SIMULTANEOUS AJ AND LPD EVALUATIONS FOR SECURE COMMUNICATION

Transcription

1 SIMULTANEOUS AJ AND LPD EVALUATIONS FOR SECURE COMMUNICATION Mu-Kig Tsay ad Chie-Hsig Lia Departmet f Cmmuicati Egieerig, Natial Cetral Uiversity, Taiwa (ROC) ABSTRACT I this paper, a systematic apprach fr evaluatig the iteractis f the lik gemetry fr ati-jammig, lw prbability f detecti, ad their cmbied applicatis are ivestigated. I may typical cases, cmmuicati desigs have t deal simultaeusly with adversary threats f bth active jammig ad passive detecti. Ad by icreasig cmmuicati pwer t cuter jammer ad ehacig ati-jammig (AJ) capability must be weighted carefully agaist the icreased threats f deteriratig lw prbability f detecti (LPD) capability ad beig detected by a uiteded iterceptr receiver. A qualitative ad quatitative apprach with sic-type atea patters beig icluded fr evaluatig bth AJ ad LPD ccurretly is reached. Ad it is ituitive t see that by spreadig sigal spectrum, cmplicatig sigal wavefrms, ad lwerig pwer ctrl ucertaity, respectively, will ehace system perfrmace accrdigly. INTRODUCTION The prblem f hw t get quatitative metrics fr evaluatig ad desigig a secure cmmuicatis system with simultaeus ati-jammig (AJ) ad lw prbability f detecti (LPD) capabilities is f great imprtace. It leads aturally t the develpmet f spread spectrum techiques. Direct sequece (DS) ad frequecy hppig (FH) are tw primary spread spectrum techiques. I additi, the receivig ad trasmissi security achieved by a cmmuicati lik depeds very strgly its lcati relative t a adversary s jammig trasmitter ad itercept receiver, which is classified as gemetry-depedet factrs. I AJ applicatis, it is very imprtat t have a accurate estimati f the prcessig gai (PG) required fr reliable cmmuicatis as a fucti f the lik gemetry. I cases f LPD, the prbability f detecti is sigificatly affected by the iteracti f the lik gemetry ad the relative lcatis f the itercept receiver ad the cmmuicati trasmitter. I may practical cases, cmmuicati desig shuld deal simultaeusly with pssible threats f bth Chug-She Shy ad Tai-Yueh Yag Cmmuicati Research Ceter, Natial Cetral Uiversity, Taiwa (ROC) active jammer ad passive iterceptr. Uder this circumstace, it s t straightfrward t make desig decisis. Wheever cmmuicati system icreases t cuter a jammer, fr example, it will icrease the threat f beig detected r eve itercepted by a uiteded iterceptr receiver accrdigly. O the ther had, wheever cmmuicati system decreases pwer t cuter the threat f a iterceptr, it will als uavidably decrease the jammig-resistat capability. The same is true if pwer is replaced with atea size. Therefre, trade-ffs have t be made if bth AJ ad LPD are eeded. Gle [1] has made a LPI aalysis ad shw the effect f sceari-depedet parameters ad detectability-threshld factrs i jammig ad -jammig evirmets. Gutma ad Presctt [] defied quality factrs based pwer gais ad lsses fr the LPI cmmuicati system t give a physical iterpretati t a grupig f terms btaied frm system lik equatis. Dillards [3] described a metric fr defiig LPD f a cmmuicati sigal by a radimeter based gai differece betwee the cmmuicati receiver ad the radimeter. Weeks et. al. [4] prpsed a methd fr defiig the LPD perfrmace f sme cmmercial-ff-the-shelf (COTS) cmmuicati systems quatitatively based pe surces specificatis ad cmmuicati theries. Mills ad Presctt [5-6] studied the LPI etwrks with multiple trasmitters based received eergy ad detecti area. But simultaeus trade-ffs f bth AJ ad LPD have t bee made frm these papers. I this paper, we will first ivestigate the relative frmulatis fr AJ, LPD, ad their cmbiatis. Firstly, the well-kw rati f jammig sigal t cmmuicati sigal (J/S) plt based a specified AJ perati gemetry will be examied. I a very similar way, the sigal-t-itercepti sigal rati (S/N) i plt will als be examied with a liear iterceptr a specified LPD perati gemetry. Thereafter, their cmbied J/S ad (S/N) i will als be examied the same perati gemetry fr trade-ffs with fixed cmmuicati sigal-t-ise rati. The BFSK ad FH spread spectrum techiques will be illustrated as a example fr the /07/$ IEEE 1f 6

