Refinement Method for Weighting Scheme of Fully Spatial Beamformer

Abstract

So far, a wideband spatial beamformer has been proposed. This kind of beamformer has a major contribution as its weighting coefficients are real valued in which they can be simply realized by attenuators or amplifiers. However, so far, the range of attenuation or amplification is relatively large which is not practical for hardware realization. Therefore, this paper proposes a concept to reduce the range of weighting coefficients hence, the hardware realization becomes practical. In this paper, a full prototype of wideband spatial beamformer is constructed to reflect the true beamforming performance of the proposed refinement method. Its radiation patterns obtained from simulation and measurement are compared. As a result, we can reduce the attenuation or amplification range while some radiation characteristic is remained.

1. Introduction

In recent years, wireless communication systems have grown rapidly and contained several technologies operating in different frequencies, for example, WiFi, WiMAX, Bluetooth, and broadband WLAN. From these reasons, the systems are demanded for better coverage, wider frequency range, higher quality of service, and more capacity [1]. So far, smart antenna systems have been envisaged to be the solution. This is because the systems can form one beam towards a desired direction and create nulls or sidelobes towards interference directions. As a result, the greatly improving system performance and also saving energy can be obtained [2]. The smart antenna systems usually consist of antenna array and a suitable signal-processing unit adjusting the weighting coefficients at individual antenna elements to ease the effect of interference signal [3]. The signal processing works according to the utilized algorithms. So far, the beamforming algorithms for smart antennas in wireless communications have concerned only narrowband operation. However, in order to support high data-rate transmission, new efforts are currently required to deal with smart antennas in wide frequency range [4]. From literatures [5–7], wideband smart antenna technology can be classified into three categories. The first category utilizes space-time signal processing, so called spatiotemporal beamformer. These systems consist of array antennas and tapped-delay line at each branch of the array to deal with the received signal in time domain [8]. However, the number of tapped-delay line increases with the bandwidth of utilized frequency. Next category relies on space-frequency signal processing. This is an alternative approach to perform wideband beam-formation without the use of tapped-delay lines. In this method the received signal will be decomposed into non-overlapping narrowband component using band-pass filter. However, for wideband signal, the systems require a large number of frequency filters [9, 10]. The last category utilizes only spatial signal processing, so-called wideband spatial beamformer. This is a new method for wideband smart antennas. In the part of signal processing, weighting coefficients are calculated using two-dimensional Inverse Discrete Fourier Transform (IDFT) technique applied to the required radiation pattern. The obtained weighting coefficients become real values which can be simply realized by attenuators or amplifiers [11–13]. The main advantage of this wideband spatial beamformer is that its design does not require phase shifters, delay circuits, or frequency filters. This is of considerable practical advantage over the other types of beamformer employing complex weighting coefficients [14]. However, the range between maximum and minimum values of weighting coefficients (attenuation or amplification) is relatively wide. This is considerably impractical. Therefore, this paper proposes the refinement method for fully spatial beamformer. The aim of the proposed method is to reduce the range of overall weighting coefficients. As a result, the implementation of this kind of beamformer is practical. A full prototype of fully spatial beamformer is developed in order to validate the proposed method.

The remainder of this paper is as follows. After brief introduction showing the background and motivation of the paper, the concept of beam formation in wide range of operating frequency band using only spatial signal processing is described in Section 2. Section 3 proposes the method to make the weights more practical, so-called refinement method. A full prototype of fully spatial beamformer is detailed and constructed in Section 4. Then, its beamforming performance is revealed with the use of proposed refinement method in Section 5. Finally, Section 6 concludes the article.

2. Wideband Beamforming Algorithm Using Fully Spatial Signal Processing

2.1. Fully Spatial Beamformer

The configuration of a wideband beamformer employing a fully spatial signal processing for beam and null steering in azimuth direction is shown in Figure 1. It is constituted by 𝑁₁×𝑁₂ where 𝑑₁ and 𝑑₂ represent array spacing in two orthogonal directions and are usually chosen as half-wavelength (𝜆/2) at the highest frequency of a given frequency band. The antenna elements are denoted by indices (𝑚₁,𝑚₂) where −𝑀₁≤𝑚₁≤𝑀₁ and −𝑀₂≤𝑚₂≤𝑀₂. The relationship between 𝑀 and 𝑁 is 𝑀_𝑖=(𝑁_𝑖−1)/2 [15].

