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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 3, JUNE 2007 1311

Cooperative Coevolutionary Genetic Algorithm forDigital IIR Filter Design

Yang Yu and Yu Xinjie, Member, IEEE

AbstractA novel algorithm for digital infinite-impulse re-sponse (IIR) filter design is proposed in this paper. The suggestedalgorithm is a kind of cooperative coevolutionary genetic algo-rithm. It considers the magnitude response and the phase responsesimultaneously and also tries to find the lowest filter order. Thestructure and the coefficients of the digital IIR filter are codedseparately, and they evolve coordinately as two different species,i.e., the control species and the coefficient species. The nondomi-nated sorting genetic algorithm-II is used for the control species toguide the algorithms toward three objectives simultaneously. Thesimulated annealing is used for the coefficient species to keep thediversity. These two strategies make the cooperative coevolution-

ary process work effectively. Comparisons with another geneticalgorithm-based digital IIR filter design method by numericalexperiments show that the suggested algorithm is effective androbust in digital IIR filter design.

Index TermsCoevolution, genetic algorithms (GAs), infinite-impulse response (IIR) digital filters, linear phase, lowest order.

I. INTRODUCTION

AN INFINITE-IMPULSE RESPONSE (IIR) filter can be

expressed in the cascade form as

H(z) = K

n

k=1

1 + bkz1

1 + akz1

m

i=1

1 + di1z1 + di2z

2

1 + ci1z1 + ci2z2(1)

where K is the gain, ak and bk for k = 1, 2, . . . , n are the first-order coefficients, and ci1, ci2, di1, and di2 for i = 1, 2, . . . , mare the second-order coefficients. The digital filter design is a

process in which a digital hardware or a program is constructed

to meet the given specification.

The traditional digital IIR filter design involves the analog

IIR filter design and the analog-to-digital transformation. When

the specification for the digital filter is given, we first change it

to the corresponding analog low-pass (LP) filter and use one of

the well-known LP filter design methods, such as Butterworth,

Chebyshev Type I, and Chebyshev Type II, to fulfill the require-

ments. Then, the analog LP filter is transferred to the digitalfilter using the bilinear transformation [1].

This method works well and has been widely used, but it

also has some disadvantages. First, the traditional digital filter

design returns only one solution, which may be unacceptable

Manuscript received February 20, 2006; revised September 12, 2006.Abstract published on the Internet January 27, 2007. This work was supportedin part by the National Natural Science Foundation of China under Project50507011.

The authors are with Tsinghua University, Beijing 100084, China (e-mail:[email protected]; [email protected]).

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

Digital Object Identifier 10.1109/TIE.2007.893063

for the real-world implementation. Second, the transformation

between the digital field and the analog field may cause inef-

ficiency. Third, the coefficient quantization errors could not be

avoided during the design. Last but not least, the lowest filter

order and the phase response requirements are very useful in

some practical applications but cannot be considered by the

traditional method.

When genetic algorithms (GAs) were introduced into the

design world, their flexibility and adaptation were very remark-

able. The initial use of GAs for filter design was reported by

Etter et al. [2]. Though the result was not impressive, theGA design method showed its distinction. It was a direct

design method in the digital field without the analog-to-digital

transformation and avoided the coefficient quantization error.

Also, it could solve the multiobjective design problem easily

and figure out more than one solution. As GAs become more

and more mature in the last few years, the work on digital filter

design using GAs has received great attention.

The normal GA design for IIR filter always assumes a

predefined topology of the filter. Only the coefficients of the

filter need to be determined. Tang et al. suggested the hierar-

chical genetic algorithm (HGA) to tackle this problem [3]. The

structure of the filter is not fixed during the design, and so it can

reach the lowest order. However, this designing method also hassome disadvantages due to its coding redundancy, which will be

discussed in Section II.

One of the drawbacks of the IIR filter is its phase response.

Linear phase response can be easily achieved by finite-impulse

response filters, which is very hard for IIR filters to implement.

How to design the approximately linear phase response of an

IIR filter becomes the focal and difficult point in some IIR

design researches.

Karaboga and Cetinkaya suggested a method for designing

an IIR filter with minimum phase [4]. Other researchers work

on the design method for minimum group delay IIR filter [5].

