The Convergence Speed of Sngle- And Mult-Obectve Immune Algorthm Based Optmzaton Problems Mohammed Abo-Zahhad Faculty of Engneerng, Electrcal and Electroncs Engneerng Department, Assut Unversty, Assut, 756, Egypt. Sabah M. Ahmed Faculty of Engneerng, Electrcal and Electroncs Engneerng Department, Assut Unversty, Assut, 756, Egypt. Nabl Sabor Faculty of Engneerng, Electrcal and Electroncs Engneerng Department, Assut Unversty, Assut, 756, Egypt. Ahmad F. Al-Aloun Haw Faculty for Engneerng Technology, Communcaton Engneerng Department, Yarmouk Unversty, Irbd, 63, Jordan. Abstract zahhad@yahoo.com sabahma@yahoo.com nabl_sabor@yahoo.com alalouna@hotmal.com Despte the consderable amount of research related to mmune algorthms and t applcatons n numercal optmzaton, dgtal flters desgn, and data mnng, there s stll lttle work related to ssues as mportant as senstvty analyss, []-[4]. Other aspects, such as convergence speed and parameters adaptaton, have been practcally dsregarded n the current specalzed lterature [7]-[8]. The convergence speed of the mmune algorthm heavly depends on ts man control parameters: populaton sze, replcaton rate, mutaton rate, clonal rate and hypermutaton rate. In ths paper we nvestgate the effect of control parameters varaton on the convergence speed for sngle- and mult-obectve optmzaton problems. Three examples are devoted for ths purpose; namely the desgn of - D recursve dgtal flter, mnmzaton of smple functon, and banana functon. The effect of each parameter on the convergence speed of the IA s studed consderng the other parameters wth fxed values and takng the average of 00 tmes ndependent runs. Then, the concluded rules are appled on some examples ntroduced n [] and [3]. Computatonal results show how to select the mmune algorthm parameters to speedup the algorthm convergence and to obtan the optmal soluton. Sgnal Processng : An Internatonal Journal (SPIJ), Volume (4): Issue (5) 47
Keywords: Immune Algorthm, Convergence, Mutaton, Hypermutaton, Populaton Sze, Clonal Selecton.. INTRODUCTION The parameters of the mmune algorthm have a large effect on the convergence speed. These parameters are the populaton sze (p s ) whch estmates the number of ndvduals (antbodes) for each generaton, the mutaton rate (p m ) whch ncreases the dversty n populaton, and the replcaton rate (p r ) whch estmates the number of antbodes chosen from the antbody populaton pool to on the algorthm operatons. Other parameters such as the clonal rate (p c ) whch estmates the number of ndvduals chosen from the antbody populaton pool to on the clonal prolferaton (selecton), as well as the hypermutaton rate (p h ) whch mproves the capabltes of exploraton and explotaton n populaton, have also great effect on the speed of convergence. In spte of the research carred out up to date, there are no general rules on how these parameters can be selected. In lterature []-[] and [3], the mmune parameters are selected by certan values (e.g. p s =00, p r =0.8, p m =0., p c =0.06, p h =0.8) wthout statng the reason for ths selecton. In ths paper we nvestgate the effect of parameters varaton on the convergence speed of the mmune algorthms developed for three dfferent llustratve examples: -D recursve dgtal flter desgn (mult-obectve problem), mnmzaton of smple functon (sngle-obectve problem), and fndng the global mnmum of banana functon. The obtaned results can be used for selectng the values of these parameters for other problems to speed up the convergence. The paper s organzed as follows. Secton descrbes the mmune algorthm behavor. In Secton 3 three llustratve examples are gven to nvestgate the effect of parameters varaton on the convergence speed of the mmune algorthm. Secton 4 dscusses the selecton crtera of these parameters to guarantee the convergence speed. In secton 5, some examples ntroduced n [3] and [] are consdered to demonstrate the effectveness of the selecton of mmune algorthm control parameters. And fnally, Secton 6 offers some conclusons.. IMMUNE ALGORITHMS BEHAVIOR Immune algorthms are randomzed algorthms nspred by mmune functons and prncples observed n nature [0]. Such algorthms begn by generatng