Figure 1 

Layout of array configuration for fully spatial beamformer.

The radiation pattern of the array as a function of angle and frequency can be given by 𝐻𝑓,𝜙=𝐺𝑓,𝜙⋅^𝑀1_{𝑚1=−𝑀1}^𝑀2_{𝑚2=−𝑀2}𝑤_𝑚1,𝑚2⋅𝑒^{𝑗(2𝜋𝑓/𝑐)(𝑑1𝑚1sin𝜙+𝑑2𝑚2cos𝜙)}, (1) where 𝑓 is the frequency variable, 𝑐 is the speed of signal, and 𝐺(𝑓,𝜙) is a function of frequency-angle-dependent gain of each antenna element. Note that signal is incident or transmitted in azimuth direction 𝜙 and 𝜃=90^∘. When 𝜙 and 𝜃 are the azimuth and the elevation angles, respectively. In order to determinate the weighting coefficients 𝑤_𝑚1,𝑚2 in (1), a modified IDFT is applied to 𝐻 as shown in (2) 𝑤_𝑚2,𝑚2=1𝑁_𝑢1⋅𝑁_𝑢2×_𝑢^0.5_1=−0.5𝑢^0.5_2=−0.5𝐻𝑢₁,𝑢₂𝐺𝑢₁,𝑢₂×𝑒^{−𝑗2𝜋𝑢1𝑚1}𝑒^{−𝑗2𝜋𝑢2𝑚2}. (2) We define two auxiliary functions as 𝑢₁=(𝑓𝑑₁/𝑐)sin𝜙 and 𝑢₂=(𝑓𝑑₂/𝑐)cos𝜙. Where 𝑁_𝑢1 and 𝑁_𝑢2 are number of sampling points in 𝑢₁−𝑢₂ plane in which 𝑁_𝑢1≅2𝑁₁ and 𝑁_𝑢2≅2𝑁₂. The 𝐻(𝑢₁,𝑢₂) is the form of Sinc function to obtain maximum gain in the desired direction [16] which can be given as 𝐻𝑢₁,𝑢₂=sin𝛼𝜋𝑢₁/𝑢₂−tan𝜙₀𝛼𝜋𝑢₁/𝑢₂−tan𝜙₀,𝑟_𝑙<|𝑟|<𝑟_ℎ,1√10,otherwise, (3) where 𝜙₀ is the direction at which the main beam to be pointed to which is assumed to be prior known [14], 𝛼 is the number related to the main beam first-null width, 𝑟_𝑙 and 𝑟_ℎ are defined by the following expressions: 𝑟_𝑙=(𝑓_𝑙/𝑐)𝑑 and 𝑟_ℎ=(𝑓_ℎ/𝑐)𝑑, respectively, when 𝑑=𝑑²₁sin²𝜙₀+𝑑²₂cos²𝜙₀,𝑓_𝑙 is lowest frequency and 𝑓_ℎ is highest frequency. Referring to the previous work presented in [14], null steering implementation is difficult to perform using IDF method. This is because the obtained weighting coefficients become real valued. Therefore, degree of freedom is relatively limited comparing the ones employing complex weighting coefficients.

To give some simulation examples, we consider 4×4-array operating over frequencies from 1.9 to 2.4 GHz and choose planar monopole having omnidirectional radiation to be an antenna element. From (1) to (3), we calculate weighting coefficients when the main beam is required to be pointed to 𝜙₀=45^∘. The outcomes of beam formation are shown in Figures 2 and 3.

Figure 2 

Simulated radiation pattern of 4×4 beamformer at 2.15 GHz, when the main beam direction is 45°.

Figure 3 

Simulated radiation pattern of 4×4 beamformer for frequencies from 1.9 to 2.4 GHz, when the main beam direction is 45°.

Figures 2 and 3 show the radiation pattern of the 4 × 4 spatial beamformer in which the objective function as shown in (3) is assumed using Sinc function with the mainbeam pointed to 45° off the array broadside direction. These figures indicate that the main beams are correctly directed towards the desired direction, and the patterns are alike throughout the desired frequency band, from 1.9 to 2.4 GHz. Also, it can be observed that as frequency increases, the array’s sidelobe levels increase, and the null locations slightly change with respect to the operating frequency. However, from the results, the validity of the wideband spatial beamforming concept can be confirmed.

