1-D Sampling Using Nonuniform Samples and Bessel Functions

Transcription

1 -D Saplig Usig Nouio Saples a Bessel Fuctios Nikolaos E. Myiis *, Mebe, IEEE,Electical Egiee, Ph.D. A & Chistooulos Chazas, Seio Mebe, IEEE, Poesso B A Cultual a Eucatioal Techologies Istitute, Tsiiski 58, Xathi, 6700 GREECE B Deocitus Uivesity o Thace, Xathi, 6700 GREECE ABSTRACT We evelop the th oe Fouie-Bessel seies expasio o -D uctios i the iteval (0,). Hece we establish the saplig theoe o a uctio with -baliite th oe Hakel taso. The latte stateet iplies that the uctio is also Fouie taso - baliite. The saples locatios ae give by the oots o th oe Bessel uctios. I aitio, the saplig istace ea the oigi iceases with the oe. Keywos: Fouie-Bessel Seies, Hakel taso, Saplig Theoe.. INTRODUCTION Saplig is a coo aspect o ivese applicatios o sigal pocessig, couicatios, ioatio stoage a epesetatio, bioeical egieeig etc. Nouio saplig ay be cucial i cetai cicustaces []. Howeve its peoace is oe coplicate tha the uio oe. The use o Bessel uctios i saplig ca be oue i the oe liteatue [], [3]. I this essay we pove the th oe Fouie-Bessel expasio o a uctio i the iteval (0,). Usig the pevious expasio we also oulate the esultig saplig theoe o a uctio which is th oe Hakel taso (HT) -baliite. The latte stateet iplies that the cosiee uctio is also Fouie taso -baliite. Fially, siulatio exaples ae give, iicatig the avatages o the heei popose Fouie-Bessel expasio a o the esultig saplig theoe.. EXPANSION OF A FUNCTION INTO A FOURIER-BESSEL SERIES OF ORDER N I ay applicatios o oe sigal pocessig the use o Bessel uctios is peee istea o sie uctios. The latte stateet iplies Fouie taso ipleetatios, saplig aspects etc. I this cotext, the zeo oe Fouie-Bessel expasio o a uctio is kow [3]. We establish heeiate the th oe Fouie-Bessel expasio o a uctio, which is the basis o the coespoig saplig theoe. Theoe : A uctio () ca be expae ito a th oe Fouie-Bessel seies i the iteval (0,a) as ( ) i ci J ( i), 0 < < () whee J () is the th oe Bessel uctio o the ist ki. I eq.() i s esult o i yi J ( i) 0 i,,... () with y i the oots o the uctio J (); the coeiciets c i ae give by ( ) J ( ) i J + i) 0 ci (3) Poo: I oe to pove this theoe -eq.()- we ollow the ext steps. - We poit out iitially that the uctios J ( i ) o a othogoal set i the iteval (0,). - We evaluate the coeiciets o the expasio, c i. * Aess all coespoece to P. C. Chazas, Deocitus Uivesity o Thace, Xathi, 6700 GREECE

2 - We o the patial su N N ( ) c J ( ) (4) a show that N () as N. We ay expess the coeiciets c i o the expasio i tes o the th oe HT, ( ω ), o (), whee ( ), 0 < < ( ) (5) 0, > The th oe HT o () is give by [3] ( ) J ( ω) (6) Fo eqs.(3) a (6) it ollows that c 0 ( ) (7) J ( ) + 3. NTH ORDER FOURIER-BESSEL NONUNIFORM SAMPLING THEOREM Nouio saplig ay use uevely space saples o a uctio o o liea uctioals o it (o exaple, saples o uctio s eivatives) [3]. Fouie-Bessel expasio o oe is the basis o the establishet o a ew schee o ouio saplig. The saples, pescibe by this saplig schee, ae locate o the oots o the th oe Bessel uctios, o the ist ki, J (). The saple uctio ust be Hakel taso a-baliite. Theoe : We cosie a uctio () which has its th oe HT -baliite, i.e, ( ω ) 0, ω > (8) I the saples o () ae kow o the oots o the th oe Bessel uctio J (), the the uctio () ca be ecovee o these saples as ( ) J ( ) J + ( ) ( ) (9) whee ae eie as i eq.(). Eq.(9) os the saplig theoe usig saples o a uctio o the oots o J (). Poo: We ca expa the uctio ( ω ) ito a Fouie-Bessel seies o oe i the iteval (0,). Iee, o eqs.() a (7) it ollows that ( ( ω ) J ( ) ( ) ω p ω J + ) ) (0) whee p (ω) is a pulse o with. The ivese th oe HT, IH {}, o the uctio J ( ω) p (ω) is [3, p.74] IH { J ( ω) p } [ J ( ) J ( ) J ( ) J ( ) ] + Sice J ( )0 it ollows that IH + { J ( ω) p } [ J ( ) J ( )] + () () Fo eq.() a the ivese th oe HT o eq.(0), eq.(9) esults. () x i Coet: It is kow that the zeos o J (x) ( ) ( ) satisy the equatio x. Theeoe the x i > i ube o saple poits equie i the iteval (0,c), c is a abitay costat, eceases with. This was aticipate sice a th oe HT - baliite uctio is also (-)th HT - baliite, i.e., the set o th oe HT - baliite uctios is a subset o the (-)th oe HT -baliite uctios. 4. NUMERICAL APPLICATION I oe to peo the pevious state Fouie-Bessel expasio o oe a the

