APPLICATION OF THEORETICAL NUMERICAL TRANSFORMATIONS TO DIGITAL SIGNAL PROCESSING ALGORITHMS. Antonio Andonov, Ilka Stefanova

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1 78 Ieraioal Joural Iformaio Theories ad Applicaios, Vol. 25, Number 1, 2018 APPLICATION OF THEORETICAL NUMERICAL TRANSFORMATIONS TO DIGITAL SIGNAL PROCESSING ALGORITHMS Aoio Adoov, Ila Sefaova Absrac: I he proposed aricle, o he basis of he fudameals of he heory of umbers, algorihms for calculaig he basic operaios i he digial sigal processig- covoluio ad discree Fourier rasform opimized i erms of speed. These operaios accou for he mos sigifica par of he eire processig volume of he primary daa received from differe ypes of iformaio measureme sysems. Keywords: digial sigal processig ITHEA Keywords: I.5.4 Sigal processig Iroducio The mai scieific issues i he field of digial sigal processig (filerig, specrum aalysis ad syhesis of sigals) is o develop ways o overcome he cosrais dicaed by available resources, capabiliies elemeal base limi of sofware ad hardware resources. Mehods for desigig isrumeaio for digial sigal processig iegraig syhesis i he specral regio wih respec o se operaig parameer values wih he devices cosiderig hese limis allow soluios o be obaied ha are close o he opimal wih respec o he crierio of miimizig he resula losses. Trasforms wih properies of cyclic covoluio (CC) Oe of he rasforms havig he propery of cyclic covoluio is he discree Fourier rasform (DFT). N-1 =0 -j2π/n X = x e, = 0,1,...,N - 1 ; (1)

2 Ieraioal Joural Iformaio Theories ad Applicaios, Vol. 25, Number 1, N-1 j2π/n =0 x = N x e, = 0,1,...,N - 1, (2) The propery of CC meas ha N-1 DFT xmh -m= DFTxDFTh (3) m=0 is fulfilled ad herefore cyclic covoluio ca be calculaed from equaio -1 (4) y = DFT DFT x DFT h. Wih deermiig he srucure of rasformaios wih cyclic covoluio propery, oly he class of liear recursive rasforms ha represe a sequece of legh N o aoher oe of he same legh is cosidered. These rasforms ca be described by a o-degeerae square marix Т wih elemes ;,m = 0,1,...,N - 1. Uderliig is used as a sig of marix. Le sequeces,m x, h deoe N- dimesioal vecors, i.e. x x ad h h. Their images will be Х, Y, H, respecively. Accordig o his symbol sysem, for he rasformed sequece i is obaied ha Accordig o his symbol sysem, for he rasformed sequece i is obaied ha Y=T у, Х=T х, H=T h (5) The as assiged is o defie he codiios ecessary o saisfy elemes m, of marix Т, so o saisfy equaios

3 80 Ieraioal Joural Iformaio Theories ad Applicaios, Vol. 25, Number 1, 2018 Y= Х H, (6) for which Y к = Х к H к, = 0,1,...,N - 1, (7) respecively. The cyclic covoluio for fiie sequeces x, h is deermied by formula N-1 m -m m=0 y = x h. (8) Sysem (5) ca be wrie i he form of N-1, =0 Y = y ; (9) N-1,m m m=0 X = x ; (10) N-1,l l l=0 H = x. (11) From (8) ad (9) for Y i is obaied ha

4 Ieraioal Joural Iformaio Theories ad Applicaios, Vol. 25, Number 1, N-1 N-1 Y = x h., m -m =0 m=0 If l=-m is subsiued, he i is obaied ha N-1 N-1 m l,m+ l m=0 l=0 Y = x h. (12) From (7), aig io accou (9), (11) ad (12), (13), i is obaied ha N1 N1 N1 N1 xh xh, (13) m l,ml m l,m,l m0 l0 m0 l0 from where i follows ha = ;,l,m = 0,1,...,N -1. (14),m+l,m,l For ay radom eleme m, i ca be wrie ha = ;...,0,0,0 = = ;,1,0,1,1,0 = = = = ; 2,2,0,2,2,0,1,1,1 = = = = = ; 2 3,3,0,3,3,0,1,2,1,1,1. m,m, 1 (15)