2.2. Chebyshev Function

In the previously section, we consider the Sinc function being as an objective function 𝐻(𝑢₁,𝑢₂). As a result, the sidelobe level (SLL) is relatively high. Thus, in this section, the objective function is created, using Chebyshev function instead as shown in (4) [17, 18]. We consider 𝐻𝑢₁,𝑢₂=1𝑋,desiredfrequencyband√10,otherwise, (4) where ()𝑋=−1^𝑁(|𝑥(),|𝑥|()cosh𝑁⋅arccosh|),𝑥<−1cos𝑁⋅arccos𝑥≤1cosh𝑁⋅arccosh𝑥,𝑥≥1,𝑥=𝑥₀cos2𝜋𝜆(𝜃)𝑑sin2𝑥₀,=coshcosh⁻¹SLL_dB𝑁−1,𝑁=𝑁₁=𝑁₂. (5)(6)(7)(8) From (6) direction of arrival on 𝑢₁−𝑢₂ plane is 𝑢𝜃=𝑎tan¹𝑢₂−𝜙₀. (9) Substituting (9) into (6) yields 𝑥=𝑥₀cos2𝜋𝜆𝑢𝑑sin𝑎tan₁/𝑢₂−𝜙₀2 (10) or 𝑥=𝑥₀𝑢cos𝜋sin𝑎tan₁/𝑢₂−𝜙₀2. (11)

In this section, it is easily done by replacing the objective function 𝐻(𝑢₁,𝑢₂) appeared in (3) by the one appeared in (4). Therefore, SLL is expected to be lower comparing to the ones shown in Figures 2 and 3. The simulation results for this assumption are shown in Figures 4 and 5.

Figure 4 

Simulated radiation pattern of 4×4 beamformer plotted at 2.15 GHz, when the main beam direction is 45°, and the objective function is Chebyshev.

Figure 5 

Simulated radiation pattern of 4×4 beamformer for frequencies from 1.9 to 2.4 GHz, when the main beam direction is 45°, and the objective function is Chebyshev.

Comparing between Figures 4 and 5 and Figures 2 and 3 reveals that the sidelobe level when objective function is assumed using Chebyshev polynomial is lower than using Sinc function for all frequencies. In Figure 5, the null location slightly changes when frequency increases as the ones appeared in Figure 3.

3. Refinement Method for Weighting Scheme of Fully Spatial Beamformer

Regarding the previous section, it has shown that we can obtain a value of weighting coefficient of smart antenna by taking IDFT to the desired radiation pattern. One example of the obtained weighting coefficients using both Sinc and Chebyshev functions is shown in Table 1. As we can see they are all real valued in which we can simply utilize attenuators or amplifiers to be weights. However, as we can notice in Table 1, they are not integer numbers, and also the range between the maximum and minimum values is relatively wide. This is considerably not practical for hardware implementation. Therefore, the modification of calculated weighting coefficients is needed while maintaining some important characteristic of radiation pattern.

Table 1 

Weighting coefficients when the required radiation pattern is created using Sinc and Chebyshev functions.

In this section, we describe the proposed refinement method for weighting scheme of fully spatial beamformer in order to make it practical. This method is implemented after initially calculating the weighting coefficients using IDFT as mentioned before. Some significant properties of radiation behavior are maintained as described below.

3.1. Refinement Algorithm

The main proposal of this algorithm is to decrease the operation range of weighting coefficients as an example shown in Table 1. This algorithm is required because simply squeezing the mentioned range affects some importance radiating characteristics such as main-beam’s direction and sidelobe levels. However, obtaining the integer numbers in weighting coefficients is byproduct when performing refinement method.