2 aalyses f the cmbied AJ ad LPD. I the mea time, a real sic type (i.e., si(x)/x) atea patter will be draw i fr real simulatis ad aalysis. Sme cclusis will be draw i the fial secti. SYSTEM MODEL The relative gemetry lcatis f ur cmmuicati system (satellite cmmuicati) ad the adversary jammer ad iterceptr are shw i Figure 1, where ur cmmuicati termial is put at the rigi, ad the latter tw are cllcated the same lcatis (x, y, 0). R c is the rage betwee ur cmmuicati system themselves (e.g., R c is assumed t be 36000kM fr a GEO satellite). R j / R i is the jammer/iterceptr rage frm their earth psiti t the satellite. As shw i this figure, e very imprtat factr fr the effectiveess f jammig r itercepti is the relative agle φ ff the mai beam patter f the victim satellite ( the psitis f (0, 0, R)) i the directi f earth jammer/iterceptr. victim satellite relatively easier, if atea shapig r ullig patter desigs are take. Figure shws the relative agles f the jammer/iterceptr ad satellite tilted with the lie f sight f earth termial ad satellite, by which they will get the advatages f the victim satellite atea patter (mailbe) t ehace their jammig r itercepti effects. The well-kw parablic atea pwer gai G (D) ad patter G(φ) (sic type) are give as fllws [7] G ( ϕ) η( πd/ λ) ( πd ( ϕ) λ) Dsi ( )/ J1 si / = π ϕ λ where D is atea diameter, λ is wavelegth, η is atea efficiecy, ad J 1 is first rder Bessel fucti. Figure 3 shws the tw 30/0GHz atea patters fr a 3m sic-type atea. It s gd fr ati-jammig desig (30GHz) but t fr LPD (0GHz) due t its wider patter. (1) Rj = Ri Y-axis(km) Rc Figure. Relative φ agles fr jammer/iterceptr Figure 1. Relative AJ ad LPD perati gemetry Geerally, ur friedly cmmuicati system (termial ad satellite) will direct mai beam patters at each ther t get maximum gai patters, ad the itetial jammers r uiteded iterceptrs will ly be able t cver frm the sidelbe directi, especially whe the peratig frequecy is higher. Figure 1 shws a basic sceari with a itercepted ad jammed satellite ad a iterceptr ad jammer the grud. Wheever the jammig ad iterceptr rage is very far away frm the victims, e.g., a GEO satellite, the iterceptr ad jammer will als get the advatage f the mai beam gai f the Gai(dB) 60 30GHz 50 0GHz Az(deg) Figure 3. A 3m sic-type Ka bad atea patter f 6

3 Jammig-t-Sigal Rati Cturs (J/S) The crdiati system t be used i develpig J/S plt is the same as shw i Figure 1. Ad the J/S rati ca be derived as fllws I this secti, i rder t csider the cmbied AJ ad LPD applicatis ccurretly, we assume that the jammig pwer spectral desity J is very large cmpared t ise pwer desity N, ad if we take BFSK mdulati ad FH spread spectrum techique with errr prbability, P fh, give by [9] ( ϕ ) J PG G R = S PG G R j js sj c t ts st j () R () f i i () 1 Pfh Eb J e 1 (4) where P j ad P t are, respectively, the jammig ad the earth termial pwer; G js ad G ts are the jammer ad earth termial atea gais i the directi f the victim satellite (receiver), respectively; G st ad G sj (φ) are the relative trasmissi atea gais i the directi f the earth termial ad jammer tilted by a agle φ agle, respectively. Equati () ca be classified as system-depedet parameters f( ) ad gemetrydepedet parameters R 1 ( ). Itercepti-t-Nise rati Cturs ((S/N) i ) where E b is bit eergy (S/R b ) ad jammig pwer desity J is give by λ 1 J = PG j jsgsj( ϕ ) J N (5) 4π R j Wss By substitutig equati (5) it (4) ad takig a iverse atural lgarithm f it, the required E b /J is give as fllws Usig the same crdiati system as i Figure 1, the cstat ctur f a liear iterceptr is give by Eb 4π S GtsGstL in i W RR ss i j = J λ N i GisG jsgsi( ϕ) Gsj ( ϕ) PjRb Rc (6) ( ϕ) S S GG LN is si t ft R b Rc = g() i R () i (3) N Li N t GtsGst Li Nfi Wii Ri where