Abstract—A 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 Terms—Coevolution, genetic algorithms (GAs), infinite-impulse response (IIR) digital filters, linear phase, lowest order.
I. INTRODUCTION
AN INFINITE-IMPULSE RESPONSE (IIR) filter can beexpressed in the cascade form as
H(z) = Kn∏
k=1
1 + bkz−1
1 + akz−1
m∏i=1
1 + di1z−1 + di2z
−2
1 + ci1z−1 + ci2z−2(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 aprocess in which a digital hardware or a program is constructedto meet the given specification.
The traditional digital IIR filter design involves the analogIIR filter design and the analog-to-digital transformation. Whenthe specification for the digital filter is given, we first change itto the corresponding analog low-pass (LP) filter and use one ofthe 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 italso has some disadvantages. First, the traditional digital filterdesign 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.
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 transformationbetween the digital field and the analog field may cause inef-ficiency. Third, the coefficient quantization errors could not beavoided during the design. Last but not least, the lowest filterorder and the phase response requirements are very useful insome practical applications but cannot be considered by thetraditional method.
When genetic algorithms (GAs) were introduced into thedesign world, their flexibility and adaptation were very remark-able. The initial use of GAs for filter design was reported byEtter et al. [2]. Though the result was not impressive, theGA design method showed its distinction. It was a directdesign method in the digital field without the analog-to-digitaltransformation and avoided the coefficient quantization error.Also, it could solve the multiobjective design problem easilyand figure out more than one solution. As GAs become moreand more mature in the last few years, the work on digital filterdesign using GAs has received great attention.
The normal GA design for IIR filter always assumes apredefined topology of the filter. Only the coefficients of thefilter need to be determined. Tang et al. suggested the hierar-chical genetic algorithm (HGA) to tackle this problem [3]. Thestructure of the filter is not fixed during the design, and so it canreach the lowest order. However, this designing method also hassome disadvantages due to its coding redundancy, which will bediscussed in Section II.
One of the drawbacks of the IIR filter is its phase response.Linear phase response can be easily achieved by finite-impulseresponse filters, which is very hard for IIR filters to implement.How to design the approximately linear phase response of anIIR filter becomes the focal and difficult point in some IIRdesign researches.
Karaboga and Cetinkaya suggested a method for designingan IIR filter with minimum phase [4]. Other researchers workon 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 thepassband. In [6] and [7], the proposed method designed filterswith approximately linear phase responses in their passbands.However, the magnitude and phase responses should be pre-scribed in these methods, and the traditional iterative algorithmsare used, including complicated matrix computations. In [8],Košir and Tasic designed the approximately linear phase IIRfilter using GA. However, their method might cause BoundedInput/Bounded Output stable problem. Extra computations arerequired to ensure the stability of the filter.
In this paper, the cooperative coevolutionary geneticalgorithm (CCGA), which is firstly suggested by Potter, is
1312 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 3, JUNE 2007
introduced into the IIR design. CCGA has the followingcharacteristics [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 combinedwith 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 meetsthe requirement of the magnitude response, but also gets theapproximately linear phase response in the passband and thetransition band and finds the lowest filter order simultaneously.The structure and the coefficients of the digital IIR filter arecoded separately, and they evolve coordinately as two differentspecies, i.e., the control species and the coefficient species. Thenondominated sorting genetic algorithm (NSGA)-II is used forthe control species, whereas the simulated annealing (SA) isused for the coefficient species. This new scheme works well inthe multiobjective digital IIR filter design problems.
Section II presents the core of this paper, where the CCGAfor digital IIR design is described in detail. Then, in Section III,a comparison of CCGA and HGA is discussed. Finally, theconclusions and discussions are given in Section IV.
II. CCGA FOR DIGITAL IIR FILTER DESIGN
A. Chromosome Coding
The coding of the suggested CCGA design algorithm isadopted from the HGA. Improvements are made by separat-ing the control genes from the coefficient genes to form twospecies, which enhances the search ability of the HGA with thesame storage space.
