analysis and design of digital iir

8/10/2019 Analysis and Design of Digital IIR

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Hindawi Publishing CorporationISRN ElectronicsVolume , Art icle ID , pageshttp://dx.doi.org/ . / /

Research Article Analysis and Design of Digital IIR Integrators andDifferentiators Using Minimax and Pole, Zero, andConstant Optimization Methods

Madhu Jain, 1 Maneesha Gupta, 2 and N. K. Jain 3

Department o Electronics and Communication Engineering, Jaypee Institute o In ormation echnology, A- , Sector ,Noida, Uttar Pradesh , India Advanced Electronics Lab, Division o Electronics and Communication Engineering, Netaji Subhas Institute o echnology,Sector , Dwarka, New Delhi , IndiaInstrument Design and Development Centre, Indian Institute o echnology, Hauz Khas, New Delhi , India

Correspondence should be addressed to Madhu Jain; ermadhu @gmail.com

Received April ; Accepted May

Academic Editors: C. W. Chiou, E. I. El-Masry, and E. lelo-Cuautle

Copyright Madhu Jain et al. Tis is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Proposed work deals with the design o a amily o stable IIR digital integrators via use o minimax and pole, zero, and constantoptimizationmethods. Firstthe minimax optimizationmethod is usedto design a amily o second-, third-, and ourth-orderdigitalintegratorsbyoptimizing themagnitude responsein a min-max senseunder thesatis actory condition o constant group delay. Tenthe magnitudeandgroup delay response is urther improvedusing pole, zero, and constant optimization method. Subsequently, by modi ying the trans er unction o all o the designed integrators appropriately, new differentiators are obtained. Simulation resultsshow that proposed approach outper orms existing design methods in terms o both magnitude and phase response.

1. Introduction

Digital integrators and differentiators are integral parts o many systems like digital signal processing, control, audio,and video processing, communication, and medical applica-tions.Frequency responseo an ideal digital integrator is 1/ando ideal digital differentiator is , where = 1andis theangular requency in radians/second.Due to increasingnumber o applications involving digital signal processing, variety o requirements that have to be met by digital integra-tors and differentiators have increased as well. Digital inte-grators and differentiators can be classi ed as nite impulseresponse (FIR) and in nite impulse response (IIR), respec-tively. For a large number o applications, where high selec-tivity and efficient processing o digital signal are required,IIR digital integrators and differentiators are pre erred.

Generally IIR digital integrators and differentiators de-signshavebeenproposedby using theNewton-Cotes integra-tion rule [ ]. Tese digital operators can be designed di-rectly or by trans ormation o analog integrators and

differentiators like impulse invariance, bilinear trans orma-tion, orward difference equation, and backward differenceequation [ ]. Rectangular, rapezoidal, Simpson / , Simp-son / , and Boole are the basic integrators proposed [ ].Schneider et al. [ ] have used parabolic and cubic numericalintegration rules in design o digital integrator. Al-Alaoui [ ]has designed digital integrator by using linear interpolationmethod (mixing o trapezoidal and rectangular integrators);since then, this method has gained immense popularity. Ngo[ ] hasdesigned an integratorby applyingthe -trans orm onone o the closed- orm Newton-Cotes integration rule. sengand Lee [ ] have used ractional delay in design o digitalintegrator. Gupta et al. [ ] have also used interpolation o various existing integrators to design digital integrators.

Optimization [ ] is a very popular design method.It is widely used to improve the per ormance o a systemby reducing its runtime, bandwidth, memory requirement,or any other property. Optimization methods such as linear,simulated annealing, genetic algorithm, and pole-zero opti-mization have been used earlier to design IIR digital


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integrators and differentiators [ ]. Papamarkos andChamzas [ ] have used linear programming optimizationmethod to design digital integrators. Al-Alaoui [ ] has alsoproposeda amilyo digital integratorsby using interpolationand simulated annealing optimization method. Upadhyay and Singh [ ] have used pole-zero optimization method in

design o digital integrators. Jiang and Kwan [ ] have usedsemide nite programming (SDP) relaxation algorithm todesign IIR digital lters in the minimax sense. Lai and Lin[ ] have used a sequential constrained least-squares methodto design IIR digital lters in the minimax sense. Hsu et al.[ ] have used genetic algorithm optimization method in thedesign o digital integrators.

Al-Alaoui [ , ] has proposed a method to design IIR digital differentiators, based on the inversion and magnitudestabilization o the trans er unction o IIRdigital integrators.Tis technique also became very popular and was used by various researcherslike Bihan [ ],Ngo[ ],Gupta etal.[ ],and Upadhyay and Singh [ ] in the design o digital di -

erentiators. Te accuracy and range in terms o percentageabsolute relative error (PARE) o the magnitude response o the designed differentiators are the same as those o integra-tors.

In this proposed work, a amily o stable IIR digitalintegrators is designed via the use o minimax [ , ] andpole, zero and constant (PZC) optimization methods. Tecombination o two optimization techniques in a design ispreviously used by Renders and Flasse and Wu et al. [ , ].Renders and Flasse [ ] have used genetic algorithm withhill-climbing optimization in order to generate a globaloptimization algorithm and Wu et al. [ ] have used particleswarm with cultural algorithm to enhance the convergencecharacteristics o the original particle swarm optimization.