5 82 Ieraioal Joural Iformaio Theories ad Applicaios, Vol. 25, Number 1, 2018 Hece =, (16),m,0,m or,0 =1. (17) Sice cyclic covoluio is calculaed, idices i (14) are summed o module N. If he wo sides of equaio (15) are raised o he power of N, i is obaied ha N mn 0,m,1 =,1 =1. (18) From above i follows ha elemes,m are equal o he N-h roo of 1 (oe). I is ow ha here are oly N differe soluios of a equaio of ype (18). Sice marix Т is o-degeerae, i follows ha all elemes,1 have o be differe ad sice here are N differe roos of equaio (18), i follows ha exacly elemes,1 mus be N i umber differe roos. Wihou reducig he se, eleme 11 ca be seleced for a roo of N-h order. The rows of he marix T ca be arraged so ha Therefore, he marix elemes mus saisfy codiio,1 = 11. Le 11 =α. m,m =α,m = 0,1,...,N - 1. (19) The symmery of he marix Т follows from (19) ad because of ha he reasoig made is also valid for is colums. Sice Т is a o-degeerae marix, here is a iverse marix, i.e. equaio Т. Т -1 =1 is fulfilled. The iverse marix elemes ca be deermied as follows:

6 Ieraioal Joural Iformaio Theories ad Applicaios, Vol. 25, Number 1, m τ,m =N α,m = 0,1,...,N - 1. (20) Hece he pair of rasform formulas for bacward rasformaios will be N-1 X = xα,=0,1,...,n-1; (21) =0-1 N-1 - =0 x =N x α, = 0,1,...,N - 1. (22) Havig compared (21), (22) ad (1), (2), i is foud ha hese rasforms have aalogous srucures. Therefore, each rasform where formulas (21), (22) are fulfilled, appears o have a DFT srucure. Hece, ay rasform wih cyclic covoluio properies mus have a DFT srucure. I ca be proved ha i he se of complex umbers he rasform ow as DFT wih roo α =exp -j2π /N is he oly oe havig cyclic covoluio properies. Theoreical umerical rasformaio Le ipu sequece x be of N elemes, which ca have oly ieger values from he fiie rig of iegers Z M = 0,1,2,...,m - 1. From (21), (22) i is obaied ha N-1 X = xα ; (23) =0 М -1 N-1 x = N x, (24) =0

7 84 Ieraioal Joural Iformaio Theories ad Applicaios, Vol. 25, Number 1, 2018 where symbols... М mea o carry ou arihmeic operaios o module M. Theoreical umerical rasformaios are defied by formulas (23), (24). The codiios ecessary ad sufficie for heir exisece are give i heorem [Pollard, 1972], [Agarwall, 1974]. The N-dimesioal rasformaio ha has a DFT srucure will possess cyclic covoluio properies whe ad oly whe: a)here is umber N i Z M, so ha NN -1 =(modm); b)here is roo of he N-h order α, i.e. N is he N smalles posiive umber for which α =1modM ; N is he divisor of umber 0(М) where T i 0( М ) =НОД P - 1, i = 1,2,..., M = P. i i=1 i There is a large class of rasformaios ha fulfill he erms of heorem. However, oly hose of hem, which have cerai advaages i regard o DFT, are of sigificace for applicaios. Those heoreical umerical sequeces (TNS), which mee he followig requiremes, are of cerai ieres: 1. The legh N of ipu sequeces mus be seleced as a umber havig a square facor i order o use fas algorihms of he id of Fas Fourier Trasform (FFT). 2. Wih calculaig FFT, i is of umos imporace ha muliplicaio wih facors is a simple operaio. Tha is possible, i case ha biary represeaio of coais a few 1s ad especially if i is a expoe of 2. I his case muliplicaio is shif of bis. 3. Calculaios are grealy simplified if he module is seleced o have a simple biary represeaio. The opios examied below are for selecig modules i such a way o saisfy he codiios meioed above. The simples opio suggesed is he selecio of modules of M=2. Accordig o a) heorem 0( М) = 2-1=2 ad he sequece bigges legh is N=1. I follows ha M=2 is pracically irreleva. Le umber M=2-1 ad le =PQ,where Р is a prime umber ad Q is a radom ieger. Number PQ 2-1 ca be preseed i he id of PQ-1 PQ-2 PQ P P 2-1= , (25) hece α Р -1 is a divisor PQ 2-1 ad he maximal legh of sequece N is deermied wih umber P 2-1. Therefore i is sufficie o cosider umbers of id Р M=2-1, he so-called umbers of Mersee. The heoreical umerical rasformaios of Mersee were iroduced by Rader [Rader,