The essence of HGA design is its coding. To represent thetransfer function of a digital IIR filter, the chromosome containstwo types of genes, namely: 1) the control gene and 2) thecoefficient gene. The control gene describes the structure ofthe filter, and the coefficient gene defines the value of thecoefficients in each block.
For example, an IIR filter, in which the maximum number ofthe first-order units and the second-order units are both two, canbe described in the cascade form, as shown in (2) at the bottomof the page.
The control gene and the coefficient gene are illustrated inFigs. 1 and 2, respectively.
The control genes are in binary bit form and decide thestate of activation for each block. In Fig. 1, the first gene ofthe chromosome represents the state of the first-order function(1 + b1z
−1)/(1 + a1z−1), where “1” means (1 + b1z
−1)/(1 +a1z
−1) 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 thestructure of IIR filter is represented by (2), the transfer function
where “∗” is the wildcard.The HGA directly connect the coefficient gene to the control
gene to form the individual chromosome. This coding makesthe structure optimization possible, but brings some inefficiencydue to the coding redundancy at the same time.
Let us take the filter described by (4) for example. Thecoefficient genes representing the second first-order block donot contribute to the fitness of the whole chromosome. Whenthey are changed during the evolutionary process, it cannot tellwhether the change is good or not, then the calculations for thatchange and the following evaluation are losing effectiveness.The evolutionary process can be more effective if the coefficientgenes and the control genes are evaluated fully. Then, a betterdirection of the evolution may be found. That is the point tochange 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, namely1) the control species C and 2) the coefficient species X.The coding for C is in binary form and in real-number formfor X. When an individual in the species C is evaluated, someindividuals from the species X need to be selected randomlyand combined with the individual from C to get the completesolutions. The values of the solutions determine the fitness ofthe individual from C. Using the same strategy, species X canbe evaluated.
In this way, for an individual in C, several individuals in Xare chosen to combine with it, respectively. So, the evaluationof the individual in C is more reliable than in HGA, where itis just combined with a fixed coefficient individual. The same
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 of X gets the same advantage. As weget more reliable evaluation of the individuals, the evolutionworks more effectively, with less good genes being lost duringthe evolution process.
B. Process of CCGA for IIR Design
As the structure and the coefficients of the filter are setto be two species separately, the evolution process is dividedinto the following two parts: 1) the evolution of the controlspecies and 2) the evolution of the coefficient species. Thewhole coevolution process is shown by the flowchart in Fig. 3.
With multiobjective optimization, Pareto solutions might beexpected to make the decision making process more reliable.An IIR filter with higher order tends to have better magnituderesponse, whereas the higher order results in more complicatedcalculations, and the linear phase response is more difficult torealize. Even when considering the IIR filters with the sameorder, the comparisons are hard to make, as some filters may besuperior in the magnitude responses, whereas others may haveapproximately linear phase responses. The decision dependson the practical uses. The suggested design algorithm can findseveral Pareto solutions, incommensurable good filters, at thesame time rather than a single “best” solution.
C. Evaluation
There are three objective functions in our suggested digitalIIR 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 designof an IIR filter, the following magnitude response conditionsare required.
• The attenuation in passband should not exceed δ1.• The attenuation in stopband should not be less than 1 − δ2.• The passband and stopband edge frequencies are repre-
sented by ωp and ωs, respectively.
The magnitude response error is calculated as follows [3]:
eHp(ω) ={
1 − δ1 −∣∣H(ejω)
∣∣ ,∣∣H(ejω)
∣∣ < 1 − δ1
0,∣∣H(ejω)
∣∣ ≥ 1 − δ1
where ω is in the passband.
eHs(ω) ={ ∣∣H(ejω)
∣∣ − δ2,∣∣H(ejω)
∣∣ > δ2
0,∣∣H(ejω)
∣∣ ≤ δ2
where ω is in the stopband.eHp(ω) and eHs(ω) are the passband and the stopband
magnitude response errors at ω, respectively. Then, the firstobjective function is
min f1 =∑ωi
eHp(ωi) +∑ωj
eHs(ωj) (5)
where ωi is the sampling frequency in the passband, and ωj isthe sampling frequency in the stopband.