Here, minimax optimization method is used to design aamily o second-, third-, and ourth-order digital integrators

by optimizing the magnitude response in a min-max senseunder the satis actory condition o constant group delay.TenthePZCoptimization is applied on minimax-optimizedintegrators to urther improve requency (magnitude andphase) response. Tis design strategy is motivated by the actthat there is a probability that minimaxoptimization does notprovide global optimum solution so the PZC optimizationis used over the result o minimax optimization to greatly increase the probability o nding the true global optimumsolution. Tereafer, by modi ying the trans er unction o designed integrators appropriately, new second-, third-, and

ourth-order digital differentiators are obtained which havethesame accuracy as thedesigned integrators. Tese designedwideband operators (integrators and differentiators) outper-

orm all the existing wideband operators in both magnitudeand phase response.

All the gures in this paper have been obtained by usingMA LAB . Tis paper is organized as ollows. Section presents the problem ormulation. Section presents thesolution methodology consisting o minimax and pole, zero,and constant (PZC) optimization. Section contains per-

ormance results by minimax and PZC optimization andcomparison o the designed integratorswith existing integra-tors. Section presents the proposed differentiators and their

comparison with the existing differentiators. Te conclusionsare given in Section .

2. Problem Formulation

It is known that IIR integrators have much better magnitude

response than FIR integrators o the same order but theirphase characteristicsare not linear which cancauseproblemsin some o the signal processing applications. In direct designmethod or IIR integrators, the desired digital trans er unc-tion is obtained directly rom the given magnitude, with orwithout a speci ed phase speci cation. Let ( ) be the trans-

er unction o an IIR digital integrator with real coefficients( , ); then

( ) = ( )( ) = =0 1 + =1

, ( )

where is theordero the integrator. ( )can also be de nedas

( ) = ( )1 + ( ), ( )

where = [0, 1, . . . , ] , = [1, 2, . . . , ] , ( ) =[1, 1, 2, . . . , ] , and ( ) = [1, 2, . . . , ] , thesuperscript is used to de ne transpose.

Te requency response ( ) o the integrator can beobtained rom its trans er unction ( ) by simply evaluatingit on the unit circle; that is,

= ( )| = = ( ). ( )

Here, | ( )|is the amplitude and ( )is the phase o thedigital integrator, respectively.

Group delay is a measure o phase distortion; it is cal-culated by differentiatingphasewithrespect to requency. Tedegree o nonlinearity o the phase indicates the deviation o the group delay rom a constant.

Te group delay o integrator ( ) is de ned as

( ) = ( ) . ( )

Te main limitation in thedesigno IIRdigital integratorsis to meet the speci ed magnitude and phase characteristics.An integrator should have linear phase response over thewideband requency range; otherwise it would introducephase distortion. Various researchers have been trying to im-prove the per ormance o digital integrators but a commondifficulty aced by them is that when the magnitude responseis improved, the phase response gets distorted and vice versa.

3. Solution Methodologies

In this paper, a amilyo second-, third-,and ourth-ordersta-ble wideband digital integratorsare designed via use o mini-maxandpole, zero, and constantoptimization. Te ow charto complete design method is shown in Figure .


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Start

Minimax optimization

I(z) =

Pole, zero, and constant optimization

Nn=0 anzn

1 + Nm=1 bmzm

Integrators ( 2nd, 3rd, and 4th order)

Very efficient IIR integrators ( 2nd, 3rd, and 4th order)

F : Flowchart o the complete design method or digital

integrator.

. . Design o IIR Digital Integrators Using Minimax Optimiza-tion. A minimax is one o the optimization techniques inwhich the task is to nd the solution with the minimumo theworst (maximum) value. It has been popularly used to ndthe unknown system parameters by minimizing the maxi-mum difference between system output and design speci ca-tions [ , ]. Te owcharto theappliedminimaxoptimiza-tion or second-order digital integrator is shown in Figure .

Te trans er unction o an integrator with unknowncoefficients is de ned as ( ). A vector containing numera-tor and denominator coefficients is de ned as = [ , ] .In order to explicitly show the dependence o the requency response o integrator ( ) on the numerator and denom-inator coefficients, the integrator is de ned as ( , ), inplace o ( ). Here, ( , ) and M int ( ) are the amplitudeo the designed and ideal integrator, respectively. Percentageabsolute relative error (PARE) or magnitude response isde ned as an objective unction and order o the lter isde ned by the value o . Here coefficients and aretaken as optimization parameters:

PARE ( , ) = int ( ) ( , )int ( ) 100. ( )

Te applied minimax optimization method can be de ned as

minimize subject to |PARE ( , )| .

( a)

In order to obtain exact approximation o the magnitude re-sponse, the PARE values should be as small as possible. Forthe case o second-order integrator ( = 2), the minimum( ), maximum ( ), and initial value ( 0) o coefficients arede ned as ollows

= 0.9999 0.9999 0.9999 0.9999 0.9, = +0.9999 +0.9999 +0.9999 +0.9999 +0.9,

0 = +0.0010 +0.0010 +0.0010 +0.0010 +0.0.( )

Te lengths o the coefficients are taken as integer and

decimal points. Te same algorithm is also applied or integer decimal and integer decimal length but thedifference in the efficiency is negligible.

