THE HARTLEY TRANSFORM IN A FINITE FIELD

Transcription

1 THE HARTLEY TRANSFORM IN A FINITE FIELD R. M. Campello de Souza H. M. de Olvera A. N. Kauffman CODEC - Grupo de Pesusas em Comuncações Departamento de Eletrônca e Sstemas - CTG - UFPE C.P Recfe - PE Brasl Phone: fax: e-mal: Rcardo@npd.ufpe.br HMO@npd.ufpe.br Abstract In ths paper the k-trgonometrc functons over the Galos Feld GF() are ntroduced and ther man propertes derved. Ths leads to the defnton of the cas k (.) functon over GF() whch n turn leads to a fnte feld Hartley Transform. The man propertes of ths new dscrete transform are presented and areas for possble applcatons are mentoned.. Introducton Dscrete transforms play a very mportant role n engneerng. A sgnfcant example s the well known Dscrete Fourer Transform (DFT) whch has found many applcatons n several areas specally n Electrcal Engneerng. A DFT for fnte felds was ntroduced by Pollard n 97 [] and appled as a tool to perform dscrete convolutons usng nteger arthmetc. Snce then several new applcatons of the Fnte Feld Fourer Transform (FFFT) have been found not only n the felds of dgtal sgnal and mage processng [-5] but also n dfferent contexts such as error control codng and cryptography [6-8]. A second relevant example concerns the Dscrete Hartley Transform (DHT) [9] the dscrete verson of the ntegral transform ntroduced by R. V. L. Hartley n []. Although seen ntally as a tool wth applcatons only on the numercal sde and havng connectons to the physcal world only va the Fourer transform the DHT has proven over the years to be a very useful nstrument wth many nterestng applcatons [-3]. In ths paper the DHT over a fnte feld s ntroduced. In order to obtan a transform that holds some resemblance wth the DHT t s necessary frstly to establsh the euvalent of the snusodal functons cos and sn over a fnte structure. Thus n the next secton the k-trgonometrc functons cos k and sn k are defned from whch the cas k (cosne and sne) functon s obtaned and used n secton 3 to ntroduce a symmetrcal dscrete transform par the fnte feld Hartley transform or FFHT for short. A number of propertes of the FFHT s presented ncludng the cyclc convoluton property and Parseval s relaton. In secton 4 the condton for vald spectra smlar to the conjugacy constrants for the Fnte Feld Fourer Transform s gven. Secton 5 contans a few concludng remarks and some possble areas of applcatons for the deas ntroduced n the paper. The FFHT presented here s dfferent from an earler proposed Hartley Transform n fnte felds [4] and appears to be the more natural one.. k-trgonometrc Functons The set G() of gaussan ntegers over GF() defned below plays an mportant role n the deas ntroduced n ths paper (hereafter the symbol := denotes eual by defnton). Defnton : G() := {a + jb a b GF()} = p r r beng a postve nteger p beng an odd prme for whch j = - s a uadratc non-resdue n GF() s the set of gaussan ntegers over GF(). Let denote the cartesan product. It can be shown as ndcated below that the set G() together wth the operatons and defned below s a feld. Proposton : Let : G() G() G() (a + jb a + jb ) (a + jb ) (a + jb ) = = (a + a ) + j(b + b ) and : G() G() G()

2 (a + jb a + jb ) (a + jb ) (a + jb ) = = (a a - b b ) + j(a b + a b. The structure GI() := < G() > s a feld. In fact GI() s somorphc to GF( ). Trgonometrc functons over the elements of a Galos feld can be defned as follows. Defnton : Let α have multplcatve order N n GF() = p r p. The GI()-valued k-trgonometrc functons of (α ) n GF() (by analogy the trgonometrc functons of k tmes the angle of the complex exponental α ) are defned as and for k =... N-. cos k ( α ) := (α k + α -k ) sn k ( α ) := (α k - α -k ) j For smplcty suppose α to be fxed. We wrte cos k ( α ) as cos k ( ) and sn k ( α ) as sn k ( ). The k- trgonometrc functons satsfy propertes P-P8 below. Proofs are straghtforward and are omtted here. P. Unt Crcle: sn k ( ) + cos k ( ) =. P. Even / Odd: cos k ( ) = cos k ( - ) sn k ( ) = - sn k ( - ). P3. Euler Formula : α k = cos k ( ) + jsn k ( ). P4. Addton of Arcs : cos k ( + t) = cos k ( )cos k ( t ) - sn k ( )sn k ( t ) sn k ( + t) = sn k ( )cos k ( t ) + sn k ( t )cos k ( ). P5. Double Arc: cos k () = + cos k ( ) sn k ( ) = cos k ( ) P6. Symmetry : cos k ( ) = cos ( k ) sn k ( ) = sn ( k ). P7. cos k ( ) Summaton: cos k ( ) = N =. P8. sn k ( ) Summaton: sn k ( ) =.

