# Deep Learning for Inverse Design of Broadband Quasi-Yagi Antenna

Authors: Wen-Ying Zhou; Zhong-lei Mei; Mai Lu; Ya-Bo Zhu
Journal: International Journal of RF and Microwave Computer-Aided Engineering
 (2023)

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## Abstract

Deep learning (DL) approaches have been increasingly adopted to design antenna autonomously. For obtaining geometry of the broadband quasi-Yagi antenna from its physic response images directly, we propose an inverse design approach based on the optimized bidirectional symmetry GoogLeNet, which can extract the required bandwidth information to redesign the geometric parameters of antenna without changing its physical structure. It demonstrates that the bandwidth of a reference quasi-Yagi antenna is improved from 0.6 GHz to 1.15 GHz through the proposed inverse design DL approach, and the measured bandwidth value of this redesigned quasi-Yagi antenna achieves 1.16 GHz, which is improved 93% actually. The numerical and measured results indicate that the proposed DL approach could significantly improve the performance of the existed quasi-Yagi antenna and present a new attempt to apply the image processing techniques in resolving physical problem.


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## Body

## 1. Introduction

The compact broadband Yagi-Uda antenna is required in the 5G multiple-in multiple-output (MIMO) technology. Since Yagi-Uda antenna was first proposed in 1926 [1], it has become a classical end-fire antenna with high gain. Huang and Densmore introduced a quasi-Yagi antenna array design in 1991 [2], in which its driven, director, and reflector elements were all printed on the substrate, and thus the planar layout was easier to integrate because of its low profile. However, the narrow bandwidth of quasi-Yagi antenna has seriously limited its development in 5G application.The physical methods to enhance bandwidth of quasi-Yagi antenna include the improvement of feeding technology [3–5] and the structure modification of the driven and director elements [6–8]. In ref. [3, 4], Wu et al. proposed two modified structures of coplanar strip for wideband enhancement. In reference [5], Yang et al. proposed the combination of a wideband feeding slot-line and a modified bowtie driver to enhance the bandwidth of quasi-Yagi antenna, its band-notched characteristics was realized by a compact interdigital capacitor loaded loop resonator with a high quality factor. Compared with the simple structure change of the driven elements [6], the combined design methods have more advantages to expand bandwidth of quasi-Yagi antenna. For example, the combination of the modified transition coplanar strip-line and dual-dipole elements [7], and the dual-bowtie dipole driven was designed with the multimode resonance operation [8]. For the application of quasi-Yagi antenna in MIMO and millimeter-wave system, the features of simpler structures and easier integral feeding are required. Chaudhari and Ray proposed a compact 3-elements quasi-Yagi antenna with a bowtie-driven element and four microstrips line-fed [9]. Rehman et al. proposed a novel high gain two-port planar antenna, which adopted the inverse concentric Yagi director around the rectangular minipatch with a stepped impedance [10].The physical design of antennas depends on the theoretical computation and empirical formula, and the accurate geometric parameters of the designed antennas are typically obtained through the electromagnetic (EM) simulation. Complex antenna design by EM simulator is prone to error and requires large amount of computational power. In order to overcome these shortcomings and improve design efficiency, deep learning (DL) can be introduced to optimize the geometric parameters of designed antennas. Through forward learning the relationship between structure geometry and its physical response, the neural network (NN) can accurately predict the chiroptical responses of the three-dimensional chiral metamaterials [11], the phase value of metasurface structure [12], the absorptivity of the design metamaterial perfect absorbers [13], the structural color of plasmonic metasurface [14], and the geometry of decoupled antenna array [15]. In order to simplify the design process, we propose an inverse DL approach to directly design the geometry of quasi-Yagi by giving physical response images.LeNet as the first generation network of convolutional neural network (CNN), it was first proposed to recognize the document in 1998 [16]. CNN, as a successful image processing DL tool, was used to accurately identify the human face with mask by distinguishing radar signals reflection [17] and classify the 1.2 million high-resolution images in the ImageNet LSVRC-2010 contest into the 1000 different classes [18]. In addition to improving image-processing applications, the CNN can also be applied in the antenna design. Harkouss applied CNN in smart antenna design to enhance the direction of arrival (DOA) estimation performance [19]. Sahedian et al. used CNN to collect spatial information from the images for plasmonic structures design [20]. Malkiel et al. published a series of researches for inverse design of the nanophotonic structures [21–23]. By continuously improving the network, their CNN approach could generate 2D images of the target nanostructures with desired spectra. As a result, it was able to design a much wider space of geometries by arbitrary predicting image.In this article, we intend to redesign the compact quasi-Yagi antenna with bent arms [24] by using an optimized bidirectional symmetry GoogLeNet network. We first extract the information form the desired bandwidth and radiation pattern images. Then, we use image information to inversely design the geometry of the quasi-Yagi antenna with broadband enhancement. By comparing the simulated and measured results of the redesigned quasi-Yagi antenna, we can demonstrate the consistency and effectiveness of our method. Different form the traditional physical method of antenna design, the proposed DL approach can quickly improve the performance of the existed antenna to satisfy the demand in more application fields.


