High-Dimensional Neural-Network Technique and Applications to Microwave Filter Modeling

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 58, NO. 1, JANUARY 2010 145

High-Dimensional Neural-Network Technique andApplications to Microwave Filter Modeling

Humayun Kabir, Ying Wang, Member, IEEE, Ming Yu, Fellow, IEEE, and Qi-Jun Zhang, Fellow, IEEE

Abstract—Neural networks are useful for developing fast andaccurate parametric model of electromagnetic (EM) structures.However, existing neural-network techniques are not suitablefor developing models that have many input variables becausedata generation and model training become too expensive. Inthis paper, we propose an efficient neural-network method forEM behavior modeling of microwave filters that have many inputvariables. The decomposition approach is used to simplify theoverall high-dimensional neural-network modeling problem into aset of low-dimensional sub-neural-network problems. By incorpo-rating the knowledge of filter decomposition with neural-networkdecomposition, we formulate a set of neural-network submodelsto learn filter subproblems. A new method to combine the sub-models with a filter empirical/equivalent model is developed. Anadditional neural-network mapping model is formulated withthe neural-network submodels and empirical/equivalent model toproduce the final overall filter model. An -plane waveguide filtermodel and a side-coupled circular waveguide dual-mode filtermodel are developed using the proposed method. The result showsthat with a limited amount of data, the proposed method canproduce a much more accurate high-dimensional model comparedto the conventional neural-network method and the resultingmodel is much faster than an EM model.

Index Terms—Computer-aided design (CAD), high-dimensionalparametric modeling, microwave filter, neural network, optimiza-tion, simulation.

I. INTRODUCTION

N EURAL networks have been recognized as useful alter-natives for device modeling where a mathematical model

is not available or time-consuming simulation is required. Theycan be utilized to model multidimensional nonlinear relation-ships. The evaluation time of a neural-network model is alsofast. For these reasons, neural networks have been used for var-ious modeling and design applications [1], [2] including pas-sive microwave structures [3], [4], electromagnetic (EM) com-puter-aided design (CAD) [5], [6], transistors [7], amplifiers [8],

Manuscript received July 07, 2009; revised October 01, 2009. First publishedDecember 22, 2009; current version published January 13, 2010. This work wassupported in part by Natural Sciences and Engineering Research Council ofCanada, in part by COM DEV Ltd., and in part by the Ontario Centers of Ex-cellence.

H. Kabir and Q.-J. Zhang are with the Department of Electronics, CarletonUniversity, Ottawa, ON, Canada K1S 5B6 (e-mail: hkabir@doe.carleton.ca;qjz@doe.carleton.ca).

Y. Wang is with the Faculty of Engineering and Applied Science, Universityof Ontario Institute of Technology, Oshawa, ON, Canada L1H 7K4 (e-mail:Ying.Wang@uoit.ca).

M. Yu is with COM DEV Ltd., Cambridge, ON, Canada N1R 7H6 (e-mail:Ming.Yu@ieee.org).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMTT.2009.2036412

antennas [9], waveguide filters [10]–[12], microwave optimiza-tion [13], [14], library of models [15], etc. Neural networks havenot only been used for developing microwave device models,but also have been used in optimization processes where theneural models are combined with full-wave simulation tools[16]–[18]. The general idea of neural-network-based CAD andoptimization is that we develop neural-network models for EMstructures and incorporate the models in circuit simulators. Thisallows circuit-level simulation speed with EM-level accuracy.

In this paper, we focus on neural-network-based modelingof microwave filters. Accurate model is essential for the firstpass design success. Conventional EM modeling method is thefirst option to obtain an accurate model. However, the modelevaluation time of this method is long, especially when repet-itive model evaluations are required. During design optimiza-tion, values of geometrical variables are required to be changedmany times and each time a complete reevaluation of the modelis required. For this reason, the EM model becomes too expen-sive. An alternative to the EM model is a neural-network modelwhose inputs are geometrical variables [1]–[3], [12], [19]–[24].The neural-network model can provide solutions quickly forvarious values of geometrical input variables.

Due to increasing complexity and variety of microwave struc-tures, the number of design variables per structure is on therise. In order to develop an accurate neural-network model thatcan represent EM behavior of filters over a range of valuesof geometrical variables, we need to provide EM data at suf-ficiently sampled points in the space of geometrical variables[1], [2]. The amount of data required increases very fast withthe number of input variables of the model. For this reason, de-veloping a neural-network model that has many input variablesbecomes challenging as data generation becomes too expen-sive. Therefore, we need an effective method to develop accu-rate high-dimensional neural-network models without requiringmassive data.

Various advanced neural-network structures have been inves-tigated for microwave modeling such as knowledge-based neuralnetworks [25], [26] for simplifying input–output relationship. Itreduces the cost of neural-network training for highly nonlinearinput–output modeling problems. However, it does not have themechanism to address the challenge of high-dimensional mod-eling problems directly. Modular neural network is an interestingtechnique, which has the potential to address high-dimensionalmodeling problem because of neural-network decomposition. Ithas been investigated within the artificial neural-network com-munity for applications such as face detection [27], [28], voicerecognition [29], pattern recognition [30], [31], directional relayalgorithm for power transmission line [32], problem simplifica-tion [33], etc. The modular concept has also been investigated for

146 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 58, NO. 1, JANUARY 2010

microwave optimization such as dielectric resonator filters [34],microstrip corporate feeds [35], power amplifiers [36], semicon-ductorprocesscharacterization[37],antennas[38],etc.Thistech-nique decomposes a complex neural network into several simplesub-neural-network modules. The modular neural-network tech-nique has been used to improve the learning capability of neuralnetworks.However, theexistingmodularneural-networkmethodisnotdirectlysuitableforhigh-dimensionalneural-networkmod-eling of microwave filters because it has not been formulated toaccommodate the knowledge of microwave filter formulas. An-other problem with the existing neural-network decompositionis the absence of connections between neural-network decompo-sition and microwave filter decomposition.