The weighting coefficient from IDFT method detailed in Section 2.1 is now defined as 𝑊_𝑜=𝑤_𝑜,1,𝑤_𝑜,2,𝑤_𝑜,3,…,𝑤_𝑜,𝑁=𝑤_𝑜,𝑖,𝑖=1,2,3,…,𝑁, (12) where 𝑁 is number of array antennas. Usually, the value of 𝑊_𝑜 is general number, then we round the weights obtained from IDFT method to integer-valued weights (𝑊_𝑟) as 𝑊_𝑟=𝑤_𝑟,1,𝑤_𝑟,2,𝑤_𝑟,1,…,𝑤_𝑟,𝑁=𝑤_𝑟,𝑖=Int𝑤_𝑜,𝑖. (13) Then, the required mainbeam direction (𝜙₀), average level of minor lobe (MLL₀) and width of main beam (𝛿₀) and weighting coefficient (𝑊_𝑟) are given. Next, we carry out the following steps.

(1) Determine the maximum of 𝑊_𝑟, that is 𝑤^(𝑘)_max=max^𝑁_{𝑟,𝑖𝑖=1}𝑤, (14) where 𝑘=1,2,3…is order of iteration.

(2) Decrease maximum of weighting coefficient (𝑤^(𝑘)_max) by decreasing a step size, where Δ𝑤>0. This results in integer number for weighting coefficients. We have 𝑤^(𝑘)_max,𝑗=𝑤^(𝑘)_max−Δ𝑤, (15) where 𝑗=1,2,3,…. Then we replace 𝑤^(𝑘)_max,𝑗 with 𝑤^(𝑘)_max in a set of 𝑊_𝑟. We consider 𝑤^(𝑘)_max,𝑗⟹𝑤^(𝑘)_max. (16)

(3) Multiply 𝑊_𝑟 by steering vector 𝜓, and we can get output as follows: 𝑦_𝑗^(𝑘)=^𝑁_𝑖=1𝑤_𝑟,𝑖𝑒^𝜓,𝑦_𝑗^(𝑘)=^𝑁_𝑖=1𝑤_𝑟,𝑖exp2𝜋𝑐𝑑₁𝑝sin𝜙+𝑑₂𝑞cos𝜙, (17)(18) where (𝑝,𝑞) is index of antenna elements.

(4) Now, we can obtain the radiation pattern from (18). Then we can check some significant parameters such as mainbeam direction (𝜙_𝑟^(𝑘)), average minor-lobe level (MLL_𝑟^(𝑘)), and beamwidth (𝛿_𝑟^(𝑘)) according to the following conditions: 𝑤^𝑘_max≤𝑤₀,𝜙−Δ𝜙≤𝜙_𝑟^(𝑘)≤𝜙+Δ𝜙,MLL_𝑟^(𝑘)≤MLL₀𝛿+ΔMLL,₀−Δ𝛿≤𝛿_𝑟^(𝑘)≤𝛿₀+Δ𝛿, (19)(20)(21)(22) where 𝑤₀ is desired maximum of weighing coefficients, Δ𝜙 is deviation of main beam angle, ΔMLL is deviation of average minor-lobe level, Δ𝛿 is deviation of beamwidth.

(5) Repeat steps 2 to 4 until the set of weighting coefficient is zero according to (19) to (22), and then we can obtain a set of new solution (𝑊_𝑟) for the refinement method.

This proposed algorithm can run in both offline or real time processing depending on the purpose of users. In case of ruing in real-time, we need a suitable DSP board to receive the important information from DFT method then perform 5 steps refinement as shown above. Afterwards, the DSP board feedbacks the weighting coefficients to control the attenuators or amplifiers.

3.2. Simulation Results

In this section we show the efficiency of the proposed refinement method using our own developed program in MATLAB. The utilized antennas are arranged in 4×4 lattice, and the operating frequency is given from 1.9 to 2.4 GHz. We also assume the decreasing step size of weighting coefficient of 1, desired maximum of weight value of 32 dB, and steering angle from 0 to 90 degrees. Some parameters are given in simulations for 6 cases as shown in Table 2.

Table 2 

Parameters given in simulation for refinement algorithm of 6 cases.