## 2. Inverse Design DL Approach

As the rapid development of CNN, its classic network structures like AlexNet [25], VGG-Nets [26], GoogLeNet [27], ResNet [28], and DenseNet [29] have appeared successively. Although the architectures of CNN are getting more and more complex, their image processing capabilities are getting stronger and stronger. Since inception module was introduced in the GoogLeNet framework, the performance of CNN has a significant improvement. Anjum et al. proposed an approach based on the GoogLeNet and ResNet101 to accurately classify the subtypes of the brain tumor [30]; Ran et al. designed a memristive GoogLeNet neural network circuits for neuromorphic computing [31].In this work, we propose a DL inverse design approach to predict the geometric parameters of the quasi-Yagi antenna according to the desired bandwidth and radiation pattern. Its architecture comprises two parts. The first part, i.e., the input layer, takes theS11 and the radiation pattern curves of the target antenna as the input data. The second part is the specific inverse design network, which can finally produces the optimized antenna parameters at the output nodes. The whole architecture is shown in Figure 1.Figure 1
Architecture of the proposed DL inverse design approach.


### 2.1. Mapping Relationship

LetG=d1i,d2j,H4x,l1m,l2n be the set of all supported geometry of the quasi-Yagi antenna, let S=si,R=rj be the set of one-dimensional (1D) images of all S11 curves and radiation pattern curves, each group geometric parameter is associated with a valid pair of S=si1,R=rj1. Therefore, S×R⟶G to be a training model that maps pair of geometric parameters associated with the radiation performance of the predictable quasi-Yagi antennas. In this study, we use the desired images of S=siD,R=rjD with 1D vector to be the input elements. Our training goal is to obtain the quasi-Yagi antenna with redesigned geometric parameters (model M), which have the optimal radiation performance, that is M≔G,SD,RD.