Recently, microwave filters have been modeled and designedusing neural-network techniques [11]. In [11], we decomposedthe filter structures into substructures and then developed in-verse submodels for the substructures. The inputs of the inversesubmodels are coupling parameters and outputs are geometricalparameters. These submodels produce filter dimensions fromgiven coupling values. The main objective of [11] was to produceneural-network inverse models of filter components. The advan-tage of the method is that we can obtain filter dimensions for agiven coupling matrix without repetitive EM model evaluation.

In this paper, we propose a new method to obtain a completeforward model for filters that have many geometrical variables.We decompose a filter structure into substructures and developforward submodels. The inputs of the submodels are geomet-rical dimensions and outputs are coupling parameters. Thesesubmodels are combined with filter equivalent-circuit model toproduce an approximate solution of the entire filter. A mappingmodel is trained and used to make the high-dimensional modelas accurate as the EM model. The main objective is to developa high-dimensional neural-network model, which is too expen-sive to develop using a conventional neural-network approach.The new method is used to develop complex filter models thathold many input variables. Results show that using the proposedmethod, we can develop accurate high-dimensional neural-net-work models in an inexpensive way. The evaluation time of theproposed neural-network model is faster than that of the EMmodel. This makes the proposed method effective and usefulfor design optimization where many geometrical design vari-ables need to be changed and EM behavior needs to be evalu-ated repetitively.

II. PROPOSED HIGH-DIMENSIONAL MODELING APPROACH

A. Problem Statement

The main objective is to obtain fast parametric models forfilters that hold many design variables, which are mainly ge-ometrical parameters. Let us assumeto be an -vector containing all the input variables of a model,e.g., iris length, cavity length, bandwidth, etc. for a filter. Let

be an -vector containing output pa-rameters such as an -parameter of the filter. A conventionalneural-network model for the problem is defined as

where defines the input–output relationship and is a neural-network internal weight vector. In this approach, we use a multi-layer perceptron or a radial-basis-function neural network [2] torepresent the entire function of (1) with represented by inputneurons and represented by output neurons. This conventionalapproach is suitable for developing simple filter models wherethe number of input variables is small. On the other hand, whena filter model has many input variables, a massive amount ofdata are required for neural-network model training to achievegood accuracy. This massive data generation and model trainingbecome too expensive and impractical. To overcome this limita-tion, we propose to use the decomposition approach to simplifythe high-dimensional problem into a set of small subproblems.Let from to represent simple subfunctions, which de-fine the input–output relationships of a set of simple functionsrepresenting various partial information of of (1). Eachof the subfunctions is defined by small number of input vari-ables and the input–output relationship becomes simpler thanthe overall high-dimensional function. In this way, cost of datageneration and model development is reduced. However, thedefinition of partial information or the formulation of neural-network submodels will not be effective unless we combine thefilter decomposition concept with neural-network decomposi-tion. Furthermore, the question of how to recombine the sub-models to form the final overall filter model and recover themissing information between subproblems must be answeredfor the neural-network decomposition.

B. Neural-Network Submodel and Filter Decomposition

We formulate neural-network decomposition together withfilter decomposition. A filter with many design variables is de-composed into several substructures, each representing a spe-cific part of the filter. Neural-network submodels are then devel-oped to represent the substructures. Let us assume that a filteris decomposed into types of substructures. Let be a vectorcontaining the design variables of the th substructure and bea vector containing the output parameters of the th substructure.As an example, the input vector contains geometrical param-eters such as length and width of an iris, and the output vector

contains electrical parameters such as coupling coefficientsof the iris. A neural-network submodel for the substructure isdefined as

where defines the geometrical to electrical relationship of theth submodel, is a vector containing neural-network weight

parameters for the th submodel, and . Thevector is a subset of the overall input vector and is expressedas

where is a selection matrix containing 1’s and 0’s in order topick corresponding inputs of submodel from the overall inputvector .

KABIR et al.: HIGH-DIMENSIONAL NEURAL-NETWORK TECHNIQUE AND APPLICATIONS TO MICROWAVE FILTER MODELING 147

In order to formulate meaningful submodels for filter appli-cations, we need to combine the filter decomposition conceptswith the submodel of (2) and (3). In microwave waveguide fil-ters, the electrical couplings between various sections of thefilter is dominantly determined by the physical/geometrical pa-rameters of the corresponding parts of the filter structure, andslightly affected by the geometrical parameters of other sections[10], [39]. Based on this concept, we use the matrix to se-lect the geometrical parameters of the relevant part of the filterignoring other parts, and use to represent the electrical cou-pling between the selected parts of the filter.

Data for each submodel is generated using an EM simulatorand neural-network submodels are then trained. Let us assume

to be the number of training samples required to developthe neural-network submodel . The submodel is developed byoptimizing the internal weight vector to minimize the errorbetween outputs and training data. The training error of sub-model is expressed as

where vector is the th sample of the training data for inputneurons of the th submodel, which contains the values of geo-metrical parameters of the th substructure, and vector is the

th sample of the training data for output neurons, which con-tains the EM solution of the th substructure. Data generation forsubmodels becomes less expensive than that for the overall filtermodel because the submodels contain fewer input variables thanthe overall filter model and the input–output relationships of thesubmodels become simpler than that of the overall filter model.