Te constraint o linear phase is applied by restricting themaximum value o absolute group delay ( ( , )) o inte-grator ( , )to be less than samples. Te requency rangeis de ned as 0 radians/second.

Te internal unction o minimax optimization in MA -LAB ( minimax) is used to obtain the coefficients or min-imum o maximum PARE( , ) over the de ned requency range.

Te ollowing command is used or this purpose:

[ ,PAREval,maxPARE]= minimax @PARE ( , ), 0,

[],[],[],[], , ,@ ( , ) < 5 ,

( )

where is thevector o coefficientshaving minimumo maxi-mum PARE or the de ned requency range.

PAREval and maxPARE are the vectors o PARE valuesover thede ned requency range andmaximum among thesecorresponding to coefficients .

Four blank square brackets are used to de ne that thereare no linear equalities and inequalities.

minimax unction works in ollowing manner.In rst iteration, a set o coefficients are selected; then

the values o objective unction (PARE) are calculated by using ( ), corresponding to these coefficients. Te selectioncriteria o the coefficients consist o two points: rst the valueo coefficient should be within de ned minimum and maxi-mum bound, and the absolute value o group delay o the cor-responding integrator should be less than samples. In nextiteration, another set o coefficients are selected and PARE values are calculated. Tis process is repeated till suitablecoefficientsare obtained.Afer last iteration,maximum PAREo each iteration is compared and coefficients corresponding


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Yes

Start

No

End

1

a = [a0, a1, . . . , aN ]T , b = [b1, b2, . . . , bN ]T , x = [aT , bT]T

f N (z) = [1, z1 , z2 , . . . , zN ]T , f M (z) = [z

1 , z2 , . . . , zN ]T

M( , x) = abs{I(ej , x)}, (ej , x)I = I(e j )

Mint ( ) = abs{1/j }, N = 2

PARE ( , x) = Mint ( ) M( , x)

Mint ( ) 100

lb = [0.9999 0.9999 0.9999 0.9999 0.9999]ub = [+0.9999 +0.9999 +0.9999 +0.9999 +0.9999]x0

x0

= [+0.001 +0.001 +0.001 +0.001 +0.001]

min = 0 and max =

max = max 0.01

max 0.9

[x, PAREval, maxPARE] = fminimax( @PARE( ,x), , [], [], [], [], lb , ub , @ g( , x)


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the previously proposed operators, where only poles andzeros have been optimized.

o obtain the poles, zeros, and constant o an integrator( ), the numerator and denominator polynomials o ( ) are

rewritten as

( ) = 0 + 11

+ 22

+ +

1 + 1 1 + 2 2 + + ( )

or

( ) = 0 + 1/ 0 1

+ 2/ 0 2 + + / 0 + 1 1 + 2 2

+ + 1.

( )

Te numerator and denominator can be actorized and ex-pressed as

( ) = 0 1 2 1 2 ,

( ) = =1 =1 ,

( )

where = 0 is a scaling constant, are the zeros, and arethe poles o integrator ( ), respectively.

For PZC optimization, the modi ed trans er unction( ) can be written as

( ) = ( + ) =1 + =1 + ,

= ( ) = .( )

Te amplitude o the modi ed digital integrator ( ) is

= ( + ) =1 + =1 + . ( )

PARE o integrator ( ) is de ned as

PARE ( ) =int ( )

int ( ) 100. ( )

Phase responseo the modi ed digital integrator is de ned as

( ) ==1 +

=1 + .

( )

Equations ( )( ) show that the optimization parame-ters , , and affect not only magnitude but phaseresponse also.

Te group delay o the modi ed integrator ( ) is de-ned as

( ) = ( ) . ( )

Various steps used in the design o PZC optimizedsecond-order IIR digital integrator are as ollows.

(i) First o all, minimax optimized second-order integra-tor is de ned in terms o poles, zeros, and constant.

(ii) Percentageabsolute relativeerror PARE ( )as de nedin ( ) is speci ed as an objective unction.

(iii) Ten the lower and upper bound, or all optimiza-tion parameters ( , , and ) are de ned as[0.0010 + 0.0010], respectively.

(iv) Te step size is taken as + . .

(v) All the optimization parameters are varied or values, and those which gives minimum PARE arede ned as optimum parameters.

(vi) Te above mentioned process is repeated or third-and ourth-order minimax optimized integrators.

In this way PZC optimized second-, third-, and ourth-orderwideband digital integrators are obtained.

4. Results after Minimax and Pole, Zero, andConstant Optimization Methods

. . Results afer Minimax Optimization and Teir Com- parison with Existing Integrators. Te second-, third-, and

ourth-order wideband digital integrators are designed orthe requency range o 0 radians/second to 0 0.90 radians/second. Coefficients and maximum PARE oreach requency range or second-, third-, and ourth-orderdigital integratorsareshown in ables , ,and , respectively.

By observing ables , , and , the minimum o max-imum PARE are obtained or the optimum coefficients asmentioned below.

Second order: 0 = 0.8650, 1 = 0.5993, 2 = 0.0539,1 = 0.4834, and 2 = 0.5145.