Table 2 shows 6 cases for the proposed refinement method. Some parameters, for example, maximum weighing coefficients, deviation of mainbeam angle (Δ𝜙), deviation of average minor lobe level (ΔMLL), and deviation of beamwidth (Δ𝛿) are differently given in each case. For case I, we set the conditions not to have any error in the parameters mentioned earlier. From running some simulations, we have found that 67 directions or cases out of 91 directions, 0°–90°, are succeeded for refinement. This means that there are only 67 directions out of 91 directions in which we can maintain no error in mainbeam direction, minor lobe level and beamwidth. For case II, we allow the slight error in mainbeam direction, minor lobe level and beamwidth as shown in Table 2. We have found that the number of success cases increases from 67 to 71 directions. Also, higher error in mainbeam direction, minor lobe level, and beamwidth is allowed as shown in Table 2 for cases III, IV, V, and VI. As we can see, the number of success cases increase as we allow more error in mainbeam direction, minor-lobe level and beamwidth. For case VI, 100% success can be achieved when we allow error in mainbeam direction of 4°, minor lobe level of 4° and beamwidth of 5°. For this case, the maximum weighting coefficients can be decreased from 54 dB to 32 dB.

Table 3 shows 16 values of weighting coefficients comparing between IDFT method and proposed refinement method when the desired mainbeam direction is given at 45°. For the refinement method, we set 𝑤₀=18 dB, Δ𝜙=0°, ΔMLL=3 dB and Δ𝛿=0°. As we can see in Table 3, we can decrease maximum weighting coefficient from 22.78 dB to 18 dB which mean the range of attenuation or amplification decreases.

Table 3 

The comparison value of attenuator factors between IDFT method and refinement method.

Figure 6 shows beamforming performance for case shown in Table 3 at frequency 2.15 GHz comparing between using IDFT method and proposed refinement method. The obtained result shows that after the refinement method is done, the mainbeam direction is still pointed to 45°, and there is no error in beamwidth. Also, the error in average minor lobe level is within 3 dB.

Figure 6 

The comparison of radiation patterns at 45° using IDFT method and Refinement method at frequency 2.15 GHz.

Figures 7 and 8 show beamforming performance of the beamformer throughout the designated band using IDFT method and proposed refinement method, respectively. As we can see, the beamforming performance for both cases is similar while we can decrease the range of attenuation or amplification up to 4.78 dB (22.78–18 dB).

Figure 7 

Radiation pattern at 45° using IDFT method from frequencies 1.9 to 2.4 GHz.

Figure 8 

Radiation pattern at 45° using proposed refinement method from frequencies 1.9 to 2.4 GHz.

4. Practical Realizations

The full prototype employing 4×4 array antennas for the proposed antenna systems is designed for the operating frequencies from 1.9 to 2.4 GHz as shown in Figure 9. The prototype is a digitally controlled analog beamforming system, and it consists of three functional blocks: antenna array, weighting network, and control devices. A block diagram of smart antenna systems starts from the received incident signals at 4×4-antenna array. The received signals are weighted in weighting network which is controlled by control devices. Then, the signals are summed at the combiner device.

Figure 9 

Diagram of smart antenna system.

4.1. Antenna

4.1.1. Single Element

A printed monopole antenna as shown in Figure 10 is chosen to be the antenna element for this paper. The reason is that this type of antenna is compact enough to be arranged to form the array having interelement spacing of half-wavelength. As appeared in Figure 10, the feed line is 25-mm in length, and ground plane is sized by 25×40 mm². The fabricated antenna is shown in Figure 11 using single-layer FR-4-based board having its thickness of 1.6 mm. This antenna was designed to be able to operate in 1.9–2.4 GHz band. Figure 12 shows the measured return loss of the antenna. As we can see, the antenna provides return loss lower than −10 dB within the designated band. Also, its radiation is omnidirectional over the designated band.

(a)

(b)

Figure 10 

Printed monopole antenna structure at (a) front view and (b) back view.

Figure 11 

Photograph of the printed monopole antenna fabricated on a 40×50 mm2 single-layer FR-4 substrate.

Figure 12 

Measured return loss of the antenna shown in Figure 11.

4.1.2. Antenna Array

Here we present the design and development of a 4×4-array antenna which is capable of beam steering in the azimuth direction over an increased operational bandwidth. The design focuses on the frequency band from 1.9 to 2.4 GHz. The photograph of constructed antenna array is presented in Figure 13 when the antenna elements are spaced by half-wavelength at 3 GHz (50 mm), which is higher than the upper frequency of the assumed band (2.4 GHz). The use of this higher frequency (3 GHz) is required in the beamforming algorithm to avoid the edge effect when finding the weighting coefficients.

Figure 13 

Photograph of fabricated 4×4-array antenna.