C. Integration of Neural-Network Submodels WithEmpirical/Equivalent-Circuit Model

The neural-network submodels should be recombined toform the overall filter model. Here, we formulate an approachwhere a filter empirical/equivalent-circuit model is used to ob-tain the solution of the overall filter by using the outputs fromthe neural-network submodels. Some of the neural-networksubmodels may be used multiple times as the same junctionmay appear several times in the overall model. For example,in a four-pole -plane filter there are three internal irises.We can develop one model of the internal iris and use it threetimes. Since the iris submodel is trained with a range of valuesof length, different iris submodels can be represented withthe same neural-network iris submodel with different valuesof . Multiple uses of submodels become a big advantageof the proposed method. In this way, we can obtain all thesubmodels needed for an overall filter model by training onlya few neural-network submodels. Let be the number ofneural-network submodels needed to form the overall filtermodel. The equivalent-circuit model is expressed in terms ofthe outputs of the neural-network submodels as

where is a vector containing approximate values of the out-puts of the overall filter, represents the empirical/equiva-lent-circuit function, and to are electrical parameters ob-tained from submodels.

The type of operation in (5) is simple and insignificant interms of computational cost. Thus, an approximate model ofthe overall filter is obtained by combining the neural-networksubmodels and the empirical/equivalent-circuit model.

D. Neural-Network Mapping Model

The outputs from the neural-network submodels providevalues of the electrical parameters (e.g., coupling matrix for afilter), which are approximate since effects of high order modesare lost due to decomposition of the overall filter. Thus, thesolution obtained from the empirical/equivalent-circuit modelis also approximate. Here, we propose an additional neural-net-work model, called the neural-network mapping model, tomap the approximate solution to the accurate EM solution ofthe overall filter. Samples of the overall filter are generated toobtain the training data for the mapping model. Based on theconcept of prior-knowledge input [26], we formulate the inputsof the mapping model using the approximate solution , andthe input variables of the overall filter, . The outputs are theaccurate solution of the overall filter that corresponds to .Thus, the neural-network mapping model is defined as

where defines the input–output relationship of the map-ping model and is a vector containing neural-network in-ternal weight parameters. Let us assume that we need sam-ples of the overall filter to train the mapping model accurately.The neural-network mapping model is developed by minimizingthe error between EM data and neural-network output by opti-mizing neural-network internal weight parameters. The trainingerror of the mapping model is expressed as

where is the th sample of training data for the output neu-rons and which is the EM solution of the overall filter.

The mapping model works well even though the number ofits inputs is higher than the original number of inputs becausethe input–output relationship (mapping task) of the mappingmodel becomes simple since an approximate solution is placedas a part of the model’s inputs. This makes the optimization ofthe neural-network internal weight parameters straightforwardduring training of the mapping model. For this reason, the map-ping model can be developed accurately with a few samples ofthe overall filter. In this way, the number of expensive EM simu-lation of the overall filter is reduced. As a result, data generationand model training in the proposed method become feasible.The mapping model can be a single model or a set of modelseach representing an individual output parameter of the overallmodel.

Fig. 1. Diagram of the proposed high-dimensional modeling structure.

E. Overall Modeling Structure

An accurate high-dimensional model representing theoverall filter is constructed by combining the neural-networksubmodels, circuit model, and neural-network mapping model.The diagram of the overall high-dimensional modeling struc-ture is presented in Fig. 1.

The neural-network mapping model, as defined in (6), can beexpressed in terms of the equivalent-circuit model of (5) as

We can further express (8) in terms of the neural-network sub-models defined in (2) as

Substituting the relationship of (3) in (9) yields

which is equivalent to

where is a vector containing neural-network internal weightparameters of the high-dimensional neural-network model.In (10), the vectors to contain weight parameters ofthe neural-network submodels and the vector containsweight parameters of the neural-network mapping model.These vectors are optimized during neural-network training ofthe submodels and the mapping model. The vector is op-timized after the optimization of the vectors to . Whenthe overall high-dimensional model is constructed combiningthe trained neural-network submodels and mapping model, thevectors to and all together become equivalent tothe vector of (11).

The relationship of (11) is equivalent to that of (1), except(11) is a combination of several simple submodels each withfew input variables, whereas (1) is a single complicated modelwith many input variables. The vector of (1) is equivalent tothe vector of (11). The difference is that the vector is opti-mized step by step through neural-network submodels and map-ping model training. Thus, in the proposed method, a combina-tion of several low-dimensional submodels, circuit model, andneural-network mapping model produces the overall high-di-mensional model.

In the proposed method, a few expensive data of the overallfilter are needed for the neural-network mapping model, as ex-plained earlier. On the other hand, in the conventional method,many expensive data are required to achieve a reasonable accu-racy because of two reasons, which are: 1) the model is a singlefunction of many input variables as defined in (1) and 2) the re-lationship of (1), which relates geometrical to circuit parametersdirectly, is complicated.

Let represent data generation time per sample of an overallfilter. Let represent the number of samples of data of theoverall filter required for the neural-network model in the con-ventional approach. The cost of data generation in the conven-tional method is expressed as

Let represent data generation time per sample for submodel .As defined before, we assume and represent the numberof samples of data required to develop neural-network submodel

and mapping model, respectively. The cost of data generationin the proposed method is expressed as

where and is the number of types of substruc-tures decomposed from an overall structure, as defined earlierin Section II-B. Data generation time per sample of the overallfilter is much more expensive than that of a submodel, i.e.,

. The proposed method requires much less data of the overallfilter, i.e., . For these reasons, the data generationcost of the proposed method becomes less than that ofthe conventional method , i.e., . Training timeincreases with the number of model input variables, number ofhidden neurons, and number of training data. The number ofinput variables for submodels is low. The input–output functionis also simple, which translates into a low number of hidden neu-rons. For these reasons, the training time for the submodel be-comes short. Thus, the total model training time of the proposedmethod becomes much less than that of the conventionalmethod , i.e., . The relationship of the total modelgeneration cost of the proposed and the conventional method isexpressed as

This describes how the total time for data generation and modeltraining of the proposed method is much less than those of theconventional method.