populaton pool (chromosome) usng real codng representaton and evaluatng the obectve values. Then, the populaton pool undergoes the algorthm operatons whch wll be descrbed n ths secton. The operatons are repeated at each generaton (gen) untl the termnaton condton s satsfed []-[]. Table () llustrates the man steps of the mmune algorthm [6].. Generaton of Antbody Populaton The antbody populaton s generated ether by usng bnary codng representaton or real codng representaton. In the bnary codng representaton, each varable s encoded as a bnary strng and the resultng strngs are concatenated to form sngle chromosome (antbody) []. However, n the real codng representaton, each antbody s encoded as a vector of floatng pont numbers, wth the same length as the vector of decson varables. Ths representaton s accurate and effcent because t s closest to the real desgn space, and the strng length represents the number of desgn varables.. Selecton for Reproducton The roulette wheel selecton s employed n mmune bases algorthms for chromosomes reproducton. Its basc dea s to determne the selecton probablty for each soluton n proporton wth the ftness value. For soluton wth ftness f, ts probablty p s defned as: Sgnal Processng : An Internatonal Journal (SPIJ), Volume (4): Issue (5) 48
p = p f s = And the cumulatve probablty q = = Where, the ftness f p, =,,...,ρ s q for each soluton s calculated as:, =,,..., ρ s f s relaton to the obectve functon value of the th chromosome. () () Gen=; Chrom=Intal_pop(); Whle (termnaton_condton) Evaluuate (Chrom); Chrom_sel=RWS_Selecton(Chrom); Chrom_rep=replcaton(Chrom_sel); Replcaton Chrom_clon=Clonng(Chrom_rep); Chrom_hyper=Hypermutaton(Chrom_clon); Chrom_tot=[ Chrom_rep, Chrom_hyper]; Chrom_chld=Mutaton(Chrom_tot); Evaluuate (Chrom_chld); Chrom=Better_selecton(Chrom, Chrom_chld); generaton gen=gen+; end % The frst generaton % Construct the ntal populaton pool % Obectve functon evaluaton % Roulette wheel selecton % Selecton of better antbodes usng % Clonal operaton % Hypermutaton operaton % Mutaton Operaton % Obectve functon evaluaton % Selecton of better antbodes for next % Increment the number of generatons TABLE (): The Immune Algorthm.3 Replcaton Operaton The replcaton operaton s used to select better antbodes, whch have low obectve values to undergo algorthm operatons. Ths s termed by clonal prolferaton wthn hypermutaton and mutaton operatons..4 Clonal Prolferaton wthn Hypermutaton Based on the bologcal mmune prncples, the selecton of a certan antbody from the antbody populaton pool to on the clonal prolferaton depends on the clonal selecton rate (p c ). Each gene, n a sngle antbody, dependng on the hypermutaton rate (p h ), executes the hypermutaton of convex combnaton. The hypermutaton rate (p h ) has an extremely hgh rate than the mutaton rate to ncrease the antbody dversty. For a gven antbody X = ( X, X,..., X, X, X k,..., X ρ ), f the gene X s determned to execute the hypermutaton and another gene X k s randomly ' ' selected to on n, the resultng offsprng antbody becomes X ( X, X,..., X, X, X ) where the new gene ' ' X s X = ( β ) X + β X k =,, and β [0, ] s a random value. k,..., X ρ.5 Mutaton Operaton Smlar to the hypermutaton mechansm, the mutaton operaton s also derved from the convex set theory [9], where each gene, n a sngle antbody, dependng on the mutaton rate (p m ), executes the mutaton of convex combnaton. Two genes n a sngle soluton are randomly chosen to execute the mutaton of convex combnaton [5]. For a gven antbody X = ( X, X,..., X, X, X k,..., X ρ ), f the genes X and X k are randomly selected for Sgnal Processng : An Internatonal Journal (SPIJ), Volume (4): Issue (5) 49
