Modification of raised cosine weighting functions family

Transcription

1 Compuol Mehod d Expermel Meureme XIV 9 Modfco of red coe eghg fuco fmly C. Lek,. Klec & J. Perk Deprme of Elecroc, Mlry Uvery of echology, Pold brc Modfco of he ko fmly of red coe eghg fuco h he poer of by ug covoluo opero h uxlry recgulr do hvg vrble duro me codered he pper. Such modfed fuco fmly derved geerl d exc form for he defed eghg fuco me d frequecy dom ell. I doe h he help of he covoluo do co legh ype prepro echque. he derved eghg fuco ppled for rdr chrp gl yhe h oler frequecy modulo (LFM). Choe exmple of he mulo reerch of he dcued modfed eghg fuco fmly feure d rdr gl of LFM ype yheed hk o re preeed he pper. Keyord: eghg fuco, rdr gl yhe, oler frequecy modulo gl, mlobe, delobe, mched flro. Iroduco Rdr gl yhe oe of he mo mpor problem of moder rdoloco. hee gl rmed o oberved pce hould hve very pecfc feure. he o-clled mched fler very pecfc pr of rdr recever. I m k o mxmze SR (Sgl o oe Ro) oupu. I h y rdr rge my be mxmzed oo. I he ce of rdr chrp gl rmo echo gl oberved he recever oupu h hpe of he very hor pule. hk o rdr rge reoluo my be good. produc of receved gl mched flro oupu gl hvg mlobe d delobe. Preece of delobe of coure ued effec becue cue ek echo gl deeco dffculy. I rco o Modellg d Smulo, Vol 48,.pre.com, ISS X (o-le) do:.495/cmem9 9 I Pre

2 3 Compuol Mehod d Expermel Meureme XIV effecve echque of delobe euo uble eghg fuco pplco. hee re o clled eghg do h modfy he mched fler rmce. reuled fler mmch ddol effec of he eghg procedure. I led o he ore vlue of he oupu SR d h y cue rdr rge horeg. pplco of rdr gl of LFM ype lole echque of delobe euo. he prcple of ory phe ued for he LFM gl yhe, deled elehere [ 3]. I uch pproch uble eghg fuco re ppled oo. reul of prcple of ory phe ug ued ccumulo of umercl error c be oberved. I order o decree he koledge cocerg exc, lycl form of he eghg fuco ppled for he gl yhe dered. here re everl ko echque ued for eghg fuco yhe hvg expeced feure mpor from he po of ve of pplco. Oe of hem bed o covoluo opero. o be more prece, here re ko mehod, uch : me covoluo of pre do, decrbed by Hrr [4] d ull [5], mulple me covoluo of recgulr do, decrbed by e [6] d D d Grech [7] or mulple me uo d cro covoluo of oher ell ko do, decrbed by Relj e l. [8, 9]. hee o clled covoluo do hve greer delobe euo d her f decy. Uforuely he oupu gl mlobe dh cree. Modfco of he ko fmly of he red coe eghg fuco by ug covoluo opero of h fuco h uxlry recgulr do hvg vrble duro me codered he pper. I eble feug of he eghg fuco propere. he exc lycl formul of he modfed fuco derved me d frequecy dom. he pper orgzed follo. he ko de of covoluo eghg fuco prepro preeed eco roducory remrk. he exc cloed formul of he modfed red coe eghg fuco fmly he couou dom of me d frequecy derved eco 3. Exmple of choe feure of he modfed red coe eghg fuco fmly re dcued eco 4. here exmple of pplco for rdr gl LFM ype yhe oo. fe eece of cocluo re re eco 5. Covoluo eghg fuco deg Geerl form of eghg fuco h fe me duro c be decrbed produc follo r here: () - codered ulmed me fuco, r () - ury recgulr do follo:, () I rco o Modellg d Smulo, Vol 48,.pre.com, ISS X (o-le) 9 I Pre