Though it is impossible to get an exactly linear phase IIRfilter, some researches tried IIR filters with linear phase in the

passband. In [6] and [7], the proposed method designed filters

with approximately linear phase responses in their passbands.

However, the magnitude and phase responses should be pre-

scribed in these methods, and the traditional iterative algorithms

are used, including complicated matrix computations. In [8],

Koir and Tasic designed the approximately linear phase IIR

filter using GA. However, their method might cause Bounded

Input/Bounded Output stable problem. Extra computations are

required to ensure the stability of the filter.

In this paper, the cooperative coevolutionary genetic

algorithm (CCGA), which is firstly suggested by Potter, is

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1312 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 3, JUNE 2007

introduced into the IIR design. CCGA has the following

characteristics [9].

1) A complete solution is divided into more than one sub-

components, which are represented by several species,

respectively.

2) When an individual is evaluated, it should be combined

with individuals in other species to form a completesolution.

3) Each species should evolve separately, using a stan-

dard GA.

The suggested novel CCGA design method not only meets

the requirement of the magnitude response, but also gets the

approximately linear phase response in the passband and the

transition band and finds the lowest filter order simultaneously.

The structure and the coefficients of the digital IIR filter are

coded separately, and they evolve coordinately as two different

species, i.e., the control species and the coefficient species. The

nondominated sorting genetic algorithm (NSGA)-II is used for

the control species, whereas the simulated annealing (SA) isused for the coefficient species. This new scheme works well in

the multiobjective digital IIR filter design problems.

Section II presents the core of this paper, where the CCGA

for digital IIR design is described in detail. Then, in Section III,

a comparison of CCGA and HGA is discussed. Finally, the

conclusions and discussions are given in Section IV.

II. CCGA FO R DIGITAL IIR FILTER DESIGN

A. Chromosome Coding

The coding of the suggested CCGA design algorithm is

adopted from the HGA. Improvements are made by separat-ing the control genes from the coefficient genes to form two

species, which enhances the search ability of the HGA with the

same storage space.

The essence of HGA design is its coding. To represent the

transfer function of a digital IIR filter, the chromosome contains

two types of genes, namely: 1) the control gene and 2) the

coefficient gene. The control gene describes the structure of

the filter, and the coefficient gene defines the value of the

coefficients in each block.

For example, an IIR filter, in which the maximum number of

the first-order units and the second-order units are both two, can

be described in the cascade form, as shown in (2) at the bottomof the page.

The control gene and the coefficient gene are illustrated in

Figs. 1 and 2, respectively.

The control genes are in binary bit form and decide the

state of activation for each block. In Fig. 1, the first gene of

the chromosome represents the state of the first-order function

(1 + b1z1)/(1 + a1z

1), where 1 means (1 + b1z1)/(1 +

a1z1) exists in the filter transfer function and 0 means the

Fig. 1. Control gene structure.

Fig. 2. Coefficient gene structure.

nonexistence. The coefficient genes are in real-number form,

which define the values of the coefficients in each block. As the

structure of IIR filter is represented by (2), the transfer function

H(z) =(1 0.1z1)(1 + 0.5z1 + 0.6z1)

(1 + 0.4z1)(1 0.9z1 + 0.1z1)(3)

has the chromosome

{1, 0, 1, 0,0.1, , 0.5, 0.6, , , 0.4, ,0.9, 0.1, , } (4)

where is the wildcard.The HGA directly connect the coefficient gene to the control

gene to form the individual chromosome. This coding makes

the structure optimization possible, but brings some inefficiency

due to the coding redundancy at the same time.

Let us take the filter described by (4) for example. The

coefficient genes representing the second first-order block do

not contribute to the fitness of the whole chromosome. When

they are changed during the evolutionary process, it cannot tellwhether the change is good or not, then the calculations for that

change and the following evaluation are losing effectiveness.

The evolutionary process can be more effective if the coefficient

genes and the control genes are evaluated fully. Then, a better

direction of the evolution may be found. That is the point to

change the HGA to the CCGA.

In the CCGA, the control genes are separated from the coef-

ficient genes. There are two species in the population, namely

1) the control species C and 2) the coefficient species X.

The coding for C is in binary form and in real-number form

for X. When an individual in the species C is evaluated, some

individuals from the species X need to be selected randomlyand combined with the individual from C to get the complete

solutions. The values of the solutions determine the fitness of

the individual from C. Using the same strategy, species X can

be evaluated.

