Efficient Algorithms and Design for Interpolation Filters in Digital Receiver

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1 Sesors & rasducers Vol. 7 Issue 5 May 4 pp. 3-4 Sesors & rasducers 4 by IFSA Publshg S. L. Effcet Algorthms ad Desg for Iterpolato Flters Dgtal Recever Xaowe u Zhmg He 3 Lwa he 4 Qag he 5 Hu Xe School of Electroc Egeerg Uversty of Electroc Scece ad echology of ha hegdu ty Schua 673 ha School of Electroc ad Iformato Egeerg hogqg hree Gorges Uversty 44 Wa Zhou ha el.: E-mal: xw455@6.com Receved: March 4 /Accepted: 3 Aprl 4 /Publshed: 3 May 4 Abstract: Based o polyomal fuctos ths paper troduces a geeralzed desg method for terpolato flters. he polyomal-based terpolato flters ca be mplemeted effcetly by usg a modfed Farrow structure wth a arbtrary frequecy respose the flters allow may pass-bads ad stop-bads ad for each bad the desred ampltude ad weght ca be set arbtrarly. he optmzato coeffcets of the terpolato flters tme doma are got by mmzg the weghted mea squared error fucto the covertg to solve the quadratc programmg problem. he optmzato coeffcets frequecy doma are got by mmzg the maxma (MMax) of the weghted mea squared error fucto. he degree of polyomals ad the legth of terpolato flter ca be selected arbtrarly. umercal examples verfed the proposed desg method ot oly ca reduce the hardware cost effectvely but also guaratee a excellet performace. opyrght 4 IFSA Publshg S. L. Keywords: Polyomal-based flter Modfed Farrow structure Iterpolato Bass fucto Varable fractoal delay.. Itroducto Wth creasg applcatos of dgtal recevers [] commucato systems the varable fractoal delay (VFD) flters have receved cosderable attetos ad these flters have may attractve features sce the Farrow structure [8 8] was troduced such as the accurate cotrol of varable frequecy characters the smple real-tme update of coeffcet values ad regular mplemetato patters etc. he polyomal-based techque s a mportat method to mprove the sgal to ose rate (SR) [5 ] whch has bee used to desg VFD flters [-4 9] by mplemetg the flters Farrow structure or ts modfcatos. hese kds of flters allow to evaluate ew sample values at arbtrary postos betwee exstg samples of a dscrete-tme sgal ad also have some other attractve features. Frstly these kds of flters have a pecewse polyomal mpulse respose. Secodly they ca be mplemeted effcetly by usg the Farrow structure or ts modfcatos. hrdly by weghtg properly the output samples of these flters t s easly to cotrol the desred tme stat for the terpolated output samples whch parallel FIR flters. Other VFD 3

2 Sesors & rasducers Vol. 7 Issue 5 May 4 pp. 3-4 techques clude the sple-based techques [] ad the Lagrage-based techques [ 7]. he desg methods of polyomal-based terpolato flters ca be maly dvded to two dfferet classes: the tme doma method ad the frequecy doma method. he Lagrage-based terpolators [ ] ad sple-based terpolators [ ] are the best kow tme doma methods. he advatages of these desg methods are that the flters coeffcets are easly avalable the closed form. However these terpolato flters become poor whe the frequecy compoets close to half the samplg rate because they eglect the frequecydoma formato of the put sgals. he secod desg method s to optmze the coeffcets of the recostructed mpulse respose frequecy doma [ ] ad the best kow frequecy doma method s the polyomal-based terpolato flters [8-]. hs desg method eables oe to obta a better flterg characterstc tha those obtaed by the methods metoed above. Such as [8] Farrow proposed a least-mea-square optmzato of the polyomal-based fractoal delay flters but these methods ot allow to optmze separately the coeffcets of the terpolato flters pass-bad or stop-bad. I [9] Harrs et al. used the reorderg of the polyomal coeffcets to obta polyomal expasos of the tme seres ad [] Hamla