The Z Transform over Finite Fields

Transcription

1 Iteratoal Telecommucatos Symosum - ITS22, Natal, Brazl The Z Trasform over Fte Felds R M Camello de Souza, H M de Olvera, D Slva Abstract Fte feld trasforms have may alcatos ad, may cases, ca be mlemeted wth a low comutatoal comlexty I ths aer, the Z Trasform over a fte feld s troduced ad some of ts roertes are reseted I INTRODUCTION Dscrete trasforms lay a very mortat role Egeerg Partcularly sgfcat examles are the well kow Dscrete Fourer Trasform (DFT) ad the Z Trasform [], whch have foud may alcatos several areas, secally the feld of Electrcal Egeerg A DFT over fte felds was also defed [2] ad aled as a tool to erform dscrete covolutos usg teger arthmetc Recetly, a dscrete Hartley trasform over fte felds was troduced [], whch has terestg alcatos the felds of dgtal multlexg ad sread sectrum [4], [] I ths aer, the fte feld Z trasform s troduced Earler attemts to deal wth Z trasforms over fte felds cosdered oly fte sequeces of elemets of a Galos feld, a clever stratagem to evade the roblem of seres covergece over a fte algebrac structure [] I order to defe a Z-trasform over a fte feld, we eed to aswer a few questos such as: What s the meag of a comlex varable z defed over a fte feld? How ca we rereset the Fte-Feld Argad- Gauss lae? What about the covergece of fte seres over fte felds? Are there coverget seres? I whch sese? Smle questos such as k : = [ 2 ] (mod 7 ) =? + k : = [ ] (mod 7 ) =? should be addressed I the ext secto some mathematcal relmares are reseted I artcular, gaussa tegers over GF() are defed ad the umodular grou of GF( 2 ) s costructed Secto troduces a reresetato for the comlex Z lae over a Galos feld GF() Ifte sequeces over fte felds are dscussed secto 4, where the roblem of seres covergece s vestgated I secto the fte feld Z trasform s troduced ad some of ts roertes are derved The dscrete-tme fte feld Fourer trasform s also defed The aer closes wth a few cocludg remarks II MATHEMATICAL PRELIMINARIES II Comlex umbers over fte felds The set G() of gaussa tegers over GF() defed below lays a mortat role the deas troduced ths aer (hereafter the symbol := deotes equal by defto) Q, R ad C deote the ratoal, real ad comlex sets, resectvely δ s the Kroecker symbol Defto : G() := {a + b, a, b GF()}, beg a odd rme for whch 2 = - (e, (mod 4)) s a quadratc oresdue GF(), s the set of gaussa tegers over GF() Let deote cartesa roduct It ca be show, as dcated below, that the set G() together wth the oeratos ad defed below, s a feld [7] Proosto : Let : G() G() G() (a +b, a 2 +b 2 ) (a +b ) (a 2 +b 2 ) = =(a +a 2 )+(b +b 2 ), ad : G() G() G() (a +b, a 2 +b 2 ) (a +b ) (a 2 +b 2 )= =(a a 2 -b b 2 ) + (a b 2 +a 2 b ) The structure GI() := < G(),, > s a feld I fact, GI() s somorhc to GF( 2 ) By aalogy wth the comlex umbers, the elemets of GF() ad of GI() are sad to be real ad comlex, resectvely Defto 2: The umodular set of GI(), deoted by G, s the set of elemets ζ=(a+b) GI(), such that a 2 +b 2 (mod ) Proosto 2: ζ ζ a + b (mod ) Proof: ζ = ( a + b) a + b (mod ), oce GI() s somorhc to GF( 2 ), a feld of characterstc Sce =4k+, * =, so that ζ a b(mod = ζ (mod ) Therefore, + * ζ ζζ = ζ a + b (mod ) Proosto : The structure <G, > s a cyclc grou of order (+) Proof: G s closed wth resect to multlcato O the other had, t s a well kow fact that the set of o-zero elemets of GF(), wth the oerato of multlcato, s a cyclc grou of order (-) (deoted here by G) [8] Therefore G s a cyclc subgrou of G ad, from roosto 2, t has order + To determe the elemets of the umodular grou t hels to observe that f ζ=a+b s oe such elemet, the so s every elemet the set Γ={b+a, (-a)+b, b+(-a), a+(-b), (-b)+a, (-a)+(-b), (-b)+(-a)} Examle : Umodular grous of GF(7 2 ) ad GF( 2 ) I each case, table I lsts the elemets of the subgrous G of order 8 ad 2, ad ther orders Deartameto de Eletrôca e Sstemas - CTG - UFPE, CODEC Commucatos Research Grou, CP 78, 7-97, Recfe - PE, Brasl E-mal: {Rcardo, hmo}@dufebr, dalos@hotlkcombr

2 TABLE I ELEMENTS OF G ζ GI(7) Order ζ GI() Order - 2-2, - 4 +, , 2+, +2, + 8, , + 8+, 8+, +, + Fgure llustrates the 2 roots of uty GF( 2 ) Clearly, G s somorhc to C 2, the grou of roer rotatos of a regular dodhecaedro ζ=8+ s a grou geerator corresodg to a couter-clockwse rotato of 2π/2 = o Symbols of the same colour dcate elemets of same order, whch occur cougate ars -8+=+ - +=-8- +8=-+8 -+= =8+ Fgure Roots of uty GF( 2 ) exressed as elemets of GI() II2 Polar Form for Gaussa Itegers a Fte Feld It s a well kow fact that, the usual comlex umber arthmetc, the so-called olar reresetato has may terestg asects that make t very attractve for may alcatos, artcularly whe oeratos such as multlcato ad exoetato are ecessary Keeg that md ad amg to create a reresetato that wll make ossble more effcet mlemetatos of arthmetc modulo, a ew reresetato for the elemets of GI() s troduced ths secto I the defto of GI(), the elemets were wrtte rectagular form ζ=(a+b) I what follows, a dfferet reresetato for the elemets of the multlcatve grou of GI() s roosed, whch allows to wrte them the form rε θ By aalogy wth the cotuum, such a form s sad to be olar Proosto 4: Let G A ad G B be subgrous, of the multlcatve grou G C of the ozero elemets of GI(), of orders N A =(-)/2 ad N B =2(+), resectvely The all ozero elemets of GI() ca be wrtte the form ζ = AB, where A G A ad B G B Proof: Sce G C s a cyclc grou ad both, N A ad N B, are dvsors of 2 -, the the subgrous G A ad G B of GI() do exst Now, the drect roduct (G A G B ) [8] has order 2 -, because, sce s of the form 4k+, the the greatest commo dvsor betwee N A ad N B satsfes GCD(N A,N B )=GCD(2k+,4(2k+2))=; e, (G A G B ) = lcm( G A, G B )=N A N B = 2 - Therefore, (G A G B ) s the multlcatve grou of GI(), ad every elemet of GF() ca be wrtte the form ζ=αβ, α G A, β G B Cosderg that ay elemet of a cyclc grou ca be wrtte as a tegral ower of a grou geerator, t s ossble 2 to set r=α ad ε θ =β, where ε s a geerator of G B The owers ε θ of ths elemet lay, some sese, the role of e θ over the comlex feld Thus, the olar reresetato assumes the desred form, ζ=rε θ Note that 2(+) lays the role of 2π I the above dscusso, t seems clear that r s gog to lay the role of the modulus of ζ Therefore, before further exlorg the olar otato, t s ecessary to defe formally the cocet of modulus of a elemet a fte feld Cosderg the ozero elemets of GF(), t