G is ad G si (φ) are, respectively, the iterceptr atea gais i the directi f the victim satellite (trasmitter) ad satellite atea gai advatage acquired i the directi f iterceptr tilted by φ agle. They are relatively the same as G js ad G sj (φ) i the J/S case i additi t differet peratig frequecies. L t ad N ft represet termial receivig lie lss (i frt f LNA) ad ise figure, respectively; L i ad N fi represet iterceptr lie lss ad ise figure; W ii ad R b are itercepti badwidth ad data rate (W ii = W ss ); (S/N) t detes the ptimum sigal t ise rati f the earth termial receiver if a pwer ctrl scheme is adpted fr fixed-value. Equati (3) ca als be classified as system-depedet parameter g( ) ad gemetry-depedet parameters R ( ). As t a geeralized liear iterceptr, its itegrati time eeded ca be give by [8] Cmbied AJ ad LPD/I Cmmuicatis (E b /J ) where (S/N) i meas the ptimum iterceptig sigal-t-ise rati f a iterceptr receiver; N i represets utput thermal ise pwer desity f iterceptr receiver ad is equal t kt 0 N fi, where k is Bltzma s cstat ( W/Hz/ K), T 0 is abslute temperature, take as 90 K, ad N fi is a iterceptr receiver ise figure. It s iterestig t te that whe all the ther system-depedet parameters ad R c is fixed, R i varies iversely with R j. Therefre, trade-ffs ca be made fr specific desig r applicati purpses. Furthermre, these tw parameters, R i ad R j, as shw i equati (6), ca be replaced by utilizig their respective expressis f equati () ad (3) as give by R R j ( / ) ( ϕ ) 1 Pj GjsGsj = Rc (7) J S P G G t ts st S N GG R N ( ϕ ) LN t ft W t is si ii i = S GG ts st LN i fi R b Li c (8) 3f 6

4 Ad after a simple maipulati, a mre direct ad cmpact expressi fr a geeralized jammer ad a liear iterceptr is available ad give by ( SN / ) ( SN / ) ( / )( / ) ( / ) Eb i t = (9) J J S S N S N Li t where J/S is iversely related with (S/N) Li with the ther parameters fixed ad (S/N) t ca be give by λ PG t trgrt S 4π Rc = N ktln W t t ft ss Equati (9) ca be further simplified as give by (10) bth AJ ad LPD capabilities. The assumed system parameters values are shw i Table 1. A maximum 7-m jammer atea size ad sic type atea patter gais f G si (φ) ad G sj (φ) are assumed i simulatis. Based equati () ad (3), the ituitive plts f J/S ad (S/N) i are shw i Figure 4 ad Figure 5, respectively, with gemetrically tilted φ agles varied with relative lcatis betwee the earth jammer/iterceptr ad earth termial. We ca see clearly that uder the assumptis f Table 1, they are maitaied a cstat slwly varyig curve withi abut 100kM rage, which shws the gai advatages (satellite mai beam) acquired by the adversaries, i.e., jammer r iterceptr. Withi this gemetric-depedet rage, the system-depedet parameters ca be adjusted furthermre fr either a apprpriate J/S r (S/N) i requiremets. Table 1 System Parameters Assumed Eb 1 1 = = J J S M M M M M ( / ) i t j i t (11) where M j, M i, ad M t are defied as J/S, (S/N) Li /(S/N) i, ad (S/N) t /(S/N) t, respectively. Ad they mea the ratis f effective jammig pwer fr the adversary jammer, effective iterceptig sesitivity fr the adversary iterceptr, ad effective pwer ctrl fr the victim earth termial. E b /J is iversely related t these three parameters. Fr example, wheever ay f these three parameters is icreased, E b /J is lwered, i.e., errr prbability is icreased. O the ctrary, wheever ay f these three parameters is decreased, errr prbability is decreased. Frm this equati (11), it is ituitive fr the victim side that by spreadig sigal spectrum r by frcig jammer t be ut f sight ( M j is decreased), cmplicatig sigal wavefrms ( M i is decreased), ad lwerig pwer ctrl ucertaity ( M t is decreased), respectively, will ehace system perfrmace (lwerig errr prbability) accrdigly. Fr the threat side, i rder t deterirate the perfrmace f this secure victim cmmuicati system, M j, M i, ad M t, shuld be tried t icrease accrdigly. J/S(dB) f / f t η P j P t G js G ts G st R c G is L t N ft L i N fi W ii R b W ss r ( S/ N) t ( S/ N) i 30/0 GHz k W 1k W 64.5 db(7m) db(3m) db(3m) km db(7m) db db db db 100 MHz 10 kbps 100 MHz 10 db -10 db -5 SYSTEM ANALYSIS I this secti, e typical Ka bad GEO satellite example (R c is assumed t be 36000kM) is illustrated thrugh the prpsed desig apprach fr a secure cmmuicati system with cmbied requiremets f Jammer-termial rage(km) Figure 4. J/S rati varyig with relative gemetry 4f 6

