Signal and zero padding to improve parameters estimation of sinusoidal signals in the frequency domain

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1 CT IEKO IN: 87X Noveer 6, Volue 5, Nuer, ignl nd zero pdding o iprove preers esiion o sinusoidl signls in he requency doin Dušn grež, Dir Ilić, Jnko Drnovšek Universiy o Ljuljn, culy o Elecricl Engineering, Trzsk 5, Ljuljn, loveni Universiy o Zgre, culy o Elecricl Engineering nd Copuing, Unsk, Zgre, Croi BTRCT This pper presens he procedure o iprove he esiion o he sic sinusoidl signl preers (requency, pliude, nd phse, respecively) in he cse o signl spling y verging in he perure ie. Prior o esiion in he requency doin y he inerpoled DT lgorihs he spled signl is pdded wih he signl verge vlues in he perure ies nd zeroes in he res o he spling inervl. We cn increse pdding poins nd nuer o he signl cycles in he whole esureen inervl nd wih his nering he errors o he level s wih esiion o he signl wihou verge spling even he spling Nyquis condiion is no ulill. ecion: REERCH PPER Keywords: spling process; spling y verging; signl pdding; zero pdding; esiion o preers; inerpoled DT Ciion: Dušn grež, Dir Ilić, Jnko Drnovšek, ignl nd zero pdding o iprove preers esiion o sinusoidl signls in he requency doin, c IEKO, vol. 5, no., ricle 8, Noveer 6, ideniier: IEKO CT 5 (6) 8 ecion Edior: Konrd Jedrzejewski, Wrsw Universiy o Technology, Polnd Received y, 6; In inl or July, 6; Pulished Noveer 6 Copyrigh: 6 IEKO. This is n open ccess ricle disriued under he ers o he Creive Coons riuion. License, which peris unresriced use, disriuion, nd reproducion in ny ediu, provided he originl uhor nd source re credied Corresponding uhor: Dušn grež, e il: dusn.grez@e.uni lj.si. INTRODUCTION In he ls decde, considerle reserch hs een crried ou on he nlysis o eicien ehods cple o ccurely esiing preers o he requency coponens o ineres [], []. In ny cses he prole o evluing he specrl perornce o given periodic signl reduces o he esiion o preers o ech specrl coponen (requency, pliude, nd phse). Preer s esiions o periodic signls osly se on spling nd cquiring digil vlues o sples y nlog-o-digil converers. In his procedure vlues o spling poins re resuls o verging in he perure ie esureen ie. This verging (or inegrion) gives reducion o noise u cuses syseic errors in esiions o he signl preers [] [5]. In his pper, lgorihs or esiion o preers y signl nd zero pdding irs nd hen inerpolion in he requency doin re presened. spling process cn e odelled wih our signls nd heir requency specr in he ie nd he requency doin: esured signl g( G( ) (igure ), ipulse response o he spling chnnel h( H ( ) (igure ), spling uncion s( ( ) (igure ), nd he window uncion o he esureen inervl w( W ( ) (igure 4) where snds or he ourier rnsorion ro ie doin o requency doin nd vice vers. g igure. esured signl. h x igure. Ipulse response o he spling chnnel. G H x CT IEKO Noveer 6 Volue 5 Nuer 47

2 The esured signl g( is lwys ilered y he ipulse response o he inpu ron circui o he esureen chnnel represened y h ( in igure 5 nd er h spled y he rel ini durion spling pulses represened y h ( convolved on he spling uncion s ( (igure : ypiclly ie uniorly disriued) nd inlly only inie nuer o sples is ken ino considerion represened y he window uncion w ( (igure 4) o ge ilered, spled nd * windowed signl g, w ( (igure 5). Considering h he equivlen o uliplicion in he ie doin is convoluion in he requency doin n s ( ) ( n ) ( g( G ( ) G( g ) d nd vice ( ) vers g( ) g ( )d G ( ) G( ) [6] he spling procedure cn e odelled s ollows (igure 5): n n igure. pling uncion in he ie nd he requency doin. w g T T T igure 4. Window uncion o he esureen inervl. W T I is eviden h he specru o he spled signl srs o chnge wih odiicion o he spling pulses h( H ( ) (igures 6 nd 7).. PDDING ND ETITION O PRETER pling y he requency o he periodic nd liied signl g coposed o coponens cn e expressed s w n sin n wih, nd s requency, pliude, nd phse o priculr coponen, respecively. In he esiion procedure one hs o ke ino