s a well-kow fact that half of them are quadratc resdues of [9] The other half, those that do ot ossess square root, are the quadratc oresdues Lkewse, the feld R of real umbers, the elemets are dvded to ostve ad egatve umbers, whch are, resectvely, those that ossess ad do ot ossess a square root The stadard modulus oerato R always gves a ostve result By aalogy, the modulus oerato GF() s gog to be defed, such that t always results a quadratc resdue of Defto : The modulus of a elemet a GF(), where =4k+, s gve by 2 a, f a (mod a : = a, f a 2 (mod Proosto : The modulus of a elemet of GF() s a quadratc resdue of Proof: Sce =4k+, t mles that (-)/2=2k+, such that ( ( ) (mod By Euler's crtero [9], f ( a a ( (mod, the a s a quadratc resdue of ; f (mod, the a s a quadratc oresdue of ( Therefore, ( a ) ( )( ) (mod, ad t follows that a s a quadratc resdue of Defto 4: The modulus of a elemet a+b GI(), where 2 2 =4k+, s gve by a + b : = a + b The er modulus sg the above exresso s ecessary order to allow the comutato of the square root of the quadratc orm a 2 +b 2, ad the outer oe guaratees that such a oerato results oe value oly I the cotuum, such exresso reduces to the usual orm of a comlex umber, sce both, a 2 +b 2 ad the square root oerato, roduce oly ostve umbers At ths ot t s coveet to substtute the grous G A ad G B deomatos for oes that are more arorate to the olar reresetato Defto : The grou of modulus of GI(), deoted by G r, s the subgrou of order (-)/2 of GI() Defto : The grou of hases of GI(), deoted by G θ, s the subgrou of order 2(+) of GI() Proosto : If ζ=a+b=rε θ, where r G r ad ε θ G θ, the r= ζ Proof: Every elemet of G r has a order that dvdes (-)/2 ( Thus, f r G r, the r (mod, ad r =r Besdes that, as show ext secto, the elemets of the grou G θ are those a+b such that a 2 +b 2 ± (mod ) Therefore, accordg to defto, such elemets have modulus equal to, whch meas that ζ = rε θ = r ε θ =r=r

3 A exresso for the hase θ as a fucto of a ad b ca be foud by ormalsg the elemet ζ (that s, calculatg ζ/r=ε θ ), ad the solvg the dscrete logarthm roblem of ζ/r the base ε over GF() Thus, the coverso rectagular to olar form s ossble The verse oe s doe smly by the exoetato oerato From the above t ca be observed that the olar reresetato beg troduced s cosstet wth the usual olar form defed over the comlex umbers The modulus belogs to GF() (the modulus s a real umber) ad s a quadratc resdue (a ostve umber), ad the exoetal comoet ε θ has modulus oe ad belogs to GI() (e θ also has modulus oe ad belogs to the comlex feld) III THE Z PLANE To defe a Z trasform over a fte feld, t s ecessary frst to establsh what s meat by the comlex Z lae GF() I order to do so, t s ecessary to troduce a secal famly of fte grous Defto 7: The sura-umodular set of GI(), deoted G s, s the set of elemets ζ=(a+b) GI(), such that (a 2 +b 2 ) 2 (mod ) Proosto 7: If ζ=a+b, the ζ 2(+) (a 2 +b 2 ) 2 (mod ) Proof: ζ =(a+b) a + b (mod ), sce GI() s somorhc to GF( 2 ), a feld of characterstc Also, sce =4k+, -, so that ζ a-b (mod ), whch meas that ζ + (a+b)(a-b) (mod ) Therefore, ζ 2(+) (a 2 +b 2 ) 2 (mod ) Proosto 8: The structure <G S,*>, s a cyclc grou of order 2(+), called the sura-umodular