2) Linear Phase Response Error: The linear phase responseis simplified as the passband and transition band linear phaseresponse. The phase response of the transition band is alsoconsidered because the magnitude response at some points oftransition band may be as high as that in the passband. If thephase response is far away from linear at these points, it couldresult in large distortion.
We sample the phase response of the digital IIR filter with thesame frequency interval and get the phase sequence as follows:
Phases = {θ1, θ2, . . . , θn}.The phase difference sequence can be counted as
The sampling of frequency is in equal interval; so in thelinear phase case, the phase sequence is an equal-differencesequence. That is to say all the elements in the ∆Phasessequence have the same value. How far a phase response is fromthe linear phase condition can be evaluated by the variances ofits ∆Phases sequence, which is then set to be the phase responseerror. Then, the second objective function is
min f2 = variance{∆θi|θi ∈ 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 =m∑
i=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)
1314 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 3, JUNE 2007
Evaluation of C and X: In the real-world implementa-tion, the magnitude response is considered to be more importantthan the phase response. So, the magnitude response error is thedominant part of the objective function.
Each individual in C is evaluated by combining with K1
random selected individuals from X. So, K1 candidate digitalIIR filters are formed for evaluating one individual in C. Themagnitude response error and linear phase response error arecalculated for all K1 candidates. Then, we get the followingrow vectors:
errmag = [mag1,mag2, . . . ,magK1 ]T
errphase = [pha1, pha2, . . . , phaK1 ]T .
If the minimum value of errmag 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 becalculated easily by using (7) for ith candidate. In this way,every individual in C gets its three objective function valuesafter the evaluation.
When an individual in X is evaluated, a similar strategy isused by combining it with K2 random selected individuals inC. Because coefficient genes do not have order value, everyindividual in X gets its two objective function values after theevaluation.
D. Evolution of the Control Species
To deal with the three objectives effectively, NSGA-II isadopted 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 everyindividual can be calculated by Deb’s method [10]. To pre-vent the crowding effect, a group insertion [3] is used beforethe elitist selection. The whole evolutionary process is givenas follows.
1) The two-point crossover and the bit-flips mutation arecarried out in Pt to form some new individuals called Qt.
2) The group insertion method in [3] is used to form a newgroup called Rt.First, all individuals in Pt and Qt are divided into sub-groups Gi according to the order, i.e., the third objectivefunction value, where the individuals with the same orderare 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 firstlychosen. If the size of Gi is smaller than K, all theindividuals in it are chosen into Rt.
3) Select N individuals from Rt into the next population.a) The nondomination rank and the crowding distance
for every individual in Rt are calculated by Deb’smethod [10].
b) Sort Rt according to the nondomination rank. LetF = {F1, F2, . . .} be the nondominated fronts of Rt,where F1 is the best nondominated set.
c) Select Pt+1 with the following procedures.For i = 1, 2, . . .If the size of Fi is smaller than N − M , where Mis the number of the individuals now in Pt+1, allmembers of the set Fi are put into Pt+1;Else all the remaining members of Fi are sorted usingthe crowding distance, and the ones with the longercrowding distance are chosen to fill the populationslots. 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 thefilter, so it plays a leading role, and the coefficient species isa subordinate part. During the evolution process, the controlgenes may change a lot. The original good coefficient genesmay not fit them any more. On the other hand, the individuals ofthe control species in one generation are different, representingdifferent kinds of filter structure. In order to find individuals inthe coefficient species to fit different kinds of filter structure andmake good filter design solutions, the diversity of the coefficientspecies needs to be kept.