et al. used the hybrd aalog/dgtal model to desg the terpolato flters ad ths method eables oe to select arbtrarly the legth of the mpulse respose but t has a hgh computatoal complexty. Motvated by the cted works above ths paper we derve a ew desg method for the terpolato flters ths desg method allows to pecewse optmze the pass bads ad stop bads of these terpolato flters the desred ampltude ad weghted fuctos ca be selected arbtrarly for each bad the legth of the terpolato flter ad the degree of the polyomals ca be chose depedetly. Accordg to the desg requremets we troduce the terpolato fuctos () t ( t ). he optmzato coeffcets of the proposed flter ca be performed ether the mmax method or the least-mea-square method. For ths proposed desg method we fd that the frst tems of the optmzato coeffcets equal to zero that s c ( )... whch mprove the hardware mplemetato ablty effectvely. he rest of ths paper s orgazed as follows. Polyomal-based terpolato flter ad ts mpulse respose are troduced secto. Secto 3 presets the polyomal bass fucto desg ad flter optmzato tme doma ad frequecy doma respectvely. Secto 4 troduces the Farrow structure of terpolato flters. I secto 5 two smulato examples are provded to demostrate effectveess of the terpolato flter. Fally coclusos are gve secto 6.. Polyomal-Based Iterpolato Flter ad ts Impulse Respose I ths paper oly the polyomal-based terpolato flters are proposed due to they ca be mplemeted effcetly usg the Farrow structure or ts modfcatos. he desg method for these kds of flters s based o the terpolato fuctos to optmze the flter coeffcets. herefore the selecto of the bass fuctos ad the optmal method for the flter coeffcets are two key problems to desg the polyomal-based terpolato flters. If fucto f (x) a eghborhood U( x ) has ( + )-order dervatve the for U( x ) f (x) has -order aylor formula ( ) f ( x)( x x) f ( x) a( x x) ()! I mathematcs a set cossts of certa gvg fuctos from set X to Y the the set ca be called a fucto space extedg the aylor formula to a fucto space oe has () f ( x) a g ( x) g ( x) ( xx ) For the fucto space V f there are a fucto sequeces { v ( x) V...} to ay fucto f( x) V f ( x) cv ( x) holds the we ca call the fucto sequeces are the bass sequeces of fucto space V. Assume the fucto space V cossts of all the effectve terpolato fuctos. ostruct the bass sequeces { v ( x) V } of the fucto space V that s { v ( x) V }. he bass sequeces ca also be expressed as v ( x ) v ( x ).... Obvously vx ( ) V cosderg the essece of terpolato method s that the orgal sgals are double-sampled processg after low-pass flterg. hus ay terpolato fuctos must have the smlar characterstcs of low-pass flterg fuctos. I the followg part based o the exstg orgal samples we wll recostruct the approxmatg sgal. o aalyze the characterstcs of terpolato fuctos we use the bass fuctos wth the ut terval s to approxmate orgal sgal sectoally. For coveece to dscuss assume s= the the power seres of the bass fuctos ca be wrtte as follows () t t () t... others (3) 33

3 Sesors & rasducers Vol. 7 Issue 5 May 4 pp. 3-4 If o other specal structos assume the bass fucto () t s bouded terval []. I Eq.(3) t s show that the overall bass fuctos cosst of the power seres ( t). For coveece the followg part we smply call the terpolato flter s based o the bass fucto () t. Assume the legth of the terpolato flter as L the the mpulse respose of the terpolato flter ca be recostructed as follows L/ L L h( t) c( ) ( t) t (4) L/ where h( t ) s the mpulse respose of the recostructed flter c () deote the flter coeffcets ( t ) deote the bass fuctos of the terpolato flter ad s the degree of the polyomals. herefore we ca use the dgtal flter theory to desg the recostructed mpulse respose h( t ) the pecewse approxmate t fally obta the optmzato coeffcets as well as realze the