3 Compuol Mehod d Expermel Meureme XIV 3 for r () rec, >, () for elehere - recgulr do me duro. Equo () he equvle of he covoluo opero he frequecy dom ( ω) [ ( ω) R ( ω) ], (3) here: ( ω) d R ( ω) re Fourer rform of he d r () repecvely. decrbed by eq (3) Le u coder eghg fuco modfco h bed o covoluo h uxlry recgulr do r hvg vrble duro me () r [ r ] r here () - modfed eghg fuco., (4) he uxlry recgulr do r defed follo for r () rec, >, (5) for elehere here - vrble duro me of he uxlry recgulr do. fcor cue h he eghg fuco mx vlue decrbed by eq (4) depede of he uxlry recgulr do duro me. I reul from he fc h he re uder r ury, decrbed by rdood []. he covoluo opero decrbed by eq (4) equl formul follo he frequecy dom ( ω) [ ( ω) R ( ω) ] R ( ω), (6) here: ( ω) d R ( ω) re he Fourer rform of he d r () repecvely. reul of covoluo opero modfed eghg fuco () dffere from zero h ervl from ( ) up o ( ) d duro me ( ). I order o keep co he fl eghg fuco duro me ( equl ) here ecey for uble clg of eq (4) d (6) me d frequecy ccordg o he Fourer rformo clg propery I rco o Modellg d Smulo, Vol 48,.pre.com, ISS X (o-le) 9 I Pre

4 3 Compuol Mehod d Expermel Meureme XIV x X ( ω), here fcor of cle chge defed follo. reul of eq (4) d (6) clg he geerl form of he modfed eghg fuco he me d frequecy dom re follo: () r r, (7) ( ω) [ ( ω) R ( ω) ] R ( ω). (8) Scled modfed eghg fuco decrbed by eq (7) dffere from zero h he ervl from up o d her duro me. he do fuco receved h y clled co-legh CO do d decrbed by Relj e l. [9]. he preeed mehod ll be ppled for ko d very ueful (mog oher for yhe of he rdr d or gl h oler frequecy modulo, LFM) fmly of coe do. 3 Modfco of red coe eghg fuco fmly geerl oluo Le u coder coe do fmly h poer of (red coe h poer of ) hvg geerl form follo ( ) () ( k k) co rec, (9) here: k - rel prmeer of he fuco, k,, - eger prmeer of he fuco,, - eghg fuco duro me. reul of ko rgoomerc dee pplco he geerl decrpo of he fuco decrbed by eq (9) h he -h poer cheved co rec ( ) (), () I rco o Modellg d Smulo, Vol 48,.pre.com, ISS X (o-le) 9 I Pre

5 Compuol Mehod d Expermel Meureme XIV 33 here: k k k k,,, x he bgge eger o hgher h x, for odd, for eve,, 3, 5,..., 4, 6,... x, x y - boml coeffce. y reul he geerl ce he eghg fuco decrbed by eq () he me dom um of co compoe d coe fuco lmed me d cled mplude. her perod ol ubmulple of he fudmel perod h equl. Coderg he eghg fuco geerl decrpo preeed he eq () he frequecy dom d bed o he eq (6) oe c ob Ug Fourer rform pr oe c cheve [ ] ( ) ( ω ) ( ω) R ( ω) δ( ω), coω δ( ω ω ) δ( ω ω ) ω δ ω δω δω fer Fourer rform of he recgulr do pplco rec ω S, here S (). deoe fuco hpe of ( x) ( x) x,. (). () S, d ug del drbuo propere poble o cheve he fl, geerl form of he coe fuco fmly h poer of he frequecy dom ω ω S S ω S ω. (3) h fuco um of fuco of he ( x) x ype h prmeer d loco o he rd frequecy x deped o he poer d he do duro me. I rco o Modellg d Smulo, Vol 48,.pre.com, ISS X (o-le) 9 I Pre

6 he geerl form of he modfed eghg fuco he me dom, defed for epre me ervl olved hk o covoluo opero of he fuco preeed he eq () h he uxlry recgulr do decrbed by eq (5): for < (), (4) for < () co S, (5) for (). (6) I order o fulfl he codo eq (4), (5) d (6) ormlzo opero doe o he coeffce h deped o S. (7) fer clg me by ug coeffce, 34 Compuol Mehod d Expermel Meureme XIV 9 I Pre I rco o Modellg d Smulo, Vol 48,.pre.com, ISS X (o-le)

