n data points is solved: This approach is based on Weierstrass approximation theorem [9-11], which stipulates that any segment a,

Transcription

1 IOSR Joural of VLSI ad Sgal Processg (IOSR-JVSP) Volume 5, Issue 6, Ver. II (Nov -Dec. 05), PP e-issn: , p-issn No. : Real-Tme Recostructo of Dscrete-Tme Sgals Hgh- Performace Dgtal Cotrol Systems: Aalytcal Evaluato of Commo Iterpolato Techques A. Mumu, F. Mumu (Electrcal/Electrocs Egeerg Departmet, Cape Coast Polytechc, Cape Coast, Ghaa) (Electrcal Egeerg Departmet, Uversty of Mes ad Techology, Tarkwa, Ghaa) Abstract: Complex computatoal problems arsg cotrol systems that employ dgtal sgal processg for cotrol tasks such as precso moto cotrol, real-tme processg of sesor data ad mache vso are tedous, requrg sgfcat amouts of calculato of mathematcal fuctos. I addto, such problems eed to be solved a strctly lmted tme, so the mathematcal computatos must be optmzed for speed. Oe of the most mportat operatos performed by such systems s the recostructo of dgtal sgals. Tradtoally, hgh-ed applcatos, ths s doe usg advaced terpolato techques. The computato-tesve requremets of such applcatos requre the formulato of the terpolato problem a way that s effcet for computer executo. However, the choce of terpolato techque s ot always straght-forward, as t depeds o the ature ad sze of the problem. The preset paper udertakes aalytcal study of the commo terpolato techques ad sytheszes correspodg algorthms for computg them. A comparatve aalyss of the varous terpolato methods provdes a accessble meas for selectg the most sutable algorthm for a gve problem. Keywords: Computatoal Complexty, Dgtal Sgal Recostructo, DSP, Iterpolato I. Itroducto I dgtal cotrol systems, very ofte, there s a eed to recostruct dgtal sgals after processg to obta orgal aalog sgals from the dscrete-tme samples. I practce, ths ca be acheved by a umber of techques. These clude smple sample-ad-hold, fractoal delay flterg, classcal terpolato techques, as well as frequecy doma methods [-]. I low-ed applcatos, recostructo of dgtal sgals s carred out usg sample-ad-hold crcuts to acheve a zero-order hold dgtal to aalog coverso [3-4]. I applcatos where hgh precso s requred, polyomal terpolato techques may be used. The smplest form of terpolato s lear terpolato or frst-order hold, whch approxmates values betwee successve samples by straght les [5]. I stuatos where lear terpolato does ot meet the requred accuracy, hgher order terpolato formulas are used [, 6]. Recostructg aalog sgals from dgtzed sgals usg hgher order polyomal terpolato methods requres a sgfcat amout of computatoal resources. I practcal applcatos, t s ofte ecessary to develop algorthms whch reduce all calculatos to a sequece of arthmetc ad logcal operatos ecessary for computer executo. At the same tme, t s also mportat to pay close atteto to the amout of computatoal tme requred to solve these problems. I ths regard, the qualty of algorthms s assessed o the bass of the order of growth of tme eeded to solve the problem. Ths fgure of mert, techcal lterature, s kow as the computatoal complexty of a algorthm. Thus, computatoal complexty, ths cotext, s the tme spet computg solutos as a fucto of the problem sze (.e. put data sze ad formato cotet). It s determed largely by the umber of elemetary operatos eeded to solve a partcular problem. Despte the creasg speed of moder computers, the complexty of advaced terpolato algorthms s stll a key ssue may egeerg applcatos [7]. Practcal mplemetato of may mathematcal solutos has bee mpeded by the computatoal complexty of the mathematcal algorthms [8]. The terpolato problem cossts of covertg dscrete data, defed oly at dscrete pots such as x, y, x, y,... x, y, to a cotuous output (Fg. ). I essece, ths amouts to fdg a smooth 0 0 curve that passes through all the dscrete pots x, y. Thus, there s the eed to determe the values of the fucto at pots whch do ot cocde wth the gve pots (.e. for x x ). Ths meas fdg a fucto f x that satsfes the terpolato codto The pots x are called odes of terpolato. DOI: / Page f x f y ()