desg of polyomal-based terpolato flter. he mpulse respose h( t) ca be expressed a pecewse terval t s desrable h() t c() () t t (5) From Eq. (5) we kow that the costructed mpulse respose each ut terval ca be pecewse optmzed. I order to make the dgtal terpolato flter has a lear phase respose the mpulse respose ht () should be a symmetrcal fucto whch meas the fuctos h (t) ad h () t are symmetrcal aroud Y axs as follows (6) h () t c ( ) () t c () () t where the fuctos () t ad ' () t are symmetrcal aroud t = / whe explotg the above symmetres the umber of optmzato coeffcets to be mplemeted ca be reduced from ( ) L to ( ) L/ thus the mpulse respose ht () ca also be recostructed as follows L/ h( t) c( ) ( t) t (7) where ( t ) s got by ( t ) ad they are symmetrcal aroud t =. he mpulse respose of the flter frequecy doma correspodgly ca be wrtte as L / H( f ) c ( ) ( f ) (8) where H ( f ) ad ( f ) are the Fourer trasform of fuctos ht () ad ( t ) respectvely. 3. Desg of Iterpolato Fuctos ad Flter Optmzato he fudametal dea for flter optmzato s to approxmate the exstg orgal sgal accordg to some tme-doma or frequecy-doma crtera ad the terpolato flter should have a effcet dgtal mplemetato structure. he desg of terpolato fuctos ca be carred out the tme doma ad frequecy doma respectvely. hus the optmzato of flter coeffcets c () ca also be dvded to two dfferet classes that s the coeffcets optmzato tme doma ad frequecy doma respectvely. 3.. Desg of Iterpolato Fuctos ad Flter Optmzato me Doma I order to obta the optmzato coeffcets of the recostructed flter tme doma we use the polyomal bass fuctos to pecewse approxmate the desred low-pass flterg fucto. I the followg we wll derve the method to get the optmzato coeffcets of the recostructed mpulse respose ht () the to solve the most weghtg coeffcets. Defe the desred low-pass flterg fucto as g() t ad defe fucto ( hg ) as the error fucto betwee ht () ad g() t that s ( hg ) s used to measure the approxmatg where degree betwee ht () ad g() t. he optmzato of the flter coeffcets ca be got va mmzg the error fucto ( hg ). I order to reduce the error at terpolato pots the mpulse respose of the recostructed flters should meet the followg codtos t h( t) (9) t L Equato (9) ca be wrtte a alteratve form as follows c () () L / c () () c () () L / () 34

4 Sesors & rasducers Vol. 7 Issue 5 May 4 pp. 3-4 From Eq.() we ca see () ad () the edpot values of the terpolato fucto determe the feasble regos of c () Let [ c() c() c() c() c() c()... c ( L/ ) c ( L/ )... c ( L/ )] [ ] L / where [ c( ) c( )... c ( )]. he where () Aeq Beq () A eq Aeq Aeq Aeq Aeq [ () () ()] Aeq [ () () ()] Beq [ ] L( ) L/ (3) I Eq.(3) deotes the degree of polyomals. herefore the optmzato of flter coeffcets ca be ca be expressed as m (hg) (4) s.t. Aeq Beq Because the flter coeffcets c () ca be pecewse optmzed each ut terval the optmzato problem above ca be dvded to L / sub-parts as follows where m (h g ) (5) s.t. Aeq Beq () () () Aeq () () () [ ] Beq [] L/ - (6) he optmzato of the flter coeffcets ca be got va mmzg the least square error fucto t s desrable ( h g ) h( t) g ( t) dt (7) Accordg to Eqs. (7) ad (7) the error fucto ca be rewrtte as follows ( h g ) c ( ) ( t) g ( t) dt c () () t dt c() cm() () m t dt m c ( ) ( t) g ( t) dt g ( t) dt c( ) ( t) g( tdt ) g ( tdt ) (8) Let [ ( ) ( ) ( )] c c c () tdt m ( ) ( ) t g t dt. eglect the costats ad further the error fucto ( h g) Eq.(8) ca be expressed ( h g ) A B (9) where ( h g) deotes the fucto ( h g) wth eglectg the costats ad A B ( ) ( ) - () he the problem Eq.(5) coverts to the quadratc programmg problem whch ca be wrtte as m A B () s.t. Aeq Beq hus gve the correspodg parameters of terpolato flters the optmzato coeffcets ca be obtaed the the polyomal-based terpolato flter ca be realzed. Further we aalyze the codtos of the quadratc programmg problem. Whe () the codtos Aeq Beq s true oly whe ad whe () the codtos ca be smplfed as follows () () () () () () () () c( ) c( ) 35

