Pascal-Interpolation-Based Noninteger Delay Filter and Low-Complexity Realization

Transcription

1 SOONTORNWONG, S CHIVAREECHA, ASCAL-INTEROLATION-BASE NONINTEGER ELAY FILTER ascal-iterpolato-based Noteger elay Flter ad Low-Complety Realzato arya SOONTORNWONG, Sorawat CHIVAREECHA ept of Telecommucato Egeerg, Faculty of Egeerg, Kg Mogut s Isttute of Techology Ladrabag, Chalogrug Rd, Ladrabag, Bago 5, Thalad s464@mtlacth, sorawat@telecommtlacth Abstract Ths paper proposes a ew method for desgg the polyomal-terpolato-type oteger-delay flter wth a ew structure formulato Sce the desg formulato ad the ew realzato structure are based o the dscrete ascal trasform T ad ascal terpolato, we call the resultg flter ascal oteger-delay flter The th-order ascal polyomal s used to pass through the gve + data pots achevg the th-order ascal flter The ascal oteger-delay flter s a realtme flter that cossts of two sectos, whch ca be realzed to the frot-secto ad the bac-secto The frotsecto cotas multplcato-free dgtal flters, ad the umber of multplcatos the bac-secto just learly creases as order becomes hgh Sce the ew ascal flter has low complety ad structure ca adjust oteger delay ole, t s more suted for fast delay tug Cosequetly, the polyomal-terpolato-type delay flter ca be acheved by usg the ascal approach wth hgh effcecy ad low-complety structure Keywords Low-complety flter structure, oteger-delay flter, dscrete ascal trasform T, polyomal-terpolato-type delay flter, ascal oteger-delay flter Itroducto Sce the tuable oteger-delay flter ca chage ts o-teger group delay or phase delay, t ca be appled to may felds of sgal processg le oe dmesoal sgal terpolato, samplg-rate coverso ad mage terpolato So far, may desg methods have bee developed the lterature [ 9] Amog those estg flters, the polyomal-terpolato-type otegerdelay flter has smple ad uque features Its ampltude respose s the mamally flat for zero frequecy The dervato ad realzato of the polyomal-terpolatotype oteger-delay flter have bee dealt wth [9 ] The most straghtforward realzato of the polyomalterpolato-type oteger-delay flter s to use the drect-form Farrow structure [] I practcal applcatos, because low frequecy compoets domate, employg the polyomal-terpolato-type oteger-delay flter ca acheve suffcetly satsfactory terpolatos as descrbed [5] The drect-form Farrow structure for realzg the polyomal-terpolato-type oteger-delay flter requres a large umber of multplcatos, but o-causal oes ca be symmetrcally realzed by eplotg symmetry trasformatos After symmetry trasformatos, the structures eed much fewer multplcatos I order to perform real-tme flterg, causal flters are eeded, but the drect-form Farrow structure of the causal polyomalterpolato-type oteger-delay flters does ot possess symmetrc or at-symmetrc structures, whch caot eplot symmetres for reducg the umber of multplcato operatos I [ ], trasformato matrces have bee proposed for trasformg causal flters to have lowcomplety structures Ths paper develops a ew desg method ad a ew realzato structure for the causal polyomal-terpolato-type delay flter wth very low hardware complety order to acheve hgh-speed flterg for real-tme applcatos As compared wth the methods proposed [ ], the method proposed ths paper requres much less multplcatos The resultg flter structures from the methods [ ] requre more multplcatos tha the ascal structure proposed ths paper As compared wth the estg methods for dervg ad realzg the causal polyomal-terpolato-type delay flters, we develop a ovel desg method ad a ew flter structure usg dfferet procedures, but the resultg structure has much lower complety The proposed desg method s related to a d of dscrete polyomal trasform called dscrete ascal trasform T that was frst proposed [], ad later the T s appled to the so-called ascal terpolato [] The ew desg method ad the ew flter structure for the polyomal-terpolato-type delay flter s based o the ascal terpolato, ad the Lagrage polyomal s chaged to ascal polyomal for fttg the data pots Therefore, ths ew type of flter s called ascal oteger-delay flter The ascal oteger-delay flter ca OI: 64/re5 SIGNALS