3 esultig saplig theoe we use the aily o uctios + J + ) ( ) (3) Fo 0 the kow sob uctio esults. We ae the uctio () sob-. The sob- uctio has th oe HT which is -baliite a give by [4] ω p (4) whee p (ω) is a pulse o with. The chose sob- uctio is the equivalet o the sic uctio use i uio saplig. We illustate the saplig theoe o 4. Cosequetly o eq.(3) the uctio J 5 ) ( (5) 5 esults, with outh oe HT 4 ω p ( ω ) (6) Fo ueical evaluatio we choose equal to π. 4. Evaluatio Iitially we acquie saples o () usig the saplig theoe. The acquie saples ae chose to be 56 a, accoig to the theoe, they ust be locate o the oots o J 4 (). I oe to iicate the valiity o the heei popose saplig theoe, the th oe Hakel taso o the itepolate uctio shoul be evaluate. We itepolate theeate the acquie saples usig eq.(9) a pouce 04 equiistat poits o the uctio. A ast HT (FHT) algoith base o FFT ipleetatio [5], gives the outh oe HT o the itepolate uctio. Theeoe we obseve the valiity o the saplig theoe o the oots o J () - eq.(9). The obseve oscillatios Gibbs pheoeo ea the equecy ωa ae ue to the tucatio. 4. Evaluatio We copae the covetioal saplig theoe with the saplig theoe o the oots o J (). We cosie the uctio o the ist evaluatio -eq.(5)- a we assue that we have oly 6 saples o it. Regaig the covetioal saplig theoe, the saples ae locate o equiistat poits, aely o k kπ/, k0,,...,5. I the case o the oot saplig the saples ae locate o y k /, k,,...,6 whee y k ae the zeos o J 4 (). The itepolate uctios obtaie via the covetioal saplig theoe a via the saplig theoe o eq.(9) ae plotte i Figs.(a),(b) espectively. We obseve that the uctio obtaie by eq.(9) is a bette appoxiatio o the oigial uctio. This was expecte sice we use the act that the uctio is outh oe HT -baliite, which iplies that the uctio is also Fouie taso - baliite. 5. CONCLUSION We evelope i this pesetatio the th oe Fouie-Bessel seies expasio o a uctio i the iteval (0,) a the coespoig saplig theoe. The latte theoe egas saples o a uctio () o the oots o the Bessel uctio J (). The uctio () is cosiee to be th oe HT -baliite. The saple poits equie i the iteval (0,c) eceases with the oe. May applicatios o this theoe ay be coute i ivese aeas o sigal pocessig, ioatio epesetatio, couicatios, etc. Fig.a) illustates the oigial uctio, sob-4, - eq.(5) - a Fig.b) its coespoig aalytical outh oe HT - eq.(6). I Fig.c) we show the outh oe HT (evaluate by a FHT) o the itepolate uctio usig the Fouie-Bessel ouio saplig theoe.

4 REFERENCES FIGURES [] N.E.Myiis & C. Chazas, "Raial Tajectoies i MRI: Uio a Bessel Nouio Saplig" 7th Meetig o ESMRMB, Pais, Face, MAGMA, vol., suppl., p.39, Septebe 000. [] N.E.Myiis & C. Chazas, Saplig o Cocetic Cicles, IEEE Tas. o Meical Iagig, vol.7, o., pp.94-99, Apil 998. [3] A.Papoulis, Systes a Tasos with Applicatios i Optics. New Yok: McGaw-Hill, 968. [4] J.D.Gaskill, Liea Systes, Fouie Tasos, a Optics. New Yok: Joh Wiley & Sos, 978. Fig.a): The oigial uctio sob- (4) [5] W.E.Higgis & D.C.Muso, A Algoith o Coputig Geeal Oe Hakel Tasos, IEEE Tas. o Acoustics, Speech, Sigal Pocessig, vol.assp-35, pp.86-97, 987. Fig.b): The aalytical outh oe HT o sob-4 uctio. Fig.c): The evaluate outh oe HT o the itepolate sob-4 uctio usig FHT a 56 iitial saples.

5 Fig.(a): The itepolate sob-4 uctio usig covetioal saplig a 6 saples. Fig.(b): The itepolate sob-4 uctio usig the saplig theoe o the oots o J 4 () eq.(9)-a 6 saples.