Compensated CIC-Cosine Decimation Filter

Transcription

1 Compensated CIC-Cosne Dematon Flter 8 ECTI TRANSACTIONS ON COPUTER AND INFORATION TECNOLOGY VOL.4, NO. AY Gordana Jovanov Doleek and Fernando Javer Treo Torres Compensated CIC-Cosne Dematon Flter Insttute INAOE, Department Eletrons, Puebla, exo Gordana Jovanov Doleek and Fernando Javer Treo Torres, Non-members ABSTRACT struture an be used [], [3]. ore detals on a Ths ABSTRACT paper presents effent modfaton CICosne dematon. The seond order ompensator omparson performanes reursve and nonreursve rate resultng mplementatons n a larger hp are area gven and n []. hgher Addtonally, data Ths paper presents effent modfaton power onsumpton espeally when dematon s ntrodued at low rate n order to mprove magntude response CIC has a hgh droop CICosne dematon. The seond order ompensator fator s hgh. In order to resolve ths problem passband nterest s ntrodued overall at low. rate The n oeffents n desred passband and low stopband attenuaton. order to non-reursve struture an be used [], [3]. mprove ompensator passband are nterest presented n overall a anonal. dgts The CSD oeffents form, and an ompensator be mplemented are usng reursve determnes and nonreursve frequeny mplementatons ω, where worst are passband ore The detals dematon on a omparson fator R performanes next dematon stage sgned only presented adders and n ashfts. anonal Consequently, sgned dgts CSD resultng form, gvendstorton n []. Addtonally, ours, and magntude frequeny response ω A, where worst s a and multplerfree an be mplemented and usng exhbts onlya adders hgh attenuaton and shfts. n CIC ase alasng hasours a hgh [4]. droop The nnormaled desred frequenes passbandand andω low A /π stopband are at attenuaton. The dematon ω /π stopband, Consequently, as well as resultng a low passband s adroop. multplerfree and exhbts a hgh attenuaton n stopband, fator R next dematon stage determnes ω π ω Al R eywords: as well asdematon a low passband, droop. frequeny ω Compensator, CIC, =, where = worst passband = dstorton =. ours, and π frequeny π ω A, where π worst ase Cosne. eywords: Dematon, Compensator, CIC alasng Several ours strutures [4]. The normaled have been frequenes proposed ωto /πdesgn CIC, Cosne. and s ω A /π are wth atmproved frequeny response as for example. INTRODUCTION [4-]. Usually methods deal to mprove er passband The. INTRODUCTION proess onvertng gven rate a sgnal ω [5-6], or stopband [7-]. nto a The dfferent proessrate onvertng s alled samplng gven rate rate onverson a sgnal π = The π. π goal = ω Al ths π = work s = to R- propose. a smple SRC. nto athe dfferent reduton rate s alled a samplng rate rate onverson s alled multplerless dematon wth no ng at hgh dematon, SRC. The and reduton s aomplshed a samplng two rate stages, s alled ng nput rate, and wth a desred hgh stopband attenuaton and dematon, downsamplng. and s aomplshed n two stages, ng and downsamplng. [] ntrodued ogenauer a very smple [] ntrodued dematon s propose wth mproved to ompensate frequeny response passband for exam- droop Several and strutures a desred have low been passband proposeddroop. to desgn To CIC ths end we ogenauer a alled very smple asaded dematon ntegrator omb alledcic asaded, ntegrator whh ple [4-]. asaded Usually CIC-osne methods dematon deal to mprove [], er ntrodung a onssts omb asaded CIC, ntegrators whh onssts and dfferentators asaded passband smple [5-6], ompensator or stopband at [7-]. low rate. The rest paper ntegrators and dfferentators separated by a downsampler, as shown n Fg.. The transfer funton tplerless dematon wth no ng at hgh The separated by a downsampler, as shown n Fg.. The s organed goal ths as work follows. s to In propose Seton a smple we propose mul- new transfer resultng funton dematon resultng dematon s gven as s gven seond order ompensator whh s llustrated wth nput rate, and wth a desred hgh stopband attenuaton and a desred low passband droop. To ths end as two examples. Seton 3 brefly ntrodues asaded CIC-osne dematon from []. The proposed [ ] - we propose to ompensate passband droop modfed ompensated CIC-osne dematon s = asaded