mutaton depend on the mutaton rate (p m ), the resultng offsprng s ' ' ' X = X, X,..., X, X, X. The resultng two genes X and X are calculated as: ( ) X ' = β + k,..., X ρ ' ( ) X β X k and X k = β X + ( β) X k where, β s selected randomly n the range [0, ]. ' ' k (3).6 Selecton Operaton The selecton operaton s generally used to select the better p s antbodes whch have low obectve values as the new antbody populaton of the next generaton. 3. ILLUSTRATIVE EXAMPLES In ths secton three dfferent examples are consdered to nvestgate the effect of parameters varaton on the convergence speed of the mmune algorthm. The frst example smulates the mult-obectve functon problem that has an nfnte set of possble solutons dffcult to fnd [7]. The second example s a sngle-obectve functon problem and t s less dffcult and the thrd example represents the famly of problems wth slow convergence to the global mnmum [6]. Example : Ths example consders the desgn of a second order -D narrow-band recursve LPF wth magntude and group delay specfcatons. The specfed magntude M ω, ω ) s shown n d ( Fgure () [], [5]. Namely, t s gven by Equaton (4) wth the addtonal constant group delay τ τ 5 over the passband ω + ω 0. π and the desgn space s [-3 3]. To solve ths d = = d problem, the frequency samples are taken at ω / π = 0, 0.0,0.04, K, 0., 0.4, K, n the ranges π ω π, and π ω π..0, ( ω, ω ) = 0.5, 0.0, for ω + ω for 0.08π < for ω + ω 0.08π ω + ω > 0.π 0.π M d (4) Example : Ths example consders the optmzaton of the exponental functon shown n Fgure () and descrbed by the followng equaton: y 9 ( x) = a x = 0 Wth the followng desred specfed values Y d (x) at x= [0,,, 3,., 0]. Y d (x) = [0. 0-0. 0 9. 998 0.7309 0 4.5587 0 5 8 9-3. 83. 8368 0 ] -4. 79 6 3.667 0 8 758. 33 7.68 0 5.904 0 9. 00 0 6 8 3.8563 0.0306 0 5. 737 0 7 9 4 4.65 0.7397 0 (5) 5. 737 0 7 8.7358 0 9 4 7.858 0 Example 3: Ths example consders a Rosenbrock banana functon that descrbed by the followng equaton [6]. Ths functon s often used to test the performance of most optmzaton algorthms [6]. The 9 Sgnal Processng : An Internatonal Journal (SPIJ), Volume (4): Issue (5) 50
global mnmum s nsde a long, narrow, parabolc shaped flat valley as shown n Fgure (3). In fact fnd the valley s trval, however the convergence to the global mnmum s dffcult. f ( x, y) = ( x) + 00( y x ) (6) FIGURE : Desred Ampltude Response ( ) M d ω,ω Of The -D Narrow-Band LPF (Example ) Sgnal Processng : An Internatonal Journal (SPIJ), Volume (4): Issue (5) 5
FIGURE : Desred Specfcatons of the Functon y ( x) (Example ) FIGURE 3: Rosenbrock Banana Functon (Example 3) Sgnal Processng : An Internatonal Journal (SPIJ), Volume (4): Issue (5) 5
4. SENSITIVITY ANALYSIS In ths secton, we examne the effect of parameters varatons on the convergence speed of the mmune algorthm for the three examples descrbed n secton 3. The number of genes (the encodng length L) for each example s defned by the number of unknown coeffcents. For the flter desgn problem, the flter transfer functon s expressed by: H, a00+ a0z+ a0z + a0z+ azz+ azz + a0z + az z+ az z = H0, a 00 ( z z ) = ( + b z+ cz + dzz)( + b z+ cz+ dzz) (7) So, 5 genes can be adusted to approxmate the specfed magntude and group delay. For the smple functon and banana functon problems, the number of genes consdered are 0 and respectvely. 4. Effect of the populaton sze (p s ) The populaton sze (p s ) s defned as the number of antbodes used n each generaton. The varatons n p s can have substantal effect on the convergence speed of mmune algorthm. If the p s s too small, the IA cannot reach to optmal soluton. However, f t s too large, the IA wastes computatonal tme effort on extra obectve values evaluatons. Here, the effect of p s on the convergence speed of the algorthm s studed by takng the average of 00 tmes ndependent runs at each p s value. The value of p s was vared from 0 to 400 wth the other parameters fxed at p r =0.8, p h =0.8, p m =0., and p c =0.06. The effect of populaton sze varatons on number of generatons requred to get the soluton for flter desgn problem, smple functon and banana functon are shown n Fgures (4-6), respectvely. The results llustrated n Fgures (4-6) show that, the speed of convergence can be measured by the number of generatons requred to reach to the optmal chromosome (global soluton). Moreover, t can be notced that the speed of convergence depends not only on the p s but also on the number of genes. Here, the p s after whch optmal chromosome s obtaned s denoted by p s *. Increasng the p s above p s * has nsgnfcant effect on speedng up the convergence. 