2 Real-Tme Recostructo of Dscrete-Tme Sgals Hgh-Performace Dgtal Cotrol Fg.. Iterpolato of dscrete data. The most straghtforward method of dog ths s polyomal terpolato, where a th order polyomal that passes through data pots s solved: ( x) P( x) C C x C x... C x () 0 Ths approach s based o Weerstrass approxmato theorem [9-], whch stpulates that ay segmeta, b of a fucto P( x) ca be suffcetly well approxmated by a aalytcal expresso whch s a P( x), algebrac polyomal. To costruct the polyomal t s ecessary to fd the coeffcets C0, C, C,... C. The coeffcets are determed from the terpolato codto P( x) y,,,...,. I practce, however, the umber of odes may be very large; leadg to extreme dffculty solvg () by classcal methods. The complexty of solvg () by Gaussa method, for example, s of the order O( 3 ) [-4]. I practcal stuatos, the desred terpolato polyomal s costructed wthout solvg ths system. A umber of techques have bee developed for dog ths [5-8]. Ths work vestgates commo terpolato techques from the pot of vew of realzg effcet algorthms for recostructg dscrete-tme sgals. The ma objectve of ths aalytcal study s to provde accessble meas of comparg the computatoal complexty of commo terpolato techques wth the vew of fdg the most approprate choce for a partcular problem. II. Classcal Iterpolato Techques Notwthstadg the prolferato of may terpolato algorthms, classcal terpolato methods are stll very popular. Oe reaso for ther popularty s that they have bee developed log ago ad are wellstuded, they are effcet, ad have prove relable for practcal applcatos. Also, may of the ewer techques, despte ther computatoal effcecy, from a practcal pot of vew of software mplemetato smplcty, have trcate formulatos, makg them less attractve tha covetoal terpolato techques... Lagrage techque A more effcet alteratve to recostructg sampled data by solvg the terpolato problem caocal form s Lagrage polyomal formula [9]. The Lagrage terpolato method may also be used to desg FIR, IIR, as well as adaptve Flters [0-]. To costruct the Lagrage polyomal for a arbtrary part of a segmet a, bcotag k x, x,..., x pots k, a algebrac polyomal Q ( x ) s chose such that, j m Q ( x ) km j. The polyomal s zero for every x except x x (.e. x, x,..., x are the roots of m k 0, j m ths polyomal). So the polyomal Q ( x ) ca be wrtte the form km j Q ( x) c( x x )( x x )...( x x )( x x )...( x x ) (3) km m m k The costat c s determed from the codto Q ( x ) by substtutg x x (3) km j m ad c Q km ( m )( m )...( m m )( m m )...( m k) x x x x x x x x x x ( x x)( x x )...( x xm )( x xm )...( x xk) ( x) (5) x x x x x x x x x x ( m )( m )...( m m )( m m )...( m k) DOI: / Page km j (4)