### 2.2. Obtaining the Input Images

In the optical research article like reference [32], the desired physical response of metasurface design was only the phase deviation with range of 0°~360°, and the map between input layer and output layer were set as a single channel. But for antennas, its S-parameters and radiation pattern are both important features. Since these two features were independent of each other, their values varied in an unfixed interval. Therefore, our proposed DL approach adopted two channels for training, which separately represent the S11 and the radiation pattern. The designed quasi-Yagi antenna could satisfy these two physical responses at the same time.The reference quasi-Yagi with bent arms has the simple structure similar to the classic planar quasi-Yagi [33] as shown in Figure 2(a), its images of bandwidth and radiation pattern are shown in Figures 2(b) and 2(c), respectively. One exciter and all the directors are printed on the top of the substrate, and another exciter and reflector are printed on the bottom of the substrate. All the elements material is copper with the thickness of 0.035 mm. The thickness of substrate material is 0.8 mm with the permittivity of 2.2.Figure 2
Geometry structure of the reference quasi-Yagi antenna and its desired radiation performance.
(a)
Layout(b)
S11(c)
Radiation patternAs shown in Figure2(a), five key geometric parameters decide the performance of quasi-Yagi antenna, i.e., l1, l2, d1, d2, and H4, which are the final optimized objects as the output elements. Their specific values are listed in Table 1. The desirable S11 curve in Figure 2(b) is artificially drawn based on the simulated result of the reference antenna, and this can greatly enhance the probability for obtaining realizable designs. The desired bandwidth has thus extended from 0.6 GHz to 1.15 GHz. The radiation pattern of the reference antenna is used as the desirable image, which is shown in Figure 2(c).Table 1
Comparison of different quasi-Yagi antennas.
Quasi-YagiMethodGain (dBi)Bandwidth (GHz) sim./mea.Geometric parameters (mm)d1d2H4l1l2WLReferenceEM sim.6.00.60/--5.104.4725.0327.0230.3164.0568.85This workInverse DL6.21.15/1.163.1720.3125.4026.2731.5362.5368.38


### 2.3. Inverse Design Process

Taking the images ofS11 and radiation pattern as the input layer, the values of l1, l2, d1, d2, and H4 as the output layer, we proposed an inverse DL approach based on the optimized bidirectional symmetry GoogLeNet. The network has 10 layers per channel and connected by the fully connected network (FCNN) at the last layer. The target broadband quasi-Yagi antenna can be obtained from the output layers. All activations are ReLU function.


#### 2.3.1. Improved Inception

The main hallmark in GoogLeNet framework is embedded the inception V1 version architecture (as shown in Figure3(a)) into CNN, which assemble multiple convolutions and pooling operations together to an independent network module. Combined the individual module, the entire GoogLeNet can be constructed. The characteristic pattern is influenced by the different convolution kernel numbers. Therefore, an inception module would provide multiple convolution kernel operations in parallel, the final convolution kernel numbers can be chosen through adjusting the parameters during the selection process. Since our images of S11 and radiation pattern are both 1D pictures, we take the inception 1D module replace of inception V1 to optimize the GoogLeNet network framework. The inception 1D network structure used in this paper is shown in Figure 3(b).Figure 3
Comparison between inception V1 and 1D.
(a)
Inception V1(b)
Inception 1D


#### 2.3.2. Training Process

The training data are from the simulated values obtained by EM simulation. In the frequency band from 1 GHz to 4 GHz with interval of 0.5 GHz, the varied ranges of five geometric parameters follow that:d1∈3mm,20mm, d2∈1mm,40mm, H4∈22mm,27mm, l1∈17mm,36mm, and l2∈30mm,40mm. These geometric parameters are simulated to obtain their S11 and radiation pattern images by EM simulation. Since the above optimization has far exceed the computational ability of EM simulation, we design an algorithm to randomly select 4900 simulated data with the optimized accuracy of 0.01 mm to train our proposed network, which breakthroughs the limitation of the EM simulation. The training time is about 116 minutes, and the predicted time is only 5 seconds. It should be noted that, because the variation of S11 and radiation pattern values have different influences to antenna geometry, the input point number ratio between S11 image and radiation pattern images is about 61 : 21 in the training network. To demonstrate the training performance of our proposed DL approach, 100 group input images are randomly selected, which is not included in the training set. Through comparing the output 5 geometric parameters (d1, d2, H4, l1, and l2) with their corresponding simulated values obtained by EM simulation, the accuracy of prediction ability is verified, and all percentage errors are basically less than 5%, as shown in Figure 4.Figure 4
Training performance in test group.
(a)
d1(b)
d2(c)
H4(d)
l1(e)
l2


## 2.1. Mapping Relationship

LetG=d1i,d2j,H4x,l1m,l2n be the set of all supported geometry of the quasi-Yagi antenna, let S=si,R=rj be the set of one-dimensional (1D) images of all S11 curves and radiation pattern curves, each group geometric parameter is associated with a valid pair of S=si1,R=rj1. Therefore, S×R⟶G to be a training model that maps pair of geometric parameters associated with the radiation performance of the predictable quasi-Yagi antennas. In this study, we use the desired images of S=siD,R=rjD with 1D vector to be the input elements. Our training goal is to obtain the quasi-Yagi antenna with redesigned geometric parameters (model M), which have the optimal radiation performance, that is M≔G,SD,RD.