Tird order: 0 = 0.8639, 1 = 0.9997, 2 = 0.2471,3 = 0.0058, 1 = 0.0220, 1 = 0.8315, and 3 =

0.1442.

Fourth order: 0 = 0.8659, 1 = 0.5426, 2 = 0.5925,3 = 0.3480, 4 = 0.0276, 1 = 0.5479, 2 = 0.1845,3 = 0.3429, and 4 = 0.2905.


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: Coefficients and maximum PARE or different requency ranges o second-order integrator.

S. no. Frequency range (radians/second) 0 1 2 1 2 Maximum PARE . + . + . + . . . . . . + . + . + . . . .

. + . + . + . . . . . + . + . + . . . . . . + . + . + . . . .

. . + . + . + . . . . . . + . + . + . . . . . . + . + . + . . . . . . + . + . + . . . . . . + . + . + . . . . . . + . + . + . . . .

: Coefficients and maximum PARE or different requency ranges o third-order integrator.

S. no. Frequency range (radians/second) 0 1 2 3 1 2 3 Maximum PARE 0.01 . . . . . . . .

0.01 0.99 . . . . . . . . 0.04 0.98 . . . . . . . . 0.05 0.97 . . . . . . . . 0 0.96 . . . . . . . . 0.01 0.95 . . . . . . . . 0.01 0.94 . . . . . . . .

0 .01 0 .93 . . . . . . . . 0.02 0.92 . . . . . . . . 0.02 0.91 . . . . . . . . 0.02 0.90 . . . . . . . .

By substituting the above mentioned optimum coeffi-cientsin( ), theresulting trans er unctionso second-, third-, and ourth-order digital integrators ( 2MO( ), 3MO( ), and

4MO( )) are obtained as

2MO ( ) = 0.8650 + 0.59931 + 0.05392

1 0.48341 0.51452 ,

3MO ( ) = 0.8639 + 0.99971 + 0.24712 + 0.00583

1 0.02201 0.83152 0.14423 ,

4MO ( )

= 0.8659 + 0.54261 + 0.59252 + 0.34803

+0.02764

1 0.54791 + 0.18452 0.34293

0.29054 1 .( )

For comparison, various existing integrators have beenconsidered. Tese are Ngo integrator ( N( )) [ ], Guptaet al. integrator ( GJK1( )) [ ], Gupta et al. integrator

( GJK2( )) [ ], Gupta et al. integrator ( GJK3( )) [ ], Al-Alaoui integrator ( A( )) [ ], and Upadhyay and Singhintegrator ( US( )) [ ]. Teir trans er unctions are

N ( )

= 1 + 2.36581 1 0.26051 + 0.04722.7925 1 1 ,

GJK1 ( )

= 0.34 1 + 2.5411 1 0.20811 + 0.038582

1 1 ,

GJK2 ( )

= 0.329 1 + 2.6631 1 0.20791 + 0.038642

1 1 ,

GJK3 ( )= 0.38125 1 + 2.2851

1 0.2471 + 0.045432

1 1 1,


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: Coefficients and maximum PARE or different requency ranges o ourth-order integrator.

S. no. Frequency range(radians/second) 0 1 2 3 4 1 2 3 4 Maximum PARE

0.01 . . . . . . . . . . 0.01 0.99 . . . . . . . . . .

0.01 0.98 . . . . . . . . . . 0.02 0.97 . . . . . . . . . . 0.02 0.96 . . . . . . . . . . 0.02 0.95 . . . . . . . . . . 0.02 0.94 . . . . . . . . . . 0.03 0.93 . . . . . . . . . . 0 .01 0 .92 . . . . . . . . . . 0.03 0.91 . . . . . . . . . . 0.02 0.90 . . . . . . . . . .

: Comparison between existing and designed integrators.

S. no. Integrator Frequency range (radians/second) Maximum PARE Mean PARE N(z) Ngo [ ] 0 . . I GJK ( ) Gupta et al. [ ] 0 . . I GJK ( ) Gupta et al. [ ] 0 0.95 . . I GJK ( ) Gupta et al. [ ] 0 . .

A( ) Al-Alaoui [ ] 0.008 0.60 . . US( ) Upadhyay and Singh [ ] 0 0.94 . .

I MO( ) 0.01 0.96 . . I MO( ) 0.01 0.93 . . I MO ( ) 0.01 0.92 . .

I MO+PZC( ) 0 0.95 . . I MO+PZC( ) 0 0.94 . . I MO+PZC( ) 0 0.91 . .

A ( )= 0.5678 + 1.0131 + 0.56292

+0.54063 + 0.15194

1 0.20971 0.21062

0.1143

0.46364 1

,

US ( ) = 0.8657 1 + 0.6811 + 0.06282

1 0.49751 0.50252 .( )

Here the sampling period o the integrator isassumed as second and the Nyquist requency is radi-ans/sample. Te PARE response o the designed integratorsand above mentioned integrators over Nyquist requency range is shown in Figure . Te use ul requency range,mean,andmaximum PAREs in that requency range orabovemen-tioned integrators are shown in able . Te group delay

response o the above mentioned integrators is shown inFigure .