Fig. 2. Flow diagram of the proposed high-dimensional neural-network mod-eling approach.

III. ALGORITHM FOR PROPOSED HIGH-DIMENSIONAL

MODEL DEVELOPMENT

We describe an overall high-dimensional modeling algo-rithm. The flow diagram of the algorithm is presented in Fig. 2.

The steps are described as follows.Step 1) Identify the parts of an overall filter that can be used

as substructures. For a waveguide filter, discontinu-ities can be decomposed into substructures. Decom-pose the overall filter into substructures.

Step 2) Generate training data of the decomposed substruc-tures using EM simulations. Standard sampling ap-proach can be employed for this purpose.

Step 3) Train and test neural-network submodels for all thedecomposed substructures.

Step 4) If the submodels are accurate, go to Step 5). Else,generate some more data of the substructures bysampling intermediate points using EM simulation,add those to the existing data, and go to the Step 3).

Step 5) Generate a few data of the overall filter using EMsimulation. Sweep the input variables and obtaincorresponding output solutions of the overallfilter.

Fig. 3. Diagram of a four-pole�-plane filter. The filter model holds eight inputvariables including five geometrical dimensions, bandwidth, center frequency,and frequency.

Step 6) Combine the neural-network submodels and the em-pirical/equivalent-circuit model.

Step 7) Supply the samples of the input variables tothe combined neural-network submodels and em-pirical/equivalent-circuit model to obtain samples ofapproximate solution of the overall filter.

Step 8) Using the concept of prior knowledge input [26], as-semble training data for the mapping model. Usethe samples of and of Step 7) as the data forthe input neurons. Use the samples of that corre-sponds to the samples of as the data for the outputneurons. Train the neural-network mapping modelusing some of the assembled data. Test the mappingmodel with the rest of the data. If accuracy is satis-fied, go to Step 9). Else, generate a few more data ofthe overall filter, add those to the existing data of theoverall filter and go to Step 7).

Step 9) Combine the neural-network submodels, empirical/equivalent-circuit model, and neural-network map-ping model, as described in Section II-E, to obtainthe overall model of the filter.

IV. EXAMPLES

A. Proposed Modeling Technique for -Plane Filters

We illustrate the proposed modeling method through a four-pole -plane filter model development. The diagram of the filteris shown in Fig. 3. The filter model has eight variables as inputs,which include five geometrical variables: iris widths , ,and , cavity lengths and , and three electrical vari-ables: bandwidth , center frequency , and frequency . Thefilter outputs are -parameters and . Thus, the input andoutput vector of the filter model is

We first decompose the waveguide filter into two typesof substructures: input–output iris and internal coupling iris.

We will develop two neural-network submodels of the two sub-structures in the next step. Each submodel contains two inputvariables: width of iris and center frequency . We use

coupling and phase length as the output parameters of the sub-models [10]. Thus, the input and output vectors of the sub-models are

where , , and represent approximate valuesof coupling parameter and phase length of the th submodel.Notice that the number of input variables of each submodel,as expressed in (17), is less than that of the overall model, asexpressed in (15).

In this step, we develop two neural-network submodels forthe two types of irises. We generate training data by simulatingthe substructures using an EM simulator based on the mode-matching method. Each substructure is composed of a rectan-gular iris and connecting waveguide sections. It can be rigor-ously analyzed since it only contains two rectangular-to-rectan-gular waveguide junctions [40]. The -parameters are then usedto calculate the coupling values and phase lengths following thesame steps and equations presented in [10]. For example, theequivalent circuit for the internal coupling iris is an impedanceinverter having a shunt reactance and series reactanceand insertion phase length on each arm (see [39, Fig. 14.18]).The following equations relate the -parameter from EM sim-ulation to circuit parameters [39]:

where is the impedance value of the inverter and is theinsertion phase length. The coupling value is then obtained bymultiplying by a factor of , where andare the free-space and guided wavelength.

We generate 35 751 samples, which cover a large range ofiris width and center frequency for each submodel. A standarddata generation scheme is followed where iris width isheld constant and the center frequency was varied over apre-specified range. Next we assign a new value for and vary

over the pre-specified range. This process is continued untilthe range of is covered. Data generation time per samplefor each of the submodels is 0.6 s, which is inexpensive as theinput–output relationships are simple and the submodels holdonly two input variables each. Training time for each submodelis less than 1 min. The average errors of the submodels are lessthan 1%. Automatic model generation module of NeuroMod-elerPlus [41] is used to develop the two neural-network sub-models.

Following Step 5) of the modeling algorithm in Section III,we generate data of the overall filter using an EM simulator.EM data are generated simulating 46 different filters. In thenext step, we combine the neural-network submodels and filterequivalent-circuit model, as shown in Fig. 4, to obtain the ap-proximate -parameter of the filter. By comparing Fig. 3 with

Fig. 4. High-dimensional modeling structure for the four-pole �-plane filter.Two neural-network submodels: input–output iris model (IO iris) and internalcoupling iris model (Co iris) are developed decomposing the filter. Five sub-models required by the overall filter, as shown in this figure, are obtained bytraining only two neural-network submodels. An equivalent-circuit model of afilter are used to obtain the approximate �-parameter. A neural-network map-ping model is then used to obtain the accurate �-parameter of the four-pole�-plane filter.

