( ) New Fastest Linearly Independent Transforms over GF(3)

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1 New Fastest Liealy deedet Tasfoms ove GF( Bogda Falkowski ad Cicilia C Lozao School of Electical ad Electoic Egieeig Nayag Techological Uivesity Block S 5 Nayag Aveue Sigaoe Tadeusz Łuba stitute of Telecommuicatios Wasaw Uivesity of Techology Nowowieska 5/9-665 Wasaw Polad Abstact New fastest liealy ideedet (L tasfoms fo teay fuctios ae itoduced i this ae The tasfoms oeate ove Galois Field ( (GF( ad have smalle comutatioal costs tha teay Reed- ulle tasfom The ew tasfoms ae built based o the kow fastest L tasfoms ove GF( ad the elatios betwee them ae show Seveal oeties fo the ew tasfoms ae eseted Exeimetal esults fo the ew tasfoms ae also listed ad comaed with the kow fastest L tasfoms ove GF( toductio Teay switchig fuctios ae maigs { } { } f : ad ca be cosideed i diffeet algebaic stuctues with vaious olyomial exasios o sectal tasfoms To develo the sectal tasfom theoy fo switchig teay fuctios oe eeds to efe to abstact hamoic aalysis that is a mathematical aoach deived fom the classical Fouie aalysis [] Develomets i hamoic aalysis o fiite Abelia gous esulted i vaious tasfoms such as Walsh Reed-ulle ad thei ossible modificatios that have may attactive featues ad ae useful i alicatios such as a ew theoy of oliea sigal ad image ocessig [4] ad may othe aeas of alied mathematics [5] Fastest liealy ideedet (L tasfoms ove Galois Field ( (GF( have the simlest buttefly diagams of all ossible L tasfoms ove GF( this ae ew fastest L tasfoms ove GF( ad thei oeties ae ivestigated ad comaed with the kow fastest L tasfoms ove GF( [6] The eseted tasfoms ad thei oeties show hee ca be used as bases fo aalysis sythesis ad testig of teay fuctios as well as ceatig thei sectal decisio diagams i a simila mae as fo othe olyomial exasios [ 7 8] Basic defiitios Defiitio Let be a N N ( N matix with ows coesodig to mitems ad colums coesodig to some teay switchig fuctios of vaiables f the sets of colums ae liealy ideedet ove GF( the has oly oe ivese i GF( ad is said to be liealy ideedet Defiitio Let be a L matix of ode N as secified i Defiitio ad F [ F F F ] T K N be the tuth colum vecto of a -vaiable teay switchig f i a atual teay odeig The A F ( ad F A ( whee A [ A K ] T eesets the sectum of fuctio ( x ( x A A N f based o is the ivese of ove GF( T deotes tasose oeato ad all the additios ad multilicatios ae efomed ove GF( Defiitio Let f ( x be a -vaiable teay switchig fuctio The by Defiitios ad the L exasio of f ( x based o a teay L tasfom ca be witte as g x f A g ( whee deotes the teay switchig fuctio whose tuth vecto is give by colum of ad A deotes the -th sectal coefficiet i the sectum of f ( x based o The additios ad multilicatios iside ( ae evaluated ove GF( Gous of teay L tasfoms with lowe comutatioal cost tha teay Reed-ulle tasfom have bee eseted i [6] whee they ae classified ito classes Z ad Z based o thei stuctue Thee

2 ae six teay L tasfoms iside each class whee fou of them have the same comutatioal costs that ae lowe tha the comutatioal cost of the othe two tasfoms i the class this ae oly those teay fastest L tasfoms with the lowest comutatioal cost ae discussed Togethe we efe to those tasfoms as kow teay fastest L tasfoms All the kow teay fastest L tasfoms eseted i [6] ae ecusive ad ca be defied i tems of the submatices o whee deotes a submatix with all its elemets beig zeo deotes the idetity submatix of size deotes the evese idetity matix