5 Sesors & rasducers Vol. 7 Issue 5 May 4 pp. 3-4 Hece f a polyomal bass fucto () t meets () ad () the c( ) holds whch ot oly reduces the dmesos of the orgal codtos but also mproves the effcecy of fdg a optmum soluto at the momet the outputs of v() are equal to zero whch ca be removed drectly t reduces the hardware costs of the Farrow structure ad made t parallel a (L )-order FIR flter. Accordg to the dscusso metoed above the expresso of terpolato fucto ca be wrtte as follows t ( ) c t (3) he terpolato fucto eeds to satsfy the followg codtos that s () (4) () c () () c () ()... L/ c ( ) ()... L/ (5) he the coeffcets of the terpolato fucto ca be derved by substtutg the Eq.(5) to the expresso of the polyomal fucto as gve by Eq.(3) c c (6) I Eq.(5) we ca see dfferet kds of flters ca be costructed by these flter coeffcets. However wth the creasg of polyomal orders the terpolato flter ot oly becomes very complex for the mplemetato of Farrow structure but also degradato the effcecy of sgal processg. A low order s a deal choce for terpolato fucto. Hece we choose the lear fucto as the bass of the proposed flter as follows () t kt b kb Rk (7) Accordg the codtos equatos (4) ad (5) t s desrable ( t) k( t ) k R k (8) From Eq.(8) we ca see for all lear bass fuctos oly the fuctos kt ( ) satsfy the codtos that we metoed above that s oly these fuctos satsfy c (). Future we ca smplfy the bass fucto whe take k the we get the proposed terpolato fucto as follows ( t) t (9) 3.. Flter Optmzato Frequecy Doma he polyomal-based terpolato flter tme doma geerally s ot a very practcal approach to the applcato of sgal processg. Because the frequecy bad of the sgal s usually kow but the tme doma characters of sgals s ukow. he proposed method metoed above ca also covert the flter optmzato problem from tme doma to frequecy doma. It s desred to desg the recostructo flter whch ca be mplemeted by usg the Farrow structure or ts modfcatos ad ot be cotrolled by the legth of the flter ad the degree of the polyomals. hus the goal s to optmze the Farrow structure coeffcets ad the recostructed mpulse respose H( f ) frequecy doma. Based o ths dea ad the demads above gve L ad a compact subset X as well as a desred fucto G( f ) that s cotuous for f X ad a weght fucto W( f ) that s postve for f X fd ( M ) / flter coeffcets c () ca be mplemeted as follows m ( H G) (3) s. t. Aef. Bef where H ad G deote the Fourer trasform of fuctos ht () ad gt () respectvely ad [ c() c()... c() c() c()... c()... c( L/) c( L/) c ( L/)] [... L/ ] Aef Aef... Aef Aef L ( ) L/ Aef [ () ()... ()] Aef [ ( j f) ( j f)... ( j f)] Bef [ ( f )...] where ( f ) (3) s the Fourer trasform of polyomal bass fucto () t. Because the recostructed 36

6 Sesors & rasducers Vol. 7 Issue 5 May 4 pp. 3-4 mpulse respose ht () s a symmetrcal fucto aroud t so the frequecy respose H( f ) s real the error fucto ( HG ) frequecy doma ca be defed as ( ) max { ( )[ ( ) ( )]} (3) H G W f H f D f df fx where the frequecy bad X cossts of the specfc pass-bads ad stop-bads. G( f) s the frequecy respose of the desred terpolato fucto. W( f ) s the weghtg fucto accordace wth the requremets of target terpolato flter the specfc frequecy bad that s the greater the weght the smaller the peak error. I order to mprove the atteuato of the flter stop-bad we ca preset a small pass-bad weght. herefore the problem Eq.