2 RAIOENGINEERING, VOL 4, NO 4, ECEMBER 5 adjust the delay parameter ole wth hgh processg speed because t uses a reduced umber of multplcatos, where the umber learly creases as the flter order creases Although the coeffcet-symmetry techques [ ] ca reduce the multplcato operatos by about 5%, the techques stll requre more multplcatos as compared wth the ascal oteger-delay flter because the umbers of multplcatos [ ] are ot lear fuctos of the flter order Cosequetly, the low-cost ad hgh-speed causal polyomal-terpolato-type flter ca be obtaed by usg the ascal approach ad the flter order ca tae both odd ad eve umbers The ascal oteger-delay flter maes ole tug wth very low hardware realzato ad hgh-speed operato possble, whch s the major progress for the polyomal-terpolato-type delay flter rect-form Structure ad Trasformed Structure Ths secto brefly revews the polyomal-terpolato-type oteger-delay flter realzed usg the drectform Farrow structure [] ad the Farrow structures usg symmetry trasformatos [ ] The polyomal-type oteger-delay flter s also termed Lagrage-type oteger-delay flter as t stems from the terpolato usg the Lagrage terpolatg polyomal [9], [ ] The output of the polyomal-terpolato-type oteger-delay dgtal flter s where y h l h l l l are the coeffcets, ad s the fractoal delay Also, s the flter order The tme-doma epresso correspods to the frequecy-doma trasfer fucto H z, h z By employg the Farrow structure, Hz, ca be realzed as Fg, where V z, =,,,, are called subflters For eample, f the flter order =, the wth H z, V z V z V z V z, V z z z V z z z, 4 Fg rect-form Farrow structure Here, the subflters have costat coeffcets, whch are fed flterg The o-teger delay of Hz, ca be adjusted through adjustg the value of the parameter The drect realzato of Hz, usg the Farrow structure requres No of multplcatos = No of addtos = O the other had, we ca see from 4 that the last subflter V z has symmetrc coeffcets, but the others do ot have ay symmetres The coeffcet symmetry of V z s tae to accout the realzato show Fg z z V z V z Fg Eample of the drect-form Farrow structure z z Fg Eample of the trasformed Farrow structure V z y I order to reduce the umber of multplcatos eeded the flterg process, t s desrable to symmetrze the coeffcets of all the subflters V z, V z,, V z I [], symmetrc structures are proposed for realzg o-causal polyomal-terpolato-type oteger-delay flters, ad [ ], trasformato approaches are proposed for realzg causal oes Fgure shows the secod-order trasformed Farrow structure, whch ca reduce the umber of multplcatos by more tha 5% As a result, the Farrow structures after trasformatos have the followg completes Odd-order case: For ay odd th-order case, the computatoal complety ca be fgured out by the same way ad ca be summarzed as follows:

3 4 SOONTORNWONG, S CHIVAREECHA, ASCAL-INTEROLATION-BASE NONINTEGER ELAY FILTER No of multplcatos = 4 No of addtos = 4 Eve-order case : For ay eve th-order case, the computatoal complety ca be summarzed as No of multplcatos = No of addtos = 4 4 It s clear that the umbers of multplcatos are stll the square fuctos of the flter order I the et secto, we wll propose a ovel desg method ad a ew flter structure called ascal otegerdelay flter for the polyomal-terpolato-type flter, whch requres much fewer multplcatos as compared to other oes We wll beg by dscussg the bacgroud of the ascal oteger-delay flter desg, cludg the geeralzed ascal oteger-delay flter ad ts flter structure realzato, the frequecy characterstc ad computatoal complety from the et secto screte ascal Trasform ad ascal Iterpolato screte ascal trasform T was proposed [], whch s tmately related to the ascal s tragle show Fg 4, Fg 4 ascal s tragle ad s classfed as oe d of dscrete polyomal trasforms The operato of T s based o ascal trasform matr that performs as a operator for sgal trasformato The dscrete ascal trasform of the sgal vector wth sze + s defed as 5 where s the ascal trasform matr wth dmeso + +, ad s the put sgal vector, ad s the trasformed output vector wth sze + Bass Fuctos of T From 5, the ascal trasform matr has the followg bass fuctos, whch are the polyomals,,,,,, 6! where + + s the sze of the ascal trasform matr ad! 7!! are called the bomal coeffcets as the bomal theorem ad are related to ascal s tragle The term 6 wll alterate the sg of the colums of the ascal trasform matr, ad, are the elemets ascal trasform matr, where correspods to the row umber ad deotes the colum umber The fucto s called fallg factoral powers defed as!,!,,, where =, ad s also called the order of bass fucto For eample, the frst four bass fuctos of T that cosst of four polyomals are,,, 6 whch ca be summarzed as the recurrece formula 9 By usg 9, we ca geerate the ascal trasform matr sze 4 4 for = as follows,, 8 The basc propertes of ths matr are as follows All the elemets of the frst colum equal The matr s a lower tragle The sum of all the elemets of each row ecept the frst row equals zero 4 The verse matr s equal to the forward matr, that s, =

4 RAIOENGINEERING, VOL 4, NO 4, ECEMBER 5 5 ascal Iterpolato Assume that L s obtaed by terpolatg the orgal sgal wth a terpolato factor of L, the L ew data pots wll be serted betwee every par of samples the orgal The ew terpolato techque usg the T [] s proposed [] The dea used ascal terpolato s smlar to the cocept of Fourer seres epaso, that s, a perodc sgal ca be epressed by the weghted-sum of epoetal bass fuctos I the case of dscrete ascal trasform T, the bass fuctos are ascal polyomals show 6 The, the dscrete-tme sgal ca be represeted by usg the T bases as, where the weghts are the ascal coeffcets that are calculated as, Here, the umber of data pots of s +, ad the order of the ascal polyomal s The ascal coeffcets are smlar to the Fourer coeffcets Fourer seres To terpolate the put wth a terpolato factor L for L = +, whch produces + + L output samples L, we scale the de by /L, e, we chage the bass fuctos from to /L Cosequetly, the resultg terpolated samples L are, y L L for L Ths equato s the mportat ey of the ascal terpolato For smplcty, the process from to /L ca be summarzed as Fg 5 Fg 5 Cocept of ascal terpolato The mechasm sde bloc dagram Fg 5 correspods to ad, respectvely For easy uderstadg, we epla the ascal terpolato by usg the followg eample Eample, gve a sequece = [ ] ad we wat to terpolate ths sequece wth L = Step : Before terpolato, the ascal coeffcets must be computed by trasformato from to usg the T, The terms represet the elemets of ascal trasform matr used for T, the ad,,,,,,, wth = The ascal coeffcet vector ca be epressed by matr equato 5, e, where =[ ] T ad =[ ] T Therefore, ths step the ascal coeffcets ca be obtaed by Step : After we obta the ascal coeffcets from Step, the terpolated sgal ca be obtaed usg I ths case, ew L = data pots wll be serted betwee each par of the put samples The, y, 6, 6 y From bass fucto 6, we get,, Therefore, y, 4 for 6 Substtutg,, ad from Step to 4 yelds y, y 4, y 5, y, y4 4 7, y5 5 8, y6