## 2.2. Obtaining the Input Images

In the optical research article like reference [32], the desired physical response of metasurface design was only the phase deviation with range of 0°~360°, and the map between input layer and output layer were set as a single channel. But for antennas, its S-parameters and radiation pattern are both important features. Since these two features were independent of each other, their values varied in an unfixed interval. Therefore, our proposed DL approach adopted two channels for training, which separately represent the S11 and the radiation pattern. The designed quasi-Yagi antenna could satisfy these two physical responses at the same time.The reference quasi-Yagi with bent arms has the simple structure similar to the classic planar quasi-Yagi [33] as shown in Figure 2(a), its images of bandwidth and radiation pattern are shown in Figures 2(b) and 2(c), respectively. One exciter and all the directors are printed on the top of the substrate, and another exciter and reflector are printed on the bottom of the substrate. All the elements material is copper with the thickness of 0.035 mm. The thickness of substrate material is 0.8 mm with the permittivity of 2.2.Figure 2
Geometry structure of the reference quasi-Yagi antenna and its desired radiation performance.
(a)
Layout(b)
S11(c)
Radiation patternAs shown in Figure2(a), five key geometric parameters decide the performance of quasi-Yagi antenna, i.e., l1, l2, d1, d2, and H4, which are the final optimized objects as the output elements. Their specific values are listed in Table 1. The desirable S11 curve in Figure 2(b) is artificially drawn based on the simulated result of the reference antenna, and this can greatly enhance the probability for obtaining realizable designs. The desired bandwidth has thus extended from 0.6 GHz to 1.15 GHz. The radiation pattern of the reference antenna is used as the desirable image, which is shown in Figure 2(c).Table 1
Comparison of different quasi-Yagi antennas.
Quasi-YagiMethodGain (dBi)Bandwidth (GHz) sim./mea.Geometric parameters (mm)d1d2H4l1l2WLReferenceEM sim.6.00.60/--5.104.4725.0327.0230.3164.0568.85This workInverse DL6.21.15/1.163.1720.3125.4026.2731.5362.5368.38


## 2.3. Inverse Design Process

Taking the images ofS11 and radiation pattern as the input layer, the values of l1, l2, d1, d2, and H4 as the output layer, we proposed an inverse DL approach based on the optimized bidirectional symmetry GoogLeNet. The network has 10 layers per channel and connected by the fully connected network (FCNN) at the last layer. The target broadband quasi-Yagi antenna can be obtained from the output layers. All activations are ReLU function.


### 2.3.1. Improved Inception

The main hallmark in GoogLeNet framework is embedded the inception V1 version architecture (as shown in Figure3(a)) into CNN, which assemble multiple convolutions and pooling operations together to an independent network module. Combined the individual module, the entire GoogLeNet can be constructed. The characteristic pattern is influenced by the different convolution kernel numbers. Therefore, an inception module would provide multiple convolution kernel operations in parallel, the final convolution kernel numbers can be chosen through adjusting the parameters during the selection process. Since our images of S11 and radiation pattern are both 1D pictures, we take the inception 1D module replace of inception V1 to optimize the GoogLeNet network framework. The inception 1D network structure used in this paper is shown in Figure 3(b).Figure 3
Comparison between inception V1 and 1D.
(a)
Inception V1(b)
Inception 1D