In this way, for an individual in C, several individuals in X

are chosen to combine with it, respectively. So, the evaluation

of the individual in C is more reliable than in HGA, where it

is just combined with a fixed coefficient individual. The same

H(z) = K(1 + b1z1)(1 + b2z1)(1 + d11z1 + d12z2)(1 + d21z1 + d22z2)(1 + a1z1)(1 + a2z1)(1 + c11z1 + c12z2)(1 + c21z1 + c22z2)

(2)

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YU AND XINJIE: COOPERATIVE COEVOLUTIONARY GENETIC ALGORITHM FOR DIGITAL IIR FILTER DESIGN 1313

Fig. 3. Flowchart of the process of CCGA for digital IIR filter design.

strategy on the evaluation ofX gets the same advantage. As we

get more reliable evaluation of the individuals, the evolution

works more effectively, with less good genes being lost during

the evolution process.

B. Process of CCGA for IIR Design

As the structure and the coefficients of the filter are set

to be two species separately, the evolution process is divided

into the following two parts: 1) the evolution of the control

species and 2) the evolution of the coefficient species. The

whole coevolution process is shown by the flowchart in Fig. 3.With multiobjective optimization, Pareto solutions might be

expected to make the decision making process more reliable.

An IIR filter with higher order tends to have better magnitude

response, whereas the higher order results in more complicated

calculations, and the linear phase response is more difficult to

realize. Even when considering the IIR filters with the same

order, the comparisons are hard to make, as some filters may be

superior in the magnitude responses, whereas others may have

approximately linear phase responses. The decision depends

on the practical uses. The suggested design algorithm can find

several Pareto solutions, incommensurable good filters, at the

same time rather than a single best solution.

C. Evaluation

There are three objective functions in our suggested digital

IIR filter design method, namely 1) magnitude response error,

2) linear phase response error, and 3) order.

1) Magnitude Response Error: When we come to the design

of an IIR filter, the following magnitude response conditions

are required.

The attenuation in passband should not exceed 1. The attenuation in stopband should not be less than 1 2.

The passband and stopband edge frequencies are repre-sented by p and s, respectively.

The magnitude response error is calculated as follows [3]:

eHp() =

1 1

H(ej) , H(ej) < 1 10,

H(ej) 1 1where is in the passband.

eHs() =

H(ej) 2,

H(ej)

> 2

0,H(ej) 2

where is in the stopband.eHp() and eHs() are the passband and the stopband

magnitude response errors at , respectively. Then, the firstobjective function is

min f1 =i

eHp(i) +j

eHs(j) (5)

where i is the sampling frequency in the passband, and j isthe sampling frequency in the stopband.

2) Linear Phase Response Error: The linear phase responseis simplified as the passband and transition band linear phase

response. The phase response of the transition band is also

considered because the magnitude response at some points of

transition band may be as high as that in the passband. If the

phase response is far away from linear at these points, it could

result in large distortion.

We sample the phase response of the digital IIR filter with the

same frequency interval and get the phase sequence as follows:

Phases = {1, 2, . . . , n}.

The phase difference sequence can be counted as

Phases = {1, 2, . . . , n1},

where i = i+1 i.

The sampling of frequency is in equal interval; so in the

linear phase case, the phase sequence is an equal-difference

sequence. That is to say all the elements in the Phasessequence have the same value. How far a phase response is from

the linear phase condition can be evaluated by the variances of

its Phases sequence, which is then set to be the phase responseerror. Then, the second objective function is

min f2 = variance{i|i passband transition band}.(6)

3) Order: When a structure is given by the control chromo-

some, the order can be formulated as follows [3]:

order =mi=1

pi + 2n

j=1

qj (7)

where m + n is the total length of control chromosome, pi andqj are the control bits governing the activation of the ith first-order block and the jth second-order block, respectively. Themaximum allowable filter order is m + 2n.

Then, the third objective function is

min f3 = order. (8)

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Evaluation ofC andX: In the real-world implementa-

tion, the magnitude response is considered to be more important

than the phase response. So, the magnitude response error is the

dominant part of the objective function.