5 6 SOONTORNWONG, S CHIVAREECHA, ASCAL-INTEROLATION-BASE NONINTEGER ELAY FILTER y = / a b From ths pot, we ow that,, ad ca be regarded as the outputs of the flters z, =,, For ay th-order the ascal coeffcets z- trasform Z{ } are obtaed by Z H z z z z 6 Fg 6 Eample of the ascal terpolato wth a factor L= The orgal dscrete-tme put sgal ad the terpolated output sgal / are show Fg 6 From the above eample, we ca see that the th-order degree ascal polyomal ca ft the + samples, the secod-order ascal polyomal fts put data pots Ths property s also detcal to the Lagrage terpolato, whch has bee used to derve the polyomal-terpolato-type oteger-delay flter Therefore, the ascal terpolato ca also be used for developg a ew structure for the polyomal-terpolato-type flter 4 ascal Noteger-elay Flter I ths secto, we wll brefly dscuss why the ascal terpolato ca be used for developg a ew structure for the polyomal-terpolato-type flter, whch s called ascal oteger-delay flter 4 Org of ascal Noteger-elay Flter Below, we develop the ascal oteger-delay flter based o ascal terpolato The ascal polyomal s used for fttg the data pots stead of the Lagrage polyomal The basc dea s drectly related to the T ad ascal terpolato, whch s descrbed the prevous secto The eample used the prevous secto s cotuously used to develop the ascal oteger-delay flter From the prevous eample Step the stage of the ascal coeffcets calculato from put data vector usg, the matr equato ca be rewrtte as 5 I the real-tme applcatos, the system must be causal The, we ca set ; preset put ; sample delayed put ; samples delayed put Therefore,,, wth H z z The et stage s to chage /L 4 to the delay parameter, whch leads to y 7 where s also the ascal oteger-delay flter order To cofrm that our ascal oteger-delay flter s correct ad ca operate, the secod-order ascal oteger-delay flter = 7 wll be tested cases: teger-delay ad oteger-delay Iteger-delay: = ; y ; o delay = ; y ; sample delayed = ; y ; samples delayed Noteger-delay: = 5 = ; y The, y Assume that =, =, ad = The, substtutg these values to the above equato obtas the output y = = 5 = 5, whch s equvalet to the oteger-delayed 5 sample delayed put sgal Fg 7 Graphcal represetatos of the put ad output of the secod-order ascal oteger-delay flter

6 RAIOENGINEERING, VOL 4, NO 4, ECEMBER 5 7 The graphcal represetato of the put ad output of the secod-order ascal oteger-delay flter are show Fg 7 for uderstadg the mechasm usg the ascal oteger-delay flter to delay the put sgal a oteger fasho The tested results show that the ascal oteger-delay flter operates properly Ay thorder ascal oteger-delay flter ad ts structure wll be descrbed the et secto Fally but more mportatly, the trasfer fucto of the secod-order ascal oteger-delay flter wll be formulated ths secto From 7, we ca get y Therefore, ˆ H z, H z, z z z z 8 The trasfer fucto 8 s equvalet to the drect form 4, whch verfes that our derved ascal oteger delay flter s the same polyomal-terpolatotype However, we wll ot use the epresso 8 to realze t Istead, the followg subsectos wll propose a more effcet realzato structure tha the Farrow structures metoed above 4 ascal Noteger-elay Flter ad Its Structure The ascal oteger-delay flterg cossts of operatos The frst oe passes the put sgal through the hghpass flters that have trasfer fuctos 6 order to obta the ascal coeffcets, =,,, The trasfer fuctos 6 ca be wrtte aga as Z H z z,,,, 9 z I the secod step, the ascal coeffcets wll go to the et operato as follows, y wth! ad I ths step, we wll cosder as the