Int. J. Electron. Commun. (AEÜ) 60 (2006) 85–92

www.elsevier.de/aeue

Component value selection for active filters using parallel tabu searchalgorithm

Adem Kalinli∗

Department of Industrial Electronics, Vocational High School, Erciyes University, 38039 Melikgazi, Kayseri, Turkey

Received 27 July 2004; received in revised form 29 October 2004

Abstract

The determined component values by means of conventional approach to the design of analogue electronic circuits, typi-cally do not coincide with all manufactured standard values and the designer selects the nearest standard value, thus causinga design deviation. In general, a better set of standard values will exist in the discrete domain of standard component values.However, this set will be in a solution space of all possible component-value combinations that is a highly complex discretesearch problem. Tabu search is a heuristic optimisation algorithm, which simulates the general rules of intelligent problemsolving and has the ability of discovering the global minima in a multi-modal search space. In this work, a novel methodbased on a parallel tabu search algorithm is described for the component value selection of the analogue active filter.

� 2005 Elsevier GmbH. All rights reserved.

Keywords: Parallel tabu search; Optimisation; Circuit design

1. Introduction

In conventional approximation, components are as-sumed as ideal and infinite values in designing analogueelectronic circuits. However, discrete components likethat resistors and capacitor are produced in approximatelogarithmic multiples of a defined number of constantvalues. Typically produced 12 preferred values (E12)which are 1.0, 1.2, 1.5, 1.8, 2.2, 2.7, 3.3, 3.9, 4.7, 5.6, 6.8,

8.2, 10, . . .. To reduce costs in design, discrete componentsare chosen from this series or other possible produced pre-ferred values. Component values that obtained conventionalapproximations do not exactly converge with the pro-duced preferred component values. Therefore, the designperformance deviates from the ideal when the nearest pre-ferred value is selected. The errors that occur due to these

∗ Tel.: +90 352 4374915; fax: +90 352 4375267.E-mail address: kalinlia@erciyes.edu.tr

1434-8411/$ - see front matter � 2005 Elsevier GmbH. All rights reserved.doi:10.1016/j.aeue.2005.03.001

deviations can be reduced by selecting nearer value com-ponents from such E24 series or using special value com-ponents that obtained connecting couple of components inseries or parallel. As a result of this, these approximationscontain errors and increase circuits dimension and cost.

Generally, conventional design methods show approxima-tions, reducing freedom degree quantity to obtain directlyapplicable formulas by selecting certain defined compo-nents equal to each other or multiples each other. The meanof this simplification in design methods is to make a de-sign as similar as the ideal one by excepting combinationsof preferred component values. There will be generally aset of produced preferred values consisting of less designerrors. To design in solution space that consists of all com-ponent values is a rather complex discrete optimisationproblem. As an example, in the eight-component circuitconsidered in this study, if components are selected fromthe E12 series over a four-decade range then the searchspace contains about 3 × 1013 points [1]. For optimum

86 A. Kalinli / Int. J. Electron. Commun. (AEÜ) 60 (2006) 85–92

design, computer-based search on all possible combinationsrequires long computation time. For this reason, to definediscrete component values, an alternative method must beapplied.

The great difficulty of optimisation problems encounteredin practical areas such as production, control, communi-cation, and transportation has motivated researchers to de-velop new powerful algorithms. The popular ones of thesenew algorithms include genetic algorithms (GAs), simu-lated annealing (SA), ant colony optimisation (ACO),and tabu search (TS) algorithm [2–4]. Although all thesealgorithms have convergence property to global optimumthey cannot always guarantee the optimum solutions forthe problem. Therefore, they are called approximate orheuristic algorithms.