### 2.3.2. Training Process

The training data are from the simulated values obtained by EM simulation. In the frequency band from 1 GHz to 4 GHz with interval of 0.5 GHz, the varied ranges of five geometric parameters follow that:d1∈3mm,20mm, d2∈1mm,40mm, H4∈22mm,27mm, l1∈17mm,36mm, and l2∈30mm,40mm. These geometric parameters are simulated to obtain their S11 and radiation pattern images by EM simulation. Since the above optimization has far exceed the computational ability of EM simulation, we design an algorithm to randomly select 4900 simulated data with the optimized accuracy of 0.01 mm to train our proposed network, which breakthroughs the limitation of the EM simulation. The training time is about 116 minutes, and the predicted time is only 5 seconds. It should be noted that, because the variation of S11 and radiation pattern values have different influences to antenna geometry, the input point number ratio between S11 image and radiation pattern images is about 61 : 21 in the training network. To demonstrate the training performance of our proposed DL approach, 100 group input images are randomly selected, which is not included in the training set. Through comparing the output 5 geometric parameters (d1, d2, H4, l1, and l2) with their corresponding simulated values obtained by EM simulation, the accuracy of prediction ability is verified, and all percentage errors are basically less than 5%, as shown in Figure 4.Figure 4
Training performance in test group.
(a)
d1(b)
d2(c)
H4(d)
l1(e)
l2


## 2.3.1. Improved Inception

The main hallmark in GoogLeNet framework is embedded the inception V1 version architecture (as shown in Figure3(a)) into CNN, which assemble multiple convolutions and pooling operations together to an independent network module. Combined the individual module, the entire GoogLeNet can be constructed. The characteristic pattern is influenced by the different convolution kernel numbers. Therefore, an inception module would provide multiple convolution kernel operations in parallel, the final convolution kernel numbers can be chosen through adjusting the parameters during the selection process. Since our images of S11 and radiation pattern are both 1D pictures, we take the inception 1D module replace of inception V1 to optimize the GoogLeNet network framework. The inception 1D network structure used in this paper is shown in Figure 3(b).Figure 3
Comparison between inception V1 and 1D.
(a)
Inception V1(b)
Inception 1D


## 2.3.2. Training Process

The training data are from the simulated values obtained by EM simulation. In the frequency band from 1 GHz to 4 GHz with interval of 0.5 GHz, the varied ranges of five geometric parameters follow that:d1∈3mm,20mm, d2∈1mm,40mm, H4∈22mm,27mm, l1∈17mm,36mm, and l2∈30mm,40mm. These geometric parameters are simulated to obtain their S11 and radiation pattern images by EM simulation. Since the above optimization has far exceed the computational ability of EM simulation, we design an algorithm to randomly select 4900 simulated data with the optimized accuracy of 0.01 mm to train our proposed network, which breakthroughs the limitation of the EM simulation. The training time is about 116 minutes, and the predicted time is only 5 seconds. It should be noted that, because the variation of S11 and radiation pattern values have different influences to antenna geometry, the input point number ratio between S11 image and radiation pattern images is about 61 : 21 in the training network. To demonstrate the training performance of our proposed DL approach, 100 group input images are randomly selected, which is not included in the training set. Through comparing the output 5 geometric parameters (d1, d2, H4, l1, and l2) with their corresponding simulated values obtained by EM simulation, the accuracy of prediction ability is verified, and all percentage errors are basically less than 5%, as shown in Figure 4.Figure 4
Training performance in test group.
(a)
d1(b)
d2(c)
H4(d)
l1(e)
l2


## 3. Discussion and Results Analyses

Through adjusting the network structure, the LeNet and FCNN can also be used to predict the broadband quasi-Yagi. The 33 groups of antenna geometries are separately obtained by LeNet and FCNN. Their corresponding physic responses ofS11 and radiation pattern are simulated by EM simulation. For comparison, the predictable performances based on the LeNet, FCNN, and our DL approach are shown in Figure 5, respectively.Figure 5
Comparison of simulated results of the predicted quasi-Yagi antennas.
(a)
S11 (FCNN)(b)
S11 (LeNet)(c)
S11 (this work)(d)
Radiation pattern (FCNN)(e)
Radiation pattern (LeNet)(f)
Radiation pattern (this work)Figure5 shows that there are some differences between predictable curves and the desirable curve in different predicted quasi-Yagi antennas obtained by three DL approaches. From Figures 5(a)–5(c), the similarity degree of S11 curves is obviously less than that of radiation pattern curves (in Figures 5(d)–5(f)). Compared with FCNN and LeNet, our proposed approaches have better predictable performance.To further quantitatively analyze the predictable performance of three DL approaches, the similarity degree ofS11 curves is calculated by mathematical models.