3 A smple example s gven to llustrate the behavor of such functons. Example - Let α = 3 a prmtve element of GF(7). The cos k () and sn k () functons take the followng values n GF(7): cos k () sn k () () (k) () j j 6j 6j j 6j j 6j 3 4 6j j 6j j 5 6j 6j j j (k) Table Dscrete cosne and sne functons over GF(7). The k-trgonometrc functons have nterestng orthogonalty propertes such as the one shown n lemma. Lemma : The k-trgonometrc functons cos k (.) and sn k (.) are orthogonal n the sense that A:= k- [cos k ( α ) sn k ( α t )] = where α s an element of multplcatve order N n GF(). Proof: By defnton N A = [ (αk + α -k ) j (αtk - α -tk )] = = 4 j k N = ( α k(+ t) - α -k(+ t) + α k(t-) - α k(-t) ). Now If = t then A = ( + + N - N ) / 4j =. If = -t then A = (N - N + - ) / 4j =. Otherwse A= ( )/4j =. A general orthogonalty condton whch leads to a new Hartley Transform s now presented va the cas k ( α ) functon. The notaton used here follows closely the orgnal one ntroduced n []. Defnton 3: Let α GF() α. Then cas k ( α ) := cos k ( α ) + sn k ( α ).

4 The set {cas k (.)}... N- can be vewed as a set of seuences that satsfy the followng orthogonalty property: Theorem : H := cas k ( α ) cas k ( α t ) = N = t t where α has multplcatve order N. Proof: From defnton 3 t follows that H = [cos k ( )cos k ( t ) + sn k ( )sn k ( t ) + +sn k ( )cos k ( t ) + sn k ( t )cos k ( )] whch by lemma s the same as H = cos k ( )cos k ( t ) + sn k ( )sn k ( t ) then t follows from property P4 that H = cos k ( - t ) and from P9 the result follows. 3. The Fnte Feld Hartley Transform Defnton 4: Let v = (v v... v N- ) be a vector of length N wth components over GF() = p r p. The Fnte Feld Hartley Transform (FFHT) of v s the vector V = (V V... V N- ) of components V k GI( m ) gven by V k := = v cas k ( α ) where α s a specfed element of multplcatve order N n GF( m ). Such a defnton clearly mmcs the classcal defnton of the Dscrete Hartley Transform [9]. The nverse FFHT s gven by the followng theorem. Theorem : The N-dmensonal vector v can be recovered from ts Hartley dscrete spectrum V accordng to v = N(mod p) V kcas k ( α ). Proof: After substtutng the V k as defned above n the expresson for v t follows that v = N(mod ) = p k r= v r cas k ( α r )cas k ( α ). Changng the order of summaton

5 N(mod p) v r r= N cas k ( α r )cas k ( α ) = = N(mod p) v r r= N = r r = v A sgnal v and ts dscrete Hartley spectrum V are sad to form a fnte feld Hartley Transform par denoted by v V. It s worthwhle to menton that the FFHT belongs to a class of dscrete transforms for whch the kernel of the drect and the nverse transform s exactly the same. Lettng now g = {g } G = {G k } and v = {v } V= {V k } denote FFHT pars of length N the followng set of useful propertes can be derved. H - Lnearty H - Tme Shft ag + bv ag + bv ab GF(). H3 - Sum of Seuence (dc term) If v = g -d then V k = cos k (d)g k + sn k (d)g -k. H4 - Intal Value V = = v. Vk = v cas N - k ( α ) = = V. N - k v = N(mod p) V k. H5 - Symmetry G Ng. H6 - Tme Reversal g - G -k. H7 - Cyclc Convoluton: If denotes cyclc convoluton then g v ½(GV + GV - + G - V - G - V - ). where G - and V - denotes respectvely the seuences {G N-k } and {V N-k }. H8 - Parseval s Relaton N N- N- g = G = k 4. Vald Spectra The followng lemma states a relaton that must be satsfed by the components of the spectrum V for t to be a vald fnte feld Hartley spectrum that s a spectrum of a sgnal v wth GF()-valued components.