3 Real-Tme Recostructo of Dscrete-Tme Sgals Hgh-Performace Dgtal Cotrol Q ( x ) are kow as Lagrage bass polyomal. km The Lagrage polyomal for the chose segmet s, thus, wrtte as k ( x x)( x x )...( x xm )( x xm )...( x xk) Pk( x) fm (6) m ( xm x)( xm x )...( xm xm )( xm xm )...( xm xk) If t s requred to terpolate the whole tervala, b, the 0, k, ad the Lagrage polyomal has the geeral form ( x x0)( x x)...( x x )( x x )...( x x) ( x xj) P( x) f f ( x x0)( x x)...( x x )( x x )...( x x) ( x xj) (7) 0 0 j 0 j Thus, the degree of the polyomal s ad for x x all terms the summato operator reduce to th zeros except the j terms, whch are f, thereby satsfyg the terpolato codto (). As the terms of the factor Q ( x) relato (7) deped oly o the choce of odes x ad the pots x, ad ot o the fucto f x, the factors f ca be computed oly oce ad used the costructo of terpolato polyomals. Computatoal complexty Lagrage terpolato algorthm The costructve valdato of Lagrage terpolato techque provdes a method to costruct the terpolatg polyomal the form expressed (7). The method s stated the followg steps: ( x xj). Computato of the Lagrage bass Q ( x) ( x xj). Multplcato of each th 3. Summato of each product term Q ( x) f 0 bass by the correspodg fuctoal value f Recallg that the Lagrage coeffcets Q( x) of a fucto y f ( x) a, b are determed oly by the odes x ad the pots x, where t s requred to compute the value of the polyomal. So f the calculatos are carred out accordg to (5) above, the calculato of each coeffcet (cosstg of odes) of the Lagrage bass requres elemetary multplcatos, dvsos, ad addtos. Obvously, t s ecessary to make teratos to compute all the polyomals (.e. P ( x), P ( x ),, P ( x ) ). 0 Cosequetly, the computato of the Lagrage bass volves: multplcatos addtos dvsos for ay gve segmet The umber of operatos step two s multplcatos, ad step three addtos. So there are, total, 3 addtos, multplcatos, ad dvsos... Newto Iterpolato Method The Lagrage terpolato polyomal s coveet cases where the odes may eed to be updated. If the task s formulated such that the addto of a extra pot would requre oly the addto of a th sgle polyomal of degree to the degree Lagrage polyomal, the ths added ca be foud by takg the dfferece of the two Lagrage polyomals of degree ad. Smple trasformatos of the Lagrage terpolato polyomal wll lead to the followg expresso: where P( x) s Lagrage polyomal of degree. ( ) 0( ) ( ) ( ) (8) P x P x P x P x DOI: / Page

4 Real-Tme Recostructo of Dscrete-Tme Sgals Hgh-Performace Dgtal Cotrol If we let Q( x) P( x) P ( x), the Q( x) s a polyomal of degree whose value s zero at x x0, x x, x x,..., x x. Therefore, the polyomal Q( x) ca be wrtte as Q( x) C( x x )( x x )...( x x ) 0, where C are coeffcets at x x0, x x, x x. Therefore, the modfed Lagrage polyomal (8) ca be rewrtte as L( x) L ( x) C( x x 0)( x x)...( x x ). Recursvely expressg all the polyomals L, L, etc. a smlar way leads to aother form of represetg the Lagrage terpolato polyomal, whch s called Newto terpolato polyomal []: ( ) 0 ( 0) ( 0)( )... ( 0)( )...( ) ( j) (9) N x C C x x C x x x x C x x x x x x C x x C, C,..., C are kow as Newto coeffcets. 0 j 0 The Newtoa bases are, ( x x0), ( x x0)( x x),...( x x0)( x x0)...( x x ). I the geeral formulato of the Newto terpolato polyomal (9) the Newto coeffcets C are computed from the terpolato codto (): j j C f C 0 Cj ( x xk) ( x xk) (0) j 0 k 0 k 0 Calculatg the coeffcets by ths approach s computatoally demadg. I practce, t s ofte approprate to compute the coeffcets usg Newto s dvded dffereces [3]. If x0 x x... xk, the, the dvded dffereces are defed as follows [4]: Zero th order dvded dfferece ( ) f [ x ] f [ x],, f x f x f, 0,,,.... The frst order dvded dfferece s f x x ad the secod order dvded dfferece s x x f [ x, x ] f [ x, x ] f x, x, x. x x It ca be see that hgher order dvded dffereces are defed recursvely usg lower-order oes. th Thus, k order dvded dfferece for the secto x, x k ca be foud through ( k ) th order dvded f [ x, x,..., x k] f [ x, x,... x k ],,,...,, x k x where k,,..., ; 0,,..., k ; degree of the polyomal. It s ofte reasoable to calculate the dvded dffereces oly for eghborg values of the dscrete data. I ths case, a hgher order dvded dfferece s calculated from two precedg oes of lower orders [5]. Hece the Newto coeffcets C 0, C, C,... C ca be expressed through dvded dffereces as follows: C 0 f 0 f [ x 0], dfferece by the recursve formula f x x x x k f f 0 f [ x] f [ x 0] C f [ x 0, x], x x0 x x 0 f f f f 0 x x x x0 f [ x, x ] f [ x0, x] C f [ x0, x, x ], x x0 x x0 ad, Ck f [ x0, x,..., xk]. Thus, for the purpose of terpolato fucto o the terval x, x the Newto polyomal, based k o dvded dffereces, ca be wrtte the form N x f f x x x x f x x x x x x x k( ), ( ),, ( )( )... f x x x x x x x x x Computatoal complexty of Newtoa terpolato algorthm,,..., k ( )( )...( k ) () Computg the value of the terpolato polyomal the Newto form ca be carred out the followg steps:. Fdg the Newto coeffcets C through dvded dffereces DOI: / Page