5 S/I(dB) Iterceptr-termial rage(km) Figure 5. (S/N) i rati varyig with relative gemetry Wheever the jammer r iterceptr culd t reach this rage, they just fall dw t the ieffective sidelbe regis f satellite atea patter. Therefre, a specific strategy culd be take t expel the apprached jammer r iterceptr beyd this rage. O the ther had, fr the adversary a eve higher jammer r a liear iterceptr ca be applied fr mre effective jammig r itercepti. Figure 6 ad Figure 7 shw the E b /J ratis plts with varyig rage ad cmbied jammig ad itercepti effects. The cmbied AJ ad LPD desig perfrmace ca be examied if a miimum 10-4 FH errr prbability is asked t maitai (E b /J =1.31dB). We ca see clearly that wheever the jammer ad iterceptr is beyd the rage (13kM), ur cmmuicati capability will t be affected if a specific strategy culd be take t expel the apprached threats. Eb/J(dB) P fh P fh Jammer/iterceptr-termial rage(km) Figure 6. Eb/J rati varyig with relative gemetry Figure 7. Eb/J ctur varyig with relative gemetry CONCLUSION I this paper, we have prpsed a systematic apprach with sic-type atea patters beig icluded fr evaluatig AJ, LPD, ad their cmbied applicatis qualitatively ad quatitatively. Ad a ituitive example with sme specific system-depedet ad gemetry-depedet parameters is illustrated, where the iteractis f key system parameters ad lik gemetries fr AJ, LPD, ad their cmbied applicatis are ivestigated. It is ituitive t see that by spreadig sigal spectrum, cmplicatig sigal wavefrms, ad lwerig pwer ctrl ucertaity, respectively, will ehace system perfrmace accrdigly. Mrever, frm this apprach, the sequetial prcedures ad may gegraphically istict results are available, which ca prvide clear ad flexible system desig ad perati trade-ffs. ACKNOWLEDGEMENT The authrs wuld be very grateful fr the research prjects grated frm Natial Sciece Cucil (NSC 95-1-E ), Taiwa. REFERENCE 5f 6

6 [1] A. B. Gle, Lw Prbability f Itercept, IEEE Cmmuicatis Magazie, Vl.1, pp.6-33, [] L. L. Gutma ad G. E. Presctt, System Quality Factrs fr LPI Cmmuicatis, IEEE AES Magazie, Vl.4, pp.5-8, Dec [3] G. M. Dillard ad R. A. Dillard, A Metric fr Defiig Lw Prbability f Detecti Based Gai Differeces, IEEE Thirty-Fifth Asilmar Cferece Sigals, Systems ad Cmputers, Vl., pp , Nv [4] G. D. Weeks, J. K. Twsed, ad J. A. Freebersyser, A Methd ad Metric fr Quatitative defiig Lw Prbability f Detecti, IEEE Military Cmmuicatis Cferece, Vl.3, pp.81-86, [5] R. F. Mills ad G. E. Presctt, Wavefrm Desig ad Aalysis f Frequecy Hppig LPI Netwrks, IEEE Tras. AES, Vl.36, pp , July 000. [6] R. F. Mills ad G. E. Presctt, Detectability Mdels fr Multiple Access Lw Prbability f Itercept Netwrks, IEEE Tras. AES, Vl.36, N.3, July 000. [7] P. C. Jai, Architectural Treds i Military Satellite Cmmuicatis Systems, Prc. Of the IEEE, Vl. 78, N0. 4, July [8] R. Schlcraft, Lw Prbability f Detecti Cmmuicatis-Wavefrm Desig ad Detecti Techiques-, IEEE Military Cmmuicatis Cferece, Vl., pp , Nv [9] B. Sklar, Digital Cmmuicatis, Pretice- Hall, d Ed, pp.18, f 6