puts of a system, ad the coeffcets are the fuctos of the delay parameter Here, we call the ascal-delayed coeffcets The coeffcets cotrol the delay of the ascal oteger-delay flter, whch play a vtal role the ascal oteger-delay flter Table shows the coeffcets of the seveth-order ascal otegerdelay flter for varous values of the delay parameter The ascal oteger-delay flter cludes systems, the frst system s show 9, ad the secod oe s gve Cosequetly, the mechasm of the ascal oteger-delay flter s summarzed Fg 8 for easy uderstadg Combg 9 ad gets the whole trasfer fucto for the ascal oteger-delay flter as Hˆ z, z z z H ˆ z, z whch ca also be wrtte matr form as ˆ H z, pˆ T z z where the ascal-delayed coeffcets vector pˆ s defed as z 4 pˆ, z z z ad s the ascal trasform matr as dscussed Sec I order to realze the ascal oteger-delay flter, we wll cosder the frst system ad the secod system separately as show Fg 8 The frst system ca be cosdered as a sgle-put, mult-output SIMO system, ad we call t the frotsecto of the ascal oteger-delay flter The secod system s a mult-put, sgle-output MISO system, ad = = = = = 4 = 5 = 6 = 7 = = = = = Tab ascal-delayed coeffcets of the seveth-order ascal oteger-delay flter for varous

7 8 SOONTORNWONG, S CHIVAREECHA, ASCAL-INTEROLATION-BASE NONINTEGER ELAY FILTER The geeralzed ascal tuable oteger-delay flter = hz, H, z, y,,,, H z z y Amptude Respose The frot-secto The bac-secto Fg 8 The mechasm of the proposed ascal otegerdelay flter Fractoal-elay Normalzed Frequecy Fg 9 Ampltude Respose of the th-order ascal oteger-delay flter = we call t the bac-secto of the ascal oteger-delay flter, whch s the ey of the ascal oteger-delay flter for adjustg the delay Therefore, from the cocept Fg 8, ad 9 to 4, we ca show the ampltude respose of the ascal oteger-delay flter as Fg 9 = I Fg 9, the value of delay parameter oteger-delay s betwee [5 5, 5 +5] The detals for realzg both frot-secto ad bac-secto of the ascal oteger-delay flter wll be descrbed the et subsectos 4 Frot-Secto Realzato The frot-secto of ascal oteger-delay flter, whch s the frst system Fg 8, s used for flterg by the flter H z 9 order to obta the ascal coeffcets I fact, the ascal coeffcets are also the tmedoma sgals After such ascal coeffcets are computed, they wll be passed to the secod system called the bac-secto I order to realze the frot-secto of ascal oteger-delay flter, we wll be cosdered by factorzg the ascal trasform matr to bary {,, } matrces The frot-secto ca be realzed wthout ay multplcatos, ad the the obtaed frot-secto of ascal oteger-delay flter wll be multplerless dgtal flters The cocept of frot-secto comes from the factorzato of the ascal trasform matr to bary {,, } matrces From 9, we ca wrte the vector of trasfer fucto H z as follows, or, H z H z H z H z z 5 z z 6 The ascal coeffcets ca be computed by usg 6, e, the T of put vector, where T = [ ] Usually, the matr trasformato requres may multplcatos ad addtos, whch s depedet o the dmeso of the used matr operator Ths paper proposes a method for factorzg the ascal trasform matr to bary {,, } matrces, that wll allow the trasformato usg matr to operate wthout multplcatos ad oly addtos are used Therefore, the hardware realzato for trasformato crcuts ca be effcetly desged by usg the so-called butterfly ut of T to establsh the whole structure Cosder the ascal trasform matr, whch ca be deoted as subscrpt s the dmeso of ad from 6, we ca get Also, from ths matr equato, we ca mae the data flow graph as Fg, Fg Butterfly ut of the T Ths flow graph s called a butterfly ut of T [] ad used as a basc ut for costructg the frot-secto of ay th-order ascal oteger-delay flter The method for factorzg the ascal trasform matr to bary {,, } matrces s based o Gaussa elmato [], whch ca be descrbed as follows The elmato matr [] s used to factorze aother d of ascal matr all elemets matr are postve, but ascal trasform matr used T alteratg sg by ; s colum de s dfferet from that [] Wth some modfcato from [], the elmato matr E for the ascal trasform matr has etry E, =, ecept E, =,ad E, =, ad all other etres are zero