(3) ca be wrtte as follows m( H G) mmax s. t. Aef. Bef fx { W( f )[ H( f ) D( f )]} df (33) Actually practcal applcato t s ofte uecessarly to get the frequecy respose of the target terpolato fucto o the overall frequecy bads ad oly eed to satsfy the characterstcs some specfc frequecy bads. hus defe the error fucto a specfc frequecy bad as ( H G). Further a pecewse frequecy bad f X let ( H G ) max fx { W ( f )[ H ( f ) D( f )]} df (34) he the problem metoed above ca be expressed as follows ( H [ max { [ W ( f ) { W max { f X G ) max { W ( f ) ( f )[ [ W ( f ) G ( f )] df } f X c ( ) f X ( f ) [ W ( f ) G ( f )] df } m c ( ) W I whch let ( f ) G ( f )]} df c ( ) c c ( ) c ( ) m [ ( f )] df } ( f ) G ( f )]} df ( f ) ( f ) G ( f ) df ( ) W c ( ) c ( )... c ( )] m ( f ) df W ( f ) ( f ) df W ( f ) ( f ) G ( f ) df (37) (38) eglect the costats ad further the error fucto ( H G ) ca be expressed ' ( H G ) A B (39) where ( H G) deotes the fucto ( HG ) wth eglectg the costats ad A B [... ( ) ( ) ] (4) where m ( H s. t. Aef G ) Bef (35) Hece the problem Eq.(35) coverts to the quadratc programmg problem whch ca be wrtte as m s. t. Aef max f X A B Bef (4) () ()... () Aef ( jf ) ( jf )... ( jf ) (36) [(f )] Bef []... L/ Accordg the Eqs. (8) ad (34) the error fucto the frequecy doma ca be wrtte as follows hus the flter optmzato coeffcets ca be got by solvg the quadratc programmg problem. Gve the legth of flter the degree of polyomals pass bads ad stop bads ad the weghtg fucto etc the flter optmzato coeffcets frequecy doma ca be obtaed by utlzg the proposed method. I secto 5 some examples wll be gve to verfy the performace of the proposed algorthm. 37

7 Sesors & rasducers Vol. 7 Issue 5 May 4 pp. 3-4 We ca also preset the weght values properly pecewse frequecy bad f accordg to the performace requremets of sgal processg. Because the polyomal-based terpolato flter ca be mplemeted effcetly by usg the Farrow structure or ts modfcatos the followg the Farrow structure wll be troduced. 4. he Farrow Structure of Iterpolato Flters If the mpulse respose ht () s a pecewse polyomal the the Farrow structure or ts modfcatos ca be mplemeted. he Farrow structure has some features to make t attractve sgal processg he umber of FIR sub-flters s ad the legth of these sub-flters s L. Flter coeffcets are determed drectly by the polyomal coeffcets of the mpulse respose. he ma advatage of the Farrow structure s that all the flter coeffcets are fxed. he resoluto of the fractoal terval s lmted oly by the precso of the arthmetc ot by the sze of the coeffcet memory. hese characters of the Farrow structure make t a very attractve structure to be mplemeted usg a VLSI crcut or a sgal processor etc. hs flter structure cossts of parallel FIR flters wth fxed coeffcet values. he desred tme stat for the terpolated output samples ca be easly cotrolled by properly weghtg the output samples of these FIR flters. I ths paper the proposed terpolato flters have a pecewse polyomal mpulse respose so they ca be mplemeted effcetly by usg the Farrow structure or ts modfcatos. Whe the terpolato fucto s () t t by substtutg t to Eq.(6) ad the flter coeffcets have the feature as follows c( ) c( ) seve (4) c( ) c( ) sodd Whe explotg the above symmetres the umber of the coeffcets to be mplemeted ca be reduced from ( ) L to ( ) L/. Accordg to the equato k k k (43) h ( u ) c ( ) ( u ) c ( ) u Whe ( k) ak b ad apply the bomal theorem m c () cm() ma b (44) m he we ca get that () t t whch has a Farrow structure ad further the Farrow structure ca be smplfed c c (45) others () () ( ) Accordg Eqs.(43) ad (45) the Farrow structure of the ( t) based terpolato flter ca be got whch s show Fg.. Fg.. Farrow structure for the ( t) based terpolato flter. 38

