### I. Introduction

*M*×

*N*PAA. This

*M*×

*N*matrix is derived as the product of the amplitude distribution vectors of two linear array antennas (LAAs) constituting the PAA. To lower the SLL while maintaining the total transmission power, we increased the amplitude of the signal applied to the central element of the array antenna and lowered the amplitude of the signal applied to the side element using the tapered amplitude distribution method [10]. Although this method reduces the sidelobe of the beam pattern, it weakens directivity owing to the widening of the beamwidth; therefore, we used the Chebyshev tapered amplitude distribution method, which is efficient and possesses the strongest directivity among distribution methods with the same maximum SLL [11]. Moreover, by limiting all sidelobes to the same level, we minimized the RF radiation power in all directions, except for the main lobe.

### II. Sidelobe Suppression Beamforming

*a*

*and*

_{m}*b*

*are the amplitude distributions of the*

_{n}*x-*and

*y-*axis LAAs,

*M*and

*N*are the numbers of

*x-*axis and

*y-*axis array elements,

*d*

*and*

_{x}*d*

*are the spacings between the elements of the*

_{y}*x*

*-*axis and

*y-*axis LAAs, and

*ψ*

*and*

_{x}*ψ*

*are the phases of the*

_{y}*x-*axis and

*y-*axis LAAs, respectively.

*k*is the wave number,

*θ*is the elevation angle, and

*φ*is the azimuth angle. The amplitude distributions of the two axes,

*a*

*and*

_{m}*b*

*, can be expressed as 1 ×*

_{n}*M*and 1 ×

*N*vectors, respectively,

*w*

*is the product of*

_{m,n}*a*

*and*

_{m}*b*

*. An*

_{n}*M*×

*N*matrix with

*w*

*, which is the amplitude distribution of the PAA, can then be expressed as follows:*

_{m,n}*ψ*

*and*

_{x}*ψ*

*) that determines the beam direction can be considered a constant; therefore, in Eq. (1), the AF of the*

_{y}*x-*axis LAA is a series of exponential functions that can be expressed as follows:

^{z}^{= }

^{e j}^{(}

^{kdx}^{ sin}

^{θ}^{cos}

^{φ}^{+}

^{ψ }

^{x}^{)}. Eq. (7) is the same as the (

*M*– 1)

^{th}order equation, and can therefore be rewritten as

*M*– 1) equation, it can be expressed as a product of (

*M*– 1) linear terms, as follows:

*z*

*are the roots in the AF,*

_{i}*β*

*are the null directions, and*

_{i}*i*ranges from 1 to

*M*– 1; hence, if the null directions are known, the roots of the AF can be found, and the AF expressed as the product of linear terms can be completed. The AF equation, which is the product of a linear term, is developed in polynomial form, and the coefficients of each term are mapped to the amplitude coefficients of the array antenna; hence, it describes the process for finding the null directions by mapping the Chebyshev polynomial [11] using the array factor. To this end, the difference in magnitude between the main lobe and the sidelobe is defined as follows:

*θ*

_{ML}*,*

*φ*

*) is the main lobe direction and (*

_{ML}*θ*

_{SL}*,*

*φ*

*) is the sidelobe direction. We then derived the null directions (*

_{SL}*β*

*) using the following equations:*

_{i}*w*

*is the weight of the*

_{xp}*p*

^{th}order term, and

*p*ranges from 1 to

*M*– 1. If

*a*

*is a coefficient that determines the overall magnitude of the amplitude distribution, then the coefficient of each order (*

_{M}*w*

*) can be expressed as a ratio; therefore, the amplitude distribution vector (1 ×*

_{xp}*M*) can be expressed as follows:

*w*

*is the weight of*

_{yq}*q*

^{th}order term, and

*q*ranges from 1 to

*N*– 1.

**, which is the Chebyshev tapered amplitude distribution for the**

*W**M*×

*N*PAA, could be obtained through Eq. (6). This process ends by mapping the elements of

**to the amplitude weights of each element of the PAA, allowing the realization of beamforming with an expected SLL. Fig. 3 illustrates the process flowchart for obtaining the Chebyshev tapered amplitude distribution for sidelobe suppression beamforming.**

*W*### III. System Design and Implementation

### 1. System Design

*zx*and

*zy*planes, respectively.

### 2. Simulation

*D*

_{taper}is the directivity of the tapered array,

*D*

_{uniform}is the directivity of the uniform array,

*w*

*is the amplitude distribution,*

_{m}*d*is the distance between elements, and

*L*is the number of antenna elements. The amplitude distribution for the beam pattern with the expected SLL can be obtained using the scheme proposed in Section II. By applying the obtained amplitude distribution, the formed beam can be expressed as the efficiency of directivity. Fig. 7 shows the taper efficiency and normalized amplitude weights for array antennas with different antenna spacings on the

*x*and

*y*axes, as explained in the previous subsection (Section III-1).