8 RAIOENGINEERING, VOL 4, NO 4, ECEMBER 5 9 For eample, the frot-secto of the thrd-order ascal oteger-delay flter ca be show as below stage stage stage Ths computato ca be realzed as Fg Notce that the matr has the sze + ; s the order of the ascal oteger-delay flter For ay thorder case, the frot-secto ca be realzed the smlar fasho ad shows the detals as Fg By cosderg the realzato dagram, we ow that the umber of addtos for the frot-secto of the ascal oteger-delay flter s No of addtos = The frot-secto does ot requre ay multplcatos Thus, the obtaed frot-secto of the ascal oteger-delay flter cotas oly multplerless dgtal flters z z z Fg A eample for frot-secto of the thrd-order ascal oteger-delay flter z z z z z Fg The frot-secto of the th-order ascal otegerdelay flter 44 Bac-Secto Realzato The bac-secto of the ascal oteger-delay flter chages the delay of the ascal oteger-delay flter ole, ad t s realzed by usg ad The frotsecto outputs, ad the are fed to the bac-secto as y 7 From ad defto of fallg factoral powers we have!! 6 y 8 Net, the Horer s rule s appled for sharg the commo terms as y 9 Cosequetly, by usg 9, the bac-secto of the ay th-order ascal oteger-delay flter ca be realzed as Fg Here, we ca summarze the computatoal complety of the bac-secto realzato as No of multplcatos = No of addtos = Sce the whole ascal oteger-delay flter cossts of both the frot-secto ad the bac-secto, ad the frot-secto cossts of oly mutplerless flters, all the multplcatos of the ascal oteger-delay flter occur oly from the bac-secto The total computatoal complety of the th-order ascal oteger-delay flter wll be cosdered the et secto ad the computatoal completes wll be also compared wth those of the drect-form Farrow structure as well as the Farrow structures after symmetry trasformatos proposed [ ] 5 Complety of the ascal Noteger-elay Flter Ths secto evaluates the ascal oteger-delay flter complety by coutg both multplcato ad addto operatos For the eample, the thrd-order ascal

9 SOONTORNWONG, S CHIVAREECHA, ASCAL-INTEROLATION-BASE NONINTEGER ELAY FILTER Type of flter No of No of Multplcatos addtos rect-form Farrow structure 5 Trasformed Farrow structure ascal oteger-delay flter 6 Tab Comparso of computatoal completes of the thrd-order polyomal-terpolato-type flters Type of flter rect-form Farrow structure Trasformed Farrow structure [ 4] Odd-order Eve-order No of No of Multplcatos addtos 4 ascal oteger-delay flter 4 5 Fg The bac-secto of the th-order ascal otegerdelay flter oteger-delay flter ca be realzed by usg the structures from frot-secto ad bac-secto as show Fg 4 The thrd-order ascal oteger-delay flter Fg 4 eeds 6 multplcatos ad addtos Table lsts the umbers of multplcatos ad addtos of the thrd-order polyomal-terpolato-type flters usg three types of structures Tab Computatoal completes of the th-order polyomal-terpolato-type delay flters Clearly, the ascal oteger-delay flter has less complety tha other types of the polyomal-terpolato-type flters, especally the umber of multplcatos s much less The complety of the th-order ascal otegerdelay flter s compared wth the polyomal-terpolatotype oteger-delay flters havg the drect-form Farrow structure ad the Farrow structures after symmetry trasformatos