TS algorithm is a form of iterative search and based onintelligent problem solving principles. The algorithm wasdeveloped by Glover [5]. TS has a flexible memory to keepthe information about the past steps of the search and uses itto create and exploit the new solutions in the search space. Abasic TS algorithm starts searching with a present solutionand constructs a set of feasible solutions from the presentone based on neighbourhood by using a history record of thesearch. It evaluates the solutions constructed and selects theone which has the highest evaluation value as the next so-lution. Having selected the next solution, the history recordis then modified. This process is repeated until a predefinedstopping criteria is satisfied. In the use of TS, the possibilityof repeatedly evaluating similar solutions decreases since TSrecords are a history of search. TS seems to be quite promis-ing for multi-modal optimisation problems, since it is basedon neighbourhood search and works iteratively and can getout of a local minima in a multi-modal search space by us-ing the history record of search [6,7]. Therefore, some re-searchers have attempted to develop modified versions of TSfor the multi-modal problems and applied to solve the prob-lems from different fields [6–9]. All algorithms described inthese works are of a conventional type, i.e. they have a serialstructure. However, a conventional TS might have a problemwith reaching the global optimum solution in a reasonablecomputation time when the initial solution is far away fromthe region where optimum solution exists. The dependenceof the convergence speed of TS on the initial solution can beweakened by introducing a parallel structure into the algo-rithm [10,11]. The parallelism helps the TS find the promis-ing regions of the search space very quickly. In our previouswork, we proposed a parallel TS (PTS) algorithm, which isbased on the crossover operator of GAs, and tested its per-formance on a set of numeric test functions and training arecurrent neural network to identify linear and non-lineardynamic plants [7].

ACO algorithm, which has global optimisation ability,is the artificial version of the natural optimisation processcarried out by real ant colonies [3]. The main features ofthe algorithm are distributed computation, positive feed-back and constructive greedy search. Therefore, the perfor-

mance of ACO algorithm is good for local search due tothe positive feedback and for global search because of thedistribution computation features. In the literature, severalmodels were proposed for ACO algorithm and the one ofthese is Touring ACO (TACO) algorithm [12]. In TACOalgorithm each solution is represented by a vector of designparameters of which each is coded with a string of binarybits, i.e. a solution is a vector of binary bits. Therefore,artificial ants search for the value of each bit in the string,in other words they try to decide whether the value of a bitis 0 or 1.

Although progress has been significantly made in au-tomating design of certain categories of digital circuits,the design of analogue circuits has not been achieved. Theheuristic algorithms, such as TS, GA, SA, and ACO algo-rithm, have attracted the attention of researchers in the fieldof analogue circuits design in recent years. Various stud-ies based on these algorithms have been realised to designanalogue circuits automatically. Horrocks et al. success-fully used GA to design active and passive filter circuits[1,13–15]. There have been many studies used for analoguecircuit design with the help of SA and GA. However, thereare few studies to minimise optimum layout and circuits di-mension in VLSI circuits related to analogue circuit designusing TS and ACO algorithm [16–21].

In this study, a novel method based on PTS algorithmhas been introduced to design analogue active filters. PTSalgorithm is used for selection of active filter circuit com-ponent values. PTS was compared to basic TS, TACO, anda conventional method in order to examine performance ofthe proposed method. Two forms of active filter with thesecond-order state variable are considered to compare theperformances. The first, resistance, capacitor and opamp,is a completely discrete circuit. The second is the semi-integrated form as exemplified by the AF100 from NationalSemiconductor [22]. In this form, the opamps and some ofthe passive components are integrated and the remaining dis-crete components are attached externally. In the examples,preferred values produced from E12 series for resistancesand capacitors are considered. Section 2 describes the basicprinciples of TS and PTS algorithms. Section 3 presentsthe state variable active filter and the conventional designprocedure. Section 4 describes how PTS can be applied tocomponent value selection for active filter. The simulationresults are given in Section 5. The work is concluded inSection 6.

2. Basic and parallel TS algorithms

2.1. Basic principles of TS algorithm

TS is a general heuristic search procedure devised forfinding a global minimum of a function f (x). The problemof searching the optimum value of x, which makes f (x)

minimum, is called the optimisation problem of f (x) and

A. Kalinli / Int. J. Electron. Commun. (AEÜ) 60 (2006) 85–92 87

Initialisation

Initial solution

Creation of the solutions Q*

Evaluation

Selection

Modification

Fig. 1. Flowchart of a basic tabu search.

can be mathematically expressed by

Minimise f (x)

Subject to x ∈ X.

The function of f (x) may be linear or non-linear and thecondition x ∈ X describes the constraints on the solution x.A step of the TS starts with a present solution xnow. xnow ∈X has an associated set of feasible solutions Q, which canbe obtained by applying a simple modification to xnow. Thismodification is called a move. In order to be able to get rid ofa local minima, a move to the neighbour, x∗, is created evenif x∗ is worse than xnow. This would cause the cycling of thesearch. In order to avoid the cycling problem, a tabu list T isintroduced. The tabu list stores all the tabu moves that cannotbe applied to the present solution, xnow. The moves storedin the tabu list are those carried out most frequently andrecently according to some criteria called tabu restrictionsand using them decreases the possibility of cycling becauseit prevents returning in a certain number of iterations to asolution visited recently. After a subset of feasible solutionsQ∗ is produced according to the tabu list and evaluatedfor f (x), the next solution is selected from it. The highestevaluation solution is selected as the next solution xnext. Thisloop is repeated until a stopping criteria is satisfied.