Each individual in C is evaluated by combining with K1random selected individuals from X. So, K1 candidate digital

IIR filters are formed for evaluating one individual in C. Themagnitude response error and linear phase response error are

calculated for all K1 candidates. Then, we get the followingrow vectors:

errmag = [mag1,mag2, . . . , m a gK1 ]T

errphase = [pha1,pha2, . . . , p h aK1 ]T.

If the minimum value oferrmag is magi, then the magnituderesponse error of the individual in C is magi, and the linearphase error is phai. If there are more than one value in thevector errmag that are equal to magi, choose the one whichhas the smallest linear phase response error. The order can be

calculated easily by using (7) for ith candidate. In this way,every individual in C gets its three objective function values

after the evaluation.

When an individual in X is evaluated, a similar strategy is

used by combining it with K2 random selected individuals inC. Because coefficient genes do not have order value, every

individual in X gets its two objective function values after the

evaluation.

D. Evolution of the Control Species

To deal with the three objectives effectively, NSGA-II is

adopted in the evolution of the control species. Suppose thepopulation size is N, and the population is Pt at generation t.The nondomination rank and the crowding distance for every

individual can be calculated by Debs method [10]. To pre-

vent the crowding effect, a group insertion [3] is used before

the elitist selection. The whole evolutionary process is given

as follows.

1) The two-point crossover and the bit-flips mutation are

carried out inPt to form some new individuals calledQt.

2) The group insertion method in [3] is used to form a new

group calledRt.

First, all individuals in Pt and Qt are divided into sub-

groupsG

i according to the order, i.e., the third objectivefunction value, where the individuals with the same order

are put together.

Then, in each subgroup Gi, K individuals are selectedinto Rt, according to the total error values f1 + f2 ofthe individuals, the ones with smallest values are firstly

chosen. If the size ofGi is smaller than K, all theindividuals in it are chosen into Rt.

3) Select N individuals fromRt into the next population.a) The nondomination rank and the crowding distance

for every individual in Rt are calculated by Debs

method [10].

b) Sort Rt according to the nondomination rank. Let

F = {F1, F2, . . .} be the nondominated fronts ofRt,where F1 is the best nondominated set.

c) SelectPt+1 with the following procedures.

For i = 1, 2, . . .If the size ofFi is smaller than N M, where Mis the number of the individuals now in Pt+1, all

members of the set Fi are put into Pt+1;

Else all the remaining members ofFi are sorted using

the crowding distance, and the ones with the longercrowding distance are chosen to fill the population

slots. Then, break the loop.

As can be seen from the aforementioned procedure, the con-

trol species can evolve toward three objectives simultaneously,

which is critical for digital IIR designer to make decision.

E. Evolution of the Coefficient Species

The control species determines the order and structure of the

filter, so it plays a leading role, and the coefficient species is

a subordinate part. During the evolution process, the control

genes may change a lot. The original good coefficient genes

may not fit them any more. On the other hand, the individuals ofthe control species in one generation are different, representing

different kinds of filter structure. In order to find individuals in

the coefficient species to fit different kinds of filter structure and

make good filter design solutions, the diversity of the coefficient

species needs to be kept.

The SA can preserve the worse individual to some extent

according to the probability. So, we use the SA in the evolution

of the coefficient species to keep the diversity. The SA needs

one value for two individuals to compare, so we sum the

magnitude response error f1 and the phase response error f2to be a total error value.

In the implementation of the SA, the heat reservation strategyand the reheating strategy are used to increase the evolu-

tionary efficiency. The settings of the parameters during the

annealing process and the original temperature are set accord-

ing to [11]. The initial temperature is set to be T = T0 =(0.1/ ln 0.5) [11].

The process of the evolution is given as follows.

1) Set k = 0.2) According to the crossover rate, some individuals are

chosen to form pairs and undergo crossover. A pair of

individuals produces one child. From every pair of par-

ents, the one with the larger error value may be replaced

by its child. The replacement happens when the child hasthe smaller value f1 + f2 or rand exp{(valueparent valuechild)/T}.

3) Choose individuals to undergo mutating according to

the mutation rate. The produced children replace their

parents if they have the smaller value f1 + f2 or rand exp{(valueparent valuechild)/T}.

4) k = k + 1.5) If k < 3, return to step 2 without changing the

temperature;

else change the temperature by

T = T 0.8.

Then, end the inner loop and turn to the next step.

