The Complex Finite Field Hartley Transform

Transcription

1 The Complex Fnte Feld Hartley Transform R. M. Campello de Souza, H. M. de Olvera, A. N. Kauffman CODEC - Communcatons Research Group Departamento de Eletrônca e Sstemas - CTG - UFPE C.P. 78, , Recfe - PE, Brasl E-mal: Rcardo@npd.ufpe.br, hmo@npd.ufpe.br, ANK@nlnk.com.br ABSTRACT Dscrete transforms, defned over fnte or nfnte felds, play a very mportant role n Engneerng. In ether case, the successful applcaton of transform technues s manly due to the exstence of the so-called fast transform algorthms. In ths paper, the complex fnte feld Hartley transform s ntroduced and a fast algorthm for computng t s suggested. 1. INTRODUCTION Dscrete transforms are a very mportant tool and play a sgnfcant role n Engneerng. A partcularly strkng example s the well known Dscrete Fourer Transform (DFT), whch has found many applcatons n several areas, specally n the feld of Electrcal Engneerng. A DFT over Galos felds was also defned [1] and appled as a tool to perform dscrete convolutons usng nteger arthmetc. Snce then several new and nterestng 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]. In both cases, nfnte and fnte, the exstence of fast algorthms (FFT) for computng the DFT has been a decsve factor for ts real-tme applcatons. Another nterestng example s the Dscrete Hartley Transform (DHT) [9], the dscrete verson of the symmetrcal, Fourer-lke, ntegral transform ntroduced by R. V. L. Hartley n 194 [1]. Although seen ntally manly 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 [11-13]. Fast Hartley transforms also do exst and play an mportant role n the use of the DHT. Recently, a new Hartley transform over fnte felds (FFHT) was ntroduced [14] whch has nterestng applcatons n the feld of dgtal multplexng [15]. However, the FFHT has the restrcton that t does not allow blocklengths that are a power of two. In ths paper, the complex fnte feld Hartley transform (CFFHT) s ntroduced. Thus, n the next secton a trgonometry for gaussan ntegers over a Galos feld s ntroduced. In secton 3 the cosne and sne (cas) functon over a fnte feld s ntroduced and some orthogonalty relatons are derved. In secton 4, the complex fnte feld Hartley transform (CFFHT) s defned usng a Galos feld gaussan nteger argument for the transform kernel whch removes the blocklength 67

2 restrcton of the FFHT. Secton 5 examnes the condton for vald spectra whch leads to results that are smlar to the conjugacy constrants for the FFFT. An effcent algorthm for computng the CFFHT s presented n secton 6. The paper closes wth a few concludng remarks and suggestons of some possble areas of applcatons.. A TRIGONOMETRY FOR GAUSSIAN INTEGERS OVER A GALOIS FIELD 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 1: G() := {a + jb, a, b GF()}, = p r, r beng a postve nteger, p beng an odd prme for whch j = -1 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 1: Let : G() G() G() (a 1 + jb 1, a + jb ) (a 1 + jb 1 ) (a + jb ) = = (a 1 + a ) + j(b 1 + b ) and : G() G() G() (a 1 + jb 1, a + jb ) (a 1 + jb 1 ) (a + jb ) = = (a 1 a - b 1 b ) + j(a 1 b + a b 1 ). The structure GI() := < G(),, > s a feld. In fact, GI() s somorphc to GF( ). In what follows ζ denotes an element of multplcatve order N n GI(), the set of gaussan ntegers over GF(), = p r, p an odd prme such that p 3 (mod 4). Trgonometrc functons over the elements of a Galos feld can be defned as follows. Defnton : Let ζ be na element of multplcatve order N n GI(), = p r, p. The GI()-valued k-trgonometrc functons of (ζ ) n GI() (by analogy, the trgonometrc functons of k tmes the angle of the complex exponental ζ ) are defned as cos k ( ζ 1 ) := ( ζ k + ζ -k ) and sn k ( ζ 1 ) := (ζ k - ζ -k ), j for, k =, 1,...,. For smplcty ζ s supposed to be fxed. We wrte cos k ( ζ ) as cos k (). The trgonometrc functons above ntroduced satsfy propertes P1-P1 below. P1. Unt Crcle: sn k ( ) + cos k ( ) = 1. Proof: sn k ( ) + cos k ( )= [ 1 (ζ k - ζ -k )] + [ 1 (ζ k + ζ -k )] = j = 1 4 (ζk - ζ -k -) + 1 (ζ k + ζ -k +) =