ccoun h vlues o sples (igure 8: line d) re represenives in he perure ie p or ypiclly verge vlues o he signl in he perure inegrion inervl (igure 8: line c). or deonsrion o spling, in igure 8 hree periods o he one coponen sine signl re presened wih duy cycle o spling D p. 4 nd wih spling rio o r T. 6. Wih spling represenives verge vlues we lose soe inorion o he signl nd especilly in he cses where he perure ie is so long h he signl chnges signiicnly g h h s g s g w g, w T g h s h w G H H W g s G g G g, w G, w igure 6. ignls in he spling process. G H H x x G W G G, w Nyq igure 5. Procedure o spling in he ie nd he requency doin. igure 7. pecr o signls in he spling process. CT IEKO Noveer 6 Volue 5 Nuer 48

3 g g x T d c p igure 8. ignls o spling y verging in he perure ie: originl signl, runced signl in he perure inervls, c verge signl vlues in he perure inegrion inervl, d verge vlue represenives o he spled signl in he perure inegrion inervl. in his inervl nd where he spling Nyquis condiion is no ulilled ( r ). One possiiliy o overcoe hese proles is o dd zeroes (known s zero-pdding echnique [7]) nd verge vlues in he perure ies wih he duy rio D p (igure 8: line c). The perure ie is posiioned syericlly round he known spling insns. By incresing he nuer o sples nd cquired signl cycles i is possile o deec signls elow he Nyquis condiion (igure 9: he whole cquired signl conins round 5 cycles o he sinusoid N 5 on N poins). n underspled sine wve sill ppers s spled sine wve u lower requency k ( k,,... ) or expressed y he relive requency k where is he relive spling requency. Wih pdding poins we pprenly increse he spling requency in relion o he signl requency. To esie preers o he ie-dependen signls conining ny periodiciy, i is preerle o use rnsorion o he signl in he requency doin. The discree ourier rnsorion (DT) o he windowed signl wk gk on N spled poins he specrl line i is given y: G d s c d c 5 s 5 s s s igure 9. pecr o signls ro igure 8: originl signl, runced signl in he perure inervls, c signl wih verge vlues in he perure inegrion inervl, d signl wih verge vlues in he perure inegrion inervl wih sinc correcion. j j -j G i W i e W i e, () where i is he requency coponen reled o he se requency resoluion T N nd consiss o n ineger pr nd he non-coheren spling displceen er inie esureen ie is source o dynic errors, which re shown s lekge prs o he esureen window specru convolved on he specru o he esured-spled signl (igure 9). The long-rnge lekge conriuions cn e reduced in ore wys: y incresing he esureen ie T, y using windows wih ser reducion o he es side loes (like he Rie-Vincen windows clss I - RV, ec. [8]), or y using he uli-poin inerpoled DT lgorihs. Dediced lgorihs re needed o oin he correc preers o he sinusoidl coponens in he signl. Preers o he esureen coponen cn e esied y ens o inerpolion 9. ro coprive sudy i cn e concluded h he key or esiing he hree sic preers is in deerining he posiion o he esureen coponen i eween he DT coeiciens G ( i ) nd G ( i ) surrounding he coponen. In esiions, he well-known expressions or he hree-poin esiions or requency (), pliude (), nd phse (4) were used. The hree-poin DT inerpolion gives opil resuls owing o: syery round he locl pek pliude DT coeicien; equl suppression o lekge coing ro oh sides; equl inil error curves s wih one-, ive- nd uli-poin inerpolions. Only he order P o he windows hs o e chnged using RV windows [9]. Gi Gi Gi Gi P () G i l P! sinπ Gi Gi Gi P π P l The single phse cn e esied wih he rguens o he rg G [9], []: hree lrges locl DT coeiciens 4 i i i i i π. (4) 6. EVLUTION O THE ETITION.. yseic ehviour We cn esie hree sic preers o priculr coponen (rue or ppren on igure 9) y he hree-poin inerpolions since we need only he locl lrges DT coeiciens. The esiion errors were copred or he requency (), pliude (), nd phse (4) esiions using he Hnn window ( P ). The solue errors o he requency esiion E ( ) nd he phse esiion es. rue E ( ) es. rue, nd he solue relive errors o he pliude esiion e ( ) es. rue re irs checked or one sine coponen in he signl wih doule scn, vrying speciic spling preers nd he phse o he signl () CT IEKO Noveer 6 Volue 5 Nuer 49