CIC-osne dematon [], ntrodung adesrbed smple ompensator n seton 3, atand low llustrated rate. The wth rest one example.. =, where s dematon fator and paper s organed as follows. In Seton we propose new seond order ompensator whh represents number asaded s. s llustrated. PROPOSED wth two COPENSATION examples. Seton FILTER 3 brefly where s dematon fator and represents ntrodues asaded CIC-osne dematon number asaded s. from []. The The transfer proposed funton modfed ompensated proposed CICosne dematon s gven by s desrbed n seton 3, and ompensaton llustrated wth one example. COP = a + b + a, 3 where a and b are real valued onstants, and s. PROPOSED COPENSATION FILTER dematon fator. Usng a multrate dentty ths Fg.: an be moved to a low rate whh s tmes less than Fg.: CIC dematon. The transfer funton proposed ompensatonhgh nput s gven rate bybeomng a seond order.. The above dematon s attratve n many The magntude response 3 s expressed as The above dematon s attratve n many applatons beause ts very low omplexty. It COP = a + b - + a -. 3 applatons s usuallybeause used at ts frstvery stagelow dematon omplexty. []. It s ω usually owever, used at ths dematon frst stage dematon exhbts several []. owever, drawbaks. dematon The ntegrator exhbts seton works several at drawbaks. hgh nputthe where a and b are real valued onstants, and s COP e = a os ω + b. 4 ths ntegrator seton works at hgh nput data rate dematon Next ssue fator. s Usng to defne a multrate oeffents dentty ths a and an be moved to a low rate whh s tmes less b. To ths The authors are wth Insttute INAOE, Department Eletrons, Puebla, exo. E- end we onsder magntude response Eq. 4 n two resultng n a larger hp area and hgher power onsumpton mal:gordana.doleek@gmal.om, espeally when gordana@eee.org, dematon fator and s than hgh nput rate beomng a seond order. frequeny ponts ω=, and ω=ω, where worst gordana@naoep.mx The magntude response 3 s expressed as hgh. In order to resolve ths problem non-reursve passband dstorton ours, defned n Eq.. In ω=

2 Compensated CIC-Cosne Dematon Flter 9 COP e ω = aos ω + b. 4 a q + b q =. Next ssue s to defne oeffents a and b. To ths end we onsder magntude response Eq. 4 n two frequeny ponts ω=, and ω = ω, where worst passband dstorton ours, defned n Eq.. In ω= desred magntude haraterst has value, resultng n followng equaton Startng wth k= values k are nreased untl ondton s satsfed. If passband droop s wthn desred lmt, orrespondng quanted oeffents a q and b q are presented n anonal sgned dgt CSD form [-5]. a + b =. 5 COP-CSD = a CSD +b CSD - +a CSD -. desred magntude haraterst has value, resultng n followng equaton START In order to ompensate passband droop at where a CSD, and b CSD are CSD representatons frequeny ω, we have from Eq. 4 quanted oeffents a a + b =. 5 q, b q proposed ompensaton 3. Compute aand b Orwse, proedure s ontnued In order to ompensate aos ω + b passband = /. droop 6 at Eq. 9 untl desred passband ompensaton s obtaned. Ths proedure s programmed n ATLAB, a and b orre- frequeny ω, we have from Eq. 4 spondng blok dagram s shown n Fg.. k = Denotng a os ω + b = /. 6 desred magntude haraterst has value, resultng Denotng Compute aqand bq n followng equaton k= k + START omp = /. 7 Eq. omp a = + / b =. 5 7 No Compute aand b we In have order to ompensate passband droop at Is Eq. 9 satsfed? frequeny ω, we have from Eq. 4 a b Yes [ ] [ ] a os k = a os = omp =. 8 ω + b = /. 6 8 os ω No Is passband omp ompensaton Denotng Compute a O? qand bq k= k + Solvng 8 we get oeffents a and b as Eq. as Yes omp = / 7 No CSD we have Is Eq. satsfed? a os ω os ω =. 9 Yes END os ω a omp os ω =. 8 os b ω No os ω Is passband omp ompensaton Fg.: Blok dagram O? ompensator desgn. In Solvng followng, 8 we get oeffents a a and and b as from 9 9 are Yes quanted as shown n next relatons The method s llustrated CSD n followng example. k aq = a / b q = b /, In followng, a os ω os ω oeffents a and b from 9 are =. 