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TABLE IDESIGN CRITERIA

TABLE IIPARAMETERS FOR GENETIC OPERATIONS

6) If the temperature is lower than 105, set the current

temperature to be T = T0, and the evolutionary processin this generation is complete.

As can be seen from the aforementioned procedure, the SA

with the heat reservation strategy and the reheating strategy

preserves the diversity of coefficient species, which is quite

useful for finding better design solutions with the evolution of

the control species.

III. EXPERIMENTAL RESULTS

In this section, the suggested CCGA is used to design some

typical IIR filters, and its performances are compared withthe performances of the HGA [3]. All the genetic operating

parameters are set exactly the same as those in [3].

The fundamental structure ofH(z) is given as

H(z) = K

3i=1

(1 + biz1)

(1 + aiz1)

4j=1

(1 + bj1z1 + bj2z

2)

(1 + aj1z1 + aj2z2). (9)

So, the highest order of the designed filter is 11. The length

of control chromosome is 7, and the length of coefficient

chromosome is 22.

Shynk summarized the stable requirements for digital IIRfilters [12]. The coefficients of the denominators in the first-

order block are limited between 1 and 1. The second-order

block coefficients of the denominators must satisfy the follow-ing equations:

1 < aj2 < 1

1 aj2 < aj1 < 1 + aj2.

When all the coefficients of a filter function are determined,

the gain value K can be determined in order to unify themagnitude response of the filter function.

Four types of the filters, namely: 1) low-pass (LP); 2) high-

pass (HP); 3) band-pass (BP); and 4) band-stop (BS), are

designed in the experiment. Parameters for the design criteriaare listed in Table I, and the parameters for genetic operations

are given in Table II.

The termination condition is that the first objective function

value f1 equals zero, and the other two objective functions areless than the given values.

We run the design process for 20 times. The best results

found by CCGA are summarized in Table III, the final filter

functions are indicated in (10)(13), as shown at the bottom of

the next page, and the magnitude and the phase responses are

shown in Figs. 47, respectively.

It can be seen from the results that the proposed design

method can fully satisfy the magnitude response requirement,

minimize the phase response error, and find the lowest order.The results are also compared with HGA in Table IV.

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TABLE IIIFILTER PERFORMANCES (DESIGNING USING CCGA)

Fig. 4. LP filter responses. (a) Magnitude response. (b) Phase response.

By studying Table IV(a)(d), we can arrive at the following

conclusions.

1) The passband and the stopband magnitude response per-

formances of the two algorithms are similar.

2) The CCGA works better than the HGA, as far as the

lowest order of the filters is concerned.

3) All phase response errors of the CCGA designed fil-

ters are smaller than that of the corresponding HGAdesigned ones.

4) The cooperative coevolutionary strategy, NSGA-II, and

the SA work well in handling the two species and the

three objectives.

IV. DISCUSSION AND CONCLUSION

GAs can design digital IIR filter directly, which is more

flexible than the traditional ways. The HGA codes the structure

of the digital filter with control genes and combines them withcoefficient genes to form a whole design. This coding has

HLP(z) = 0.1823 (1 + 0.6430z1)(1 1.0019z1 + 0.9958z2)

(1 0.3888z1)(1 1.1631z1 + 0.6501z1)(10)

HHP(z) = 0.2150 (1 0.4521z1)(1 + 0.9479z1 + 0.9374z2)

(1 + 0.3117z1)(1 + 1.1656z1 + 0.6154z2)(11)

HBP(z) = 0.1990 (1 1.6727z1 + 0.9964z2)(1 + 1.6536z1 + 0.9948z2)

(1 + 0.5414z1 + 0.5351z2)(1 0.6039z1 + 0.5134z2)(12)

HBS(z) = 0.4251 (1 0.2977z1 + 0.9124z2)(1 + 0.2191z1 + 0.9068z2)

(1 0.8265z1 + 0.4912z2)(1 + 0.7727z1 + 0.4599z2)(13)

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Fig. 5. HP filter responses. (a) Magnitude response. (b) Phase response.

Fig. 6. BP filter responses. (a) Magnitude response. (b) Phase response.

Fig. 7. BS filter responses. (a) Magnitude response. (b) Phase response.

the shortcoming of redundancy, which may cause computation

ineffectiveness.