It is seen that the maximum deviations rom constantgroup delay in case o 2MO( ), 3MO( ), and 4MO( ) are .samples, . samples and . samples, respectively, while, themaximum deviations rom constant group delay in case o existing integrators Ngo [ ], Gupta et al. [ ], Gupta et al. [ ], Gupta et al. [ ], Al-Alaoui [ ], and Upadhyay andSingh [ ] are . samples, . samples, . samples, . sam-

ples, . samples and . samples, respectively.It is veri ed rom Figures and and able that theminimax optimized designed second-, third-, and ourth-order integrators outper orm all the existing integrators overentire Nyquist requency range.

. . Results afer PZC Optimization and Teir Comparisonwith the Minimax Optimized Integrators. Te requency response o the minimax optimized digital integrators isthen improved by using pole, zero, and constant (PZC) opti-mization. For this minimax optimized integrators ( 2MO( ),

3MO( ), and ( 4MO( )) ( ) are de ned in terms o pole, zeroand constant. As all the coefficients o polynomials in ( ) are


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1.2

1.4

1.6

1.8

2

P e r c e n

t a g e a b s o

l u t e r e

l a t i v e e r r o r

( P A R E

)

Normalized frequency w/ (radians/sample)0

00.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

1

1

I2MOI3MOI4MOIGJK1IGJK2

INIAIUSIGJK3

F : Percentage absolute relative error response o designedintegrators; 2MO ( ), 3MO ( ), and 4MO( ), Ngo integrator N( ) [ ],Guptaet al. integrator GJK1( )[ ], Gupta et al. integrator GJK2( )[ ], Gupta et al. integrator GJK3( )[ ], Al-Alaoui integrator A( )[ ], and Upadhyay and Singh integrator US( ) [ ].

real, there ore thepoles,andzeroswill be either real orappearin complex conjugate pairs:

2MO ( ) = 0.8650( + 0.1063) ( + 0.5865)( 0.9986) ( + 0.5152)

,

3MO ( ) = 0.8639( + 0.8172) ( + 0.3139) ( + 0.0261)( 0.9989) ( + 0.7954) ( + 0.1815) ,

4MO ( ) = 0.8659( + 0.5382) ( + 0.0929)

0.79821.5680 0.7982 1.5680

( 0.9987) ( + 0.4565) 0.79831.5672

0.7983 1.5672 1.( )

For PZC optimization, the modi ed trans er unction can bewritten as

2MO ( ) = (0.8650 + ) + 0.1063 + 1 + 0.5865 + 2

0.9986 + 1 ( + 0.5152 + 2) 1,

3MO ( ) = (0.8639 + ) + 0.8172 + 1 + 0.3139 + 2 + 0.0261 + 3

G r o u p

d e l a y

( s a m

p l e s

)

2

1.5

1

0.5

0.5

0

1

1.5

2

Normalized frequency w/ (radians/sample)0.20.1 0.4 0.50.3 0.6 0.7 0.8 0.9 1

I2MOI3MOI4MOIGJK1IGJK2

INIAIUSIGJK3

F : Group delay response o designed integrators; 2MO ( ),3MO ( ), and 4MO ( ), Ngo integrator N( ) [ ], Gupta et al.

integrator GJK1(z) [ ], Gupta et al. integrator GJK2( ) [ ], Guptaet al. integrator GJK3(z) [ ], Al-Alaoui integrator A( ) [ ], andUpadhyay and Singh integrator US( ) [ ].

0.9989 + 1 + 0.7954 + 2 ( + 0.1815 + 3) 1,

4MO ( ) = (0.8659 + ) + 0.5382 + 1 + 0.0929 + 2 0.79821.5680 + 3 0.7982 1.5680 + 4

0.9987 + 1 + 0.4565 + 2 0.79831.5672 + 3

( 0.7983 1.5672 + 4)1.

( )

Afer optimization, the optimum values are obtained asollows:

or second order:

= +0.0005, 1 = 0.0003, 2 = +0.0008, 1 =0.0004, and 2 = +0.0004;

or third order:

= +0.0007, 1 = 0.0006, 2 = +0.0009, 3 =0.0008, 1 = 0.0008, 2 = 0.0007, and 3 =0.0008;


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or ourth order:

= +0.0004, 1 = 0.0006, 2 = +0.0010, 3 = +0.00091.5680, 4 = +0.0009 1.5680, 1 =0.0008, 2 = +0.0009, 3 = +0.00041.5672, and 4 = +0.0004 1.5672.

By substituting all these values in ( ), respectively, the new integrators are obtained as

2MO+PZC ( ) = 0.8655( + 0.1060) ( + 0.5873)( 0.9990) ( + 0.5156) ,

3MO+PZC ( ) = (0.8646( + 0.8166) ( + 0.3148) ( + 0.0253))

(( 0.9997) ( + 0.7947)

( + 0.1807))1

,4MO+PZC ( ) = 0.8663( + 0.5376) ( + 0.0939)

0.79731.5680

0.7973 1.5680

( 0.9995) ( + 0.4574)

0.79791.5672

( 0.7979 1.5672) 1 .

( )

All the designed integrators (( ) and ( )) are stable astheir respective poles are inside the unit circle. Te PAREresponse o the designed integrators 2MO( ), 3MO( ),

4MO( ), 2MO+PZC( ), 3MO+PZC( ), and 4MO+PZC( ) isshown in Figure . Te comparison o designed and existingintegrators on the basis o mean PARE, maximum PARE anduse ul requency range is shown in able .