of dimesio ad deotes a submatix with all its elemets beig zeo excet fo oe elemet located at oe coe of each matix deedig o the locatio of Thei ecusive defiitios have the followig geeal fom: (4 ( (5 ( ( (4 whee each submatix ( } {45 has a dimesio of ad cotais oe ecusive equatio which is eithe X o whee X o Defiitio 4 Thee ae fou teay fastest L tasfoms with the lowest comutatioal cost i class [6] Thei fowad tasfoms ae defied ecusively as (5 (6 4 (7 ad 5 (8 whee ad have bee defied befoe Thei ivese tasfoms ca be obtaied by simly elacig ad i the fowad tasfom with ad esectively as show i (9( (9 ( 4 ( 5 ( Defiitio 5 The oeato R o a matix is defied as efomig 9 couteclockwise otatios twice ivolvig 9 8 submatices each of ode ( Examle Let The R R Defiitio 6 The oeato R o a matix is defied as ecusively alyig oeato R o fo The squae of oeato R is secified as R R R Defiitio 7 The fowad ad ivese teay fastest L tasfoms ad ca be calculated by fast tasfom by eesetig them i the followig factoized fom K (

3 ad ( K (4 whee K ad K deote the -th factoized tasfom matix of ( ad ( esectively ( {45 } Poety gives the geeal fomulae fo K ad K Poety The factoized tasfom matices of ( ad ( ( {45 } ca be deived by usig (5( as follows: K K (5 K whee ( K K (6 K K K X (7 K ( K X K 4 R ( K 4 R K 5 R ( K R K K (8 K (9 K ( K ( 5 K ( K eesets the idetity matix of size with bottom left elemet elaced by ( K eesets the idetity matix of size with bottom left elemet elaced by ( { } ad + if X Note that othewise ad i (5(8 ae the same as i (5( Fig shows the fowad ad ivese buttefly diagams fo fast comutatio of the teay fastest L tasfoms ( based o Defiitio 7 ad Poety The same buttefly diagams fo the L tasfom ( ae show i Fig both figues the solid ad dashed lie eesets the values ad esectively (a Figue Buttefly diagams of ( tasfom; (b vese tasfom (a Figue Buttefly diagams of ( tasfom; (b vese tasfom (b : (a Fowad (b : (a Fowad Poety [6] The umbe of additios equied to comute the secta of teay fastest L tasfoms ( {45 } is ( Teay fastest L tasfoms with emutatio this ae we wat to idetify ew teay fastest L tasfoms that have the same comutatioal cost as the kow teay fastest L tasfoms ad ca also be calculated efficietly by fast tasfoms while offeig the ossibility of moe comact olyomial eesetatios ie have the smalle umbe of ozeo tems e of the simlest ways to do that is by emutig the kow teay fastest L tasfoms Such class of teay fastest L tasfoms is defied i this Sectio t should be oted that due to the elatios betwee the kow teay fastest L tasfoms the L exasios based o all the teay fastest L tasfoms with emutatio cove the L exasios based o all the kow teay fastest L tasfoms As such the miimum umbe of ozeo sectal coefficiets i the secta of teay fastest L tasfoms with emutatio is always smalle tha o equal to the miimum ozeo sectal coefficiet umbe i the secta of all the kow teay fastest L tasfoms

4 Pemutatio matices ae matices that cotai exactly oe i each ow ad colum As such thee ae six ossible emutatio matices of size ad 6 emutatio matices of size that ca be deived fom the Koecke oduct of the emutatio matices Defiitio 8 Let the six emutatio matices be deoted by ρ ρ ρ ρ ρ 4 ad ρ 5 whee ρ ρ ρ ρ ρ 4 ad ρ 5 The P is defied as the emutatio matix of size with emutatio umbe ( 6 that is calculated by P ρ ( whee < > 6 < K > is the -digit sixvalued eesetatio of ad deotes Koecke oduct [ 4 5 7] Due to the oety of Koecke oduct the ivese of P deoted by ( P is simly P ρ (4 whee ρ ad ρ 4 ae iveses of each othe