A tabu restriction is activated if the reverse of a moverecorded in the tabu list occurs within a predetermined num-ber of iterations or it occurs with a certain frequency over alonger range of iterations. The former produces a recency-based restriction and the latter a frequency-based restriction.Tabu restrictions might sometimes prevent the search to findthe solutions, which have not been visited yet, or even causeall available moves to be classified as tabu. Therefore, itshould be possible to forget the tabu constraints when a free-dom is required for the search. A criterion called aspirationcriteria is employed to determine which moves should befreed in such cases.

The flowchart of a basic TS is presented in Fig. 1. In theinitialisation unit, a random feasible solution xinitial ∈ X for

the problem is generated, and the tabu list and other param-eters are initialised. In the neighbour production unit, a fea-sible set of solutions is produced from the present solutionaccording to the tabu list and aspiration criteria. The eval-uation unit evaluates each solution x∗ produced from thepresent one xnow. After the next solution xnext is determinedby the selection unit, in the last unit the history record of thesearch is modified. If the next solution determined is betterthan the best solution found so-far xbest the next solution isreplaced with the present best solution. In this work, a stringof binary numbers is used to represent a possible solutionand TS employs the following two constraints that are basedon recency and frequency memories as the tabu constraints:

recency(x∗)�restriction_period,

frequency_ratio(x∗)�frequency_limit,

where restriction_period = recency factor× length of solu-tion string and frequency_limit = frequency factor ×average_frequency.

The recency of a move is the difference between the cur-rent iteration count and the last iteration count at which thatmove was created. The frequency measure is the frequencyratio whose numerator represents the count of the number ofoccurrences of a specific move and whose denominator rep-resents the average numerator value over all possible moves.

In the algorithm used, the highest evaluation move is se-lected as the next solution. The aspiration by default is em-ployed as the aspiration criteria. According to this criteria,the least tabu solution is selected as the next solution. Theleast tabu solution is the solution that loses its tabu clas-sification by the least increase in the value of the presentiteration number.

2.2. Parallel TS algorithm

There are mainly four sources of parallelism in TS algo-rithm [7]: parallelism (a) in the cost function evaluation; (b)in the neighbourhood examination; (c) in the problem de-composition; and (d) in the solution domain exploration bycarrying out different searches. In the last type parallelism,different independent TS algorithms are executed from dif-ferent initial solutions. TS algorithms being executed inparallel have the ability of exchanging information. If thecommunication between the algorithms is carried out at thepredetermined moments, then parallel algorithms are calledsynchronous. If this information exchange is realised at dif-ferent moments, these type of algorithms are called asyn-chronous. The parallelism used in this work is of the lasttype. The information exchange process between the basicTS algorithms executed in parallel is based on the crossoveroperation used in GAs.

The crossover operator employed by GAs is used to createtwo new solutions (children) from two existing solutions(parents) in the population formed by the selection operation.Depending on the representation of the problem in the string

88 A. Kalinli / Int. J. Electron. Commun. (AEÜ) 60 (2006) 85–92

initial solutions

1 2 3 h….

TS ….

Crossover

TS TS TS

final solutions

….

Fig. 2. Flowchart of the PTS.

form, a proper crossover operator must be chosen. Whenthe problem is represented in the binary string form, thesimplest crossover operation can be applied as follows: twosolutions are randomly selected as parent solutions from thepopulation and cut at a randomly selected point. The tails,which are the parts after the cutting point, are swapped andtwo new solutions are produced. A crossover operation canthus yield better solutions by combining the good featuresof parent solutions. An example of this simple crossoveroperator is given below

PresentSolution1 111100|001111 → Tail,PresentSolution2 110001|111111 → Tail,

NewSolution1 111100|111111,

NewSolution2 110001|001111.

The flowchart of the proposed model is depicted in Fig. 2.Blocks from 1 to h are each identical to the flowchart shownin Fig. 1. These blocks represent h TSs executed independentof one another. The initial solutions for TSs at the first levelare created using random number generator with differentseeds.