4 priculr relive requencies ecuse he long-rnge lekges re requency- nd phse-dependen (igures o :, nd π π, π 8 ). irs, he spling rio r ws chnged o ind inervls where he inerpolion lgorih cn e used (igures o 9). The solue xiu error vlues (ro 9 ierions chnging phse) given spling rio re copred using he duy rio o D p. 4. In igure, we cn see h i is possile o esie he requency o one coponen even eer hn i we hve coplee signl lso when he spling rio r is in inervl eween nd (he spling condiion is no ulilled). The viciniies round he ineger vlues nd where we cnno esie preers s in he cse o he originl signl depend on he nuer o signl cycles in he whole esureen inervl T. The lrges vlue gives eer requency resoluion nd orders coe closer o ineger vlues nd (igures,, nd ). The widh o he error esiion in-loe round ineger vlues depends on he posiion o he invesiged coponen nd inerspcing eween neighouring coponens i we hve enough spling poins in one period (4 poins re used in siulions ro nlysis in igures nd ). I we hve 5 cycles in he esureen inervl (igure ) we ge sic in resoluion 5 nd his resoluion gives error in-loe orders.85 r. 5 nd.7 r. round inegers where he requency esiion errors increse due o lekge inluence on he invesiged coponen ro is replics k k, (igures 9 E x - - nd ). I cn e lso noiced h here re error peks which nuer increses elow r due o replics k wih higher vlues o k, 4,.. (igure ). Decresing he in resoluion o increses unusle inervls o.8 r. nd.5 r. 5 (igure ). Going in opposie direcion y incresing he in resoluion o (igure ) reduces unusle inervls o.9 r. nd.85 r.5, nd he requency cn e ccurely esied lso in he inervl o spling rio.. 85 wh is elow he spling condiion. In he cse o he pliude esiion (igures nd 5) we need o correc he esied pliude or he coplee pliude DT specru (igure 9, line d) y he well-known sinc correcion kcorr-sinc π p T sinπ p T []. We see he se ehviour o he errors in he cses o pliude nd phse esiions s wih he requency esiion (igures o 6). in resoluion o 5 reduces unusle inervls.85 r.5 nd.8 r. or he phse esiion s or requency esiion (igure 4) nd even eer or he pliude esiion (igure : unusle inervls.9 r. nd.7 r. ). resoluion o urher reduces unusle inervls o.9 r. 8 nd.85 r.5 (igures 5 nd 6) E x... k k k igure. solue xiu vlues o errors o he requency esiion in relion o he spling rio: originl signl, signl wih verge vlues; N k, 5Hz 6Hz, 5. E x igure. solue xiu vlues o errors o he requency esiion in relion o he spling rio: originl signl, signl wih verge vlues; N 8k, Hz 64Hz,. igure. solue xiu vlues o errors o he requency esiion in relion o he spling rio: originl signl, signl wih verge vlues; N 4k, Hz Hz, e x.5.5 igure. solue xiu vlues o relive errors o he pliude esiion in relion o he spling rio: originl signl, signl wih verge vlues; N k, 5Hz 6Hz, 5. CT IEKO Noveer 6 Volue 5 Nuer 5

5 - E x deg - -5 E x k ; k ; - k ; k ; igure 4. solue xiu vlues o errors o he phse esiion in relion o he spling rio: originl signl, signl wih verge vlues; N k, 5Hz 6Hz, 5. igure 7. solue xiu vlues o errors o he requency esiion in relion o he spling rio: originl signl, signl wih verge vlues; N k, 5Hz 6Hz, 5. e x e x igure 5. solue xiu vlues o relive errors o he pliude esiion in relion o he spling rio: originl signl, signl wih verge vlues; N 4k, Hz Hz,. igure 8. solue xiu vlues o relive errors o he pliude esiion in relion o he spling rio: originl signl, signl wih verge vlues; N k, 5Hz 6Hz, 5. E x deg E x deg igure 6. solue xiu vlues o errors o he phse esiion in relion o he spling rio: originl signl, signl wih verge vlues; N 4k, Hz Hz,. igure 9. solue xiu vlues o errors o he phse esiion in relion o he spling rio: originl signl, signl wih verge vlues; N k, 5Hz 6Hz, 5. lgorihs were nlysed lso y he uli-coponen signl. The second hronic coponen s he closes nd he os disuring coponen ws dded wih pliude ( THD. ) nd phse, oher preers o siulions re he se