5 Real-Tme Recostructo of Dscrete-Tme Sgals Hgh-Performace Dgtal Cotrol. Computg the value of the polyomals (9) usg the Newto coeffcets C 0, C, C,... C To perform the frst operato, oe eeds to carry out addtos ad dvsos for each ode x. Ths requres teratos each for each of the arthmetc operatos. Hece, the umber of addtos would be ( ) ( ), ad the umber of dvso operatos would also be same. Calculato of the polyomal P (step ) requres addtos through teratos, plus addtoal summatos for the terms C x x j. Thus, for terpolato of odes, the umber of addto operatos: ( ) ( 3) ( ). Smlarly, there would be multplcatos through teratos: ( ) ( ). Thus, total umber of each operato s dvsos. addtos, multplcatos, ad.3. Sple Iterpolato The Sple s a form of pecewse polyomal method used for terpolatg dscrete pots. I a pecewse polyomal terpolato the segmet a, b s dvded to sub-segmets havg a smaller umber of terpolato odes. Lower degree terpolato polyomal ca the be used o each of these segmets. The most wdely used pecewse terpolato method s cubc sple terpolato. Cubc sple has a feature that each partal terval s represeted by a thrd degree polyomal, ad o the whole terval of terpolato, t s cotuous together wth ts frst ad secod dervatves [6-8]. O each terval [ x, x ] the cubc sple has the form [9] S ( x) a b ( x x ) c ( x x ) d ( x x ) f, x [ x, x] () 3 Thus, the sple satsfes the Lagrage terpolato codto S ( x ) f, 0,,,... (3) The sple () o each of the segmets[ x, x],,,... s defed by four coeffcets, ad, therefore, for ts costructo o the whole terval a, b 4 equatos are eeded to determe the coeffcets. Codto (3) yelds equatos. As oted, the sple S( x ), together wth ts frst ad secod dfferetals, s cotuous all teral odes. The codto of cotuty of the dfferetals S ( x) ad S( x) gves ( ) addtoal equatos. Thus, there are ( ) 4 equatos. The two mssg codtos are determed dfferet ways. They may be obtaed from the so-called boudary codtos [30-3], whch are assged o the bass of physcal propertes or other cosderatos related to the terpretato of the features of the dscrete data. These clude referece to frst or secod dervatves of the extremes of the tervals, or perodcty codtos. To obta the remag two ukows, here we use the Free Boudary Codtos: S( a) S( b) 0 (4) The result s a system of 4 a, b, c, d, lear equatos for determg the ukow coeffcets,,...,. Prelmary aalyss of these equatos ad a umber of smple trasformatos lead to a farly smple sequece of operatos for fdg the values of these coeffcets as follows. O the terval [ x, x] the secod dervatve of the sple S ( x) s a straght le wth slope m, ad ca be wrtte the form DOI: / Page