After a given number of iterations (maxit1) the execu-tion of h TSs is stopped. maxit1 is normally chosen to besufficiently large to allow the search to complete the localsearching. The h solutions found by these TSs at the firstlevel represent the ‘preliminary’ solutions. An operation isapplied to the solutions found by h parallel TSs at the firstlevel to create the initial solutions for TSs at the secondlevel. Various methods could be used for this purpose; forexample, the best solution found by the TSs at the first levelcan be selected as one of the initial solutions for the nextlevel always. The others can be produced from the prelimi-nary solutions by using a suitable crossover operator or allthe initial solutions can be created using crossover opera-tor. Another procedure to form initial solutions could be thefollowing: the best preliminary solution is selected directly,some can be created by applying crossover operator to thepreliminary solutions and the rest can be generated by a ran-dom number generator. In this work, the best preliminary

R3

R4

-

+input

C1

-

+

-

+

C 2

R5

R2

R6R1

VHP

VBP

VLP

Fig. 3. Basic design for a state variable active filter.

solution is directly selected and the rest of initial solutionsof TSs are created by applying crossover operation to the so-lutions obtained at the previous level. At all levels, TSs areexecuted in the same way. The best solution found throughthe whole search process is taken as the optimal solution forthe problem. Each TSs might have different control param-eter values.

3. The state variable active filter

The state variable active filter (SVAF) is a different typeof multiple-feedback filter, using three or four opamps. It hasone input and three outputs providing low-pass, high-pass,and band-pass filtering simultaneously. The basic design cir-cuit for a SVAF is illustrated in Fig. 3. Detailed analysis ofthe circuit can be found in the literature, for example [1,23].The low-pass output is assumed here to be the desired out-put. The response of a second-order low-pass circuit is com-pletely specified by the three basic filter parameters: cut-offfrequency (w0 = 2�f0), quality factor (Q), and passbandgain (H). These quantities are given, in terms of the passivecomponent values, by Eq. (1).

In this work, the specification chosen was w0 = 10 000 ÷2� = 1591.55 rad/s, and Q = √

2 = 1.41421. The passbandgain, H, can be easily compensated for by other cascadedanalogue circuits and it is not very critical in most appli-cations. In the conventional design procedure, H is fixed atsome value; however, for the PTS method described belowit is unconstrained:

H = R2(R3 + R4)

R3(R1 + R2),

w0 =√(

R4

R3

) (1

C1C2R5R6

),

Q = R3(R1 + R2)

R1(R3 + R4)

√C1R4R5

C2R3R6. (1)

3.1. Conventional design procedure

In this work, the design method which is used by Horrocksand Spittle in [1] is considered as a conventional procedure.The explanation of this method is given below.

A. Kalinli / Int. J. Electron. Commun. (AEÜ) 60 (2006) 85–92 89

For the fully discrete circuit there are six resistors andtwo capacitors. Its values are to be chosen related to thespecified values for w0 and Q; H is also important. Theconventionally proposed method arbitrarily chooses to makeboth capacitors equal to C and all resistors except R2 equalto R. Thus, Eq. (1) reduces to the following:

w0 = 1

RC, (2)

R1 = R3 = R4 = R5 = R6 = R,

C1 = C2 = C, R2 = (2Q − 1)R. (3)

Firstly, the procedure is to choose a pair of values for R andC to satisfy Eq. (2), then to calculate the circuit componentsusing Eq. (3). The resulting passband gain is H = 1. For anexact design, a reasonable way is to choose a preferred valuefor R in the middle of the range. The values for the remainingR2 and C then follow from Eqs. (2) and (3). In general,these values have to be special non-preferred values, if anexact design is to be obtained. For a design using preferredvalues, only the exact values can be rounded to the nearestpreferred values [1].

For the semi-integrated circuit AF100, components R1,R5, and R6 are connected externally, and the other five com-ponents, having fixed values, are integrated in the device.In the conventional design, the user specifies the three per-formance parameters, H, w0 and Q, and solves for the threeexternal resistor values by means of Eq. (1).

4. Application of the TS algorithm to theproblem

For the application of TS, each passive component ofthe considered circuit was represented with one locationin the solution vector. Thus, for the fully discrete circuitproblem eight locations in solution vector were used (seeFig. 4). Each of these locations was allocated a group of bitsto identify a particular preferred value for that component.The number of bits allocated depends on the representationmethod. For the one-bit approach, one bit is allocated, whichallows a choice from two preferred values that are nearest tothe exact design values that emerged from the conventionaldesign procedure described above. The number of bits isprogressively increased in the two-bit, three-bit, and four-bit approaches. In the four-bit approach, four bits allow 16preferred values spread on either side of the exact values tobe offered for selection by the TS algorithm. The purposeof these methods is to explore the effect of increasing therange of the search space.