7 he me ervl border d he fl geerl form of he modfed eghg fuco he me dom re follo: for < (), (8) for < () co S, (9) for < (). () he ormlzo coeffce decrbed by eq (7) he me. From he eq (8), (9) d () reul h he modfed eghg fuco lope he me dom re creed by he um of he co compoe, ler fuco d e fuco. Modfco of he eghg fuco cerl pr me clg mplude. reul of he modfed eghg fuco exc vero lye he frequecy dom d bed o he eq (6) oe c ob ω ω ω R. () Compuol Mehod d Expermel Meureme XIV 35 9 I Pre I rco o Modellg d Smulo, Vol 48,.pre.com, ISS X (o-le)

8 36 Compuol Mehod d Expermel Meureme XIV Ug Fourer rformo of he uxlry recgulr do ( ω) eq (3) d fer he ormlzg coeffce pplco ( ) ( ω) R d oe c cheve ω ω ω S S S S ω. () ω S S ω he fl geerl decrpo of he modfed eghg fuco fer clg opero ug follo ( ) ( ω) ω ω S S ω S S ω ω S S ω.(3) From he eq (3) reul h he uxlry eghg do repoble for mplude modulo of he modfed eghg fuco epre compoe d cue chge of her lobe dh d loco o he rdl frequecy x. 4 Smulo reul mulo model of he modfed eghg fuco geeror d he pule rdr gl geeror prepred for eg. I doe Lbdo CVI (ol Irume) evrome. hk o he model boh eghg fuco d geered rdr LFM gl feure e re poble. he fluece of he duro me of he uxlry recgulr do o he geered gl chrcerc he m k of reerch. reul umber of chrcerc ere obed. fe of hem re preeed belo. ormlzed mgude pecr of he modfed eghg fuco for dffere vlue of he poer of preeed o fg.. Vlue of he prmeer k d ere fud he erve y order o cheve m level of delobe. ormlzed mgude pecr of he o-modfed eghg fuco d recgulr do re preeed oo o commo fgure for compro. he fr oe prepred for he me vlue of he prmeer d for opmum vlue of he prmeer k h gve he mmum level of delobe. cocluo c be dr h hk o he modfco opero he delobe level depedece o he vlue, decree from bou 4 d up o 7 d h relo o he ce of he o-modfed eghg fuco. he delobe level euo cheved he co of he mlobe brodeg. For exmple, comprble delobe level c be obed for he modfed do h d he o-modfed do h 6, bu he ecod ce he mlobe dh -3 d level.35 me der. I rco o Modellg d Smulo, Vol 48,.pre.com, ISS X (o-le) 9 I Pre

9 Compuol Mehod d Expermel Meureme XIV 37.. ormlzed mgude pecr [d] ormlzed mgude pecr [d] recgulr o modfed modfed ormlzed frequecy recgulr o modfed modfed ormlzed frequecy ormlzed mgude pecr [d] recgulr o modfed modfed ormlzed frequecy ormlzed mgude pecr [d] ormlzed mgude pecr [d] ormlzed mgude pecr [d] recgulr o modfed modfed ormlzed frequecy recgulr o modfed modfed ormlzed frequecy recgulr o modfed modfed ormlzed frequecy Fgure : ormlzed mgude pecrum of he modfed, o-modfed, d recgulr do for dffere. Deg of he rmed LFM rdr gl doe he ex reerch ep. he mched fler oupu gl gve by he pu fler gl uocorrelo fuco. he gl uocorrelo fuco deermed by he vere Fourer rform of he eergy pecrl dey. So o geere LFM gl h uform mplude oe c ue he gl h he mgude pecrum decrbed by qure roo of he eghg fuco, red coe h poer of for exmple. Fdg he oler frequecy l ug he ory-phe prcple uggeed by Fole [], Cook d erfeld []. h mehod ppled h pper. reul ome chrcerc ere obed. umber of hem re preeed here. ormlzed mgude of he LFM gl oberved he mched fler oupu veru ormlzed o pule duro me re depced fg. d fg. 3. I rco o Modellg d Smulo, Vol 48,.pre.com, ISS X (o-le) 9 I Pre