### 3.1. Fréchet’s Distance Analysis

The Fréchet distance [34] is a measure of similarity degree between different curves in mathematics. We use the Fréchet distance to measure the similarity degree of the S11 curves (Pi) of each group of predicted antennas (33 groups in total) with the desirable S11 curves (Dj) in our work. Pm and Dn are any point on two curves, as shown in Figure 6. The Fréchet distance error is the difference value between the predicted values and the desired values.Figure 6
Fréchet distance description.Then the minimum network distance betweenPi and Dj can be calculated by the following:
(1)dPm,Dn=mindPi,Dj+lengthPm,Pi+lengthDj,Dn,dPi+1,Dj+lengthPm,Pi+1+lengthDj,Dn,dPi,Dj+1+lengthPs,Pi+lengthDj+1,Dn,dPi+1,Dj+1+lengthPm,Pi+1+lengthDj+1,Dn.The comparison results of Fréchet’s distance obtained by three DL approaches are shown in Figure7. The abscissa shows the different error intervals of the Fréchet distance, the number “0” represents the minimum error, the number “10” represents the maximum error. The ordinate represents the occurrence frequency of the three DL approaches in different intervals. From Figure 7, among the 33 groups of antennas predicted by FCNN, the Fréchet distances error which are lower than 2 have the ratio of 3%, and the other 97% are in the interval [2, 4]. Among that antennas predicted by LeNet, the Fréchet distances error which are lower than 2 only take the proportion of 0.06%, 67% are in the interval [2, 4], and 27% are bigger than 4. Among that, antennas predicted by our proposed DL approach, the Fréchet distances which are lower than 2 take the proportion of 27% and 68% are in the interval [2, 4]. It demonstrates that the Fréchet distances of our proposed method have lower occurrence frequency in smaller error intervals.Figure 7
Fréchet’s distance comparison.


### 3.2. Hausdorff’s Distance Analysis

Hausdorff’s [35] distance measures how far two subsets of a metric space are from each other in mathematics. It is the longest of all the distances from a point in one set to the closest point in the other set, as shown in Figure 8.Figure 8
Hausdorff’s distance description.AssumeX and Y be two subsets of a geometric distance (M, d), their Hausdorff’s distance can be defined by
(2)dHX,Y=maxsupx∈Xdx,Y,supy∈YdX,y,where sup represents the supremum, inf means the infimum, and dx,B=infb∈Bda,b quantifies the distance from a point a∈X to the subset B⊆X. In this work, Y represents the predictable S11 curve, which is the simulated result of each predicted antennas, and X represents the desirable S11 curve. The Hausdorff distance error is the difference value between the predicted values and the desired values.We compare the sum of 33 Hausdorff distances separately obtained by three DL approaches, and the results are shown as Figure9. It reveals that, among the 33 groups of antennas predicted by FCNN, the Hausdorff distances error which are lower than 2 have the ratio of 6%, and the other 94% are in the interval [2, 4]. Among that antennas predicted by LeNet, the Hausdorff distances error which are lower than 2 take the proportion of 45%, 12% are in the interval [2, 4], and the other 24% are bigger than 4. Among that antennas predicted by our proposed DL approach, the Hausdorff distances error which are lower than 2 take the proportion of 30% and the other 61% are in the interval [2, 4]. It demonstrates that the Hausdorff distances of our proposed method have lower occurrence frequency in smaller error intervals.Figure 9
Hausdorff’s distance comparison.The above comparison results demonstrate that the predicted antenna which obtained by our proposed DL approach owns the highest similarity ofS11 curves.