The suggested CCGA borrows the idea of structure coding

from HGA and separates the control genes and the coefficient

genes into two species. When the genes in one species are

evaluated they are combined with several randomly selected

genes from the other species to form several complete solutions.So, these genes can be evaluated more thoroughly, which means

the CCGA uses the same memory space as HGA but do a much

in-depth space searching.

The suggested CCGA for digital IIR filter design considers

the magnitude response error, phase response error, and lowest

order simultaneously. The control species determines the search

direction. So, the NSGA-II has been used to maintain the

diversity in the three objectives. The coefficient species needsto keep the diversity to ensure that the evolving control genes

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TABLE IVFILTER PERFORMANCES COMPARISON (HGA AN D CCGA). (a) LP FILTER. (b) HP FILTER. (c) BP FILTER. (d) BS FILTER

can always find the proper combination parts. So, the SA with

the heat reservation strategy and the reheating strategy hasbeen used.

The design results for LP, HP, BP, and BS digital IIR filters

show that the suggested CCGA can handle magnitude response

error, phase response error, and lowest order requirements

properly.

REFERENCES

[1] M. D. Lutovac, D. V. Tosic, and B. L. Evans, Filter Design for SignalProcessing Using Matlab and Mathematica. Upper Saddle River, NJ:Prentice-Hall, 2001.

[2] D. M. Etter, M. J. Hicks, and K. H. Cho, Recursive adaptive filter designusing an adaptive genetic algorithm, in Proc. IEEE Int. Conf. ASSP,1982, pp. 635638.

[3] K. S. Tang, K. F. Man, S. Kwong, and Z. F. Liu, Design and optimizationof IIR filter structure using hierarchical genetic algorithms, IEEE Trans.

Ind. Electron., vol. 45, no. 3, pp. 481487, Jun. 1998.[4] N. Karaboga and B. Cetinkaya, Design of minimum phase digital IIR

filters by using genetic algorithm, in Proc. 6th IEEE Nordic SignalProcess. Symp. (NORSIG), Espoo, Finland, Jun. 911, 2004, pp. 2932.

[5] M. Nilsson, M. Dahl, and I. Claesson, Digital filter design of IIR filtersusing real valued genetic algorithm, WSEAS Trans. Circuit Syst., vol. 3,no. 1, pp. 2934, Jan. 2004.

[6] M. C. Lang, Least-squares design of IIR filters with prescribed mag-nitude and phase responses and a pole radius constraint, IEEE Trans.Signal Process., vol. 48, no. 11, pp. 31093121, Nov. 2000.

[7] W. S. Lu, Design of stable IIR digital filters with equiripple passbandsand peak-constrained least squares stopbands, IEEE Trans. CircuitsSyst. II, Analog Digit. Signal Process., vol. 46, no. 11, pp. 14211426,Nov. 1999.

[8] A. Koir and J. F. Tasic, Genetic algorithm and filtering, in Proc. 1st Int. Conf. Genetic Algorithms Eng. Syst.: Innovations and Appl., 1995,pp. 343348. G1.

[9] M. A. Potter and K. A. DeJong, A cooperative coevolutionary approach

to function optimization, in Proc. PPSN, 1994, pp. 249257.[10] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, A fast and elitist

multiobjective genetic algorithm: NSGA-II, IEEE Trans. Evol. Comput.,vol. 6, no. 2, pp. 182197, Apr. 2002.

[11] L. Wang and D. Z. Zheng, Simulated annealing with the state generatorbased on Cauchy and Gaussian distributions, Tsinghua Sci. Technol.,vol. 40, no. 9, pp. 109112, 2000.

[12] J. J. Shynk, Adaptive IIR filtering, IEEE ASSP Mag., vol. 6, no. 2,pp. 421, Apr. 1989.

Yang Yu was born on September 8, 1982, in Fujian,China. She received the B.Sc. degree in electronicscience and engineering from Nanjing University,Nanjing, China, in 2004. She is currently working

toward the Ph.D. degree in electrical engineering atTsinghua University, Beijing, China.Her current research interests are simulations and

analyses of the very fast transient overvoltage in GIS.

Yu Xinjie (M01) received the B.S. and Ph.D.degrees in electrical engineering from TsinghuaUniversity, Beijing, China, in 1996 and 2001,respectively.

He is an Associate Professor of Electrical Engi-neering at Tsinghua University. His research inter-ests include all aspects of computational intelligenceand computational electromagnetics.

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