10 38 Compuol Mehod d Expermel Meureme XIV he chrcerc re ho for he mgude pecrum decrbed by omodfed d modfed eghg fuco d for o dffere vlue of he produc. ormlzed oupu gl mgude [d] ormlzed me ormlzed oupu gl mgude [d] b ormlzed me Fgure : ormlzed mgude oupu mched fler gl for he omodfed (. k. 3, ) d modfed (b. k. 53,.535 ) eghg fuco, d. ormlzed oupu gl mgude [d] ormlzed me ormlzed oupu gl mgude [d] b ormlzed me Fgure 3: ormlzed mgude oupu mched fler gl for he o modfed (. k. 9, ) d modfed (b. k. 94,.45 ) eghg fuco,. reul of he recurve fdg of he mmum delobe level by vlue of he k d prmeer chgg he.8 d mproveme of h level ere obed for d.8 d for 3 repecvely. 5 Cocluo he m gol of he pper dervo of he geerl, lycl formul of he modfed eghg fuco d pplco for LFM ype rdr gl yhe. he modfco eece he ko eghg fuco fmly of red coe ype covoluo h he uxlry recgulr do hvg vrble chgebly duro me. he recgulr uxlry do duro me eleco gve he opporuy for feure fe ug. reul decree of delobe level from 4 d up o 7 d, h muleou d expeced mlobe dh cree for choe prmeer I rco o Modellg d Smulo, Vol 48,.pre.com, ISS X (o-le) 9 I Pre

11 Compuol Mehod d Expermel Meureme XIV 39 cheved. Moreover he LFM rdr gl prmeer mproveme oced he oupu of he mched fler. he cle of h effec relvely lo d deped o he produc. I orh o dd h he decrbed mehod my be ppled for fe ug oher ype of eghg fuco. he modfed eghg fuco decrbed he pper my be ued o oly for rdr gl yhe bu for hrmoc ly d fler deg oo. ckoledgeme h ork uppored by he ol Ceer for Reerch d Developme for he yer 7- uder Commoed Reerch Projec PZ-MS- DO-4/I/7. Referece [] Fole, E.., he Deg of FM Pule Compreo Sgl. IEEE rco o Iformo heory, I-, pp. 6-67, 964. [] Cook, C.E. & erfeld, M., Rdr Sgl, Iroduco o heory d pplco, rech Houe, oo d Lodo, pp , 993. [3] Levo,. & Mozeo, E., Rdr Sgl, Joh ley & So, Hoboke, pp , 9, 9, 4. [4] Hrr, F.J., O he Ue of do for Hrmoc ly h he Dcree Fourer rform, Proceedg of he IEEE, 66, pp. 5-83, 978. [5] ull,.h., Some do h Very Good Sdelobe ehvour, IEEE rco o couc, Speech, d Sgl Proceg, SSP-9, pp. 84-9, 98. [6] e, P., F d Hgh-Preco Meureme of Dored Poer ed o Dgl Flerg echque, IEEE rco o Irumeo d Meureme, 4, pp , 99. [7] D, X. & Grech, R., Qu-Sychroou Smplg lgorhm d pplco, IEEE rco o Irumeo d Meureme, 43, pp. 4-9, 994. [8] Relj, I.S, Relj,.D., Ppć, V.D. & Koć, P., e do Fuco Geered by Me of me Covoluo - Specrl Lekge Error, Proc. of he 9h Coferece MELECO, el vv, pp , 998. [9] Relj, I.S, Relj,.D. & Ppć, V.D., Exremely Fl-op do for Hrmoc ly, IEEE rco o Irumeo d Meureme, 56, pp. 5-4, 7. [] rdood D., Fourer rform Rdr d Sgl Proceg. rech Houe, oo d Lodo, pp. 4-53, -5, 3. I rco o Modellg d Smulo, Vol 48,.pre.com, ISS X (o-le) 9 I Pre