3 P. Even / Odd: cos k ( ) = cos k ( - ) sn k ( ) = - sn k ( - ). Proof. cos k (- )= 1 (ζ-k + ζ k ) = cos k ( ); sn k ( - ) = 1 (ζ -k - ζ k ) = -sn k ( ). j P3. Euler Formula : ζ k = cos k ( ) + jsn k ( ). Proof. cos k ( ) + jsn k ( ) = 1 (ζ-k + ζ k ) + 1 (ζk - ζ -k ) = ζ 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 ( ). Proof. Clearly cos k ( + t ) = 1 (ζ(+t)k + ζ -(+t)k ) = 1 (ζ)k ζ t)k + ζ -)k ζ -t)k ) = = 1 {[cos k ()+jsn k ()][cos k (t)+jsn k (t)]+[cos k ()-jsn k ()][cos k (t)-jsn k (t)]} = cos k ()cos k (t)-sen k ()sen k (t). The proof for the sn(.) functon s smlar. P5. Double arc: cos k () 1 + cos = k ( ) ; sn k ( ) = 1 cos k ( ) Proof. Accordng to P4, cos k ( ) = cos k ( ) - sn k ( ) = cos k ( ) - [1 - cos k ( )] = cos k ( ) - 1. The proof for the sn(.) functon s smlar. P6. Symmetry: cos k ( ) = cos ( k ) sn k ( ) = sn ( k ). Proof. Follows drectly from defnton. P7. Perodcty: cos k ( + N) = cos k ( ) e sn k ( + N) = sn k ( ). Proof. cos k ( + N) = 1 (ζ(k+n) + ζ -(k+n) ) = 1 (ζk ζ N + ζ -k ζ N ) = cos k ( ), snce the order of ζ s N. P8. Complement: cos k ( ) = cos k ( t ) where tk and +t = N sn k ( ) = -sn k ( t ) where tk and +t = N. Proof. [cos k ( )-cos k ( t )] = (ζ k +ζ -k -ζ tk -ζ -tk ) ζ kt kt = ζ ζ-k -ζ tk = (ζ -k -ζ tk ) ζ ζ P9. cos k ( ) summaton: N, = cos k ( ) =., -kt -kt =. 69

4 Proof. Let σ := σ = 1 [ 1 ( ζ N ) 1 cos k ( ) = ( ζ N ) 1 ζ 1 ζ 1 P1. sn k ( ) summaton: sn k ( ) =. Proof. Let σ:= = 1 j [ 1 ( ζ N ) 1 sn k ( ) = 1 j (ζ k + ζ -k ). If = then σ = N. Otherwse ] = 1 [ + ] =. (ζ k - ζ -k ). If = then σ =. Otherwse σ - 1 ( ζ N ) 1 ] = 1 [ - ] =. ζ 1 ζ 1 j A smple example s gven to llustrate the behavour of such functons. Example 1: Let ζ = j, an element of order 4 n GI(3). The cos k () and sn k () functons take the followng values n GI(3): Table 1 Dscrete cosne and sne functons over GI(3). cos k ( ) sn k ( ) 1 3 ( ) 1 3 ( ) ( k ) ( k ) 3. ORTHOGONALITY RELATIONS The trgonometrc functons (defnton ) have nterestng orthogonalty propertes, such as the one shown n lemma 1. Lemma 1: The k-trgonometrc functons cos k (.) and sn k (.) are orthogonal n the sense that [cos k ( ζ ) sn k ( ζ t )] =, where ζ s an element of multplcatve order N n GI(). Proof. From P4, we have cos k ( ζ ) sn k ( ζ t ) 1 = [sn k (t + ) + sn k (t - )], and therefore [cos k ( ζ ) sn k ( ζ t )] = 1 k N Then, from P1, the result follows. 1 = [sn k (t + ) + sn k (t - )]. 7