6 Real-Tme Recostructo of Dscrete-Tme Sgals Hgh-Performace Dgtal Cotrol S ( x) m( x x), where s a costat. Sce S ( x) s a lear fucto o[ x, x], ts form s completely determed by ts two extreme values m, So h x x x x S ( x) for x [ x, x], where h x x, S ( x). h h Itegratg S ( x) twce gves aother form of the cubc sple represetato Where T T,,...,, ad f f f f 0 f f f f q,...,. h h h h DOI: / Page ad ( x x) ( x x ) S x A x x B x x 6h 6h at the eds of the terval [ x, x]. 3 3 ( ) (5) A ad B are the costats of tegrato. Accordg to the terpolato codto S( x) f, h cosequetly, lettg x x, yelds S( x) f B h or B 6 Also f we let x x ad otg that S( x ) S ( x ), we obta A f h Substtutg h. 6 A ad B equato (5) gves f h. h 6 ( h ) ( ). Thus, 6h 3 S x f A h 3 3 x x x x ( x x) h ( x x) ( x x ) h ( x x ) S ( x) f f (6) 3 h h 6h 6h From (6) the oe-sded dervatves for x, x,..., x usg property that the sple s cotuous over [ x, x ] ca be obtaed: h h f f S x 6 3 h ( 0) (7) h h f f S x 3 6 h Equatg (7) ad (8) for,,...,, we obta equatos of the form ( 0) (8) h h h h f f f f h h Ths equato ca be wrtte the matrx form [3] A q wth ukows (,,..., ), where h h h h h h h A h h h h h h h 0 6 3, (9)

7 Real-Tme Recostructo of Dscrete-Tme Sgals Hgh-Performace Dgtal Cotrol Thus, matrx form the equato s a b q c a b q c a b q c a q (0) h h h h f f f f c, a, b, q h h Solvg the sple terpolato requres solvg the trdagoal matrx A [33]. The soluto of such systems ca be orgazed a way that does ot clude the zero elemets of the matrx, thereby savg memory ad decreasg the requred amout of computato. Ths method mathematcs lterature s called Thomas algorthm [34]. Thomas algorthm cossts of two stages - forward substtuto ad backward substtuto. Forward substtuto cossts calculatg the auxlary coeffcets ad, whereby each ukow s expressed terms of : ( ),,,..., () From the frst equato of (0), b q. But from () a a. Thus, equatg the two relatos equatos gves b a q a Now puttg the secod equato of (0): c a b q Thus, ca be expressed through 3 :. 3 q b 3 c or 3, c a b q c. ad so c a ad c a Smlarly, all auxlary coeffcets ad ca be calculated by the recursve formulas b If we let e a c, the e b c a q c a c q c ad e. (3) () Backward substtuto cossts of recursvely calculatg. Ths volves, frst, fdg usg equato () ad the last equato of the trdagoal matrx (0). N N (4) N N DOI: / Page

8 Elmatg N Real-Tme Recostructo of Dscrete-Tme Sgals Hgh-Performace Dgtal Cotrol from the two equatos gves q a c N N N N N q N a N N N c N N N The, usg the formula (), ad the prevously calculated coeffcets ad (3), we ca successvely calculate all ukows,,..., (5). Computatoal Complexty Of Cubc Sple Iterpolato Algorthm (6) ad accordg to formulas () From the above aalyss, the soluto of the sple S(x) requres oly fdg h, A, B, q ad. The steps volved calculatg ca be summarzed as follows: h. Computato of requres addtos. Computato of A ad B both requre ( ) dvsos, multplcatos ad addtos each 3. Computato of q requres ( ) addtos ad dvsos 4. Computato of volves a umber of steps: Computato of ad as per (). Ths requres dvso operatos Computato of ad (, 3..., ) as per (3). Ths requres ( ) ( ) multplcato, ad ( ) dvsos addtos, Computato of as per (6). Ths requres addtos, multplcatos, ad dvso operato Computato of (,,..., ) as per (). Ths requres addtos, ad multplcato Hece, the total umber of operatos s 7 6 addtos, 5 3 multplcatos, ad 7 4 dvsos. Ths gves a total operato cout of9 3. Notg that the coeffcets ad e are defed solely by the choce of odes ad do ot deped o the fuctoal values y, computato ca be reduced f there s the eed to solve a seres of problems volvg dfferet fuctoal values at predetermed odes. I ths stuato, ad e are calculated oly oce. Oly are computed for subsequet problems. Ths reduces the umber of operatos by 3( ), resultg total umber operatos. Thus, the total umber of operatos volved computg the sple S x 6 0 s8 7. If we are carryg out a seres of aalyss volvg the same odes, subsequet computato are reduced to5 4. III. Results Ad Dscusso Table summarzes the results of the foregog aalyss. The last colum s the total cout of all operatos, N. Ths s plotted agast put data sze, Fg. wth a logarthmc scale o both axes. It s worth otg that most practcal cases, addto, multplcato ad dvso operatos do ot cost the same. But sce the duratos of these basc operatos are kow, t s usually straght-forward to compute the total cost terms of the umber of mache cycles. Table. Summary of computatoal complexty of classcal terpolato methods Addtos Multplcatos Dvsos Total Lagrage 3 Newto Cubc Sple DOI: / Page