grou of GI() Proof: G S s closed wth resect to *, sce that f (a+b) ad (c+d) belog to G S, e, f (a 2 + b 2 ) 2 (c 2 +d 2 ) 2 (mod ), the e+f=(a+b)*(c+d)=(ac-bd)+(ad+bc) So that (e 2 +f 2 ) 2 =(a 2 c 2-2abcd+b 2 d 2 +a 2 d 2 +2abcd+b 2 c 2 ) 2 =((a 2 +b 2 )*(c 2 +d 2 )) 2 =(a 2 +b 2 )*(c 2 +d 2 ) 2 (mod ), whch meas that (e+f) G S Now, G S s a closed subset of a cyclc grou (the multlcatve grou of GI()), therefore G S s a cyclc subgrou Besdes that, from roosto 2, ζ G S satsfes ζ 2(+) (mod ) Thus, ζ s oe of the 2(+)th roots of uty GI() There exsts 2(+) such roots e therefore G S has order 2(+) The elemets ζ=a+b of the sura-umodular grou G S satsfy (a 2 +b 2 ) 2 (mod ), e, a 2 +b 2 ± (mod ), ad all have modulus, as the umodular grou G However, G S has a larger order tha G It s mortat to observe that, due to the fact that a cyclc grou has oly oe subgrou of a gve order [8], G S s recsely the grou of hases G θ troduced defto The roblem of fdg a grou geerator for G S, s dealt wth roosto 9 below Proosto 9: If s a Mersee rme (=2 -, >2), the elemets ζ=a+b such that a 2 +b 2 - (mod ) are the geerators of G S Proof: Let ζ G S have order N Sce a 2 +b 2 - (mod ), N dvdes 2(+)=2 + However, ζ s ot umodular, so that N does ot dvde +=2 Therefore, N=2 + =2(+), ad ζ s a geerator of G S Examle 2: Let =, a Mersee rme, ad ζ=+ From 2 2 defto : r = + 7 (mod ), β of order N, such that N 2, ca be foud takg 2( + ) / N 4 / N β = ε = ε A geerator ε of the sura-umodular must be used to costruct the Z lae over GF() From the owers of ε those elemets that are o the ut crcle are obtaed By multlyg those by the members of the grou of modules, the remag elemets of GI(7) are detfed o other crcles o the Z lae Proosto : I the Z comlex lae over GF() there are 2(+) elemets each crcle Proof: There are as may crcles the Z lae as the order of the grou of modulus G r Therefore, deotg by m the umber of elemets of a gve modulus, t s ossble to wrte m G r = 2 - ad the result follows Examle : The Z lae over GF(7) Let =7, ad ζ=+4 From defto, 2 2 r = (mod 7 ), so that ε=ζ/r=+2 ad a 2 +b 2 = - (mod ) Therefore ε has order 2(+)=, so t s a geerator of the grou G S The Z lae over GF(7) s dected fgure 2 below The ozero elemets of GF(), amely ±,±2,±, are located o the horzotal axs, the rght or left sde, accordg f they are, resectvely, quadratc resdues or quadratc o-resdues of =7 (resectvely, ostve or egatve umbers, the usual comlex Z lae) There are three crcles, of radus,2 ad 4, corresodg to the (-)/2= elemets of the grou of modules G r A smlar stuato occurs for the elemets of GI(7) of the form b (corresodg to magary elemets the cotuous case) The elemets o the ut crcle corresod to the elemets of G S ad are obtaed as owers of ε The eve owers corresod to the elemets of G (a 2 +b 2 (mod 7)) ad the odd owers to the elemets satsfyg a 2 +b 2 - (mod 7) The remag 2 elemets of the Z lae are obtaed smly by multlyg those o the ut crcle by the modulus 2 ad 4 Observe that the elemets o the straght le y=±x over a fte feld also ossesses the usual terretato assocated to tgθ=± Table II relates the ots of fgure 2 wth ther resectve orders as elemets of GI(7) TABLE II THE NUMBER OF ELEMENTS OF A GIVEN ORDER IN THE Z PLANE OVER GF(7) so that ε=ζ/r=2+2 ad a 2 +b 2 = (mod ) Therefore ε has order 2(+)=4 (a geerator) A umodular elemet