9 Example : Consder ENDdesgn ompensator for quanted as shown os n ω next relatons omp asaded CIC dematon wth =4, =4, and where k s quantaton fator, a q and b q are dematon fator seond stage s R= 8. The passband orrespondng a q = quanted -k a/ -k oeffents, b q = -k b/and -k os ω os ω Fg.: Blok dagram ompensator desgn. droop Fg.: Blok CIC dagram at normaled ompensator frequeny desgn. ω /π s. x represents nteger part x. The relaton 5 must also hold for In followng, nteger part oeffents x. The relaton a and 5 b must from also 9 are.9579db. quanted values a and b resultng n ondton holdquanted for as quanted shown n values next relatons a and b resultng n owever, we want to have a resultng passband droop ondton The method less The than method s.db. llustrated llustrated n n followng followng example. example. a q + kb q =. a From Eq. 7, t follows q = a / bq = b /, Example : Consder omp = desgn ompensator for 3 asaded CIC dematon wth =4, =4, and Startng where k wth s k= quantaton values fator, k are anreased q and b q untl are dematon ondton s satsfed. If passband droop s wthn From 9 fator we have seond stage s R= 8. The passband orrespondng quanted oeffents, and x represents droop CIC at normaled frequeny ω /π s desred nteger lmt, part orrespondng x. The relaton quanted 5 must also oeffents hold for

3 are nts for n n nts D ed ed s k Fg.: Blok dagram ompensator desgn. ECTI TRANSACTIONS ON COPUTER AND INFORATION TECNOLOGY VOL.4, NO. AY The method s llustrated n followng example. Example Example : Consder : Consder desgn ompensator for for asaded asaded CIC CIC dematon wth =4, =4, and and dematon dematon fator fator seond seond stage stage s R= s R= 8. The 8. The passband passband CIC droop CIC at normaled at normaled frequeny frequeny ω /π s droop ω /π s.9579db..9579db. owever, we want to have a resultng passband owever, we want to have a resultng passband droop droop less than.db. less than From.dB. Eq. 7, t follows From Eq. 7, t follows omp = omp = From 9 we have From 9 we have The passband detals CIC and and proposed The ompensated passband CIC detals CICs s are are shown CIC shown n and Fg. n Fg. 3. proposed 3. ompensated There exsts CIC s a trade-f are shown between n Fg. 3. desred pass- a = orrespondng CSD terms for quanted oeffents and and ompensaton stopband attenuaton wth orrespondng -6.53dB. CSD The terms transfer funton s gven by ompensaton quanted oeffents wth s gven by orrespondng CSD Startng orrespondng wth k= CSD we terms fnd for quanted oeffents values shown and 4 ompensaton terms quanted wth oeffents orrespondng s gven by CSD terms n Table transfer I. funton s gven by quanted oeffents Startng wth4k= we 3fnd 5 quanted values shown 5 The nondton COP _ CSD Table I. = s satsfed + only + for + + values. COP _ CSD = + + COP-CSD s = gven by k=3 and The k=5, ondton shown bold n stable satsfed I. 5 only for values COP _ CSD = COP _ CSD = + + owever, k=3 and for k=3 k=5, shown passband bold ndroop Tabledoes I. not satsfy desred owever, spefatons. for k=3 For passband k =5 droop desred does passband Table not satsfy s desred satsfed. spefatons. Therefore usng For k k =5 =5 we desred The passband detals for CIC, proposed I: Quanted Fators and Coeffents. spefaton k a fnd ompensated CIC, and ompensated CIC from [5] are q b q A db The The passband passband detals detals for forcic, proposed passband Table I: spefaton Quanted Fators s satsfed. and Coeffents. Therefore usng k ompensated gven n Fg.4. CIC, and ompensated CIC from [5] are =5 we k fnd a q orrespondng b.5 q CSDA db terms for quanted oeffents and.5.5 transfer funton s gven by gven n nfg COP-CSD 4.35 = GandB GandB CIC CICCompensated CIC, k=5 Compensated CIC, k=5 Next example ompares proposed ompensaton wth that reently proposed n lterature [5]. Example : Consder desgn a CIC dematon for desgn parameters =, =5, [5]. From [5], normaled passband and stopband frequenes nterest are ω /π= and ω Al /π= The orrespondng passband droop and stopband attenuaton CIC are -.788dB and A Al = dB, respetvely. The ompensaton from [5] has orrespondng passband droop.597db, and stopband attenuaton dB, respetvely. Usng proposed ompensator for quanted fator k=8, we got passband droop -.475dB Gan db Gan db CIC Flter -.5 CIC Flter CIC Flter+Roll-Off Compensator, [5] -.5 CIC Flter+Roll-Off CIC