ad ρ ρ ρ ad ρ 5 ae self ivese Defiitio 9 Let ( deote teay fastest L tasfom matix of size with emutatio umbe ( 6 The ( ad its ivese ( ae defied as tasfom matix ( P K (5 (6 esectively Poety Thee ae altogethe + ozeo ad ( K ( P ad elemets iside both ( ( All the ozeo elemets i ( ae s wheeas iside ( ( of the ozeo elemets ae s ad the est ae s Poety 4 Fom Defiitio 9 ad the elatios betwee the teay fastest L tasfom matices of classes ad Z eseted i [6] it ca be established that Poety 5 Let ( x ( (7 ( 6 ( 4 (6 ( Z 6 ( Z4 (5 6 P P (8 (6 (9 ( f be a -vaiable teay switchig fuctio with the tuth vecto F The thee ae sectal coefficiets i the sectum of f ( x based o ( whose values ae equal to the values of the tuth vecto elemets ie thei values ca be diectly obtaied fom F without ay additios o multilicatios Futhemoe if F is defied as the subset of the tuth vecto elemets whose values affect the values of the sectal coefficiets that eed to be calculated F has elemets Poety 6 All ossible teay fastest L matix with emutatio ca be divided ito gous of size such that if S ( is defied as the set of tuth vecto elemets that ae diectly fowaded to the sectal coefficiets of ( the all ( i the same gou have idetical set S ( Theefoe the umbe of elemets iside S ( that have ozeo values gives the miimum umbe of ozeo elemets fo the coesodig gou of ( Let the matix Z be defied as Z whee the ow ad colum idex umbes stat fom zeo The ay two fastest L matices with emutatio ( a ad belog to the same gou if ( b Z Z fo K a b ad Z Z a b fo ( whee < > < K > ad a 6 a a a

5 < b > < K > 6 b b b 4 Geealized teay fastest L tasfoms The teay fastest L tasfoms with emutatio defied i Sectio ca be futhe exteded ito a wide set of teay fastest L tasfoms by allowig the emutatio to be located eithe i oe side of the buttefly diagams o betwee the buttefly diagam stages ad by allowig the buttefly diagam stages to be eodeed such as it has bee doe fo biay fastest L tasfoms [9] The esultig teay L tasfoms ae called geealized teay fastest L tasfoms As eodeig ad emutatio do ot icu ay additioal cost the comutatioal cost of the geealized teay fastest L tasfoms ae the same as the kow teay fastest L tasfoms Defiitio Let ( ϕσ deote a geealized teay fastest L tasfom of dimesio with odeig ϕ emutatio ositio σ ( σ + ad 6 The emutatio umbe ( ϕσ is defied as σ + K ϕ P K if ϕ σ σ ( ϕ σ P K ϕ othewise ( whee {45 } ad K ad P have bee defied i Poety ad Defiitio 8 esectively Poety 7 The odeig ϕ is a -digit stig i which evey digit takes values fom to ad o two diffeet digits i it ae allowed to have the same values ϕ < ϕ ϕ K ϕ > ( whee ϕ adϕ ϕ iff i i i { } Poety 8 Clealy the ivese of ( ϕσ ( ϕ σ σ i K ( P ( K ϕ K ( P ϕ - σ ϕ is simly if σ + othewise (4 Poety 9 Ay two geealized teay fastest L ϕ σ ϕ σ ae matices ad idetical whe σ σ ad S ϕ σ } S ϕ σ } { { Poety By (9 ( ad ( it ca be deived that 4 ϕ σ R ϕ σ ' (5 5 ad ( σ R ( ϕ σ ' ϕ (6 whee < > 6 < K > ' 4 ad 5 if 4 ad 5 esectively ad ' > < ' ' K > < 6 ' 5 Exeimetal esults The calculatio of the secta based o all ( ad geealized teay fastest L tasfoms ( ϕσ have bee imlemeted i ATLAB ad u fo seveal biay bechmak fuctios that have bee modified to eeset teay fuctios The taslatio fom biay to teay cases has bee doe by chagig evey two iut (outut bits i biay files to a iut (outut symbol i teay files f the umbe of iut ad/o outut vaiables is odd the a zeo bit is fist added behid the biay cubes to make it eve Fo iut (outut is coveted to is coveted to is coveted to is coveted to ad is igoed (coveted