The SA can preserve the worse individual to some extentaccording to the probability. So, we use the SA in the evolutionof the coefficient species to keep the diversity. The SA needsone value for two individuals to compare, so we sum themagnitude response error f1 and the phase response error f2
to be a total error value.In the implementation of the SA, the heat reservation strategy
and the reheating strategy are used to increase the evolu-tionary efficiency. The settings of the parameters during theannealing 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 ofindividuals produces one child. From every pair of par-ents, the one with the larger error value may be replacedby 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 tothe mutation rate. The produced children replace theirparents 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.
YU AND XINJIE: COOPERATIVE COEVOLUTIONARY GENETIC ALGORITHM FOR DIGITAL IIR FILTER DESIGN 1315
TABLE IDESIGN CRITERIA
TABLE IIPARAMETERS FOR GENETIC OPERATIONS
6) If the temperature is lower than 10−5, set the currenttemperature to be T = T0, and the evolutionary processin this generation is complete.
As can be seen from the aforementioned procedure, the SAwith the heat reservation strategy and the reheating strategypreserves the diversity of coefficient species, which is quiteuseful for finding better design solutions with the evolution ofthe control species.
III. EXPERIMENTAL RESULTS
In this section, the suggested CCGA is used to design sometypical IIR filters, and its performances are compared withthe performances of the HGA [3]. All the genetic operatingparameters are set exactly the same as those in [3].
The fundamental structure of H(z) is given as
H(z) = K
3∏i=1
(1 + biz−1)
(1 + aiz−1)
4∏j=1
(1 + bj1z−1 + bj2z
−2)(1 + aj1z−1 + aj2z−2)
. (9)
So, the highest order of the designed filter is 11. The lengthof control chromosome is 7, and the length of coefficientchromosome 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-orderblock 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), aredesigned in the experiment. Parameters for the design criteriaare listed in Table I, and the parameters for genetic operationsare given in Table II.
The termination condition is that the first objective functionvalue f1 equals zero, and the other two objective functions areless than the given values.
We run the design process for 20 times. The best resultsfound by CCGA are summarized in Table III, the final filterfunctions are indicated in (10)–(13), as shown at the bottom ofthe next page, and the magnitude and the phase responses areshown in Figs. 4–7, respectively.
It can be seen from the results that the proposed designmethod 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.
1316 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 3, JUNE 2007
TABLE IIIFILTER PERFORMANCES (DESIGNING USING CCGA)
By studying Table IV(a)–(d), we can arrive at the followingconclusions.
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 thelowest 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, andthe SA work well in handling the two species and thethree objectives.
IV. DISCUSSION AND CONCLUSION
GAs can design digital IIR filter directly, which is moreflexible than the traditional ways. The HGA codes the structureof the digital filter with control genes and combines them withcoefficient genes to form a whole design. This coding has
the shortcoming of redundancy, which may cause computationineffectiveness.
The suggested CCGA borrows the idea of structure codingfrom HGA and separates the control genes and the coefficientgenes into two species. When the genes in one species areevaluated they are combined with several randomly selectedgenes 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 muchin-depth space searching.
The suggested CCGA for digital IIR filter design considersthe magnitude response error, phase response error, and lowestorder simultaneously. The control species determines the searchdirection. So, the NSGA-II has been used to maintain thediversity in the three objectives. The coefficient species needsto keep the diversity to ensure that the evolving control genes
1318 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 3, JUNE 2007
TABLE IVFILTER PERFORMANCES COMPARISON (HGA AND CCGA). (a) LP FILTER. (b) HP FILTER. (c) BP FILTER. (d) BS FILTER
can always find the proper combination parts. So, the SA withthe heat reservation strategy and the reheating strategy hasbeen used.
The design results for LP, HP, BP, and BS digital IIR filtersshow that the suggested CCGA can handle magnitude responseerror, phase response error, and lowest order requirementsproperly.
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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 workingtoward the Ph.D. degree in electrical engineering atTsinghua University, Beijing, China.
Her current research interests are simulations andanalyses of the very fast transient overvoltage in GIS.
Yu Xinjie (M’01) 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.