It is seenthat thePZC optimized integrators 2MO+PZC( ),3MO+PZC( ), and 4MO+PZC( ) have PARE 0.31 over

0 0.95 radians / second, . over 0 0.94 radians / second, and . over 0 0.91radians / second, respectively. Te group delay response o thedesigned integrators 2MO( ), 3MO( ), 4MO( ), 2MO+PZC( ),

3MO+PZC( ), and 4MO+PZC( ) is shown in Figure . It is ver-i ed rom Figures and and able that PZC optimizationimproves requency (both magnitude and phase) response o minimax optimized integrators. In this way, a per ect amily o second-, third-, and ourth-order wideband stable digitalintegrators are designed with superior requency response.

Te square wave response o designed and existingintegrators is shown in Figures and , respectively. It is seenthat the response o designed integrators 2MO( ), 3MO( ),

4MO( ), 2MO+PZC( ), 3MO+PZC( ), and 4MO+PZC( ) is exacttriangular wave, while the response o existing integrators isdistorted triangular wave.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Normalized frequency w/ (radians/sample)0 0.2 0.4 0.6 0.8 1

I2MOI3MOI4MO

I2MO + PZCPZC

PZC

I3MO +I4MO +

P e r c e n

t a g e a b s o

l u t e r e


( P A R E

)

F : Percentage absolute relative error response o designedintegrators; 2MO( ), 3MO ( ), 4MO ( ), 2MO+PZC ( ), 3MO+PZC ( ),and

4MO+PZC ( ).


1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

G r o u p

d e l a y

( s a m

p l e s

)

I2MOI3MOI4MO

I2MO + PZCPZCPZC

I3MO +I4MO +

F : Group delay response o designed integrators; 2MO ( ),3MO ( ), 4MO ( ), 2MO+PZC ( ), 3MO+PZC ( ), and 4MO+PZC ( ).

5. Design of IIR DigitalDifferentiators and Their Comparison withthe Existing Differentiators

A digital differentiator can also be designed by using trans erunction o digital integrator in a similar way to that used

in the design o analog differentiator, as suggested by Al-Alaoui [ ]. Tis method consists o our design steps. In the

rst step, an integrator is designed that has the same range


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A m p

l i t u

d e

0.6

0.4

0.2

0

0

0.5 1 1.5 2

0.2

0.4

0.6

2.5 3 3.5 4 4.5 5Time (s)

Input

Time offset: 0 (s)

I2MOI3MOI4MO

I2MO + PZCPZC

PZC

I3MO +I4MO +

F : Square wave response o designed integrators; 2MO ( ),3MO ( ), 4MO( ), 2MO+PZC ( ), 3MO+PZC ( ), and 4MO+PZC ( ).

A m

p l i t

u d

e

0.6

0.4

0.2

0

0

0.5 1 1.5 2

0.2

0.4

0.6

2.5 3 3.5 4 4.5 5Time (s)

Input

Time offset: 0 (s)IGJK1IGJK2IN

IAIUS

IGJK3

F : Square wave response o existing integrators; Ngo N( )[ ], Gupta et al. GJK1( ) [ ], Gupta et al. GJK2( ) [ ], Gupta etal. GJK3( ) [ ], Al-Alaoui A( ) [ ], and Upadhyay and Singhintegrator US( ) [ ].

and accuracy as the desired differentiator. Ten, the trans erunction o this integrator is inverted and stabilized by re ecting the poles that lie outside theunit circle to inside theunit circle, andin the last step, the magnitude is compensatedappropriately.

In this section, a amily o second-, third-, andourth-order digital differentiators have been obtained

by inverting the trans er unctions o designed integrators2MO( ), 3MO( ), 4MO( ), 2MO+PZC( ), 3MO+PZC( ), and4MO+PZC( ) de ned in ( ) and ( ), respectively.

Te resultant trans er unctions are

2MO ( ) = ( 0.9986) ( + 0.5152)0.8650( + 0.1063) ( + 0.5865),

3MO ( ) = (( 0.9989) ( + 0.7954) ( + 0.1815))

(0.8639( + 0.8172)

( + 0.3139) ( + 0.0261))1,

4MO (z) = ( 0.9987) ( + 0.4565)

0.79831.5672

0.7983 1.5672

(0.8659) ( + 0.5382)

( + 0.0929)

0.79821.5680

( 0.7982 1.5680) 1,

2MO+PZC ( ) = ( 0.9990) ( + 0.5156)0.8655( + 0.1060) ( + 0.5873),

3MO+PZC ( ) = (( 0.9997) ( + 0.7947) ( + 0.1807))

(0.8646( + 0.8166)

( + 0.3148) ( + 0.0253))1,

4MO+PZC ( ) = ( 0.9995) ( + 0.4574) 0.79791.5672

0.7979 1.5672

(0.8663) ( + 0.5376)

( + 0.0939)

0.79731.5680

( 0.7973 1.5680) 1.( )

As all the poles o designed differentiators (see ( ))are inside the unit circle, there ore stabilization and com-pensation o these trans er unctions are not required. ocompare the efficiency o these designed differentiators, vari-ous recently proposed differentiators have been considered.Tese are Ngo differentiator ( N( )) [ ], Gupta et al. differentiator ( GJK1( )) [ ], Gupta et al. differentiator( GJK2( )) [ ], Gupta et al. differentiator ( GJK3( )) [ ],Al-Alaoui differentiator ( A( )) [ ], and Upadhyay andSingh differentiator ( US( )) [ ].