are show Tab The computatoal completes from the secod-order to th-order of types of polyomal-terpolatotype delay flters are show Fg 5 ad Fg 6, respectvely By usg Fg 5, we ca compare the umbers of multplcatos of types of realzato structures As compared to the drect-form Farrow structure [], the structures usg symmetry trasformatos [ ] ca reduce the umber of multplcatos by about 5% However, the umber of multplcatos requred by the method z z Number of Multplcatos Farrow structure after trasformato ascal tuable oteger-delay flter rect-form Farrow structure z th-order of tuable oteger-delay flter Fg 4 The thrd-order ascal oteger-delay flter Fg 5 Number of multplcatos of types of the polyomal-terpolato-type delay flters

10 RAIOENGINEERING, VOL 4, NO 4, ECEMBER 5 Number of Addtos ascal tuable oteger-delay flter rect-form Farrow structure Farrow structure after trasformato th-order of tuable oterger-dalay flter Fg 6 Number of addtos of types of the polyomalterpolato-type delay flters [ ] s ot lear wth respect to the order, but t s a square fucto The ascal oteger-delay flter requres fewer multplcatos tha the Lagrage-type wth trasformed Farrow structures [ ] The umber of multplcatos s lear wth respect to the order I Fg 6, we compare the umbers of addtos of types of polyomal-terpolato-type delay flters, the ascal oteger-delay flter requres less addtos tha the drect-form Farrow structure [] Obvously, sce the ascal oteger-delay flter requres the lowest computatoal complety, t s more sutable for fast flterg 6 Cocluso I ths paper, we have derved a ew flter realzato structure for the polyomal-terpolato-type delay flter Sce t s formulated from the dscrete ascal trasform T ad ascal terpolato, we call t ascal oteger-delay flter The resultg ascal oteger-delay flter has a low-complety structure that cotas two sectos frot-secto ad bac-secto The former cossts of multplerless flters oly, ad the latter requres less multplcatos Because the proposed ascal structure ca ole adjust the delay wth less multplcatos, t s suted for fast tug applcatos Moreover, we have compared the umbers of multplcatos of types of the polyomal-terpolato-type delay flters, ad cocluded that the ascal structure requres the smallest umber of multplcatos Cosequetly, the polyomal-terpolato-type delay flter ca be acheved by usg the ascal approach wth hgh effcecy ad the lowest complety flter structure Refereces [] FARROW, C W A cotuously varable dgtal delay elemet I roceedgs of IEEE Iteratoal Symposum o Crcuts Systems ISCAS Espoo Flad, 988, vol, p OI: 9/ISCAS [] ENG, T-B scretzato-free desg of varable fractoaldelay FIR dgtal flters IEEE Trasactos o Crcuts ad System II: Aalog gtal Sgal rocessg,, vol 48, o 6, p OI: 9/8947 [] ZHAO, H, YU, J-B A smple ad effcet desg of varable fractoal delay FIR flters IEEE Trasactos o Crcuts ad System II: Epress Brefs, 6, vol 5, o, p 57 6 OI: 9/TCSII [4] TSENG, C-C esg of varable fractoal delay FIR flters usg dfferetator ba I roceedgs of IEEE Iteratoal Symposum o Crcuts Systems ISCAS hoe Arzoa, USA,, vol 4, p 4 44 OI: 9/ISCAS48 [5] ENG, T-B, NAKAGAWA, Y SV-based desg ad ew structure for varable fractoal-delay dgtal flters IEEE Trasactos o Sgal rocessg, 4, vol 5, o 9, p 5 57 OI: 9/TS489 [6] ENG, T-B, LIAN, Y Weghted-least-squares desg of varable fractoal-delay FIR flters usg coeffcet symmetry IEEE Trasactos o Sgal rocessg, 6, vol 54, o 8, p 8 OI: 9/TS [7] ENG, T-B, QIN, W Coeffcet relato-based mma desg ad low-complety structure of varable fractoal-delay dgtal flters Sgal rocessg,, vol 9, o 4, p 9 9 OI: 6/jsgpro4 [8] ENG, T-B, QIN, W Improved b-equrpple varable fractoaldelay flters Sgal rocessg, 4, vol 94, o, p 7 