4 N N N Fgure 4 Order traectory for ζ=+, a elemet of order N=24 of GI(7), o the Z Plae over GF(7) N 4 Fgure 2 The Z Plae over the Galos Feld GF(7) Whe calculatg the order of a gve elemet, the termedate results geerate a traectory o the Z lae, called the order traectory I artcular, If ζ has order N, the traectory goes through N dstct ots o the Z lae, movg a atter that deeds o N Secfcally, the order traectory touches o every crcle of the Z lae (there are G r of them), order of crescet modulus, always returg to the ut crcle If t starts o a gve radus, say R, t wll vst, couter-clockwse, every radus of the form R+kr, where r=( 2 -)/N ad k=,,2,,n- Examle 4: Table III lsts some elemets ζ GI(7) ad ther orders N Fgures - show the order traectores geerated by ζ TABLE III SOME ELEMENTS AND THEIR ORDERS IN GI(7) N ζ N N Fgure Order traectory for ζ=2, a elemet of order N=2 of GI(7), o the Z Plae over GF(7) N N Fgure Order traectory for ζ=+4, a elemet of order N=48 of GI(7), o the Z Plae over GF(7) IV INFINITE SERIES OVER FINITE FIELDS It s straghtforward to coceve fte sequeces whose elemets belog to a fte feld For stace, gve a sequece of tegers { x[ ]}, t s ossble to geerate a sequece over GF() smly by cosderg { x[ ] ( mod ) } I geeral, x[] may eve be a sequece of ratoal elemets: ay elemet r/s Q ca be maed over GF(), assumg values [r (mod )][s (mod )] - The focus here cocers erodc fte sequeces over a fte feld For stace, cosder the followg rght-sded GF(7)-valued sequeces: Examle : Let ) ) { } = { } = { x[ ]}, x[] GF() For stace: over GF(7), e, { } over GF(7), e, { } We woder whether a Z-trasform of ths sequece ca be de-? fed by = + X ( z ): x[ ]z, z GI() Is ths seres coverget? The ma terest here s the covergece of fte = seres, because mortat trasforms, such as Fourer ad Z trasforms, volve a sum of ftely may terms It s a well kow fact that the fte seres

5 + ( ) = dverges the classcal sese = However, Euler ad others otced that the arthmetc mea of the artal sums coverges to /2 The artal sums of ths seres are S =, S 2 =, S =, S 4 = ad the arthmetc mea σ : = S k forms a sequece (σ ) that coverges to /2 k = Whe a seres coverges the sese that the arthmetc mea of the artal sums coverges, t s sad to be Cesàro-summable (Eresto Cesàro (89-9)) Every coverget seres the usual sese s Cesàro-summable ad the seres sum ( the usual sese) s equal to the lmt of the sequece of the artal sums arthmetc mea That shows the Cesàro summablty cocet s useful, sce t ca make dverget seres summable A (ew) covergece crtero, sutable for seres over fte felds, whch s derved from the Cesàro summablty, s ow troduced Gve { x[ ]}, the artal sums S[] are defed accordg to: S[ ] : = x[ k ] k = Defto 8: The Cesàro sum over a fte feld s defed by σ : = S[ k ], where S[k] GF() are terreted as k = tegers If { x[ ]} s a erodc sequece over GF(), so s { S[ ]} Let P deote the erod of the latter sequece Therefore lm S[ k ] = k= lm / P P S[ k ] + S[ k ] k = k = ( mod P ) The secod term vashes ad lm σ lm / P P = S[ k ] k = lm But / P = P lm P so that S[ k ] = S[ k ] ad fally k = P k = lm