R 1 R 2 R 3 R 4 R 5 R 6 C1 C 2

Fig. 4. Representing the component values in the string form.

In the full evaluation approach eight groups of six bits areused to specify any preferred value. Two of these bits signifythe decade in the range of 103–106 � for the resistors and10−9– 10−6 F for the capacitors. Values outside these rangeswere judged to lead to unwanted practical effects such asstray capacitance effects or large signal currents. The re-maining four bits are used to signify any of the 12 preferredvalues in the decade range. For the semi-integrated circuitthe full evaluation approach has been applied using the samedata structure as for the fully discrete circuit, but three ratherthan eight groups of six bits to specify the three externallyconnected components. Since each component value is rep-resented with six bits in this work, the length of a solutionin the form of a binary string is 48 for the first circuit and18 for the second circuit.

As a result of choosing preferred values for the compo-nents, the cut-off frequency and quality factor will deviatefrom the specification by �w and �Q, respectively. The aimis to keep these deviations as low as possible. The error cri-terion adopted here is

error = a1|�w|w0

+ a2|�Q|Q

, (4)

where a1 and a2 are constants and their values were cho-sen as 0.5. Thus, the acceptable design tolerances for cut-off frequency and quality factor are equal. In this case, thecomponents values of the circuits are successively adjustedby TS algorithm until the error is minimised.

5. Simulation results

The results obtained by using the conventional designmethod for the first circuit is shown in Table 1 . The designerror achieved from the conventional method was 7.1824%[1]. The errors achieved from PTS for the fully discrete cir-cuit with various bits are shown in Table 1 as well. Fromthe results it is clearly evident that a very large reductionin design error was obtained. Even highly restricted one-bit approach PTS produces a 16 fold improvement over theconventional method. For the full-evaluation approach, thedesign error is about 0.0011%. The performance of PTS wasalso compared with the results of basic TS and TACO algo-rithm. The results of TACO algorithm have been taken from[24]. The design results of these methods are given in Table2. It is clearly seen that the designs obtained by using PTSalgorithm have lower error values.

In order to show the robustness of the proposed approach,the frequency histogram of the results were obtained for 50different runs with different initial solutions by using TS, andPTS algorithms were obtained. Fig. 5 shows the histogramsfor the fully discrete circuit. In the simulations, basic TSdescribed in Section 2.1 was executed for 150 000 evalua-tions for each trial. After that, the simulation results wereobtained for the PTS. The number of the basic TSs runningparallel was three (h = 3). Each basic TS at the first level

90 A. Kalinli / Int. J. Electron. Commun. (AEÜ) 60 (2006) 85–92

Table 1. Component values and performances of the conventional and PTS methods

Parameters Conventional [1] PTS Algorithm

One bit Two bit Three bit Four bit Six bit

Q 1.37234 1.42119 1.41655 1.41264 1.41414 1.41421w0 1773.05 1585.86 1591.52 1591.52 1591.22 1591.51R1 4700 5600 3300 3900 22 000 120 000R2 8200 10 000 6800 8200 18 000 1 800 000R3 4700 4700 4700 5600 6800 150 000R4 4700 5600 3300 8200 3300 18 000 000R5 4700 4700 3300 4700 3300 560 000R6 4700 5600 5600 8200 2200 4700C1 1.2 × 10−7 1.5 × 10−7 1.5 × 10−7 1.5 × 10−7 2.2 × 10−7 1.2 × 10−8

C2 1.2 × 10−7 1.2 × 10−7 1.0 × 10−7 1.0 × 10−7 1.2 × 10−7 1.5 × 10−6

Error (%) 7.1824 0.4253 0.0836 0.0568 0.0126 0.0011

Table 2. Error values for various design methods

Methods Errors (%)

One Bit Two bit Three bit Four bit Six bit

TS 0.7259 0.2909 0.1771 0.0799 0.0332PTS 0.4253 0.0836 0.0568 0.0126 0.0011TACO [24] 0.4253 0.1442 0.0617 0.0355 0.0129Conventional [1] 7.1824

was run for 50 (maxit1) iterations. The best of the solutionsfound by TSs at the first level was directly taken as the ini-tial solution for a TS at the second level. Two other initialsolutions required by the second-level TSs were created byapplying the crossover operator, described in Section 2.2,to the preliminary solutions found by TSs at the first level.At each level, two crossover operations were carried out.After each crossover process, two new solutions were pro-duced and one of these new solutions was randomly chosenand replaced with a present solution. The total number ofiterations made at the first level was 150(hxmaxit1). Thisprocess was repeated through 150 000 evaluations. The fre-quency histogram of the results that belongs to TACO al-gorithm has been taken from [24] and the total evaluationnumber is 500 000 for TACO algorithm.