Fig. 4, we can see that the input–output iris model is used torepresent the irises at the input and output ports of the four-polefilter of Fig. 4. The three internal coupling irises of Fig. 4 are rep-resented by the internal coupling iris model. Thus, the two typesof neural-network submodels are concatenated to represent thefour-pole filter. The IO iris 1 produces and IO iris 2 produces

. The three coupling iris models produce , , and. These coupling parameters are then used for producing

approximate -parameters of the four-pole filter using the filterequivalent-circuit equation of (20). Note that the input–outputiris model is used twice and the internal coupling iris modelis used three times to represent the overall four-pole filter, i.e.,

. In other words, the five submodels required in the filterare obtained by training only two submodels. The neural-net-work submodels produce approximate coupling matrix and sub-sequently, the circuit model generates approximate -parame-ters of the four-pole filter using the following equation [39]:

in which , is the filter order,and in this case, is a identity matrix, is the

approximate coupling matrix, is a matrix withall entries zero, except and , andand are approximate values of the filter’s input and outputcoupling parameters, respectively.

In Step 7), we supply the geometrical values of the 46 filtersused in Step 5) to the combined neural-network submodels andfilter empirical/equivalent model and obtain approximate -pa-rameter by sweeping frequency from 10.95 to 13.05 GHz witha 1-MHz step. The center frequency is held constant at 12 GHzand bandwidth is swept from 50 to 500 MHz with a 10-MHzstep. The model outputs at this stage are

TABLE ICOMPARISON OF TEST ERRORS OF FOUR-POLE �-PLANE FILTER

MODELS DEVELOPED USING CONVENTIONAL AND PROPOSED

HIGH-DIMENSIONAL MODELING APPROACH

where the superscript denotes that the values are approximate.As described in Step 8) of the modeling algorithm in

Section III, we assemble training data for the input neuronsof the mapping model using the input samples of Step 5)and approximate output samples, obtained in Step 7) ofthe 46 filters. The training data for the output neurons are theaccurate -parameter of the 46 filters generated using EMsimulation in Step 5). These data are then used to train and testneural-network mapping model, which maps the approximate

-parameter to the accurate -parameter. Four different setsof training and testing data, as shown in Table I, are used todevelop four mapping models. In Set 1, we use data of 23 filtersfor training and data of 23 other filters for testing. Trainingsamples are reduced and testing samples are increased in thesubsequent sets. The training error of the mapping models areless than 0.5%.

After the mapping model is trained, we construct the com-plete model of a four-pole filter using the neural-networksubmodels, circuit model, and mapping model in NeuroModel-erPlus, as shown in Fig. 4. The model is then used for testingpurposes. For comparison, we develop four neural-networkmodels following the conventional method and using the samefour sets of data used in the proposed method. In the conven-tional method, the neural-network model is trained to learn thecomplicated relationship between geometrical variables and

-parameter directly. The results are summarized in Table I,which show that the proposed method produces a more accurateresult than the conventional method. The amount of data is notenough to produce the 8-D parametric model of the -planefilter in the conventional method. On the other hand, the pro-posed method converts the overall model into a set of simplesubmodels, and thus is able to produce the accurate model withthose limited training data. The error of Table I is calculatedusing the least square error method [1]. The error is calculatedtaking the normalized differences between the magnitudes ofthe -parameter (real and imaginary parts) of the neural-net-work model and the magnitudes of the -parameter (real and

Fig. 5. Comparison of approximate solution with accurate EM solution of afour-pole�-plane filter. The approximate solution is obtained without using themapping model of the proposed method. The similarity between the solutionsconfirms that a simple mapping using a few training data of overall filter canmap the � to accurate EM solution. Filter geometry: � � �� cm, � �

�� cm,� � �� cm,� � �� cm, � � �� cm, and � � ��GHz.

imaginary parts) of EM simulation. The worst case error is theworst error among all the test structures of a particular data set.

In Fig. 5, we compare the approximate -parameter of an-plane filter with its accurate -parameter. The approximate

solution obtained from the neural-network submodels and em-pirical/equivalent-circuit model combined is fairly close to theaccurate EM solution. For this reason, the input–output relation-ship of the mapping model becomes simpler than the originalmodeling relationship between geometrical variables and -pa-rameters. Fig. 6 shows four-pole filter responses from the con-ventional neural-network model, proposed model and EM simu-lation of two different geometrical configurations. In both cases,the proposed method produces a more accurate result than theconventional method.

B. Proposed Modeling Technique for Side-Coupled Filters

We apply the proposed high-dimensional modeling methodto develop a neural-network model of a complex filter knownas a side-coupled circular waveguide dual-mode filter [42],[43]. Fig. 7 shows a physical diagram of the filter. Unlikethe conventional longitudinal end-coupled configuration, thefilter input–output coupling and coupling between the circularcavities are realized at the sides of the circular cavities. Thistype of filter offers significant performance improvement andfinds its applications in the satellite multiplexers with extremelystringent mass, size, and thermal requirements. However, thedesign and simulation becomes more difficult due to the struc-tural complexity [43].

The filter contains 15 design variables including 12 geomet-rical parameters, bandwidth, center frequency, and frequency.By using a conventional neural-network approach to representthis 15-D problem, i.e., 15 input neurons, data generation, andneural-network training would be prohibitive. Here we apply theproposed neural-network decomposition method to simplify thehigh-dimensional modeling problem into a set of low-dimen-sional modeling problems. As will be shown in the following,for such complex filters, responses based on submodels aloneare not satisfactory. Instead of direct mapping of -parametersof the EM simulator and neural-network model, a circuit modelbased on the coupling matrix is adopted as the modeling objec-tive. In doing so, the difficulty in the alignment or mapping of

Fig. 6. Comparison of �-parameter of conventional neural network andproposed model of a four-pole �-plane filter. (a) Filter geometry 1:� � �� cm, � � �� cm, � � �� cm, � � �� cm,� � �� cm, and � � �� GHz. (b) Filter geometry 2: � � �� cm,� � �� cm, � � �� cm, � � �� cm, � � �� cm, and� � �� GHz. Output of the conventional model is not accurate because theamount of data used for training is not enough for the conventional method.However, the same data is enough for the proposed method.

full EM and neural-network model responses is significantly re-duced, enabling the accurate modeling of complex filters witha minimum number of full EM simulations. Once the accuratecoupling matrix is achieved, a circuit simulator can be used toobtain the accurate -parameter for any frequency range.