Flter+Proposed Compensator, Compensator, [5] k=8 -. CIC Flter+Proposed Compensator, k= Normaled.5. Frequeny.5 ω/π.3.35 Normaled Frequeny ω/π Fg. 4: Example. -.3 Fg.4: 4: Example Note that proposed ompensator results n a slghtly -.4 Note better that that passband proposed proposed haraterst. ompensator ompensator results results n a slghtly n a -.4 better slghtly passband betterharaterst. passband haraterst CIC-COSINE DECIATION FILTER CIC-COSINE DECIATION FILTER.5. Normaled.5Frequeny. ω/π.5.3 Normaled Frequeny ω/π 3. CIC-COSINE Consderng DECIATION ase when down-samplng FILTER fator Fg.3: 3: Example. Consderng. an Consderng be expressed ase ase when when down-samplng down-samplng fator fator Fg. 3: Example. an be an expressed be expressed as as = 3... N 7 band There droop exsts and a trade-f number between CSD desred termspassband oeffents and exsts a and a number trade-f b. between CSD terms desred oeffents passband a we an rewrte Eq. = as 3... N. 7 = 3... N we an rewrte Eq. as 7 droop There droop and b. and number CSD terms oeffents a N and b. N = = Next example ompares proposed ompensaton = = Next wth example that reently ompares proposed proposed n lterature ompensaton [5]. = wth that reently proposed n lterature [5]. = 8 where 8

4 better passband haraterst..3 passband ffents a ton ematon stopband 55 and roop and 88dB and spondng nuaton - quanted 75dB and B. The 3. CIC-COSINE DECIATION FILTER Compensated CIC-Cosne Dematon Flter Consderng ase when down-samplng fator an be expressed as = 3... N 7 we an rewrte Eq. as we an rewrte Eq. as [ N N Q ] At frst we desgn modfed CIC-osne = = - = mccos [] n order to satsfy desred stopband attenuaton. Next, we fnd passband droop = =. 8 8 n frequeny nterest ω = ω desgned modfed CIC-osne. Usng 7 we ompute where omp and solvng equatons 8 we get oeffents a and b ompensator. Fnally, oeffents = = a and b are expressed n CSD form as explaned n = ; =. 9 Seton. The method s llustrated n Example 3. = Example 3: We desgn dematon for =3, whh has passband droop at frequeny nterest less than.db, and stopband attenuaton Usng Eqs. 8 and 9 we express modfed 9 CIC n frequeny nterest at least db. Frst we m as, [] desgn CIC-osne hoosng =, =4 and Usng Eqs. 8 and 9 we express modfed 3 =4 yeldng k k kn N CIC m = m as, [] N, where k s number asaded s. m = k.k... k N N... N- =., = 4, We use osne pres ntrodued n [6] to mprove 3 8 = where k s number asaded s. In next, we asade s, and 3 8 as shown n followng equaton We use osne pres ntrodued n [6] to mprove frequeny haraterst modfed m = CIC. CCOS = COS N. = The stopband attenuaton ths s furr mproved ntrodung asade expanded osne s where [6]. CCOS = COS 8 4 COS 4 4 COS. 8 COS N =.5 + -N + -N. frequeny haraterst modfed CIC where CCOS = In general we have [] COS N COS =.5 COS + -N COS. + -N, 8. 9 where N CCOS = COS N =, =. 3 N N N COS = where [6]. From usng R=8 passband and stopband N N N From frequenes usng nterest R=8 are passband and stopband COS The transfer =.5 funton + + asaded. modfed CIC frequenes nterest are and osne pres s ω ω /π = / π =.3965 In general we have []. 3 mccos = m n COS N ω ω A /π = A / π = N. 4 =. 3 + Stopband attenuaton n frequeny nterest s = Stopband attenuaton n frequeny nterest s A Al = dB,.e., t s satsfed. owever A Al = dB,.e., t s satsfed. owever The transfer funton asaded modfed CIC and where n s number asaded osne pres, passband osne pres s passband droop s droop -.36dB. s -.36dB. In order In order to to mprove mprove and N =. passband, passband, we desgn we desgn ompensaton ompensaton as desrbed as desrbed n For more detals see []. n N Seton = n. Seton. mccos m COS, 4 Solvng = Solvng 9 we 9 get we getoeffents oeffents where n 4. PROPOSED DECIATION FILTER s number asaded osne pres, and N =. The proposed s asade modfed a For more CICosne detals see []. mccos [] and ompensator =. 3. ntrodued n Seton PROPOSED DECIATION FILTER G = mccos COP-CSD.. 5 Table II shows orrespondng quanted values 3 a q The proposed s asade modfed CICosne mccos [] and ompensator and b q. ntrodued n Seton. Table II: Quanted Fators and Coeffents. G = mccos COP _ CSD. 5 k a q b q