4. Effect of the Replcaton Rate (p r ) The replcaton rate (p r ) estmates the number of antbodes chosen from the antbody populaton pool to on the algorthm operatons. The effect of p r on the speed of convergence of the IA s studed by takng the average of 00 tmes ndependent runs at each p r value. The value of p r was vared from 0. to wth the other parameters fxed at p s =00 p h =0.8, p m =0., and p c =0.06. The effect of p r varaton on the number of generatons requred to produce the soluton for flter desgn problem, smple functon and banana functon are shown n Fgures (7-9), respectvely. These fgures show that, the hgh values of replcaton rate have a sgnfcant effect on speedng up the convergence, but the computatonal tme ncreases as the p r ncreases. It s also notced that the values of p r greater than p r * have no further effect on speedng up the convergence. Sgnal Processng : An Internatonal Journal (SPIJ), Volume (4): Issue (5) 53
FIGURE 4: The Effect of Populaton Sze on the Speed of Convergence of the Flter Desgn Problem. FIGURE 5: The Effect of Populaton Sze on the Speed of Convergence for Smple Functon Mnmzaton Sgnal Processng : An Internatonal Journal (SPIJ), Volume (4): Issue (5) 54
Fgure 6: The Effect Of Populaton Sze On The Speed Of Convergence For Fndng The Global Mnmum Of Banana Functon. FIGURE 7: The Effect of Replcaton Rate on the Speed of Convergence for Flter Desgn Problem. Sgnal Processng : An Internatonal Journal (SPIJ), Volume (4): Issue (5) 55
FIGURE 8: The Effect of Pr on the Speed of Convergence for Smple Functon Mnmzaton. FIGURE 9: The Effect of Pr on the Speed of Convergence for Fndng the Global Mnmum of Banana Functon. Sgnal Processng : An Internatonal Journal (SPIJ), Volume (4): Issue (5) 56
4.3 Effect of the Clonal Selecton Rate (p c ) The clonal selecton rate (p c ) estmates the number of antbodes that can be chosen from the antbody populaton pool to on the clonal prolferaton. The effect of p c on the speed of convergence of the IA s studed by takng the average of 00 tmes ndependent runs at each p c value. The value of p c was vared from 0.0 to wth the other parameters fxed at p s =00, p r =0.8, p h =0.8, and p m =0.. The effect of p c varaton on the number of generatons requred to produce the optmal soluton for flter desgn problem, smple functon and banana functon are shown n Fgures (0-), respectvely. From these fgures, we can conclude that low values of p c (0.05 p c <0.) have sgnfcant effect on speedng up the convergence. It s also notced that the use of hgh values of p c (p c p c *) have an effect of slowng down the convergence. Ths s manly due to the nfeasble selected ndvduals whch oned to the clonal prolferaton. 4.4 Effect of the Hypermutaton Rate (p h ) The hypermutaton rate (p h ) s used to mprove the capabltes of exploraton and explotaton n populaton. The effect of p h on the convergence speed of the IA s evaluated by takng the average of 00 tmes ndependent runs at each p h value. The value of p h was vared from 0.0 to wth the other parameters fxed at p s =00, p r =0.8, p c =0.06, and p m =0.. The effect of hypermutaton varaton on the number of generatons requred to produce the soluton for flter desgn problem, smple functon and banana functon are shown n Fgures (3-5), respectvely. The results gven n Fgures (3-5) show that, the value of p h depends on the problem doman. The values of p h for the three llustratve examples are 0.5, 0.5, and 0.7, respectvely. The p h should be n the range (0.5 p h <) to speed up the convergence of small number of genes problems (example 3) and t s about 0.5 for other ones. FIGURE 0: The Effect of Clonal Rate on the Speed of Convergence for Flter Desgn Problem. Sgnal Processng : An Internatonal Journal (SPIJ), Volume (4): Issue (5) 57
FIGURE : The Effect of Clonal Rate on the Speed of Convergence for Smple Functon Mnmzaton. FIGURE : The Effect of Clonal Rate on the Speed of Convergence for Fndng the Global Mnmum of Banana Functon. Sgnal Processng : An Internatonal Journal (SPIJ), Volume (4): Issue (5) 58
FIGURE 3: The Effect of Hypermutaton Rate on the Speed of Convergence for Flter Desgn Problem. FIGURE 4: The Effect of Hypermutaton Rate on the Speed of Convergence for Smple Functon Mnmzaton. Sgnal Processng : An Internatonal Journal (SPIJ), Volume (4): Issue (5) 59