### 3.3. Comparison of the Predictable Antenna with the Optimal Performance

Through comparing the 33 groups of geometries of antennas, obtained by the above three DL approaches, we select 3 groups optimal antennas to simulate their physical response. The obtainedS11 and radiation results are compared with the desirable curves, as shown in Figure 10.Figure 10
Performance comparison of the selected optimal quasi-Yagi antennas.
(a)
S11(b)
Radiation patternFigure10(a) shows that, although the best bandwidth of the selected antennas, obtained by the three DL algorithms, has the same value of 1.15 GHz, the predicted quasi-Yagi antenna obtained by our proposed DL approach has the optimal S11 curve with the smallest values, which cannot be obtained from FCNN and LeNet. From another perspective, the quasi-Yagi antennas predicted by our proposed DL approach cannot only get the similar S11 images with the desired one but also can obtain the best one. From Figure 10(b), it can be seen that the radiation patterns obtained by the three DL algorithms are similar to the reference antenna.


## 3.1. Fréchet’s Distance Analysis

The Fréchet distance [34] is a measure of similarity degree between different curves in mathematics. We use the Fréchet distance to measure the similarity degree of the S11 curves (Pi) of each group of predicted antennas (33 groups in total) with the desirable S11 curves (Dj) in our work. Pm and Dn are any point on two curves, as shown in Figure 6. The Fréchet distance error is the difference value between the predicted values and the desired values.Figure 6
Fréchet distance description.Then the minimum network distance betweenPi and Dj can be calculated by the following:
(1)dPm,Dn=mindPi,Dj+lengthPm,Pi+lengthDj,Dn,dPi+1,Dj+lengthPm,Pi+1+lengthDj,Dn,dPi,Dj+1+lengthPs,Pi+lengthDj+1,Dn,dPi+1,Dj+1+lengthPm,Pi+1+lengthDj+1,Dn.The comparison results of Fréchet’s distance obtained by three DL approaches are shown in Figure7. The abscissa shows the different error intervals of the Fréchet distance, the number “0” represents the minimum error, the number “10” represents the maximum error. The ordinate represents the occurrence frequency of the three DL approaches in different intervals. From Figure 7, among the 33 groups of antennas predicted by FCNN, the Fréchet distances error which are lower than 2 have the ratio of 3%, and the other 97% are in the interval [2, 4]. Among that antennas predicted by LeNet, the Fréchet distances error which are lower than 2 only take the proportion of 0.06%, 67% are in the interval [2, 4], and 27% are bigger than 4. Among that, antennas predicted by our proposed DL approach, the Fréchet distances which are lower than 2 take the proportion of 27% and 68% are in the interval [2, 4]. It demonstrates that the Fréchet distances of our proposed method have lower occurrence frequency in smaller error intervals.Figure 7
Fréchet’s distance comparison.


## 3.2. Hausdorff’s Distance Analysis

Hausdorff’s [35] distance measures how far two subsets of a metric space are from each other in mathematics. It is the longest of all the distances from a point in one set to the closest point in the other set, as shown in Figure 8.Figure 8
Hausdorff’s distance description.AssumeX and Y be two subsets of a geometric distance (M, d), their Hausdorff’s distance can be defined by
(2)dHX,Y=maxsupx∈Xdx,Y,supy∈YdX,y,where sup represents the supremum, inf means the infimum, and dx,B=infb∈Bda,b quantifies the distance from a point a∈X to the subset B⊆X. In this work, Y represents the predictable S11 curve, which is the simulated result of each predicted antennas, and X represents the desirable S11 curve. The Hausdorff distance error is the difference value between the predicted values and the desired values.We compare the sum of 33 Hausdorff distances separately obtained by three DL approaches, and the results are shown as Figure9. It reveals that, among the 33 groups of antennas predicted by FCNN, the Hausdorff distances error which are lower than 2 have the ratio of 6%, and the other 94% are in the interval [2, 4]. Among that antennas predicted by LeNet, the Hausdorff distances error which are lower than 2 take the proportion of 45%, 12% are in the interval [2, 4], and the other 24% are bigger than 4. Among that antennas predicted by our proposed DL approach, the Hausdorff distances error which are lower than 2 take the proportion of 30% and the other 61% are in the interval [2, 4]. It demonstrates that the Hausdorff distances of our proposed method have lower occurrence frequency in smaller error intervals.Figure 9
Hausdorff’s distance comparison.The above comparison results demonstrate that the predicted antenna which obtained by our proposed DL approach owns the highest similarity ofS11 curves.