5 4 ers, and ed CICor 5 e topband n odfed solvng b are =3, rest less equeny -osne GandB Seton. Solvng 9 we get oeffents a = ECTI TRANSACTIONS ON COPUTER AND INFORATION TECNOLOGY VOL.4, NO. AY Table TableII IIshows shows orrespondng quanted quanted values values a q and a q band b q. The ondton s satsfed only for q. Table II: Quanted Fators and Coeffents. Table : Quanted Fators and Coeffents. k a q b q The ondton s satsfed only for k=7 resultng n k=7 resultng orrespondng passband orrespondng droop passband -.73dB, droop whh -.73dB, satsfes whh desred satsfes spefaton. desred Consequently, spefaton. present Consequently, quanted oeffents we present for quanted k=7 n CSD oeffents terms we for k=7 n CSD terms COP _ CSD = Usng multrate denttes we have effent struture shown n Fg oeffents an ontrol desred passband droop overall dematon. The overall exhbts a low passband droop, and hgh stopband attenuaton at frequenes nterest. Addtonally, usng polyphase deomposton, s at frst stage. 3 an be moved at lower rate. Therefore, re s no ng at hgh nput rate. GandB CIC-osne Compensated CIC-osne, k= Normaled Frequeny ω/π GandB GandB a. Overall gan responses. CIC-osne Compensated CIC-osne, k=7. 6 Overall magntude response s shown n Fg. 6.a Usng multrate denttes we have effent struture -.5 along wth that CIC-osne shown n Fg. 5. Normaled Frequeny ω/π x -3 Overall Correspondng magntude passband response oom s shown s shown n Fg. n6.a Fg. b. Passband ooms. along 6.b. wth that CIC-osne. Fg. 5: Effent struture for Example 3. Correspondng Fg. 6. llustrates passband attenuaton oom s shown n n neghborhood frst ero. Fg. 6.b. equatons, and Fg. expressed 6. llustrates n CSD attenuaton terms whh n neghborhood an be and mplemented frst usng ero. only adders and shfts. As a result - CIC-osne proposed s a multpler free. Compensated CIC-osne, k=7 There s 5. CONCLUSIONS a trade-f between desred -4 ompensaton 5. CONCLUSIONS passband droop, and 7 omplexty We proposed CSD here representaton. multstage Number dematon -6 furr adders based may We be on proposed furr CIC-osne redued here usng multstage dematon sub-expresson dematon and CSE basedseond for order CSD CIC-osne represented ompensator. dematon oeffents The oeffents and [5]. In ontrast ompensator symmetr methods seond are order obtaned [5-6], ompensator. solvng proposed two The oef- smple - -8 osne elmnaton symmetr ompensaton fents desgn ompensator s smpler, areand obtaned solvng two oeffents smple an equatons, ontrol and desred expressed passband n CSD droop terms whh - overall andematon be mplemented. The usng overall only adders exhbts and shfts. As -4 a low passband a result droop, proposed and hgh stopband s attenuaton a multpler at free. There nterest. s a trade-f Addtonally, between usng desred om- Normaled Frequeny ω/π frequenes polyphase pensaton deomposton, passband s droop, at and frst stage omplexty. Stopbands. an be moved at CSD lower representaton. rate. Therefore, Number re s adders no may ng at behgh furr nput redued rate. usng sub-expresson elmnaton Fg.6: Example3. 3. CSE for CSD represented oeffents [5]. In ontrast to methods CIC-osne [5-6], proposed ompensaton desgn s smpler, Compensated and CIC-osne, k=7-5 oeffents 5. ACNOWLEDGEENT an ontrol desred passband droop over- dematon. The overall exhbts a low 6. ACNOWLEDGEENT Ths work s part CONACYT proet -all No Ths work s part CONACYT proet passband droop, and hgh stopband attenuaton at No frequenes nterest. Addtonally, usng polyphase deomposton, s at frst stage Referenes 7. REFERENCES -an be moved at lower rate. Therefore, re s [] E. B. ogenauer, An eonomal lass no ng at hgh nput rate. [] E. dgtal B. ogenauer, s for An dematon eonomal and lass nterpolaton, dgtal -5 s IEEEfor Transatons dematon onand Aousts, nterpolaton, Speeh, IEEE and Sg- Transatons on Aousts, Speeh, and Sgnal Proessng, vol. ASSP-9, No., pp.55-6, Aprl a. Normaled Frequeny ω/π 98. Overall gan responses. [] Y. Gao, L. Ja, J. Isoaho and. Tenhunen, A omparson desgn omb demators for sgma ACN Ths No REFER [] E. B. s Transa Proes 98. [] Y. G omp delta Integ, p [3]. Ab. o onve Syste Proe [4] A. Appl ntegra on Sg 997.