FIGURE 5: The Effect of Hypermutaton Rate on the Speed of Convergence for Fndng the Global Mnmum of Banana Functon. 4.5 Effect of the Mutaton Rate (p m ) The mutaton rate (p m ) s one of the most senstve mmune algorthm parameters, snce t ncreases the dversty n populaton. The choce of mutaton rate s essentally a tradeoff between conservatsm and exploraton [4]. The effect of p m on the convergence speed of IA s studed by takng the average of 00 tmes ndependent runs at each p m value. The value of p m was vared from 0.0 to wth the other parameters fxed at p s =00, p r =0.8, p c =0.06, and p h =0.8. The effect of mutaton rate varaton on the number of generatons requred to produce the soluton for flter desgn problem, smple functon and banana functon are shown n Fgures (6-8), respectvely. From these fgures, we can conclude that the low values of mutaton rate (p m p m *) have sgnfcant effect on speedng up the convergence. Also, t s notced that to guarantee the convergence speed, the p m should be between / p s and /L, where p s s the populaton sze and L s the encodng strng length. From above studyng, we can conclude that the general heurstcs on IA parameters to guarantee the convergence speed are: ) the populaton sze should be greater than 00; ) the replcaton rate should be hgher than 0.; 3) the clonal rate should be small n the range (0.05 p c <0.); 4) the hypermutaton rate should be hgh n the range (0.5 p h <); and 5) the mutaton rate should be between / p s and /L. Sgnal Processng : An Internatonal Journal (SPIJ), Volume (4): Issue (5) 60
FIGURE 6: The Effect of Mutaton Rate on the Speed of Convergence for Flter Desgn Problem (Ps=00 and L=5). Sgnal Processng : An Internatonal Journal (SPIJ), Volume (4): Issue (5) 6
FIGURE 7: The Effect of Mutaton Rate on Speed of Convergence for Smple Functon Mnmzaton (Ps=00 and L=0). FIGURE 8: The Effect Of Mutaton Rate On Speed Of Convergence For Fndng The Global Mnmum Of Banana Functon (Ps=00 And L=). Sgnal Processng : An Internatonal Journal (SPIJ), Volume (4): Issue (5) 6
5 RESULTS AND DISCUSSION In ths secton, some examples ntroduced n [3] and [] are consdered to llustrate the effect of mmune algorthm parameters on the convergence speed. Example 4: Ths example s consdered n [3] for solvng system dentfcaton problem. It s repeated here to demonstrate the effectveness of the selecton of mmune algorthm control parameters. In ths example, t s requred to approxmate second-order system by frst-order IIR flter. The secondorder system and the flter are descrbed respectvely by the followng transfer functons [3]: 0.05 0.4z a0 H p ( z ) = and ( z ) =.34z + 0.5z H f (8) b z In Table (), the control parameters selected based on the study descrbed n prevous secton and that used n [3] are gven. Table (3) llustrates the transfer functon, the number of functon evoluton and NMSE of the resultng IIR flter and that s descrbed n [3]. The NMSE s calculated usng the followng equaton: NMSE = N k= Where, M d ( k) and ( k) N ( M( k) M d( k) ) ( M d( k) ) k= M are the magntude responses of the nd order system and that of the desgned flter respectvely calculated at N=000 samplng ponts. IA Parameters The selected parameters based on the above study (9) The selected parameters n [3] Populaton sze 00 50 Replcaton rate 0.85 0.80 Mutaton rate 0. 0.05 Clone rate 0.05 Not used n ths method Hypermutaton rate 0.8 Not used n ths method TABLE : The IA Control Parameters Of Examples And IIR flter obtaned usng proposed parameters values 0.453 0.8645z IIR flter obtaned usng parameters values stated n [3] 0.3 0.906z Transfer Functon H f ( z ) = H ( ) f z = NMSE 0.0796 0.77 Number of functon evaluatons to fnd the global optmal soluton 056 30 TABLE 3: The Transfer Functon, Number Of Functon Evolutons And NMSE Of Both Resultng IIR Flter And IIR Flter Descrbed In [3]. Fgure (9) shows the magntude responses of the second-order system, the resultng IIR flter and IIR flter descrbed n [3]. From Fgure (9) and Table (3), notced that the resultng IIR flter Sgnal Processng : An Internatonal Journal (SPIJ), Volume (4): Issue (5) 63