*η*

*) and the normalized amplitude of the central and side elements of the array antenna were presented according to*

_{T}*S*, which is the SLL. As

*S*increases, the taper efficiency is saturated at a specific value, and the amplitude deviation of the central and side elements is also saturated. Therefore, we studied three cases: one for the uniform amplitude distribution (

*W***), another for a taper efficiency of 0.9 (**

_{1}

*W***), and the last for the moment of entering the region within 5% of the saturation section of the efficiency curve (**

_{2}

*W***). Table 1 lists the SLL, amplitude distribution matrix (**

_{3}

*a***,**

**) and taper efficiency (**

*b**η*

*) for each case. Fig. 8 shows the directivity when the amplitude distributions for each case were applied to the designed array antenna.*

_{T}### 3. Implementation

*S*

_{11}) of the single antenna was −29.8 dB, measured with a network analyzer (Keysight P5007A), and the directivity of the 4 × 4 array antenna was determined as 21 dBi.

### IV. Experimental Verification

*zx*and

*zy*planes were measured. As shown in Fig. 11, this was based on the relative power by rotating the Tx array antenna in place while fixing the position of the Rx at the center. Fig. 12 illustrates the normalized radiation pattern plotted by measuring the received RF power from −90° to resemble that shown in Fig. 8, which resulted from the simulation. Compared with the SLL with

*W***, which was a uniform distribution, the SLL while applying**

_{1}

*W***decreased by 7.5 dB from −11.6 dB to −19.1 dB on the**

_{3}*zx*plane and 4.4 dB from −11.5 dB to −15.9 dB on the

*z*

*y*plane. However, we confirmed that the received RF power at the central position (0°) decreased by only 0.6 dB, from 21.0 dBm to 20.4 dBm. Furthermore, we verified that the rectified DC power also decreased by only 9.3 mW, from 67.8 mW to 58.5 mW.

^{2}) where the Rx could exist. The phase values for each channel of the main beam direction were adjusted by (5), giving the AF of the far field approximation. The main beam direction was set as (

*θ*,

*φ*), with the center of the Tx array antenna as the origin in spherical coordinate systems. The experiment was conducted for three main beam directions, when (

*θ*,

*φ*) was (0°, 0°), (20°, 135°), and (20°, 220°), respectively. Therefore, nine experiments in total were conducted for the three cases of amplitude distribution and three cases of phase distribution. Table 2 shows the distributions except for

*W*

_{1}and (0°, 0°) where all the amplitudes and phases of the channels were identical. Fig. 13 shows a color map obtained by measuring the 15 × 15 points of normalized received RF power for the nine experiments. We confirmed that the region with the sidelobe had smaller power in

*W***and**

_{2}

*W***than in**

_{3}

*W***. Additionally, we verified the widening of the main lobe with an increase in the area occupied by power in the direction of the main lobe. Table 3 summarizes the results of the second experiment, including the received RF power and SLL for the nine measurements. Furthermore, we calculated Δpower and ΔSLL (the differences between the received power and the SLL) from the uniform amplitude distribution (i.e.,**

_{1}

*W***). In (0°, 0°)—the case with central beamforming—received power was the same as in the first experiment, but the SLL decreased further. This occurred because the origin of the measured 15 × 15 points was 1 m away from the Tx, but the remaining points were at greater distances than 1 m, meaning that the received power was lower. Therefore, in the case of beamforming toward the center, we confirmed ΔSLL to be 9.3 dB at the maximum. In (20°, 135°)—the case with the second direction—ΔSLL and Δpower were 5.9 dB and 0.8 dB in**

_{1}

*W***, respectively. Additionally, in (20°, 220°)—the case with the third direction—ΔSLL and Δpower were 6.8 dB and 0.5 dB in**

_{3}

*W***, respectively. These results proved that the received power barely decreased and that the SLL decreased significantly.**

_{3}*zx*plane in Fig. 12, which resulted from the first experiment. Studies [3–5] focused on the beamforming technique without considering lowering the sidelobe, and, in most cases, demonstrated high SLL. Additionally, [6–9] designed and manufactured a Tx device using a technique to lower the SLL, but different elements may be required to achieve a minimum SLL for other beam directions. The proposed sidelobe suppression beamforming system is suitable for safe wireless power transmission with high received power and low SLL.

### V. Conclusion

*M*×

*N*). Furthermore, to minimize decreased directivity due to the non-uniform amplitude distribution and to lower the SLL, we used the Chebyshev taper for the amplitude distribution.

*zx*plane, and by 4.4 dB to −15.9 dB on the

*zy*plane, compared with uniform distribution. Furthermore, we verified that the received RF power was 20.4 dBm, even with sidelobe suppression. This amounted to a reduction of only 0.6 dB compared with the uniform distribution case. Through a color map, we demonstrated that the area occupied by the sidelobe decreased in the three-directional beamforming experiment. Moreover, compared with other studies, the proposed system has independent and flexible amplitude and phase control. Consequently, the adaptive beamforming system presented in this paper should be capable of delivering efficient and safe MPT by accurately forming a designed beam pattern.