7 COMPLETE LC LADDER BESSEL FILTER TABULATION Fu-Nia Ku, Chie-Hsig Lia, ad Mu-Kig Tsay Departmet f Cmmuicati Egieerig, Natial Cetral Uiversity, Taiwa (ROC) ABSTRACT Bessel filter has bee widely used i idustry fr aalg filter circuit desig with its maximally flat time-delay r least grup delay ripple perfrmace. Nevertheless, there has bee ly e set f tabulati per rder (LC elemet umber) available fr the realizati f a specific Bessel filter thrugh LC ladder architecture i cvetial literature. T sythesize a LC ladder Bessel filter thrughly, especially fr higher rder ad mre tabulatis ccers, the prblem f calculati accuracy shuld be precisely ccered t avid eglectig the relatively sigificat but smaller cefficiet values. I this paper we have btaied a cmplete set f tabulati per rder, which ca be geeralized as ( + ) / fr a th rder LC ladder Bessel filter etwrk with duality beig icluded. INTRODUCTION I this paper we are ccered with the prblem f hw t get mre tabulated elemet values per rder (LC elemet umber) fr a specific LC ladder Bessel filter desig due t limitati f ly e set f them listed available fr desig purpse i cvetial literature [1-9]. Althugh the basic Bessel filter desig has bee preseted i may publicatis ad wrked ut by may surces ad preseted i the frm f tabulated elemet values, Jhs et al. als have bee presetig may variatis f the Bessel plymials fr eve better filter perfrmace. Nevertheless, there is still at mst ly e set f tabulated elemet values per rder fr specific filter etwrk desig purpse [10-14]. The basic prcess f trasfrmig a ideal filter it a real filter etwrk is thrugh the well kw prcedures f apprximati ad sythesis. Hwever, it shuld be emphasized that the techical literature cverig these spas several decades i histry ad is vast i scpe ad frmidable i theries t egieers. Therefre, e set f rmalized elemet values per rder fr differet classes f filter etwrk architectures is listed i may hadbks ad a straightfrward arithmetical calculati is btaied fr filter etwrk Chug-She Shy ad Tai-Yueh Yag Cmmuicati Research Ceter, Natial Cetral Uiversity, Taiwa (ROC)) desig. Our ivestigati ad treatmet give fr mre tabulati is quite selective ad limited t LC ladder Bessel filter fr its maximally flat time-delay r least grup delay ripple perfrmace, which is widely used i idustry fr passive ad active aalg filter circuit desigs [15-17]. Furthermre, if fr sychrus basebad digital trasmissi [18], it als has superir iter-symbl iterferece perfrmace, mre rbust t samplig clck jitter, ad higher data rate icrease withut icreasig filter badwidth tha the ther cvetial liear filter classes, e.g., Butterwrth r Chebyshev, fr rectagular iput pulse shape. T sythesize a LC ladder Bessel filter, especially fr higher rder ad mre tabulati ccers, the prblem f calculati accuracy shuld be precisely csidered. The demiatr f Bessel plymial at higher rder has very wide rage f cefficiet values, e.g., 1 ad fr a Bessel plymial f degree 9, s the least sigificat digits shuld be cuted wheever fudametal arithmetic peratis f these cefficiet values are i prcess i rder t avid eglectig the relatively sigificat but smaller cefficiet values. This paper is rgaized it fur sectis. Secti 1 is itrducti. I Secti, the characteristics f the lssless Bessel filter etwrk with fixed resistaces termiated at bth eds will be addressed fr bth matched ad umatched cases ad frmulatis fr cmplete tabulati will be ivestigated as well. I Secti 3, elemet values tabulati fr bth matched ad umatched cases will be listed ad examples will be illustrated fr verificatis as well. Cclusi is draw i the fial Secti 4. LC LADDER BESSEL FILTER SYNTHESIS A lssless filter etwrk with resistace-termiated at bth iput ad utput eds, i.e., R i, ad R, is shw i Fig. 1. The trasfer fucti f this filter etwrk is give by /07/$ IEEE 1 f 6

8 V H ( s) = (1) Vi where V i ad V are iput ad utput vltage fuctis, respectively. Fig. 1. Lssless filter etwrk with resistace termiatis at bth eds Such resistace-termiated etwrks are fte used i matchig prblems where it is desired t maximize the lad pwer at specified frequecies. It is therefre atural t measure the quality f the match i terms f the reflecti pwer P r ad trasmissi pwer P t, the latter beig defied as the maximum pwer that ca be delivered by a vltage surce V s with a iteral resistace R i. I additi, the iput impedace Z 1 lkig it the filter etwrk is t realize a lssless