σ P (mod S [ k ] (mod P (mod k = The heart of the matter s to take frst the lmt, ad the evaluate the result after reducg t modulo Defto 9: A seres over a fte feld s sad to be Cesàro coverget to σ f ad oly f lm σ σ : = (mod GF() Corollary: Every erodc seres over a fte feld, wth a ozero erod, that s, P (mod ), s Cesàro coverget Examle (Revsted): ) {S[k]}= { }, P= (mod 7) Therefore, σ = ( ) (mod 7 ) Let us ow retur to the seres [ + k k ] (mod 7 ) = + k : = [ ] (mod 7 ) = (mod 7 ) k If we wrte Z over GF(7) ad assume Z=, we fd k = Z +? k (mod 7 ) ( 2 ) (mod 7 ) Ths seres over GF(7) coverges exactly to the Cesàro sum σ! + = ) The seres x [ ] derved from { x [ ] } { } s ot k = coverget over GF(7) Remark that {S[k]}= { }, P (mod 7) ad σ s +? k udefed (ubouded) I ths case, Z = has Z= as k = Z a ole, so that + k dverges over GF(7) Now we are able to k = vestgate the followg: "Gve a erodc sequece { x [ ]} + over GF(), what s the rego of Cesàrocovergece o the Z-Fte-Feld lae for the seres + x [ ]Z, Z GF( 2 )?" = V THE FINITE FIELD Z-TRANSFORM V Basc Sequeces The Z trasform troduced ths aer deals wth sequeces x[], -<<, defed over the Galos feld GF(), whch are obtaed from basc tyes of sequeces: δ[], u[], A(a ) ) The fte feld mulse (Galos mulse), deoted δ[], s the, = sequece x[] defed by x[ ] = δ [ ] : =, As t haes wth real sequeces, ay fte feld sequece x[] ca be exressed as a sum of shfted ad scaled Galos mulses ) The ut ste over GF() (Galos ut ste) s gve by (mod, x [ ] = u[ ] : =, < ) The exoetal sequece s x[]=a(a), A ad a GF() Ths sequece s erodc wth erod P, whch s the multlcatve order of a (mod ) V2 The Z trasform Defto : The fte feld Z trasform (FFZT) of a sequece x[] over GF(), s the GI()-valued fucto = X ( Z ) : x[ ]Z, Z GI() = I the fte seres of the above defto, covergece s cosdered the sese of defto 9 For ay gve sequece, the set of values of Z GI() for whch X(Z) coverges s called the rego of covergece (ROC) A class of mortat ad useful fte feld Z trasform s that for whch X(Z) s a ratoal fucto P(Z)/Q(Z), where P(Z) ad Q(Z) are olyomals Z The roots of P(Z) ad Q(Z) are called the zeros ad oles of X(Z), resectvely For the Z lae over GF(), the ROC of X(Z) corresods to the elemets of GI() that are ot oles of X(Z) Examle : Rght-sded exoetal sequece over GF() Let x[]=a u[], a GF() ad u[] beg the Galos ut ste I ths case, sce x[] s o-zero oly for, X ( Z ) = a u[ ]Z = ( az ) = =

6 Comutg the artal sums: S = S 2 = + az - S = + az - + (az - ) 2 S N- = + az - + (az - ) 2 +(az - ) ++(az - ) N-2 S N = + az - + (az - ) 2 +(az - )++(az - ) N-2 +(az - ) N- Now, deotg by N the multlcatve order of (az - ), the erod of the sequece S[k] s recsely N Therefore, t s ossble N N ) to wrte σ = ( N )( az N =, whch s equal to N ( az ) N σ N = ( az ) Sce (az - ) has multlcatve order N, σ az N = Z N N = a Z, whch s the same N = Z N d Z N d as σ N = ( a Z ) dz, or σ N = ( ( az ) ) dz N = N = After some drect maulato, ths s seem to be equal to σ N =, whch s the desred Z trasform X(Z) The az oly ole of X(Z) s Z=a, so that the ROC s Z a The ROC the fte feld case has a dfferet structure from the ROC of the usual Z-trasform ad, besdes that, there exsts may regos of covergece for