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: Comparison between existing and designed differentiators.

S. no. Differentiator Frequency range (radians/second) Maximum PARE Mean PARE N( ) Ngo [ ] 0 . .

DGJK ( ) Gupta et al. [ ] 0 . . DGJK ( ) Gupta et al. [ ] 0 0.95 . . DGJK ( ) Gupta et al. [ ] 0 . . A( ) Al-Alaoui [ ] 0.04 . .

US( ) Upadhyay and Singh [ ] 0 0.94 . . D MO( ) 0.01 0.96 . . D MO( ) 0.01 0.93 . . D MO( ) 0.01 0.92 . .

D MO+PZC( ) 0 0.95 . . D MO+PZC( ) 0 0.94 . . D MO+PZC( ) 0 0.91 . .

1.2

1.4

1.6

1.8

2

P e r c e n

t a g e a b s o

l u t e r e


( P A R E

)

Normalized frequency w/ (radians/sample)

00.20

0.2

0.4

0.4

0.6

0.6

0.8

0.8

1

1

2MO

3MO

4MOGJK1

NA

US

D DDD

DGJK2D

GJK3D

DD

F : Percentage absolute relative error response o designeddifferentiators; 2MO( ), 3MO ( ), and 4MO ( ), Ngo differentiator

N( ) [ ], Gupta et al. differentiator GJK1( ) [ ], Gupta et al. differentiator GJK2( ) [ ], Gupta et al. differentiator GJK3( )

[ ], Al-Alaoui differentiator A( ) [ ], and Upadhyay and Singhdifferentiator US( ) [ ].

Teir trans er unctions are

N ( )= 2.7925 1 1

(2.3658) 1 + 0.42261

(1 0.26051 + 0.0472) 1 ,

0

1

2

3

4

2

1


3MO

4MO

2MOGJK1

NAUS

DDDD

DGJK2D

GJK3D

D

D

G r o u

p d

e l a y

( s a m

p l e s

)

F : Group delay response o designed differentiators;2MO ( ), 3MO( ), and 4MO ( ), Ngo differentiator N( ) [ ],

Gupta et al. differentiator GJK1( ) [ ], Gupta et al. differentiatorGJK2( ) [ ], Gupta et al. differentiator GJK3( ) [ ], Al-Alaoui

differentiator A( ) [ ], and Upadhyay and Singh differentiatorUS( ) [ ].

GJK1 ( )= 1 1

0.34 (2.541) 1 + 0.39351

(1 0.20811 + 0.038582) 1 ,


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Normalized frequency w/ (radians/sample)0 0.2 0.4 0.6 0.8 1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

P e r c e n

t a g e a b s o

l u t e r e


( P A R E

)

D2MOD3MOD4MO

2MO

3MO

4MO

D + PZCD + PZCD + PZC

F : Percentage absolute relative error response o thedesigned differentiators; 2MO ( ), 3MO ( ), 4MO( ), 2MO+PZC ( ),

3MO+PZC ( ), and 4MO+PZC ( ).

GJK2 ( )= 1 1

0.329 (2.663) 1 + 0.37551

(1 0.20791

+ 0.038642

)1

,GJK3 ( )= 1 1

0.38125 (2.285) 1 + 0.43761

(1 0.2471 + 0.045432) 1 ,A ( )

= 0.01903 0.029051 + 1.1232 1.1813

1 + 0.18461 0.0017482 + 0.034843 ,

US ( )

= 0.5805 1 + 0.991 1.992

1 + 0.6811 + 0.06282 .( )

Te PARE response o the designed differentiators 2MO( ),3MO( ), 4MO( ), and above mentioned existing differen-

tiators ( or = 1) is shown in Figure . Te comparisono designed and existing differentiators on the basis o meanPARE, maximum PARE, anduse ul requency range is shownin able .


0

0.2

0.1

0.4

0.5

0.3

0.6

0.7

0.8

0.9

1

G r o u p

d e l a y

( s a m

p l e

s )

D2MOD3MOD4MO

2MO

3MO

4MO


F : Group delay response o the designed differentia-tors; 2MO( ), 3MO ( ), 4MO( ), 2MO+PZC ( ), 3MO+PZC ( ), and

4MO+PZC ( ).

Te group delay response o the designed differentiators2MO( ), 3MO( ), 4MO( ), and above mentioned existing

differentiators is shown in Figure . It is seen that themaximum deviation rom constant group delay in caseo 2MO( ), 3MO( ), and 4MO( ) are . samples, .samples and . samples, respectively. While, the maximumdeviation rom constant group delay in case o the existingdifferentiators Ngo [ ], Gupta et al. [ ], Gupta et al. [ ],Gupta et al. [ ], Al-Alaoui [ ], and Upadhyay and Singh[ ] are . samples, . samples, . samples, . samples,

. samples, and . samples, respectively.It is veri ed rom Figures and and able that the

minimax optimized designed second-, third-, and ourth-order differentiators outper orm all the existing differentia-tors over entire Nyquist requency range.