Thus, the input and output vectors of the model are, respec-tively,

In (22), and represent lengths of input iris and outputiris, respectively, , , and represent lengths ofthree screws of cavity 1, , , and represent threescrews of cavity 2, represents the length of the sequentialcoupling iris, represents the length of the cross-couplingiris, and represent the lengths of cavity 1 and cavity 2,respectively, represents bandwidth, represents the centerfrequency, and represents the frequency. In (23), andrepresent input and output coupling bandwidth, toare self-coupling bandwidths, and , , , andrepresent sequential and cross-coupling bandwidths.

In the first step, we decompose the filter into three types ofsubstructures , called the input–output iris, internalcoupling iris, and coupling and tuning screw [10] for whichthree neural-network submodels will be developed. The inputs

Fig. 7. Diagram of a side-coupled filter showing various dimensional variablesof the filter. (a) Perspective view, (b) side view, and (c) top view of a side-coupledcircular waveguide dual-mode filter.

of the input–output iris model are iris length and . The out-puts are coupling bandwidth and phase representing theloading effect of the internal coupling iris. The inputs of the in-ternal coupling iris model are lengths of the sequential couplingiris , cross coupling iris and , and outputs are sequen-tial coupling , cross coupling , and phases and .Phases and are the loading effect of the internal couplingirises on the two orthogonal modes, respectively. The inputs ofcoupling and tuning screw model are screw lengths , ,and and . The outputs are coupling bandwidth for

, , and . Note that the number of input variables ofeach substructure is much less than that of the overall filter.

Next we combine neural-network decomposition with theside-coupled filter decomposition scheme. Following Step 2) ofthe modeling algorithm, we generate training data to developneural-network submodels for each of the substructures. Sinceeach of the substructures has few design variables, e.g., theinput–output iris has only two variables, we can generate manydata in a short time. This allows us to develop very accuratesubmodel. Each substructure is simulated using EM simulatorbased on mode-matching method, as described in [42]. Forexample, the substructure representing the input iris is com-posed of a rectangular-to-rectangular waveguide junction and a

circular-to-rectangular side-coupled waveguide T-junction. Therigorous mode-matching technique for simulating circular tomultiple off-center rectangular side-coupled waveguide T-junc-tions presented in [42] is used for analyzing this structure. Thefilter input–output couplings are obtained using the group-delaymethod and the inter-resonator couplings are calculated usingeigenvalue calculation [39]. We generate 423 samples of datafor the input–output iris model and the model testing erroris 0.5%. We also generate 4930 data samples to develop theinternal coupling iris model and less than 0.2% average testingerror is achieved for this model. For the coupling and tuningscrew model, we generate 36 015 samples of data and averagemodel testing error is 0.51%. Training times of the three sub-models are less than 1 min, approximately 3 min, and 2 h,respectively. The submodels are trained using the automaticmodel generation module of NeuroModelerPlus [41].

In Step 5), full EM data are generated by simulating the en-tire side-coupled filter with 64 different combinations of geo-metrical values. The bandwidth and center frequency are variedfrom 27 to 54 MHz and 11 to 11.7 GHz, respectively. As men-tioned earlier, instead of using the -parameters generated usingthe EM simulator directly, coupling parameters are used as themodeling objectives. We extract 64 coupling values using the

-parameter extraction technique, as presented in [44].In Step 6), we combine the neural-network submodels to rep-

resent the filter structure. Both the input–output iris model andcoupling and tuning screw model are used twice and the in-ternal coupling iris model is used once to represent the filter,i.e., . The neural-network submodels are used to pro-duce cross-couplings and empirical models are used to computeself-couplings.

As described in Step 7) of the modeling algorithm inSection III, we produce approximate coupling values using thesame samples of geometrical parameters of Step 5). Followingthe procedure as described in Step 8), we assemble training dataof the mapping model. Since individual coupling parametersare a function of specific geometrical dimensions rather than afunction of all the dimensions, we produce a separate mappingmodel for each of them. Thus, ten mapping models for the tencoupling parameters, as described in (23), are developed. Themapping models are defined as

where represents the th coupling parameter, rep-resents the th approximate coupling parameter obtainedfrom the neural-network submodel, is a subset of ,

. Four different sets of EM data of the overallfilter, as listed in Table II, are used to develop four sets ofmapping models. In Set 1, data from 44 filter geometries areused for training and data from 20 other filter geometries areused for testing. The number of filter geometries is reducedfor training in the subsequent three sets and listed in Table II.Training time of the ten neural-network mapping models areless than 5 min.

We construct an accurate model of the side coupled filter byconnecting the ten neural-network mapping models with the

TABLE IICOMPARISON OF TEST ERRORS OF SIDE-COUPLED CIRCULAR WAVEGUIDE

DUAL-MODE FILTER MODELS DEVELOPED WITH CONVENTIONAL AND

PROPOSED HIGH-DIMENSIONAL MODELING APPROACH

submodels and empirical models used to produce an approxi-mate coupling matrix. The overall model is then tested using thetest data, as listed in Table II. For comparison, four neural-net-work models are also trained using the same four data sets inthe conventional method, which relates geometrical variables tothe coupling matrix directly. The average errors of Table II arecalculated taking the differences between the magnitudes of the

-parameter (real and imaginary part) from the neural networksand the magnitudes of the -parameter (real and imaginary part)of the EM simulation. The worst case error is obtained by se-lecting the worst average error among all data of respective testdata set.