s or igure 8. We ge he os disuring replics o he second hronic posiions r. 5 nd due o he lekge eec k ( k, ) (igures 7 o 9). iulion resuls show h he proposed esiion procedures give very good resuls when he spling condiion is ulilled, u eyond he Nyquis requency he THD hs o e low (. ). I we chnge he duy rio D p in he spling inervl (chnging he perure ie ixed spling requency) he esiion errors do no chnge very uch in coprison o he esiion o he originl signl i we correc he esiion y sinc correcion []. In igures nd he duy rio ws chnged los in he whole possile inervl D..998 r. 6 wih N 4 P CT IEKO Noveer 6 Volue 5 Nuer 5

6 E x 5 E x igure. solue xiu vlues o errors o he requency esiion in relion o he duy rio: originl signl, signl wih verge vlues in he perure inervl. p igure. solue xiu vlues o errors o he requency esiion in relion o he nuer o spling poins in he period: originl signl, signl wih verge vlues in he perure inervl. N p.4. e x 5 - e x p igure. solue xiu vlues o relive errors o he pliude esiion in relion o he duy rio: originl signl, signl wih verge vlues in he perure inegrion inervl wih sinc correcion N p sples in one period (oher preers were he se s in igure 8). The esiion lso uch depends on he nuer o poins in he period N P nd in he whole esureen inervl. Pdding wih ore poins (verge signl vlues nd zero vlues) will iprove esiions since inerpolion equions (), (), nd (4) re derived or lrge nuer o poins N [9]. In igures nd, he nuer o poins in he period N P ws chnged ro N P o ore hn N P 5 (oher preers were he se s in igure 8). We cn see h he requency esiion does no dier signiicnly i we hve reduced he inorion o he signl (igure : curves nd ) nd oh esiions errors decrese wih incresing nuer o poins. The pliude esiion uch ore depends on he nuer o poins (igure ) u er hving ore hn N 8 P poins per period he esiion errors drop o he level s wih he esiion o he originl signl wihou verging in he perure ie... Noise propgion The price or he eecive lekge reducion is in he increse o he esiion uncerinies reled o he unised Crér-Ro ounds ixed y he signl-o-noise-rio or priculr coponen NR corruped y whie noise wih sndrd unceriny 4. In igures 4 nd 6, here re sndrd uncerinies o he requency, pliude, nd phse esiions reled o he CR ounds, respecively. NR N CRB,, (5) igure. solue xiu vlues o relive errors o he pliude esiion in relion o he nuer o spling poins in he period: originl signl, signl wih verge vlues in he perure inegrion inervl wih sinc correcion. CRB,, (6) NR N NR N CRB,. (7) oving wy ro inegers o he relive requency, wh is he cse when he spling rio is elow, he sndrd uncerinies increse in relion o he inil inle vlues (igure 4 or he requency esiion, igure 5 or he pliude esiion, nd igure 6 or he phse esiion) u hese chnges cn e negleced. In he cse o pliude esiion he sndrd deviions even decrese i he requency is esied irs. CRB, igure 4. ndrd unceriny o he hree poin displceen esiion () reled o he CR ound (5). CT IEKO Noveer 6 Volue 5 Nuer 5

7 .6.4 CRB, - - E x igure 5. ndrd unceriny o he pliude hree poin esiion () reled o he CRB (6); is esied, is known. CRB, igure 7. solue xiu vlues o errors o he requency esiion in relion o he spling rio; n s.p. 8, N k, khz, e x igure 6. Rios o he uncerinies o he phse hree poin esiion (4) reled he CRB (7); is esied, is known. 4. EXPERIENTL REULT To deonsre he proposed lgorihs in reducing oh esiion errors (phse nd noise conriuions dieren requencies) in rel esureen environen we use digiizing voleer o cquire signl, gilen 458 [5] nd sle volge generor, Keysigh 5B [6] o genere noinl sine volge wih pliude V u, n ( THD.4% ) nd chnging requency. In opposie o siulions, he spling requency ws se khz μs nd he signl requency ws chnged ro khz down o.5 khz o chieve he se spling rio r... The ixed perure ie p 4 μs deerined he duy cycle o spling D p.4 nd wih n s.p. 8 cquired spling poins he esureen ie ws T ns.p. 8 s wih requency resoluion o T 5Hz. s in he siulions, N 4 pdd pdding poins (or signl nd zero pdding) ws used round he cquired poins, nd logeher N