filter etwrk with ladder circuit cfiguratis very straightfrward ad cveietly, which will be addressed later. Sice the filter etwrk is lssless, it is clear that at real frequecies all pwer eterig the etwrk is dissipated i the utput lad R ad ca be give by ( ) ( ) V V V Pt = = = Vi Ri + R s i Ri + R Ri Ri R V Ri R R Fr simplicity ad lss f geerality, equati (1) ca be replaced it equati (), ad we ca defie a real-value trasmissi plymial f degree, e (s), ad express the trasmissi pwer P t as equati (3) ad (4), respectively, as fllws ( R + R ) 1 i e () s = H () s R R i () (3) V Pt = e() s e( s) (4) R Similarly, we ca express the reflecti pwer P r as equati (5), where the left side it is equal t the trasmissi pwer mius the lad dissipated, ad defie a real-value reflecti plymial f degree, f (s), as give i equati (6), respectively V V ( () ( ) 1 ) r ( () ( )) R R P = e s e s = f s f s (5) f () s f ( s) = e () s e ( s) 1 (6) Equati (6) is a basic expressi fr sythesizig a matched (R i =R ) r umatched (R i R ) ladder filter etwrk, where its cmplexity is depedet the trasfer fucti selected, e.g., Bessel plymial, ad the rder required. Nevertheless, i rder t sythesize a ladder filter etwrk mre cveietly, the reflecti cefficiet ρ (s) ca be defied as the rati f trasmissi plymial e (s) ad reflecti plymial f (s), which is give by f() s f() s ρ() s = = (7) e() s e() s Ad the iput impedace Z 1 f the filter etwrk ca be expressed as fucti f the predefied e (s) ad f (s) fuctis by replacig equati (7) it the left side f equati (8) as fllws 1 ρ( s) Z1 = Ri (8) 1 + ρ( s) Bessel Plymial The lw-pass Bessel filter is a all-ple filter havig maximally flat time delay ad th rder trasfer fucti give by H ( s) ( )! B ( 0)! a = = = B ( ) ( )! s + k k q() s s k ( k)! k! k = 0 where B (s) is a Bessel plymial f degree i the variable s ad q (s) is defied as a geeral descedig rder Bessel plymial as give by k = 0 (9) k q() s = ak s (10) Cmplete Tabulati f Bessel Filter The ivestigati f a lssless filter etwrk ad lw-pass Bessel Filter characteristics are addressed as afremetied, i rder t sythesize ad realize a filter etwrk by usig equati (6) ad Bessel trasfer fucti, i.e., equati (9), either fr matched r umatched case, we ca relate the cefficiets f the trasmissi plymial e (s) ad reflecti plymial f (s) t Bessel plymial cefficiets, i.e., a k,ad K parameter as give by equati f 6

9 (11) ad (1), respectively, fr apprpriate sythesis () R + R 1 Z () s Z () s Z () s i 4 1 H s = R R Ri + Z1() s Z3() s Z5() s Z() s (15) e () s = k = 0 a k s K k (11) ( R + R ) 1 s k i f() s = + cks + 1 K k = 1 4 Ri R where c k ca be related t a k ad give by ( k)! a c c c = K ( / ) k m= 1 1 k m k m k k! (1) m+ 1 ( ) k ( ) ( / ) (13) ( ) ( + ) where c 0 = c (-j) = c (+j) =0, j=0,1,, meas lwer limit, ad K is defied ad give by K = 4 Ri R i ( R + R ) a (14) Therefre, it is clearly that the umber f real-value slutis fr the cefficiets f f (s) will be ( ) /. Nevertheless, i rder t sythesize the filter etwrk, we ecuter the calculati accuracy prblem. Because the demiatr f the Bessel plymial at high rder has a very wide rage f cefficiet values, e.g., = 9, B 90 = 1 ad B 99 = , the fudametal arithmetic peratis f ay tw cefficiet umbers, either fr trasmissi r reflecti plymial, shuld be cuted t the least digit t avid eglectig the lwer but critical cefficiet values. The umber f real-value slutis available fr th rder filter etwrk after sythesizig ca be geeralized as ( ) / ( is dd r eve), e.g., 8(eight) slutis will be available fr = 8 r 9. LC Ladder Bessel Filter Sythesis By usig the derived equatis ad sythesized results with mre slutis tha ever befre, the iput impedace r admittace see it this filter etwrk is available fr realizig a ladder etwrk, which is desirable as a filter fr its straightfrward aalysis ad simple structure readily be see. Fig. shws a typical T-type LC ladder etwrk fr dd, where the series ad shut braches f a ladder are labeled