Te PARE response o the designed differentiators2MO( ), 3MO( ), 4MO( ), 2MO+PZC( ), 3MO+PZC( ),

and 4MO+PZC( ) is shown in Figure . Te designed differ-entiators 2MO+PZC( ), 3MO+PZC( ),and 4MO+PZC( )havePARE . over 0 0.95 radians / second, . over0 0.94 radians / second and . over 0 0.91radians / second, respectively. Te group delay responseo the designed differentiators 2MO( ), 3MO( ), 4MO( ),

2MO+PZC( ), 3MO+PZC( ), and 4MO+PZC( ) is shown inFigure . It is veri ed rom Figures and and able that PZC optimization improves requency (both magnitudeand phase) response o minimax optimized differentiators.In this way, a per ect amily o second-, third-, and ourth-order wideband stable digital differentiators are designedwith superior requency response.

Te triangular wave response o designed and existingdifferentiators is shown in Figures and , respectively.It is seen that the response o designed differentiators


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0

0.5 1 1.5 2

0.2

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2.5 3 3.5 4 4.5 5Time (s)Time offset: 0 (s)

D2MOD3MOD4MO

2MO

3MO

4MO


Input

F : riangular wave response o the designed differentia-tors; 2MO ( ), 3MO ( ), 4MO ( ), 2MO+PZC ( ), 3MO+PZC ( ), and

4MO+PZC ( ).

A m p

l i t u

d e

0.60.8

1

0.40.2

0

0

0.5 1 1.5 2

0.20.4

0.80.6

2.5 3 3.5 4 4.5 5Time (s)Time offset: 0 (s)

GJK1

NA

US

DDD

DGJK2DGJK3D

Input

F : riangular wave response o existing differentiators; NgoN( ) [ ], Gupta et al. GJK1( ) [ ], Gupta et al. GJK2( ) [ ],

Gupta et al. GJK3( ) [ ], Al-Alaoui A( ) [ ], and Upadhyay and Singh US( ) [ ].

2MO( ), 3MO( ), 4MO( ), 2MO+PZC( ), 3MO+PZC( ),and 4MO+PZC( ) is exact square wave, while the response o existing differentiators is distorted square wave.

6. Conclusion

Tispaper is ocusedon theuseo twooptimizationmethods,namely, minimax and pole, zero, and constant (PZC). Ithas been proved that the efficiency o minimax optimizedintegrators is remarkably improved by PZC optimization.Subsequently, by modi ying the trans er unction o thesedesigned integrators appropriately, new differentiators are

obtained which have the same accuracy as the designedintegrators. Te amily o designed recursive integrators anddifferentiators are important when excellent magnitude andlinear phase response is required at the same time.

References

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[ ] S. K. Mitra, Digital Signal Processing a Computer Based Approach, ata McGraw Hill, .

[ ] A. M. Schneider, J. . Kaneshige, and F. D. Groutage, Higherorder s-to-z mapping unctions and their application in digitiz-ing continuous-time lters, Proceedings o the IEEE, vol. , no.

, pp. , .[ ] M. A. Al-Alaoui, Novel digital integrator and differentiator,

Electronics Letters, vol. , no. , pp. , .

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[ ] M. Gupta, M. Jain, and N. Jain, A new ractional orderrecursivedigital integratorusing continued ractionexpansion,in Proceedings o the IEEE International Con erence on Power Electronics (IICPE ), pp. , Delhi, India, .

[ ] N. Papamarkos and C. Chamzas, A new approach or thedesign o digital integrators, IEEE ransactions on CircuitsSystem Part I: Fundamental Teory and Applications , vol. , no.

, pp. , .[ ] M. A. Al-Alaoui, Class o digital integrators and differentia-

tors, IE Signal Processing , vol. , no. , pp. , .[ ] D. K. Upadhyay and R. K. Singh, Recursive wideband digital

differentiator and integrator, Electronics Letters, vol. , no. ,pp. , .

[ ] A. Jiang and H. K. Kwan, Minimax design o IIR digital ltersusing SDP relaxation technique, IEEE ransactions on Circuitsand Systems I: Regular Papers, vol. , no. , pp. , .


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[ ] X. P. Lai and Z. P. Lin, Minimax design o IIR digital ltersusing a sequential constrained least-squares method, IEEE

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[ ] C. C. Hsu, W. Y. Wang,andC. Y. Yu, Geneticalgorithm-deriveddigital integrators and their applications in discretization o continuous systems, in Proceedings o the CEC Congress onEvolutionary Computation , pp. , Honolulu, Hawaii,USA, .

[ ] J. M. Renders and S. P. Flasse, Hybrid methods using geneticalgorithms or global optimization, IEEE ransactions onSystems, Man, and Cybernetics, Part B, vol. , no. , pp.

, .[ ] Y. Wu, X. Z. Gao, X. L. Huang, and K. Zenger, A hybrid opti-

mization method o particle swarm optimization and culturalalgorithm, in Proceedingso the th International Con erence onNatural Computation (ICNC ), pp. , August .

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