4.2. Weighting Network

The weighting networks consist of four major components: digital attenuators, digital switching, phase shifters, and power combiners. Each component operates very well over wide range of frequency covering frequencies from 1.9 to 2.4 GHz. The detail of each component is shown as follows.

(1) The utilized attenuator is a 50-Ω RF digital step attenuator offering an attenuation range up to 31.5 dB with 0.5 dB step controlled by 6 bit serial interface. The operating frequency covers from DC to 2.4 GHz.

(2) The digital switching is 50-Ω high isolate SPDT RF switch designed for wireless application, covering a broad frequency range from DC up to 3 GHz with low insertion loss.

(3) Phase shifters using FR-4 substrate were designed from CST microwave studio. These components are utilized to 180°-shift the phase of signal in order to produce the minus sign for some weighting coefficients.

(4) The power combiners are separated into two different types which are 2 : 1 operating in frequency range from 1.6 to 3.3 GHz and 16 : 1 operating for frequencies from 1.8 to 2.6 GHz.

4.3. Control Devices

Twelve ATMEGA328 microprocessors are utilized to digital attenuator and digital switching. The reason to choose this type of control device is that it is low of cost but be able to meet the requirement in this paper.

4.4. Spatial Beamformer

The prototype of 4×4 spatial beamformer is constructed to test the validation of the proposed method as shown in Figure 14. The power supply is located on the bottom of the shelf. This supports needed power to microcontrollers, digital attenuators, and digital switches. The control devices as mentioned in last section are located at the 2nd shelf followed by the weighting network which is placed on the 3rd shelf as we can see in Figure 14. The top of the shelf is where we put the 4×4 antenna array as mentioned in last section. Also, the power combiners are placed on the top shelf and pass the output signal to the computer.

Figure 14 

Photograph of full prototype for 4×4 spatial beamformer consisting of antenna array, weighting network, control devices, and power supply.

5. Experimental Results

5.1. Experimental Setup

The constructed prototype is tested in anechoic chamber as shown in Figure 15. The obtained results in terms of beamforming performance will be compared to the ones obtained from computer simulation. The measurement is taken place at single frequency from 1.9 to 2.4 GHz. All the process starting from giving desired mainbeam direction performs automatically. However to show some example, the desired directions to be chosen are 7 directions: 2°, 7°, −17°, ±34°, 57°, and 75°.

Figure 15 

Testing the constructed prototype in an anechoic chamber.

5.2. Experimental Results

In this section we validate the beamforming performance of the proposed refinement method through the prototype measurement described in last section. The performance is compared between employing IDFT method and proposed refinement method. There are 3 cases for this section as follows. Please note that the operating frequencies are assumed from 1.9 to 2.4 GHz.

Case A. the beamforming performance using the proposed refinement method is tested. In this case, the mainbeam direction is chosen at 7°. The parameters for this case are as 𝑤₀=4 dB, Δ𝜙=5°, ΔMLL=−11 dB, and Δ𝛿=0°. After the refinement process is done, we have found that the range of attenuation or amplification for weighting coefficients can be reduced from 16 to 4 dB.

The beamforming performance in terms of radiation pattern at frequency 2.15 GHz when using proposed refinement method according to condition for Case A is shown in Figure 16. In the figure, comparison between the result obtained from computer simulation and experiment is shown. As we can see, both results have a good agreement to each other. Figure 17 shows the beamforming obtained from simulation for Case A throughout the designated band. We can see that the mainbeam direction does not deviate from the given direction (7° for this case).

Figure 16 

Simulated and measured radiation pattern using refinement method for frequency 2.15 GHz, when the main beam direction is 7°.

Figure 17 

Simulated radiation pattern using refinement method for frequencies from 1.9 to 2.4 GHz, when the main beam direction is 7°.

Case B. For this case, the mainbeam is supposed to be pointed to −34°. Also the parameters for refinement are as follows: 𝑤₀=4 dB, Δ𝜙=2°, ΔMLL=−15 dB, and Δ𝛿=0°. After the refinement process is done, we have found that the range of attenuation or amplification can be reduced from 30 dB to 4 dB. This is considerable practical.