8 Sesors & rasducers Vol. 7 Issue 5 May 4 pp. 3-4 I Fg. the legth of terpolato flter s L the degree of polyomals s. It s show that the proposed terpolato flter cossts of L adders L multplers ad L ( ) delays ad practcal applcato the delays ca be shared [] that s oly L delays are eeded. I the followg part some examples wll be gve to compare the performace of the proposed terpolato flter wth the Lagrage-based terpolator. 5.. Example ake the terpolato fucto () t t ad utlze the proposed optmal method to approxmate the rased-cose flters tme doma t s show Fg ad the optmzato coeffcets of the flter tme doma are show able. I Fg. t has show that the proposed terpolato flters have a good approxmatg performace to the orgal lowpass flters tme doma. he parameter deotes the roll off factor of rased-cose flter. Whe explotg the symmetres Eq. (6) the umber of the optmzato coeffcets to be mplemeted ca be halved thus able oly shows halves of the optmzato coeffcets. he legth of the terpolato flter s L = 6 the degree of the polyomals s = 3. able. Optmzato coeffcets of the proposed flter versus =.5. (a) Orgal low-pass flter c () c () c () c () 3 = = = Example For some lear modulato trasmtted sgals such as PAM PSK QAM the receved sgals of the recevers ca be expressed as follows j () t x() t a g( t () t ) e () t (b) (t-)-based terpolato flter Fg.. he approxmatg effect of proposed terpolato flter to the orgal low-pass flter. 5. umercal Examples hs secto provdes two desg examples to llustrate the flexblty of the metoed MMax method the least-mea-square sythess method ad the performaces of the proposed terpolato flters. I example we wll dscuss the approxmatg performace of the proposed terpolato flter. I example we wll compare the error rate of the proposed terpolato flter over those obtaed usg the Lagrage desg method ad the smulato sgals are three kds of lear modulato sgals.e. MSK 3QAM 56QAM. where a deotes the sedg complex data s symbol perod () t deotes the tmg error fucto () t s the carrer phase dfferece fucto g() t s mpulse respose fucto of the system ad t () deotes the zero mea Gaussa whte ose. he fuctos () t ad () t usually chage slowly so we assume they are costats a short tme. he smulato below assumes that the respose of system s a rased-cose fucto wth the roll off factor.5 preset the tmg error.5 ad carrer phase dfferece. Gve these parameters a smulato s carred out to aalyze three kds of modulato sgals metoed above that s MSK QAM3 ad QAM56 respectvely. he receved sgals are sampled by four tmes of the symbol rate. Whe the tmg error s obtaed the use the ( t) based terpolato flter ad Lagrage-based terpolator to flter the samplg sgals respectvely. Fg. 3 Fg. 4 ad Fg. 5 show the costellato charts of tmg sychrozato of the output sgals. 39

9 Sesors & rasducers Vol. 7 Issue 5 May 4 pp. 3-4 I Fg. 3 Fg. 4 ad Fg. 5 t s show that the costellato charts of the output sgals by the proposed terpolato flter s much smaller tha by the Lagrage-based terpolator whch meas that the proposed terpolato flter has a much lower error rate tha the Lagrage-based oe. hus the proposed terpolato flter has a better flterg performace tha the Lagrage-based terpolator. Assume the flter legth s L the polyomals degree s the comparg wth the Lagrage-based terpolato flter ths proposed oe saves ( ) L adders ad ( ) L multplers due to c ( )... ad the symmetres Eq.(6). I most practcal applcatos the recostructo pulse respose ht () ca get a good approxmatg to the orgal sgal whe the degree of polyomal s small [7]. Hece ths case the ( t) based terpolato flter reduces the hardware costs greatly. able ca expla t clearly. So comparg wth Lagrage-based terpolato flter the proposed oe ot oly reduces the hard costs greatly but also has a better flterg performace. able. Hardware costs for the terpolato flters. t- Lagrage t- Lagrage t- Lagrage t- Lagrage t- Lagrage t- Lagrage L Multplers( a ) Adders( b ) 8 (7.4%) 79 (74.7%) (73.9%) 59 (74.8%) (65.4%) 79 (66.5%) 3 36 (66.%) 359 (66.6%) (6.8%) 399 (4.6%) (6.%) 799 (6.5%) a ad b deote the percetage of savg multplers ad adders respectvely. MSK sgal by polyomal Lagrage-based terpolato flter I-Phase MSK sgal by polyomal (t-)-based terpolato flter I-Phase (a) Lagrage-based terpolato flter (b) (t-)-based terpolato flter Fg. 3. ostellato charts of output sgal (MSK) by the flters. QAM3 sgal by polyomal Lagrage-based terpolato flter I-Phase QAM3 sgal by polyomal (t-)-based terpolato flter I-Phase (a) Lagrage-based terpolato flter (b) (t-)-based terpolato flter Fg. 4. ostellato charts of output sgal (QAM3) by the flters. 4