to The esultig umbes of ozeo sectal coefficiets iside the secta of each teay iut fuctio based o ( ( 6 (6 ad (5 6 ae listed i Table Recall that those teay fastest L tasfoms with emutatio coesod to the kow teay fastest L tasfoms additio the umbe of ozeo sectal coefficiets fo each iut fuctio based o all ( ae comaed ad the miimum umbe is show i the ightmost colum of Table Based o the umbes i Table it ca be see that fo some teay fuctios ( educes the umbe of tems equied to eeset them which leads to faste calculatio of the outut value fo examle fo co d84 9sym ad alu4 Table the esultig miimum umbes of ozeo sectal coefficiets that ca be obtaied by each tye of ( ϕσ ae show Comaig the umbes i Tables ad it ca be obseved that fo some teay fuctios ( ϕσ ca give moe comact eesetatios tha ( i tems of smalle umbe of ozeo sectal coefficiets Sice ( is a secial

6 case of ( ϕσ coefficiets based o all ( ϕσ that based o ( Table the miimum umbe of sectal is eve lage that This ca be clealy see fom Table Numbe of ozeo sectal coefficiets fo ( ut Numbe of ozeo sectal coefficiets fileame ( ( 6 (6 (5 6 timum ( xo5 9 co squa z5x ic d misex ex sym cli aex ex alu misex Table iimum umbe of ozeo sectal coefficiets fo ( ϕ σ ut ( ϕ σ ( ϕ σ 5 ( ϕ σ ( ϕ σ fileame xo co squa z5x ic d misex ex sym 7 7 cli aex ex Coclusio Extesio of the kow teay fastest L tasfoms [6] to geeate ew classes of teay fastest L tasfoms with the same lowest comutatioal cost have bee eseted Seveal oeties ad elatios fo the tasfoms have also bee give The eseted oeties ad elatios ca be used to educe the time ad comutig esouces equied to obtai the most comact olyomial eesetatio fo a teay fuctio based o all the teay fastest L tasfoms The theoy eseted i this ae may be of iteest ot oly to eseaches wokig i the aea of teay fuctios but also i othe aeas whee mathematical models of teay exasios ad fuctios ae used The coesodig olyomial exasios ove GF( ca be used ot oly as bases fo bio-othogoal systems but also as the mathematical aaatus to aalyze the stability of fiite automata givig moe flexibility tha the kow esults fo biay dyamic systems [5] A uified aoach to the geeatio of buttefly stuctues eseted hee ca also be of iteest fo eseaches develoig multiesolutio digital sigal ocessig systems usig ucovetioal alicatios of buttefly decomositio techiques [] 7 Refeeces [] R S Stakovic R Stoic ad S Stakovic Recet Develomets i Abstact Hamoic Aalysis with Alicatios i Sigal Pocessig Sciece Publishe Belgade 996 [] S Agaia Astola ad K Egiazaia Biay Polyomial Tasfoms ad Noliea Digital Filtes acel Dekke New ok 995 [] Astola ad R S Stakovic Sigal Pocessig Algoithms ad ultile-valued Logic Desig ethods Poc 6th EEE t Sym o ultile-valued Logic Sigaoe ay 6 CD ublicatio [4] L P aoslavsky ad Ede Fudametals of Digital tics Bikhause Bosto 996 [5] G P Gavilov ad A A Saozheko Poblems ad Execises o Couse of Discete athematics Sciece oscow 99 i Russia [6] B Falkowski ad C Fu Fastest Classes of Liealy deedet Tasfoms ove GF( ad Thei Poeties EE Poc Comutes ad Digital Techiques vol 5 o Set 5 [7] P Davio P Deschams ad A Thayse Discete ad Switchig Fuctios Geoge ad cgaw-hill New ok 978 [8] R S Stakovic ad T Astola Sectal teetatio of Decisio Diagams Sige-Velag New ok [9] S Rahada B Falkowski ad C C Lozao Fastest Liealy deedet Tasfoms ove GF( ad Thei Poeties EEE Tas o Cicuits ad Systems vol 5 o Set 5 [] A Dygalo Buttefly thogoal Stuctue fo Fast Tasfoms Filte Baks ad Wavelets Poc 7th EEE t Cof o Acoustic Seech ad Sigal Pocessig Sa Facisco USA ach