ns.p. Npdd k ws used in he inerpolion DT lgorihs. Ech spling sequence o 8 poins ws synchronized using he sync ou erinl o he generor nd he exernl rigger inpu o he voleer. xil esiion errors re shown in igures 7 o 9, where reerence vlues were hose se on he generor (, khz.5 khz nd π π, π 8 ).5.5 igure 8. solue xiu vlues o he relive errors o he pliude esiion in relion o he spling rio; n s.p. 8, N k, khz, E x deg.5.5 igure 9. solue xiu vlues o errors o he phse esiion in relion o he spling rio; n s.p. 8, N k, khz, 8 5. nd wih 5 rils ech requency nd phse. The esiion error ehviours re very close o hose in he siulions. I cn e noiced h syseic error conriuions ehves s expeced nd very sll second hronic coponen is presened in he esureen syse. The requency esiion (igure 7) cn e copred wih he siulion resuls ro igure excep he noise loor is higher nd 4 he level o E x( ) even in he inervl elow he spling condiion The resuls o he CT IEKO Noveer 6 Volue 5 Nuer 5

8 pliude (igure 8) nd he phse esiions (igure 9) re worse in coprison o he siulion resuls (igures nd 4) due o inccure vlues he oupu o he generor nd inccurcy o he spling voleer, u he syseic conriuions o errorss conir he expeced ehviour like wih he requency esiion. 5. CONCLUION The pper proposes lgorihs or he esiion o sic sinusoidl preers (requency, pliude, nd phse o he requency coponen, when he cquired spling poins do no ulil he spling condiion nd soe signl inorion is los due o signl verging in he perure ie. In he proposed procedure, he epy spce eween successive rel spling poins is virully pdded y verge vlues o he rel spling poin in he inervl o knowing perure nd zeroes in he res o he spling inervl. This procedure wih suile lrge cquired cycles in he whole esureen inervl iproves he requency resoluion, nd he inerpoled DT esiion lgorihs cn e doped or priculr requency coponens. In ny cses he nuer o spling poins is liied u y peroring he lgorihs on copuer we cn increse pdding poins nd wih his nering he errors o he level s wih esiion on he heoreiclly originl signl wihou verging in he perure ie nd wih he nuer o poins equl o ll pdding poins. iulion nd experienl resuls show h he preers esiions re possile lso even in he inervl elow he spling condiion. REERENCE [] G. D non,. errero, Digil ignl Processing or esureen yses. Theory nd pplicions, pringer cience, 6. [] D. grež, "Dynics o requency esiion in he requency doin", IEEE Trns. Insr. es. 56 (7) pp. -8. [] R. L. werlein, " pp ccure digil C esureen lgorih", Proc. NCL Workshop yp., 99, pp [4] G.. Kyrizis, "Exension o werlein s lgorih or C Volge esureen in he requency Doin", IEEE Trns. Insr. es. 5 () pp [5] H. E. vn den Bro, E. Houzger,. Verhoeckx, Q. E. V. N. rin, G. Rieveld, "Inluence o pling Voleer Preers on R esureens o Josephson epwise- pproxied ine Wves", IEEE Trns. Insr. es. 58 (9) pp [6] W. cc. ieer, Circuis, ignls nd yses, The IT Press, cgrw-hill, Cridge, New York, 986, pp [7] G. Leus,. oonen, "ei-lind chnnel esiion or lock rnsissions wih non-zero pdding", Records o Con. on ignls, yses nd Cop.,, pp [8]. J. Hrris: "On he use o windows or hronic nlysis wih he discree ourier rnsor", Proc. IEEE 66 (978) pp [9] J. Šreelj, D. grež, "Nonpreric esiion o power quniies in he requency doin using Rie-Vincen windows", IEEE Trns. Insr. es. 6 () pp [] J. choukens, R. Pinelon, H. Vn He, "The inerpoled s ourier rnsor: coprive sudy", IEEE Trns. Insr. es. 4 (99) pp. 6-. [] T. J. ewr, "pecrl lekge errors when using n gilen 458 o esure phse ins power requencies", Proc. NCL Workshop yp. 99, pp [] D. grež, D. Ilić, J. Drnovšek, "Esiion o periodic signl preers y signl nd zero pdding", in Proc. XXI IEKO World Congress/5, Prgue, Czech Repulic, 5, TC4: []. oschi, P. Crone, "Crer-Ro lower ound or preric esiion o qunized sinewves", Proc. IEEE ITC/4, Coo, Ily, 4, pp [4] Widrow B, Kollr I, Qunizion Noise, Cridge Univ. Press, 8. [5] 458 ulieer - User's Guide, Ed. 5, 988-,, gilen Technologies. [6] 5B eries Wveor Generors D hee, EN, 5, Keysigh Technologies. CT IEKO Noveer 6 Volue 5 Nuer 54