10 Sesors & rasducers Vol. 7 Issue 5 May 4 pp. 3-4 QAM56 sgal by polyomal Lagrage-based terpolato flter QAM56 sgal by polyomal (t-)-based terpolato flter I-Phase I-Phase (a) Lagrage-based terpolato flter (b) (t-)-based terpolato flter Fg. 5. ostellato charts of output sgal (QAM56) by the flters. 6. oclusos he ma cotrbuto of ths paper was a proposed geeral desg for the polyomal-based terpolato flter. he mmax method or leastmea-square method to optmze the flter coeffcets tme doma ad frequecy doma respectvely. he legth of the terpolato flter the degree of the polyomals the pass-bads ad stopbads the desred respose ad weghtg fucto were used to optmze the flter coeffcets. We also foud the assocato betwee the proposed terpolato fuctos ad the Farrow structure further t wll exted the ablty to meet dfferet sgal processg evromets. We have aalyzed the decomposto expresso of the recostructed mpulse respose realzed the flter Farrow structure by usg the proposed polyomal fucto. We foud that the frst tems of the optmzato coeffcets of the proposed terpolato flter equal to zero. Actually most practcal applcatos the recostructed mpulse respose ca get a good approxmatg performace to the desred oe by usg a low approxmato order (geerally 3). I ths respect the proposed terpolato flter saves the hardware costs greatly. If the legth of flter s L the degree of polyomals s the compares wth the Lagrage-based terpolator the proposed oe saves ( ) L adders ad ( ) L multplers. Examples dcated that the proposed terpolato flter ot oly has a good flterg performace but also reduces the mplemetato complexty of the Farrow structure. Ackowledgemets hs work was supported part by the atoal atural Scece Foudato of ha ( ) to a certa degree ad t also beefted by the Fudametal Research Fuds for the etral Uverstes (ZYGXJ3) ad Opeg opc Fud for Key Laboratory of omputer Archtecture (ARH3). hs work was also supported by echology Project Foudato of hogqg Educato ommttee (KJ3) ad Key Laboratory of sgal ad formato processg hogqg hree Gorges Uversty. Refereces []. A. Frack K. Bradeburg A closed-form descrpto for the cotuous frequecy respose of Lagrage terpolators IEEE Sgal Processg Letters Vol. 6 Issue 7 9 pp []. A. Bhadar P. Marzlao Fractoal delay flters based o geeralzed cardal expoetal sples IEEE Sgal Processg Letters Vol. 7 Issue 3 pp [3]. H. H. Dam Desg of all pass varable fractoal delay flter IEEE rasactos o Sgal Processg Vol. 59 Issue pp [4].. B. Deg Frequecy-doma weghted-least squares desg of quadratc terpolators IE Sgal Processg Vol. 4 Issue pp. -. [5].. B. Deg Robust structure trasformato for causal Lagrage-type varable fractoal-delay flters IEEE rasactos o rcuts ad Systems Vol. 56 Issue 8 9 pp [6].. B. Deg oeffcet-symmetres for mplemetg arbtrary-order Lagrage-type varable fractoaldelay dgtal flters IEEE rasactos o Sgal Processg Vol. 55 Issue 8 7 pp [7].. B. Deg losed-form desg ad effcet mplemetato of varable dgtal flters wth smultaeously tuable magtude ad fractoal delay IEEE rasactos o Sgal Processg Vol. 5 Issue 6 4 pp [8].. W. Farrow A cotuously varable dgtal delay elemet Proceedgs of the IEEE Iteratoal Symposum o rcuts ad Systems ISAS 88 Espoo Flad 988 pp [9]. F. Harrs Performace ad desg cosderatos of Farrow flter used for arbtrary resamplg Proceedgs of the 3 th Iteratoal oferece o Dgtal Sgal Processg Sator Greece 997 pp

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