converge to the second-order system after smaller number of obectve functon evaluatons wth smaller NMSE compared to that gven n [3]. So, the good selecton of the IA control parameters speeds up the algorthm convergence. FIGURE 9: The magntude responses of second-order system and IIR flter Example 5: Ths example s also consdered n [3] for solvng system dentfcaton problem. It s requred to approxmate a second order system by IIR flter wth the same order. The system and the flter are descrbed respectvely by the followng transfer functons [3]: H ( z ) = and ( ) z = p.z + 0.6z H f (0) b z Sgnal Processng : An Internatonal Journal (SPIJ), Volume (4): Issue (5) 64 b z Usng the same control parameters of example, the optmal soluton (b = -.966, b = -0.595) s obtaned after 503 obectve functon evaluatons wth MSE=0.393x0-3. However, the soluton n [3] s obtaned after 3000 obectve functon evaluatons wth MSE=0.5x0-3. Example 6: Ths example s consdered n [], for fndng the global soluton of the followng test functon: N N x f 4 = x cos + () 4000 = = The proposed IA s used to solve ths functon wth 30 dmensons (.e. N=30) n soluton space [- 600, 600]. In Table (4), the control parameters selected based on the study descrbed n prevous secton and that used n [] are gven.
IA Parameters The selected parameters based on the above study The selected parameters n [] Populaton sze 00 00 Replcaton rate 0. 0. Mutaton rate 0.0 0.0 Clone rate 0.06 0.0 Hypermutaton rate 0.8 0.0 Table 4: The IA Control Parameters Of Example 3 Usng the proposed IA, the soluton s obtaned after 30 functon evaluatons; however n [] s reached after 5743 functon evaluatons. So, the IA control parameters are havng sgnfcant effect on the convergence speed. 6 CONCLUSIONS In ths paper, general rules on speedng up the convergence of the IA are dscussed. The convergence speed of the IA s mportant ssues and heavly depends on ts man control parameters. In spte of the research carred out up to date, there are no general rules on how the control parameters of the IA can be selected. In lterature []-[3], the choce of these parameters s stll left to the user to be determned statcally pror to the executon of the IA. Here, we nvestgate the effect of the parameters varaton on the convergence speed by adoptng three dfferent obectve optmzaton examples (-D recursve flter desgn, mnmzaton of smple functon, and banana functon). From the studed examples, the followng general heurstcs on mmune algorthm parameters that guarantee the convergence speed are concluded: ) the populaton sze should be greater than 00; ) the replcaton rate should be hgher than 0.; 3) the clonal rate should be small n the range (0.05 p c <0.); 4) the hypermutaton rate should be hgh n the range (0.5 p h <); and 5) the mutaton rate should be between / p s and /L. These heurstcs are appled to study cases solved n [3] and [] to show effect of control parameter selecton on the IA performance. Numercal results show that the good selecton of the control parameters of the IA have sgnfcant effect on the convergence speed of the algorthm. 7 REFERENCES. J. T. Tsa, W. H. Ho, J. H. Chou. Desgn of Two-Dmensonal Recursve Flters by Usng Taguch Immune Algorthm. IET sgnal process, ():0-7, March 008. J. T. Tsa, J. H. Chou. Desgn of Optmal Dgtal IIR Flters by Usng an Improved Immune Algorthm. IEEE Trans. sgnal processng, 54(): 458 4596, December 006 3. A. Kalnl, N. Karaboga. "Artfcal mmune algorthm for IIR flters desgn". Engneerng Applcatons of Artfcal Intellgance, 8(8): 99-99, December 005 4. Alex A. Fretas, Jon Tmms. Revstng the Foundatons of Artfcal Immune Systems for Data Mnng. IEEE Trans. on Evolutonary Compuaton, (4): 5-540, August 007 5. A. H. Aly, M. M. Fahmy. "Desgn of Two Dmensonal Recursve Dgtal Flters wth Specfed Magntude and Group Delay Characterstcs". IEEE Trans. on Crcuts and Systems, 5(): 908-96. November 978 6. Roy Danchck. Accurate numercal partals wth applcatons to optmzaton, Appled mathematcs and computaton, 83(): 55-558, December 006 Sgnal Processng : An Internatonal Journal (SPIJ), Volume (4): Issue (5) 65
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