Table II compares the model error between the two methods,which shows that the proposed method is much more accu-rate than the conventional method for all data sets. By usingthe proposed method, we can produce good accuracy with alimited amount of data because the mapping function becomessimple after obtaining approximate couplings from submodels(trained with inexpensive data) and the empirical circuit model.On the other hand, the conventional method is inaccurate be-cause the amount of training data is insufficient to produce a15-D side-coupled filter model. If we were to improve the ac-curacy of the conventional method, we would have to use a lotmore data, which would be expensive and difficult to generate.

In Fig. 8, we plot responses of two different filter configura-tions obtained from the proposed model. It shows that the modelcan be used to obtain responses for various filter geometries.

Fig. 9 shows the effectiveness of the mapping model. Theapproximate filter response, which is generated from the ap-proximate coupling matrix without using the proposed mappingmodels, is not satisfactory. The mapping models then provideaccurate couplings, which leads to the response very close tothe accurate EM response.

Fig. 10 shows a plot of the average model test error versusthe number of filter geometry used for model training. The plotshows that the model test error of the proposed method is lowand decreases consistently with the number of filters used fortraining. On the other hand, the error of the conventional methodstays high at approximately 20%. To reduce the error of theconventional method, we need to use massive training data.

Fig. 8. Reflection coefficients of two different side-coupled circular wave-guide dual-mode filters obtained using the proposed model. Geometry 1:� � �� cm, � � �� cm, � � �� cm, � � �� cm,� � �� cm,� � ��cm,� � ��cm,� � �� cm,� � �� cm, � � �� cm, � � �� cm, � � �� cm,� � �� MHz, � � �� GHz. Geometry 2: � � �� cm,� � ��cm,� � ��cm,� � ��cm,� � ��cm,� � ��cm,� � ��cm,� � ��cm,� � ��cm,� � �� cm, � � �� cm, � � �� cm, � � MHz,� � �� GHz.

Fig. 9. Reflection coefficient of a side-coupled circular waveguide dual-modefilter with � � � MHz, � � �� GHz showing the effectiveness of theneural-network mapping in the coupling parameter space.

Fig. 10. Comparison of average model test error versus the number of filtergeometry used for model training in conventional and proposed method of theside-coupled circular waveguide dual-mode filter.

In Table III, we list the model evaluation time of two com-monly used EM modeling methods and compare it with the eval-uation time of the proposed high-dimensional neural-networkmodeling method. Full EM simulation of the entire filter needsapproximately 6 min using a mode-matching-based EM simu-lator [42] and 45 min using a finite-element-based EM simulatorsuch as the High Frequency Structure Simulator (HFSS) [45].The comparison clearly shows that the proposed method is sig-nificantly faster than the EM methods, enabling fast design andoptimization.

TABLE IIICOMPARISON OF CPU TIME OF EM AND NEURAL-NETWORK MODEL

OF A SIDE-COUPLED CIRCULAR WAVEGUIDE DUAL-MODE FILTER

In order to develop an accurate model, e.g., less than 2% ofmodel testing error, using the conventional method, we need tosample sufficiently the specified range for all input variables.For example, if we sample three values for each of the ten ge-ometrical variables, seven values for , and four values for ,we need a total samples of theoverall filter. The data generation time per sample of the overallfilter is min. The total data generation time for this 15-Dneural-network model of the conventional method using (12) isestimated to be min years, which istoo expensive. The model training time using the massivetraining data would also be too expensive.

We now calculate data generation time of the proposedmethod. Data generation time per sample of input–output iris,internal coupling iris, and coupling and tuning screw substruc-tures are s, s, and s, respectively.To cover the same range of the input geometrical space thatis used in this example, we need samples of theinput–output iris, samples of the internal couplingiris, and samples of the coupling and tuning screwsubstructures. In order to achieve less than 2% of model testingerror using the proposed method, we also need approximately

samples of the overall filter for the training of themapping model. The total data generation time of the proposedmethod using (13) is calculated to be

h (25)

The model training time of three submodels and ten map-ping models all together is less than 10 min and is, therefore,insignificant. Thus, an accurate neural-network model of theside-coupled filter, which is very expensive to develop using theconventional neural-network method, becomes feasible usingthe proposed method.

It is worth mentioning that the methodology is not limited towaveguide filters. It can be readily applied to other types of fil-ters such as coaxial cavity, dielectric, and planar circuit filters.Take the well-known capacitive-gap coupled microstrip filter asan example [39]. Neural-network submodels will be developedto model capacitive gaps, which can be represented by admit-tance inverters. The capacitive discontinuity can be simulatedusing a method-of-moment-based EM simulator and the sim-ulation results are then related to parameters of the equivalentcircuit, i.e., an admittance inverter. Development of the com-plete filter model follows the general steps and flow diagramin Section III. Therefore, the implementation for each type offilter will only differ in how the overall filter is decomposed,

the EM method suitable for the type of structure, and the empir-ical/equivalent model of choice.

V. CONCLUSION

We have proposed an effective neural-network modelingtechnique for filters that hold many design variables. It is im-practical to develop a neural-network model for such structuresin the conventional neural-network approach. We propose anew formulation to integrate neural-network decompositionwith filter structure decomposition and then incorporate cir-cuit knowledge to obtain a complete filter model. The filterstructure is decomposed into substructures, which reduces thenumber of variables per submodel. Neural-network submodelsare then developed for each of the substructures. Empir-ical/equivalent-circuit models are combined with neural-net-work submodels to produce an approximate solution of thefilter. Another neural-network model is then trained to mapapproximate solution to the accurate solution of the filter. Theresult shows that the proposed method can be used to producehigh-dimensional models with few full EM training data, whichare usually expensive to generate, compared to the conventionalneural-network technique. The method is very useful for de-veloping neural-network models of microwave filters that havemany design variables. The developed neural-network modelsbecome very useful for fast design optimization of those filters.