5 A general orthogonalty condton, whch leads to a new Hartley Transform, s now presented va the cas k ( ζ ) functon. The notaton used here follows closely the orgnal one ntroduced n [1]. Defnton 3: Let ζ GI(), ζ. Then cas k ( ζ ) := cos k ( ζ ) + sn k ( ζ ). The cas k (.) functon satsfes propertes C1-C5 below: C1. Addton of Arcs: ) cas k ( + t) = cos k ( ) cas k ( t ) + sn k ( ) cas k ( -t ). ) cas k ( - t) = cos k ( ) cas k ( -t ) + sn k ( ) cas k ( t ). Proof: ) By defnton cas k ( + t) = cos k ( + t) + sn k ( + t), so that from P and P4, cas k ( + t) = cos k ( )cos k ( t ) - sn k ( )sn k ( t ) + sn k ( )cos k ( t ) + sn k ( t )cos k ( ) = = cos k ( )[cos k ( t ) + sn k ( t )] + sn k ( )[cos k ( -t ) + sn k ( -t )] = cos k ( )cas k ( t ) + sn k ( )cas k ( -t ). The proof for () s smlar. C. Product: cas k ( ) cas k ( t ) = cos k ( - t ) + sn k ( + t ) Proof: cas k ( ) cas k ( t ) = [cos k ( )+sn k ( )][cos k ( t )+sn k ( t )] = cos k ( )cos k ( t )+ +sn k ( )sn k ( t ) + sn k ( )cos k ( t ) + sn k ( t )cos k ( ), and, from P and P4, the result follows. C3. Symmetry: cas k ( ) = cas ( k ) Proof: Drect from P6. C4. Quadratc Norm: Let cas( ζ ), wth argument ζ = α GF(). Then [cas k ( )] +1 = cas k ( ) = cos k ( ). Proof: Wth cas k ( ) = a+jb, then (cas k ( )) = a + j b = a jb. Therefore [cas k ( )] +1 = cas k ( ) = [cos k ( )] - [sen k ( )] = cos k ( ) (P1 and P5). C5. Perodcty: cas k ( +N ) = cas k ( ). Proof: Drect from P11. The set {cas k (.)}, 1,...,, can be vewed as a set of seuences that satsfy the followng orthogonalty property: Theorem 1. order N. Proof. cas k ( ζ )cas k ( ζ t ) = From C t follows that [cas k ( ζ )cas k ( ζ t )] = Now, usng P1, we obtan [cas k ( ζ )cas k ( ζ t )] = N, = t, t [cos k ( - t) + sn k ( + t)]., where ζ has multplcatve [cos k ( - t)], and, from P9, the result follows. 4. THE COMPLEX FINITE FIELD HARTLEY TRANSFORM Let v = (v, v 1,..., v ) be a vector of length N wth components over GF(). The Complex Fnte Feld Hartley Transform (CFFHT) of v s the vector V = (V, V 1,..., V ) of components V k GI( m ), gven by 71

6 V k := v cas k ( ζ ) where ζ s a specfed element of multplcatve order N n GI( m ). Such a defnton extends the defnton of the Fnte Feld Hartley Transform. A sgnal v and ts dscrete Hartley spectrum V are sad to form a complex fnte feld Hartley transform par, denoted by V = H (v) or v V. The nverse CFFHT s gven by the followng theorem. Theorem : The N-dmensonal vector v can be recovered from ts spectrum V accordng to N 1 v = 1 V k cas k ( ζ ). N(mod p) Proof. N v = 1 1 v r cas k ( ζ r )cas k ( ζ ), N(mod p) r= nterchangng the summatons N v = 1 1 v r cas k ( ζ r )cas k ( ζ ), N(mod p) r= whch, by theorem 1, s the same as N v = 1 1 N, = r v r N(mod p) r=, r = v. Therefore, as t happens n the contnuous Hartley Transform, the CFFHT s symmetrcal n the sense that t uses the same kernel for the drect and nverse transforms. 5. CONJUGACY CONSTRAINTS Theorem 3 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. Theorem 3: The vector V = {V k }, V k GI( m ), s the spectrum of a sgnal v = {v }, v GF(), f and only f V k = VN-k, where ndexes are consdered modulo N,, k =, 1,..., and N ( m - 1). Proof: From the CFFHT defnton and consderng that GF(), = p r, has characterstc p, t follows that Vk = ( vcask ()) = ( v cask ()) If v GF(), then v = v. The fact that j = -1 GF() f and only f s a prme power of the form 4s + 3, mples that j = -j. Hence, 7

7 V k = vcas N-k () = VN-k. On the other hand, suppose V k = VN-k. Then v cas N-k () = vcas N-k () Now, let N-k = r. Snce GCD( m -1, ) = 1, k and r ranges over the same values, whch mples v casr () = vcasr () r =, 1,...,. By the unueness of the CFFHT, v = v so that v GF() and the proof s complete. The cyclotomc coset partton nduced by ths relaton s such that an element and ts recprocal modulo N belongs to the same class, whch mples that the number of CFFHT components that need to be computed to completely specfy the spectrum V s approxmately half of the number needed for the Fnte Feld Fourer Transform. Example - Wth = p = 3, r = 1, m = 5 and GF(3 5 ) generated by the prmtve polynomal f(x) = x 5 + x 4 + x + 1, a FFHT of length N = 11 may be defned by takng an element or order 11 (α 198 s such an element). The vectors v and V gven below are an FFHT par. v = (, 1,, 1, 1,,,,, 1, 1 ) V = (, α 15 + jα 46, α 41 + jα 51, α jα 138, α 33 + jα 96, α 39 + jα 3, α 39 + jα 153, α 33 + jα 17, α jα 17, α 41 + jα 17, α 15 + jα 167 ). 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 1. Ths can be verfed smply by calculatng the cyclotomc classes nduced by lemma 1 whch, n ths case, are C = () and C 1 = (1, 8, 9, 6, 4, 1, 3,, 5, 7). 6. COMPUTING THE CFFHT A well known transform defned over fnte felds s the Fnte Feld Fourer Transform (FFFT)[1]. Let v = (v, v 1,..., v ) be a vector of length N wth components over GF() GI(), = p r. The FFFT of v s the vector F = (F, F 1,..., F ) of components F k GF( m ) GI( m ), gven by F k := 73 v α k. where α s a specfed element of multplcatve order N n GF( m ). There s a close relaton between the FFFT and the FFHT, as t s shown n proposton. Proposton - Let v = {v } V= {V k } and v = {v } F = {F k } denote, respectvely, a CFFHT and an FFFT par. Then