OI: 6/jsgpro74 [9] LIU, G-S, WEI, C-W A ew varable fractoal sample delay flter wth olear terpolato IEEE Trasactos o Crcuts ad System II: Aalog gtal Sgal rocessg, 99, vol 9, o, p 6 OI: 9/8588 [] ENG, T-B Coeffcet-symmetres for mplemetg arbtraryorder Lagrage-type varable fractoal-delay dgtal flters IEEE Trasactos o Sgal rocessg, 7, vol 55, o 8, p 478 to 49 OI: 9/TS [] ENG, T-B Symmetrc structures for odd-order mamally flat ad weghted-least-squares varable fractoal-delay flters IEEE Trasactos o Crcuts ad System I: Regular apers, 7, vol 54, o, p 78 7 OI: 9/TCSI [] ENG, T-B Trasformato matr for odd-order Lagrage-type varable fractoal-delay flters I roceedgs of 6 th Iteratoal Coferece o Iformato Commucatos ad Sgal rocessg ICICS 7 Sgapore, 7, p 5 OI: 9/ICICS [] ENG, T-B Trasformato matr for eve-order Lagrage-type varable fractoal-delay dgtal flters I roceedgs of Iteratoal Coferece o Itellget ad Automato Systems ICIAS 7 Kuala Lumpur Malaysa, 7, p 79 8 OI: 9/ICIAS [4] EI, S-C, TSENG, C-C A comb flter desg usg fractoalsample delay IEEE Trasactos o Crcuts ad System II: Aalog gtal Sgal rocessg, 998, vol 45, o 6, p OI: 9/86765 [5] ENG, T-B Hgh-resoluto mage terpolato usg twodmesoal Lagrage-type varable fractoal-delay flter I roceedgs of Iteratoal Symposum o Nolear Theory ad Applcatos NOLTA 5 Bruges Belgum, 5, p 4 7 [6] SHYU, J-J, EI, S-C, CHAN, C-H Mma phase error desg of allpass varable fractoal-delay dgtal flters by teratve weghted least-squares method Sgal rocessg, 9, vol 89, o 9, p OI: 6/jsgpro9

11 SOONTORNWONG, S CHIVAREECHA, ASCAL-INTEROLATION-BASE NONINTEGER ELAY FILTER [7] TSENG, C-C Closed-form desg of dgtal IIR tegrators usg umercal tegrato rules ad fractoal sample delays IEEE Trasactos o Crcuts ad System I: Regular apers, 7, vol 54, o, p OI: 9/TCSI [8] SHYU, J-J, EI, S-C, HUANG, Y- Two-dmesoal Farrow structure ad the desg of varable fractoal delay - FIR dgtal flters IEEE Trasactos o Crcuts ad System I: Regular apers, 9, vol 56, o, p OI: 9/TCSI888 [9] TSENG, C-C esg of - ad - varable fractoal delay allpass flters usg weghted least-squares method IEEE Trasactos o Crcuts ad System I: Fudametal Theory Applcatos,, vol 49, o, p 4 4 OI: 9/TCSI86 [] ABURENE, M F, GOOMAN, T J The dscrete ascal trasform ad ts applcatos IEEE Sgal rocessg Letters, 5, vol, o 7, p OI: 9/LS [] GOOMAN, T J, ABURENE, M F Iterpolato usg the dscrete ascal trasform I roceedgs of 4 th Aual Coferece o Iformato Sceces ad Systems CISS 6 New Jersey USA, 6, p 79 8 OI: 9/CISS68666 [] SKORAS, A N Effcet computato of the dscrete ascal trasform I roceedgs of 4th Europea Sgal rocessg Coferece EUSICO 6 Florece Italy, 6, p 4 OI: 9/CISS68666 [] EELMAN, A, STRANG, G, USA ascal Matrces pages [Ole] Cted Avalable at: About the Authors arya SOONTORNWONG was bor Chobur, Thalad He receved hs BSIdEd Telecommucato Egeerg from Kg Mogut s Isttute of Techology Ladrabag KMITL ad M Eg Electrcal Egeerg from rce of Sogla Uversty SU 6 He s a Eg caddate studet Electrcal Egeerg Kg Mogut s Isttute of Techology Ladrabag KMITL support by Tha Govermet Scece ad Techology Scholarshp, Mstry of Scece ad Techology MOST Scholarshp Hs research terests clude dgtal flter desg ad dgtal sgal processg S Sorawat CHIVAREECHA was bor Naorpathom, Thalad He s a Assstat rofessor wth the epartmet of Telecommucato Egeerg, Faculty of Egeerg, Kg Mogut s Isttute of Techology Ladrabag KMITL He receved hs B Eg Telecommucato Egeerg from Suraaree Uversty of Techology SUT 998, M Eg ad Eg Electrcal Egeerg from Kg Mogut s Isttute of Techology Ladrabag KMITL, ad 8, respectvely Hs research terests clude dgtal flter desg ad mplemetato, VLSI for dgtal sgal processg, formato scece ad satellte egeerg