8 Ieraioal Joural Iformaio Theories ad Applicaios, Vol. 25, Number 1, ], who proved ha N =2P for α =-2 is saisfied for sequece legh N. Mersee's heoreical umerical rasforms are of lile ieres i pracice because hey require a large legh of regiser, ad also N is iappropriae for usig a fas algorihm of Fas Fourier Trasform (FFT) ype. The mos appropriae umbers o choose a module are hose of Ferma, which have he id of b N b F = 2 +1,b = 2. (26) Accordig o he heorem quoed, i order o have a heoreical umerical rasformaio of Ferma (TNTF) wih legh N, i is ecessary ha N is a divisor of 0F. Sice he umbers of Ferma F 0 o F 4 are prime, 0F 2 b m, i.e. i is possible o form TNTF for radom legh N 2,mb. These prime b umbers of Ferma allow a maximum legh of N=2 for α =3. The roo α =2 is of order 2 1 ad maes possible o rasform a sequece of legh N =2b. This paricular case is called Rader's rasformaio. I ca be proved [Agarwall, 1974] ha if is chose from formula 2 α c/2 c -1 4b =2 2-1,c=2, (2) roo α 4b is of order 2 +2 ad he bigges row legh is N =4b. If roo α 4b = 2 is used, he legh of sequeces subjeced o covoluio will icrease wice as compared o roo α =2 a he same legh of regiser b (b is he umber of regiser bis). The mai advaages of TNTF are as follows: (a) o muliplicaio operaio is required i calculaio ; b) i is possible o apply a algorihm of FFT ype as i is a expoe of 2. The quesio, which arises, is wha codiios sequeces x sequece of repors x ad h be oupu for a aalogue-o-digial coverer ad h have o saisfy. Le he is a weighig fucio of a digial filer. Le calculae heir covoluio accordig o formula (8) by module M i z M rig. If he resuls of covoluio calculaed i modular arihmeic have o coicide wih hose obaied by usig Euclidea (regular) arihmeic, i is ecessary o depic boh posiive ad egaive umbers of sequeces x ad h o he rig ad i iervals 0,M / 2 / ad /M / 2,M - 1 respecively, aig

9 86 Ieraioal Joural Iformaio Theories ad Applicaios, Vol. 25, Number 1, 2018 io accou he correc ierpreaio of resuls. If sequeces x ad h are prelimiarily ow, i is possible o calculae he maximum value of resul of y usig formula max 1 N max max 0 y x h. (28) This codiio ca serve o choose he value of module М. From i, uder he codiio ha y max <M/ 2, i is obaied ha M>2y. (29) max Coclusio Based o he examied srucures, DFT ad codiios of TNT (heoreical umerical rasformaios exisece, i is possible o selec roo i regard o a seleced module M ad sequece legh N or o selec appropriae module M ad roo for a give sequece of deermied legh N. Of all possible TNT, hose seleced require simple arihmeic ad ca use algorihms of fas DFT ype for pure calculaio. These requiremes are fulfilled by TNTF. The heoreical umerical rasformaio ca be applied paricularly o digial filers wih fiie impulse feaures similar o he applicaio of FFT. Bibliography [Agarwall, 1974] Agarwall R., C. Burrus. Fas covolusio usig Ferma umber rasforms wih applicaios o digial filerig. IEEE Tras. O Acousics, Speech ad Sigal Processig, ASSP- 22,1974, 2, [Li, Reod, 1977] Li K., I.Reod. Fas algorihms for complex ieger rasforms. IEEE Tras. O Acousics, Speech ad Sigal Processig, ASSP-25,1977, 5,

10 Ieraioal Joural Iformaio Theories ad Applicaios, Vol. 25, Number 1, [Mari, 1976] Mari S., Micropocessors implemeaio of umber heoreic rasforms. IEEE, CAS-26, 1976, 3, [Pollard, 1972] Pollard J., The fas Fourier rasformaio i fiie field. Mahemaics of Compuaio, 1972, , [Rader, 1972] Rader C., Discree covoluios via Mersee rasforms. IEEE Tras. O Compuers, C- 22, 1972,12, [Vegh, Leibowiz, 1976 ]Vegh E., L. Leibowiz. Fas complex covoluio i fiie rigs. IEEE Tras. O Acousics, Speech ad Sigal Processig, ASSP-24,1976, 4, Auhors' Iformaio Aoio Adoov Uiversiy of Telecommuicaios ad Pos, Sofia, Bulgaria, e- mail:a_adoov@abv.bg Major Fields of Scieific Research: radioechical sysems, echical cybereics Ila Sefaova Uiversiy of Telecommuicaios ad Pos, Sofia, Bulgaria, e- isefaova24@abv.bg Major Fields of Scieific Research: discree-ime sigal processig, digial sigals of Walsh