a gve aalytcal exresso for X(Z) Evaluatg the FFZT o G s results the dscrete-tme Fourer trasform over GF() Defto : The fte feld dscrete-tme Fourer trasform of a sequece x[] over GF() s the GI()-valued fucto X ( θ + θ ε ) : = x[ ] ε, = where ε G s has multlcatve order 2(+) The above fte feld Fourer trasform s Cesàro-coverget f the ROC of the FFZT X(Z) cludes G s It should be comared wth the revous defto by Pollard [2] V The verse FFZT Lemma : 2 Z = ( ) δ, Z GI ( Proof: For = the summato s clearly equal to 2 - From roosto 4 Z ( r) ( ) Z GI ( r Gr = θ ε θ Gθ But 2( + ) ε θ ε =, ad therefore ε θ G = θ 2( + ) = G G δ Z r θ, Z GI( Theorem : The verse fte feld Z trasform (IFFZT) s x[ ] = X( Z )Z 2 Z GI( Proof: By defto, = X ( Z ) x[ k ]Z k Multlyg k = both sdes by Z ad summg over Z GI(), oe obtas X( Z )Z = ( Z GI( Z GI( k = x[ k ]Z Iterchagg the order of summato, the rght-had sde (RHS) becomes RHS k = x[ k ]( Z k = Z GI( but the er summato, from lemma, s ozero oly for k=, so that RHS = ( 2 )x[ ] ad the result follows It s terestg to observe that, although the drect trasform s a fte summato, defed so to deal wth fte sequeces, the IFFZT requres oly a fte sum over the feld GI() The Z trasform troduced ths aer satsfes most roertes of the usual Z trasform defed over the comlex feld, such as learty, tme-shft, multlcato by exoetal ad so o VI CONCLUSIONS Ths aer dscusses the roblem of costructg a Fte Feld Z Trasform, caable of hadlg fte sequeces over the Galos Feld GF() To deal wth such sequeces, the cocet of Cesàro covergece was used ad t was show that erodc sequeces over GF() coverge to the arthmetc mea of the Cesàro sums of the sequece The Z lae over GF() was costructed ad a Fte Feld Z Trasform (FFZT) was defed A geeral verso formula for the FFZT was determed The fte feld dscrete-tme Fourer trasform was also troduced, whch corresods to the FFZT evaluated o the suramodular grou (the ut crcle) The trasforms troduced here are also able of hadlg fte sequeces ad therefore they are a useful tool to deal wth both FIR ad IIR flters defed over fte algebracal structures REFERENCES [] A V Oehem, R W Schafer e J R Buck, Dscrete-Tme Sgal Processg, Pretce-Hall 999 [2] J M Pollard, The Fast Fourer Trasform a Fte Feld, Math Comut, vol 2, No 4, -74, Ar 97 [] R M Camello de Souza, H M de Olvera ad A N Kauffma, Trgoometry Fte Felds ad a New Hartley Trasform, Proc of the IEEE It Sym o Ifo Theory, 29, Cambrdge, MA, Aug 998 [4] H M de Olvera, R M Camello de Souza ad A N Kauffma, Effcet Multlex for Bad-Lmted Chaels: Galos-Feld Dvso Multle Access, Proceedgs of the 999 Worksho o Codg ad Crytograhy - WCC '99, 2-24, Pars, Ja 999 [] H M de Olvera, R M Camello de Souza, Orthogoal Multlevel Sreadg Sequece Desg, Codg, Commucatos ad Broadcastg, 29-, Eds P Farrell, M Darell ad B Hoary, Research Studes Press / Joh Wley, 2 [] T Cooklev, A Nshhara ad M Sablatash, Theory of Flter baks over Fte Felds, IEEE Asa Pacfc Coferece o Crcuts ad Systems, APCCAS'94, 2-2, 994 [7] R E Blahut, Fast Algorthms for Dgtal Sgal Processg, Addso Wesley, 98 [8] H S Stoe, Dscrete Mathematcal Structures ad ther Alcatos, Scece Research Assocates, 97 [9] DM Burto, Elemetary Number Theory, Ally ad Baco, 97 ), )Z