Figure 19 shows the beamforming performance obtained from simulation and measurement for Case B at frequency 2.15 GHz. As we can see, after refinement method is done, the error in mainbeam direction is 2°, and the error in minor lobe level is within −15 dB. Also, there is no error in beamwidth. This error occurred because we allow some error in the refinement method. Comparing with the results obtained from Case A, we can obtain higher range reduction of weighting coefficients while expecting some error in mainbeam direction, minor-lobe level and beamwidth.

Figures 20 and 21 reveal the beamforming performance for Case B throughout the designated band, both from simulation and measurement, respectively. We can observe more error in mainbeam direction, average minor-lobe level and beamwidth compared with the ones presented in Figures 17 and 18. As expected, the ones obtained from measurement are likely to have higher minor-lobe level.

Figure 18 

Measured radiation pattern using refinement method for frequencies from 1.9 to 2.4 GHz, when the main beam direction is 7°.

Figure 19 

Simulated and measured radiation pattern using refinement method for frequency 2.15 GHz, when the main beam direction is 34°.

Figure 20 

Simulated radiation pattern using refinement method for frequencies from 1.9 to 2.4 GHz, when the main beam direction is −34°.

Figure 21 

Measured radiation pattern using refinement method for frequencies from 1.9 to 2.4 GHz, when the main beam direction is −34°.

Case C. For this case, the beamforming performance using proposed refinement method comparative with IDFT method is pointed out. The parameters for Case C are as follows: 𝑤₀=4 dB, Δ𝜙=2°, ΔMLL=−11 dB, and Δ𝛿=0°. Please note that the mainbeam direction for this case is changed to 2°. As a result, this case allows us to reduce the range of weighting coefficients from 8 dB to 4 dB.

Figure 22 shows the beamforming performance at 2.15 GHz using IDFT method for both simulation and measurement. Figures 23 and 24 show radiation pattern throughout the designated band, 1.9 to 2.4 GHz, obtained from simulation and measurement, respectively. As we can see, its beamforming behavior looks stable throughout the band. However, the ones obtained from measurement have slightly higher minor lobe level.

Figure 22 

Simulated and measured radiation pattern using IDFT method at frequency 2.15 GHz, when the main beam direction is 2°.

Figure 23 

Simulated radiation pattern using IDFT method for frequencies from 1.9 to 2.4 GHz, when the main beam direction is 2°.

Figure 24 

Measured radiation pattern using IDFT method for frequencies from 1.9 to 2.4 GHz, when the main beam direction is 2°.

Figures 25 to 27 show the performance similar to the ones from Figures 22 to 24 but the proposed refinement method is taken place after we obtain the weighting coefficients from IDFT method. Its beamforming at 2.15 GHz is shown in Figure 25 while Figures 26 and 27 show it performance throughout the designated band via simulation and measurement, respectively. As we can see, the beamforming behavior looks similar to the cases using IDFT method with narrower range of weighting coefficients.

Figure 25 

Simulated and measured radiation pattern using refinement method at frequency 2.15 GHz, when the main beam direction is 2°.

Figure 26 

Simulated radiation pattern using refinement method for frequencies from 1.9 to 2.4 GHz, when the main beam direction is 2°.

Figure 27 

Measured radiation pattern using refinement method for frequencies from 1.9 to 2.4 GHz, when the main beam direction is 2°.

6. Conclusion

This paper has presented the method to reduce the range of weighting coefficients for fully spatial beamformer, so-called refinement method. This is considerably useful as the weighting coefficients for this kind of beamformer are real valued; therefore, narrower range of weighting coefficients is advantage. As a result, the choice of attenuators or amplifiers becomes practical. This refinement method is taken place after weighting coefficients are calculated using IDFT method. This proposed method can be succeeded in every mainbeam direction when error in some parameters is allowed. Those parameters are mainbeam direction, average minor lobe level, and beamwidth. The more errors are allowed, the more success cases can be achieved. A full prototype of fully spatial beamformer was constructed to test the validation of the proposed refinement method. The results in terms of radiation pattern obtained from simulation and measurement are compared. The obtained results reveal that we can reduce the range of attenuation or amplification using the proposed refinement method while some significant radiation behaviors are remained.

Acknowledgment

This work was supported by Suranaree University of Technology (SUT) and by the Office of the Higher Education Commission under NRU project of Thailand.