9 Real-Tme Recostructo of Dscrete-Tme Sgals Hgh-Performace Dgtal Cotrol I hgh performace dgtal cotrol systems wth real-tme requremets, the terpolato problem usually eeds to be solved wth a strctly short durato, say a few mllsecods. The choce of a good terpolato algorthm allows moder DSP systems ot oly to provde hgh accuracy ad performace, but also exted the magtude of tasks that ca be performed. For stace, s f a task requred 00 mllsecods ad a dgtal sgal processor wth processg speed of mllo floatg pot operatos per secod (MFLOPS) s employed, the maxmum umber of dscrete puts that ca be terpolated would be 0000, 400, ad 500 for the sple, Lagrage ad Newto, respectvely (Fg. ). It s obvous that for the sple algorthm, creasg processor speed by a factor of 00 results a crease problem sze the order of 00. For the Lagrage ad Newto terpolators, a speed crease by 00 tmes results oly a 0 tmes crease problem sze. Fg.. Comparso of computatoal complexty of commo terpolato methods. O the bass of ths aalyss, t ca be sad that the Cubc Spe terpolato algorthm s the best choce (for data pots greater tha 8). The Lagrage formula also outperforms Newto s. However, as oted above, f there would be the eed to update terpolatg data, the Newto s would be a better choce. I geeral, the complexty of a algorthm s evaluated order p of magtude of put data, ad s p deoted O( ), so called bg-o otato [35-36]. I ths represetato, oly the fastest growg (hghest order) term s cosdered; all terms of lower order are gored. For example, the tme complexty of the Lagrage algorthm s equal to 4 5, but ts computatoal complexty s of order ad s expressed as O( ). Lkewse, Newto terpolato formula s O( ) ad the cubc sple s O( ) However, there are stuatos where t s useful to kow the exact relatoshp of the depedece of complexty o put data. Ths s especally so whe a small umber of put data s to be terpolated, or whe the terpolato formula cotas a large costat whch ca have a sgfcat effect o the qualty of the algorthm. O the graph of Fg., t ca be observed that the Lagrage formula s the best for a task volvg to 8 data puts. Other factors, such as avalable memory resources, requred accuracy, ad the ature of sgals, may also fluece the choce of algorthm. For stace, f the sgal s to be sampled at a much hgher rate tha the aalog sgal frequecy, the the Lagrage terpolato formula s preferred to the Sple [6]. Also, some classes of algorthms may be less stable for certa types of problems. IV. Cocluso The study shows that a umber of terpolato methods ca be used to costruct a approxmate aalog output from dgtal sgals. The requred accuracy ca be acheved by all the above methods of polyomal terpolato. However, the practcal mplemetato of the dfferet terpolato techques requres dfferet computatoal resources. So addto to the accuracy of approxmato of the dgtzed sgals, oe should take to accout the CPU tme requred to costruct these sgals. The preset study attempted to aalyze ad compare the most popular classcal terpolato methods used dgtal sgal processg ad dgtal cotrol systems for the purpose of recostructg dscrete-tme sgals. The results of the study ca be used as a bass for selectg the most approprate algorthm for a partcular terpolato task. It s also expected that ths study wll serve as a bass for further vestgato to the ssues of effectve use of terpolato algorthms for recostructg dscrete data. For example, further theoretcal ad expermetal study DOI: / Page

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