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Humayun Kabir received the B.Sc. degree in elec-trical and electronic engineering from the BangladeshUniversity of Engineering and Technology, Dhaka,Bangladesh, in 1999, the Masters degree in electricalengineering from the University of Arkansas, Fayet-teville, in 2003, and the Ph.D. degree in electrical andcomputer engineering from Carleton University, Ot-tawa, ON, Canada in 2009.

From 2001 to 2003, he was a Research Assistantwith HiDEC, where he was involved with microwavepassive and active circuit design, fabrication, and

testing. In 2005, he was a Co-Op Student with COM DEV Ltd. He was also aTeaching and Research Assistant with the Department of Electronics, CarletonUniversity. He is currently a Research Assistant with Carleton University. Hisresearch interests include RF/microwave modeling, design, and optimization.

Ying Wang (M’05) received the B.Eng. and Mastersdegrees from the Nanjing University of Science andTechnology, Nanjing, China, in 1993 and 1996, re-spectively, and the Ph.D. degree in electrical engi-neering from the University of Waterloo, Waterloo,ON, Canada, in 2000.

From 2000 to 2007, she was with COM DEV Ltd.,where she was involved in development of CAD soft-ware for design and simulation of microwave circuitsfor space application. In 2007, she joined the Fac-ulty of Engineering and Applied Science, University

of Ontario Institute of Technology, Oshawa, ON, Canada, as an Assistant Pro-fessor. Her research interests include RF/microwave CAD, microwave circuitsdesign, and radio wave propagation modeling.

Ming Yu (S’90–M’93–SM’01–F’09) received thePh.D. degree in electrical engineering from theUniversity of Victoria, Victoria, BC, Canada, in1995.

In 1993, while working on his doctoral dissertationpart time, he joined COM DEV Ltd., Cambridge,ON, Canada, as a Member of Technical Staff. Hewas involved in the design of passive microwave/RFhardware from 300 MHz to 60 GHz for both space-and ground-based applications. He was also a prin-cipal developer of a variety of COM DEV Ltd.’s core

design and tuning software for microwave filters and multiplexers, includingcomputer-aided tuning software in 1994 and fully automated robotic diplexertuning systems in 1999. His varied experience also includes being the Managerof Filter/Multiplexer Technology (Space Group) and Staff Scientist of Corpo-rate Research and Development (Research and Development). He is currentlythe Chief Scientist and Director of Research and Development, COM DEV Ltd.He is responsible for overseeing the development of the company’s researchand development Roadmap and next-generation products and technologies,including high-frequency and high-power engineering, EM-based CAD andtuning for complex and large problems, and novel miniaturization techniquesfor microwave networks. He is also an Adjunct Professor with the Universityof Waterloo, Waterloo, ON, Canada. He has authored or coauthored over 90publications and numerous proprietary reports. He holds eight patents with sixpending.

Dr. Yu is the vice chair of MTT-8 and served as chair of TPC-11. He is amember of the Editorial Board of many IEEE and IET publications. He is anIEEE Distinguished Microwave Lecturer from 2010 to 2012. He was the re-cipient of the 1995 and 2006 COM DEV Ltd. Achievement Award for the de-velopment of a computer-aided tuning algorithms and systems for microwavefilters and multiplexers. He holds a Natural Sciences and Engineering ResearchCouncil (NSERC) Discovery Grant (2004–2013).

Qi-Jun Zhang (SM’84–M’87–SM’95–F’06)received the B.Eng. degree from the Nanjing Univer-sity of Science and Technology, Nanjing, China, in1982, and the Ph.D. degree in electrical engineeringfrom McMaster University, Hamilton, ON, Canada,in 1987.

From 1982 to 1983, he was with the System Engi-neering Institute, Tianjin University, Tianjin, China.From 1988 to 1990, he was with Optimization Sys-tems Associates (OSA) Inc., Dundas, ON, Canada,where he developed advanced microwave optimiza-

tion software. In 1990, he joined the Department of Electronics, Carleton Uni-versity, Ottawa, ON, Canada, where he is currently a Full Professor. He has au-thored or coauthored over 200 publications. He authored Neural Networks forRF and Microwave Design (Artech House, 2000). He coedited Modeling andSimulation of High-Speed VLSI Interconnects (Kluwer, 1994). He contributed tothe Encyclopedia of RF and Microwave Engineering (Wiley, 2005), Fundamen-tals of Nonlinear Behavioral Modeling for RF and Microwave Design (ArtechHouse, 2005), and Analog Methods for Computer-Aided Analysis and Diag-nosis (Marcel Dekker, 1988). He was a Guest Co-Editor for the “Special Issueon High-Speed VLSI Interconnects” for the International Journal of Analog In-tegrated Circuits and Signal Processing (Kluwer, 1994), and a two-time GuestEditor for the “Special Issues on Applications of ANN to RF and MicrowaveDesign” for the International Journal of RF and Microwave CAE (Wiley, 1999and 2002). He is a member of the Editorial Board of the International Journal ofRF and Microwave Computer-Aided Engineering and the International Journalof Numerical Modeling. He is an Associate Editor for the Journal of Circuits,Systems, and Computers. His research interests are microwave CAD, neural net-works, and optimization methods for high-speed/high-frequency circuit design.

Dr. Zhang is a Fellow of the Electromagnetics Academy. He is a member onthe Editorial Board of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND

TECHNIQUES. He is a member of the Technical Committee on CAD (MTT-1) ofthe IEEE Microwave Theory and Techniques Society (IEEE MTT-S).

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