Ideal scan-direction dependent and realized reflection coefficients on the complex plane marked as blue diamonds and red circle, respectively.The reflection coefficients are scaled with log 10 (r n ).

Fig. 6 .
Fig. 6.(a) Front view with dimensions and (b) back view of the manufactured parasitically terminated prototype.

Fig. 7 .
Fig. 7. Analytical, simulated, and measured embedded element patterns of the center element of the parasitically loaded array.

Fig. 8 .
Fig. 8.Total array radiation pattern when each element is driven in same phase and amplitude.(a) E-plane cut.(b) H-plane cut.

Fig. 10 .
Fig. 10.Envelope of the main beam realized gain as a function of beam-steering direction.The GL-free scan window is highlighted.(a) E-plane cut.(b) H-plane cut.

Fig. 11 .
Fig. 11.TARCs of the three simulated arrays on the two principal cutting planes.

TABLE I COMPARISON
OF THE MAIN CHARACTERISTICS IN THE STATE-OF-ART