as L 1, C, L 3,, C -1, ad L sequetially frm left t right, with Z i (i=1 t ) defied as the iput impedace f the ladder lkig it each brach ad the Z 1 the iput impedace f the whle ladder etwrk. The trasfer fucti f this ladder etwrk is shw i equati (15) by usig geeral vltage ad curret divider laws. This equati ca be equated t the Bessel filter expressi f equati (9). Fig. T-type LC ladder etwrk (=dd) Frm Fig. we develp a iterative prcedure fr fidig Z 1 by cvertig the impedace it a partial fracti expasi ad usig the well kw ctiued fracti expasi f successively ivertig ad addig t fid each relative rmalized elemet value. Equati (16) is a cmpacted expressi fr realizati by directly read ff each rmalized brach impedace (z ) ad admittace (y -1 ) values, where z = sl, y -1 = sc -1, ad symbl meas ctiued summati f each brach elemet values thrugh ctiued fracti Z1 = z1 (16) y z3 y 1 z + z Duality Oe very represetative T-type LC ladder etwrk cfigurati fr =dd is ivestigated as abve fr realizig a lw-pass filter etwrk. Nevertheless, mre ameable etwrk cfiguratis ca be btaied by usig duality characteristic f a give etwrk. A less severe defiiti f duality states that a -prt etwrk is the dual f a secd -prt etwrk if the impedace matrix f e etwrk is the iverse f the impedace matrix f the ther. I ther wrd, tw dual etwrks will have the same rder ad perfrmace, but differet cfiguratis, iterchaged elemet ad trasfrmed elemet values Z i (=k/z +1-i ), where k is equal t R /R i. Fr example, if is dd, the dual etwrk f a T-type etwrk is a π-type etwrk, ad if is eve, the dual etwrk f a Г-type etwrk is a type -ך (iverse Г) etwrk. Whe Z i is a shut capacitace, L i = C +1-i k, ad whe Z i is a series iductace, C i = L +1-i / k. Therefre, the umber f ameable selectis available fr a give th rder filter etwrk will be geeralized t be ( + ) /. th COMPLETE TABULATIONS The characteristics f a lssless filter etwrk fr a rder give ladder LC etwrk have already bee 3 f 6

10 ivestigated i previus secti. Mre ameable selectis fr each th rder etwrk are available fr desigs. I this secti, tabulatis f elemet values f each th rder fr bth matched ad umatched LC ladder Bessel filter etwrk will be listed. Oe simple desig example will be als fially illustrated fr verificati. Table 1 Elemet values tabulati fr matched case T& ך L 1 C L 3 C 4 L 5 C 6 L 7 C 8 L 9 C 10 N π & Γ C 1 L C 3 L 4 C 5 L 6 C 7 L 8 C 9 L 10 Matched LC Ladder Netwrk Table 1 lists ly the ( ) / elemet values tabulati fr a matched LC ladder Bessel etwrk (R i = R ) available fr th rder frm t 10. The labels tp ad at bttm rws shw etwrk cfiguratis fr each th rder. They share the same elemet values listed betwee them. I reality, ameable selectis available will be + / if duality is icluded. ( ) Umatched LC Ladder Netwrk Table a ad Table b list the cmplete elemet values tabulatis fr the umatched LC ladder Bessel filter etwrk (R /R i =1.5) available fr th rder frm t 9 with duality beig csidered, i.e., ( + ) /. The frmer is fr T- ( dd) ad -ך type ( eve) etwrk, ad the latter is fr π- ( dd) ad Г-type ( eve) etwrk, respectively. The assumed R /R i =1.5 rati is based the resistace termiatis usually adpted i radi ad electrics egieerig, i.e., 50Ω r 75Ω. Fr example, if =9, ( 9+ )/ =3 selectis altgether will be available frm these tw tables. Table a. Elemet values tabulati fr umatched case ) ך& (T ך &T L 1 C L 3 C 4 L 5 C 6 L 7 C 8 L 9 N f 6