## 3.3. Comparison of the Predictable Antenna with the Optimal Performance

Through comparing the 33 groups of geometries of antennas, obtained by the above three DL approaches, we select 3 groups optimal antennas to simulate their physical response. The obtainedS11 and radiation results are compared with the desirable curves, as shown in Figure 10.Figure 10
Performance comparison of the selected optimal quasi-Yagi antennas.
(a)
S11(b)
Radiation patternFigure10(a) shows that, although the best bandwidth of the selected antennas, obtained by the three DL algorithms, has the same value of 1.15 GHz, the predicted quasi-Yagi antenna obtained by our proposed DL approach has the optimal S11 curve with the smallest values, which cannot be obtained from FCNN and LeNet. From another perspective, the quasi-Yagi antennas predicted by our proposed DL approach cannot only get the similar S11 images with the desired one but also can obtain the best one. From Figure 10(b), it can be seen that the radiation patterns obtained by the three DL algorithms are similar to the reference antenna.


## 4. Analysis of Experimental Results

Through our proposed DL approach training, the reference quasi-Yagi antenna geometry is redesigned to the desirable broadband quasi-Yagi antenna, and their specific values are listed in Table1. Because d4 effect on the antenna performance is little which has been proved in our previous study, the redesigned antenna can be further optimized by reducing d4 from 22.75 mm to 8 mm. Its geometric parameters are shown in Figure 11(a), and the fabrication prototype is shown in Figure 11(b).Figure 11
Redesigned broadband quasi-Yagi antenna and the fabrication prototype.
(a)
Layout(b)
Photograph of the fabrication prototypeThe fabrication of the redesigned quasi-Yagi antenna is measured to prove the reliability of our inverse design. The measured values ofS11 and radiation pattern are compared with their simulated values, and the results are shown in Figure 12.Figure 12
Comparison of simulated and measured values of the predicted antenna.
(a)
Comparison ofS11(b)
Simulated value of radiation pattern(c)
Measured value of radiation patternFigure12(a) shows that the experimental value of the bandwidth is 1.16 GHz. Figures 12(b) and 12(c) show that the measured results of radiation pattern are basically agreed with the simulated results. The errors of measured values are mainly caused by the machining precision and the substrate material.The compared results between the reference and the redesigned quasi-Yagi antenna are listed in Table1.Table1 shows that the total geometry of the redesigned quasi-Yagi antenna is a bit smaller than the reference antenna, but the bandwidth is actually improved 93%.


## 5. Conclusions

Different from the traditional inverse design method of antenna, we propose an inverse DL approach to directly redesign the reference quasi-Yagi antenna according to the desirable radiation performance images. It means that the relationship between antenna geometry and its physical response could be translated to the images learning problem. In this article, both theS11 and radiation pattern images are learned at the same time to predict the geometric parameters of the required broadband quasi-Yagi antenna, which improves the design efficiency. The bandwidth of the redesigned quasi-Yagi is improved 92% with no radiation pattern deterioration. The analysis of Fréchet’s distance error and Hausdorff’s distance error confirms that our proposed bidirectional symmetry GoogLeNet can obtain the optimal quasi-Yagi antenna with the best radiation performance. The measured results of the redesigned quasi-Yagi antenna demonstrate the effectiveness of our proposed inverse design DL approach. Our work highlights the convenience of inverse design from physical response images to antenna geometry with desired performance, which can greatly shorten the design time in the complicated structure design.



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*Source: 7819156-2023-02-09.xml*