6 Compensated CIC-Cosne Dematon Flter 3 Fg.5: 5: Effent struturefor for Example GandB GandB equatons, and expressed n CSD terms whh an be mplemented nal Proessng, usng only vol. adders ASSP-9, and shfts. No., As pp.55-6, a result proposed Aprl 98. s a multpler free. [] There Y. Gao, s a L. trade-f Ja, J. Isoaho between and. desred Tenhunen, ompensaton A omparson desgn passband droop, omb and demators omplexty for sgmadelta CSD analog-to-dgtal representaton. Number onverters, adders Analog may be Integrated furr redued Cruts usng and sub-expresson Sgnal Proessng, elmnaton CSE for CSD represented oeffents vol., pp. 5-6, 999. [5]. In ontrast to methods [5-6], proposed [3]. Aboushady, Y. Dumontex,.. Loerat, ompensaton desgn s smpler, and oeffents an and ontrol. ehre, desred passband Effent droop polyphase overall deomposton dematon. 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Gao, L. Ja, J. Isoaho and. Tenhunen, A omparson desgn omb demators for sgmadelta analog-to-dgtal onverters, Analog Integrated Cruts and Sgnal Proessng, vol. Gordana Jovanov Doleek reeved a BS degree from Depart-, pp. 5-6, 999. [3]. Aboushady, Y. ment Dumontex, Eletral. Engneerng,. Loerat, and Unversty Saraevo, polyphase andeomposton S degree from. ehre, Effent Unversty Belgrade, and a PhD degree from Faulty Eletral En- omb dematon s n Σ-Δ analog-to-dgtal onverters, IEEE gneerng, Transatons Unversty on Cruts Saraevo. & She Systems II: wasanalog pressorand at Dgtal Faulty Sgnal Eletral Proessng, vol. Engneerng, 48, pp , Unversty Otober Saraevo. untl 993, and she was wth [4] A. wentus, Z. Insttute Jang, halo and A. Pupn, Wllson, Belgrade. Jr., In 995 Applaton she oned Insttute INAOE, sharpenng Department to for asaded Eletrons, Puebla, ntegrator-omb exo, wheredematon she works ass, a pressor IEEE and researher. Trans. Durng Gordana - Jovanov she Doleek was at reeved Department a BS Eletral Computer degree on Engneerng, Sgnal from Proessng, Department Unversty vol.45, Calforna, Eletral pp , SantaFebruary Barbara, as vstng Engneerng, 997. researher Unversty on a sabbatal Saraevo, an leave. S She s author three degree books, from Unversty edtor one Belgrade, book, and anda PhD author more than papers. degree from er researh Faulty nterests Eletral nlude Engneerng, dgtal sgnal proessng and dgtal ommunatons. She s a Senor member IEEE, Unversty member Saraevo. exan She was Aademy pressor at Sene, and member Faulty Natonal Eletral Researher Engneerng, System Unversty SNI exo. Saraevo untl 993, and she was wth Insttute halo Pupn, Belgrade. In 995 she oned Insttute INAOE, Department for Eletrons, Puebla, exo, where she works as a pressor and researher. Durng - she was at Department Eletral & Computer

7 4 ECTI TRANSACTIONS ON COPUTER AND INFORATION TECNOLOGY VOL.4, NO. AY Fernando Javer Torres reeved a BS degree from Department Eletral Engneerng, Unversty Las Ameras, Puebla, 5, exo and an S degree from Insttute INAOE, Puebla, exo, 7. s researh nterest s dgtal sgnal proessng and ommunatons. Bography: Fernando Javer Torres reeved a BS degree from Department Eletral Engneerng, Unversty Las Ameras, Puebla, 5, exo and an S degree from Insttute INAOE, Puebla, exo, 7. s researh nterest s dgtal sgnal proessng and ommunatons.