In practical realisations, manufacturing tolerance in thecomponents might mask these low error values. The de-sign of active filters generally requires accurate components.Typically, resistors with 1% or 5% tolerances are used indiscrete circuits, rarely, in less critical applications 10% or20% resistors will suffice. On the other hand, capacitorswith 10% or 20% tolerances are more readily available andare preferred to save cost [23]. One hundred different trialswere performed by adding tolerances maximum ±10% and±1% to components values, which are obtained with con-ventional and PTS approaches in order to examine the effectof the manufacturing tolerances on design errors. Averagevalues of the quality factors, cut-off frequencies, and errors

0

5

10

15

20

25

30

0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

error (%)

freq

uenc

y (%

)

0

5

10

15

20

25

30

0.001 0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

error (%)

freq

uenc

y (%

)

0

5

10

15

20

25

30

0.01 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7

frequ

ency

(%

)

error (%)

(a)TS

(b)PTS

(c)TACO

Fig. 5. Histograms drawn from the results obtained for thefully-discrete circuit.

A. Kalinli / Int. J. Electron. Commun. (AEÜ) 60 (2006) 85–92 91

Table 3. The results by considering manufacturing tolerances in the fully-discrete circuit

Parameters Tolerance (10%) Tolerance (1%)

Conventional PTS Conventional PTS

Average Q 1.32457 1.43333 1.37106 1.41350Average w0 1786.57 1593.54 1773.23 1590.79Average error (%) 10.1665 3.3173 7.2331 0.4528

Table 4. Component values and error values for AF100 circuit

Parameters Conventional [1] TS PTS TACO [24]

Q 1.32008 1.40262 1.41521 1.41521w0 1610.80 1601.28 1610.81 1610.81R1 10 000 68 000 560 000 560 000R2 (100 000) (100 000) (100 000) (100 000)R3 (100 000) (100 000) (100 000) (100 000)R4 (10 000) (10 000) (10 000) (10 000)R5 8200 390 000 820 000 820 000R6 470 000 100 000 47 000 47 000C1 (1.0 × 10−9) (1.0 × 10−9) (1.0 × 10−9) (1.0 × 10−9)

C2 (1.0 × 10−9) (1.0 × 10−9) (1.0 × 10−9) (1.0 × 10−9)

Error (%) 3.9055 0.7156 0.6403 0.6403

obtained from these trials are given in Table 3. As seen inTable 3, lower design errors obtained with PTS and also incase manufacturing tolerances are considered. It is also pos-sible to realise designs close to the ideal case by selectingcomponents with tolerances as low as 1%.

Simulation results obtained by different methods for thesemi-integrated AF100 circuit are shown in Table 4. This cir-cuit has five fixed internal components, indicated by paren-thesis. The remaining three component values are availablefor choice by the PTS. Because this is more restrictive thanthe fully discrete circuit, the error obtained is larger. Evenso, the error value achieved from PTS is less than basic TS,and is the same as TACO algorithm.

In the PTS evolution of these designs, the passband gain,H, was unconstrained. The resulting passband gains, takento be H = 1 in the conventional design range from 0.36187to 2.4352 in the PTS designs. These values would be accept-able for many applications. In the AF100 an extra opamp isincluded, which can be used for adjusting the passband gain.

6. Conclusions

A new method based on parallel TS algorithm has beenpresented for the analogue active filter design. The perfor-mance of the proposed method was also compared with aconventional design method, basic TS and TACO algorithms.Significant reductions in design errors can be achieved withrespect to the conventional approach and two other heuristicalgorithms. Simulation results have shown that the parallel

TS algorithm could be efficiently used for the analogue ac-tive filter design. It is apparently seen that expanding thesearch space helps achieve lower design errors and selectingcomponents with the lower tolerances series provides closerapproximation to the ideal case in the parallel TS designmethod.

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