11 Table b Elemet values tabulati fr umatched case (π &Г) π & Γ C 1 L C 3 L 4 C 5 L 6 C 7 L 8 C 9 N Table 3: Nrmalized agular frequecy (ω 0 =-3dB) ω Example 1: A th rder lw-pass Bessel filter with half pwer (-3dB) physical frequecy respse at f p = 5 MHz ly is t be realized. The rati f lad resistace R ad surce resistace R i is set equally (matched case). Frm Table 1, the d set f each th rder frm 4 t 10 is selected t realize the specified Bessel lw-pass filters. If is dd (=5, 7, 9), it is a T- r a π-type etwrk; if is eve (=4, 6, 8, 10), it is a -ך r a Г-type etwrk, they bth share the same values. Fig. 3a ad 3b shw their respective basic iserti lss ad grup delay characteristics. Frm Fig. 3a, it ca be shw clearly that it makes big differeces fr the iserti lss characteristic wheever rder is higher tha 7 (seve); frm Fig. 3b, as predicted, the grup delay is lger wheever rder is higher. Iserti Lss (db) d Set Bessel LPF ( =4 t 10) f (MHz) Fig. 3a. The d set iserti lss 5MHz (=4 t 10) =10 =9 =8 =7 =6 =5 =4 Desig Example 10 d Set Bessel LPF ( =4 t 10) I this secti, simple example is illustrated t verify the cmpleted tabulati. The rmalized agular frequecy (ω 0 ) values at half pwer (-3dB) are listed i Table 3 accrdig t basic Bessel filter characteristics. Mrever, by usig the available rmalized elemet values frm Table 1, Table a ad Table b, the physical LC elemet values with specified physical agular frequecy (ω p ) ca be btaied by multiplyig with the relative iductace L 0 ad capacitace C 0, respectively, where L 0 = R i ω 0 /ω p, C 0 = R i -1 ω 0 /ω p, ad R i =50 hm. Grup Delay (s ) f (MHz) Fig. 3b. The d set f grup delay 5MHz (=4 t 10) =10 =9 =8 =7 =6 =5 =4 5 f 6

12 CONCLUSION Cvetially, there is at mst ly e set f tabulated elemet values per rder listed available fr aalg passive r eve active Bessel filter etwrk desig selecti. I this paper, bth the matched ad umatched Bessel filter etwrks have bee ivestigated thrughly fr mre tabulati with duality beig icludig simultaeusly. The cmplete tabulati btaied fr a specific th rder LC ladder Bessel filter etwrk ca be geeralized as ( ) / fr ameable selectis r fur times f this, i.e., ( + ) /, if duality is icluded as well. ACKNOWLEDGEMENT The authrs wuld be very grateful fr the research prjects grated frm Natial Sciece Cucil (NSC 95-1-E ), Taiwa. REFERENCE [1] A. B. Williams ad F. J. Taylr, Electric Filter Desig Hadbk, McGraw-Hill, 4 th Ed, pp , 006. [] Sidey Darligt, A histry f etwrk sythesis ad filter thery fr circuits cmpsed f resistrs, iductrs, ad capacitrs, IEEE Trasactis Circuit ad Systems-I: Fudametal Thery ad Applicatis, Vl. 46, N. 1, Ja [3] H. Y. F., Lam, Aalg ad Digital Filters: Desig ad Realizati, Pretice-Hall, p. 74, [4] D. E. Jhs, Itrducti t Filter Thery, Pretice-Hall, pp , [5] L. Weiberg, Netwrk Aalysis ad Sythesis, Hutigt, N.Y [6] A. I. Zverev. Hadbk f Filter Sythesis, Jh Wiley & Ss, pp , [7] L. Strch, Sythesis f cstat-time-delay ladder etwrks usig Bessel plymials, Prceedigs f the IRE, vl. 4,. 11, pp , Nv [8] I. E. Wd, Nte maximal flat delay etwrks, IRE Tras. PGCT, Vl. CT-5, N. 4, pp , Dec [9] W. E. Thms, Maximal-flat delay etwrks, IRE Tras. PGCT, Vl. CT-6, N., p. 35, Ju [10] A. H. Marshak, David E. Jhs, ad Jhy R. Jhs, A Bessel ratial filter, IEEE Trasactis Circuits ad Systems, Nv [11] David E. Jhs, Jhy R. Jhs, ad Al Eskadar, A mdificati f the Bessel filter, IEEE Trasactis Circuits ad Systems, Aug [1] J. R. Jhs, D. E. Jhs, P. W. Budra, Jr., ad V. P. Stkes, Filters usig Bessel plymials, IEEE Trasactis Circuits ad Systems, Vl. CAS-3, N., Feb [13] J. R. Jhs, D. E. Jhs, ad R. J. Lacara, Filters with variable rder ad virtually idepedet time delay, IEEE Trasactis Circuits ad Systems, Vl. CAS-6, N. 8, Aug [14] R. Kaszyski, J. Piskrwski, Bessel filters with varyig parameters, IMTC 005- Istrumetati ad Measuremet Techlgy Cferece, Ottawa, Caada, May 17-19, 005. [15] S.-Hyu Yag, K.-H Kim, C.-Ke Chegt, ad K.-Rk Ch, Desig f a ew liear OTA with a mbility cmpesati techique, 003 Suthwest Sympsium Mixed-Sigal Desig, pp , Feb [16] S. Takeda, Y. Shigeka, A ptical thi film Bessel filter fr 40Gbit/sec-100GHz spacig D-WDM system, 00 8th Eurpea Cferece Optical Cmmuicati, Vlume 5, pp.1-, Sep. 00. [17] J. M. Khuv, Desig f a 15-MHz CMOS ctiuus-time filter with -chip tuig, IEEE Jural f Slid-State Circuits, Vl. 6, N. 1, Dec [18] H. Leib, ad S. Pasupathy, Digital trasmissi perfrmace f stadard aalg filters, IEEE Trasactis Cmmuicatis, Vl. 40, N. 1, Ja f 6