6 Lemma : The vector V= {V k } V k GI( m ) s the spectrum of a sgnal v = {v } v GF() = p r f and only f k V = V N-k where ndexes are consdered modulo N k =... N- and N ( m - ). Proof: From the FFHT defnton and consderng that GF(p r ) has characterstc p t follows that Vk = v cask ( α )) = ( v cas ( α )). ( = = k If v GF() then v = v. The fact that j = - GF() f and only f s a prme power of the form 4s + 3 mples that j = -j. Hence On the other hand suppose V k = V N k. Then = v cas N - k ( α ) = = v cas N - k ( α ). Now let N-k = r. Snce GCD( m - ) = k and r ranges over the same values whch mples = v cas ( α ) = r = v cas ( α ) r r =... N-. By the unueness of the FFHT v = v so that v GF() and the proof s complete. Example - Wth = p = 3 r = m = 5 and GF(3 5 ) generated by the prmtve polynomal f(x) = x 5 + x 4 + x + a FFHT of length N = may be defned by takng an element or order (α s such an element). The vectors v and V gven below are an FFHT par. v = ( etc etc...) V = ( etc etc etc...) The relaton for vald spectra shown above mples that only two components V k are necessary to completely specfy the vector V namely V and V. Ths can be verfed smply by calculatng the cyclotomc classes nduced by lemma whch n ths case are C = () and C = ( ). 5. Conclusons In ths paper trgonometry for fnte felds was ntroduced. In partcular the k-trgonometrc functons of the angle of the complex exponental α were defned and ther basc propertes derved. From the cos k ( α ) and sn k ( α ) functons the cas k ( α ) functon was defned and used to ntroduce a new Hartley Transform the Fnte Feld Hartley Transform (FFHT).

7 The FFHT seems to have nterestng applcatons n a number of areas. Specfcally ts use n Dgtal Sgnal Processng along the lnes of the so-called number theoretc transforms (e.g. Mersenne transforms) should be nvestgated. In the feld of error control codes the FFHT mght be used to produce a transform doman descrpton of the feld therefore provdng possbly an alternatve to the approach ntroduced n [6]. Dgtal Multplexng s another area that mght beneft from the new Hartley Transform ntroduced n ths paper. In partcular new schemes of effcent-bandwdth code-dvson-multple-access for band-lmted channels based on the FFHT are currently under development. Acknowledgements The authors wsh to thank Prof. James Massey for hs suggestons and nsghtful comments whch mproved the fnal verson of ths paper. References [] J. M. Pollard The Fast Fourer Transform n a Fnte Feld Math. Comput. vol. 5 No. 4 pp Apr. 97. [] C. M. Rader Dscrete Convoluton va Mersenne Transforms IEEE Trans. Comput. vol. C- pp Dec. 97. [3] I. S. Reed and T. K. Truong The Use of Fnte Feld to Compute Convolutons IEEE Trans. Inform. Theory vol. IT- pp. 8-3 Mar [4] R. C. Agarwal and C. S. Burrus Number Theoretc Transforms to Implement Fast Dgtal Convoluton IEEE Proc. vol. 63 pp Apr [5] I. S. Reed T. K. Truong V. S. Kwoh and E. L. Hall Image Processng by Transforms over a Fnte Feld IEEE Trans. Comput. vol. C-6 pp Sep [6] R. E. Blahut Transform Technues for Error-Control Codes IBM J. Res. Dev. vol. 3 pp May 979. [7] R. M. Campello de Souza and P. G. Farrell Fnte Feld Transforms and Symmetry Groups Dscrete Mathematcs vol. 56 pp [8] J. L. Massey The Dscrete Fourer Transform n Codng and Cryptography accepted for presentaton at the 998 IEEE Inform. Theory Workshop ITW 98 San Dego CA Feb 9-. [9] R. N. Bracewell The Dscrete Hartley Transform J. Opt. Soc. Amer. vol. 73 pp Dec [] R. V. L. Hartley A More Symmetrcal Fourer Analyss Appled to Transmsson Problems Proc. IRE vol. 3 pp Mar. 94. [] R. N. Bracewell The Hartley Transform Oxford Unversty Press 986. [] J.-L. Wu and J. Shu Dscrete Hartley Transform n Error Control Codng IEEE Trans. Acoust. Speech Sgnal Processng vol. ASSP-39 pp Oct. 99. [3] R. N. Bracewell Aspects of the Hartley Transform IEEE Proc. vol. 8 pp Mar [4] J. Hong and M. Vetterl Hartley Transforms Over Fnte Felds IEEE Trans. Inform. Theory vol. IT- 39 pp Sep. 993.