8 V k = 1 [(Fk + F N-k ) + j(f N-k - F k )] = F e + jf o where F e and F o denote the even and odd parts of F respectvely. Proof: so that F N-k = 1 [(Fk + F N-k ) + j(f N-k - F k )] = = v α (N-k) = v α -k, v cos k ( α ) + v cas k ( α ) = V k v sn k ( α ) = Based on ths result an effcent scheme can be devsed to compute V as shown below. It s necessary only to compute the FFFT of v, whch can be done va a Fast Fourer Transform algorthm. Even(.) Re(V) v FFT Odd(.) Im(V) Fg. 1- Computng the CFFHT The exstence of fast algorthms (FFT) for computng the CFFHT s a decsve factor for ts real-tme applcatons such as dgtal multplexng, whch makes t attractve for DSP mplementatons. 7. CONCLUSIONS In ths paper, a trgonometry for gaussan ntegers over a Galos feld was ntroduced. In partcular, the k-trgonometrc functons of the angle of the complex exponental ζ were defned and some of ther basc propertes derved. From the cos k ( ζ ) and sn k ( ζ ) functons, the cas k ( ζ ) (cosne and sne) functon was defned. It was then shown that the set {cas k (.)}, 1,...,, can be vewed as a set of seuences that satsfy an orthogonalty relaton, whch n turn was used to ntroduce a new Hartley Transform, the Complex Fnte Feld Hartley Transform (CFFHT). Two mportant relatons that have mplcatons as far as the computaton of the CFFHT were establshed. Frstly t was shown that the CFFHT components satsfy 74

9 the so-called conjugacy constrants, whch mples that only the leaders of the conjugacy classes need to be computed. Secondly, a smple relaton between the CFFHT and the Fnte Feld Fourer Transform was establshed, whch meant that FFT type algorthms can be used to compute the CFFHT. The CFFHT 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 CFFHT 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. REFERENCES [1] J. M. Pollard, The Fast Fourer Transform n a Fnte Feld, Math. Comput., vol. 5, No. 114, pp , Apr [] C. M. Rader, Dscrete Convoluton va Mersenne Transforms, IEEE Trans. Comput., vol. C-1, pp , Dec [3] I. S. Reed and T. K. Truong, The Use of Fnte Feld to Compute Convolutons, IEEE Trans. Inform. Theory, vol. IT-1, pp. 8-13, 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 [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 1998 IEEE Inform. Theory Workshop, ITW 98, San Dego, CA, Feb [9] R. N. Bracewell, The Dscrete Hartley Transform, J. Opt. Soc. Amer., vol. 73, pp , Dec [1] R. V. L. Hartley, A More Symmetrcal Fourer Analyss Appled to Transmsson Problems, Proc. IRE, vol. 3, pp , Mar [11] R. N. Bracewell, The Hartley Transform, Oxford Unversty Press, [1] J.-L. Wu and J. Shu, Dscrete Hartley Transform n Error Control Codng, IEEE Trans. Acoust., Speech, Sgnal Processng, vol. ASSP-39, pp , Oct [13] R. N. Bracewell, Aspects of the Hartley Transform, IEEE Proc., vol. 8, pp , Mar

10 [14] R. M. Campello de Souza, H. M. de Olvera and A. N. Kauffman, Trgonometry n Fnte Felds and a New Hartley Transform, Proceedngs of the 1998 Internatonal Symposum on Informaton Theory, p. 93, Cambrdge, MA, Aug [15] H. M. de Olvera, R. M. Campello de Souza and A. N. Kauffman, Effcent Multplex for Band-Lmted Channels, Proceedngs of the 1999 Workshop on Codng and Cryptography - WCC '99, pp , Pars, Jan