Progress In Electromagnetics Research C, Vol. 26, , 2012

Transcription

1 Progress I Electromagetcs Research C, Vol 6, 67 79, 0 THE CLOSE-FORM SOLUTION FOR SYMMETRIC BUTLER MATRICES C Leclerc,, *, H Aubert,, A Al,, A Aab 3, ad M Romer 4 CNRS; LAAS; 7 Aveue Du Coloel Roche, Toulouse F-3077, Cedex 4, Frace Uversté de Toulouse; UPS, INSA, INP, ISAE, UT, UTM, LAAS, Toulouse F-3077, Cedex 4, Frace 3 COBHAM; 7 Chem de Vaubesard, Dourda 940, Frace 4 CNES; 8 Aveue Edouard Bel, Toulouse 340, Cedex 9, Frace Abstract The desg of a Butler matrx s usually based o a teratve process I ths paper, recurrece relatos behd ths process are foud, ad the close-form solutos, e, o-recursve fuctos of, are reported These solutos allow the drect dervato of the scatterg matrx coeffcets of symmetrc ad large Butler matrces INTRODUCTION Whe used beam-formg applcatos, the Butler matrx s a mult-put/mult-output devce wth puts (beam ports) ad outputs (atea array ports) that allows sytheszg radatg beams Itroduced 96 by Butler ad Lowe [], ths passve multport devce s composed of 3 db-couplers, phase shfters ad cross-overs (see Fgure ) Several artcles dscuss procedures for desgg Butler matrces I 964, Moody [] reported the desg of a symmetrc Butler matrx based o a teratve process Shelto ad Kelleher [3] proposed 96 a reduced scatterg matrx whch has aalogous propertes to those of the Butler scatterg matrx (wth respect to some codtos) whle Alle [4] aalyzed the orthogoalty of the Butler matrx I 967, Jaeckle [5] reported a alteratve desg wth dfferet values Receved 4 November 0, Accepted 9 December 0, Scheduled December 0 * Correspodg author: Céle Leclerc (leclerc@laasfr)

2 68 Leclerc et al Fgure Archtecture of a 3 3 symmetrc Butler matrx of phase shfts More recetly, 987 Macamara [6] publshed a detaled ad systematc procedure for desgg asymmetrc Butler matrces wth 3 db/80 couplers Besdes the theory of Butler matrx, research works have bee reported for reducg sze [7] or solvg problems due to cross-overs [8] I order to remove the cross-overs, ew techologes have bee recetly appled such as the Substrate Itegrated Wavegude (SIW) techology o oe layer [9] or o two layers [0] However, all these works, cludg the precursor aalyss of Butler ad Lowe [], are based o a teratve costructo of the Butler matrx For large matrx dmesos, such dervato may be fastdous ad tme-cosumg To the authors kowledge the closeform soluto of the recurrece relatos behd the teratve process case of symmetrc ad lossless Butler matrx has ot bee reported yet I ths paper, the recurrece relatos used for sytheszg Butler matrces are derved, ad ther close-form solutos, e, o-

3 Progress I Electromagetcs Research C, Vol 6, 0 69 recursve fuctos of, are reported for the frst tme These solutos allow the drect calculato of the scatterg parameters of Butler matrces RECURRENCE RELATIONS FOR CALCULATING THE SCATTERING PARAMETERS OF LOSSLESS AND SYMMETRIC BUTLER MATRICES The scatterg matrx (S-matrx) of recprocal devces havg M ucoupled puts ad N ucoupled outputs wth all ports matched s gve by Eq () [3, 4]: 0 0 S (M) S (MN) 0 0 S m(m) S m(mn) 0 0 S [S] = M(M) S M(MN) S (M) S (M)M 0 0 S (M) S (M)M 0 0 S (MN) S (MN)M 0 0 () Let [S r ] desgates the N M o-zero S-matrx appearg at the bottom left quarter of the S-matrx of Eq (): S (M) S (M)M [S r ] = () S (MN) S (MN)M If [S r ] s utary, that s, f [S r ] [S r ] T = [I] where [S r ] T s the cojugate traspose matrx of [S r ] ad [I] s the ut matrx wth dmeso N N, the the S-matrx gve by Eq () s also utary ad ca be rewrtte as follows [4]: [ ] [0] [S r ] T [S] = (3) [S r ] [0] Moreover, whe M = N =, the matrx gve Eq () may represet the S-matrx of a Butler matrx As a example,

4 70 Leclerc et al followg [] ad [], the S-matrx [S r ] assocated wth the Butler matrx show Fgure s gve by: [S r ] = e j5 8 e j9 8 e j6 8 e j0 8 e j5 8 e j9 8 e j8 8 e j 8 e j6 8 e j 8 e j 8 e j7 8 e j8 8 e j4 8 e j5 8 e j 8 e j7 8 e j 8 e j0 8 e j4 8 e j 8 e j5 8 e j6 8 e j0 8 e j8 8 e j4 8 e j5 8 e j 8 e j4 8 e j0 8 e j3 8 e j9 8 e j9 8 e j3 8 e j0 8 e j4 8 e j 8 e j5 8 e j4 8 e j8 8 e j0 8 e j6 8 e j5 8 e j 8 e j4 8 e j0 8 e j 8 e j7 8 e j 8 e j5 8 e j4 8 e j8 8 e j7 8 e j 8 e j 8 e j6 8 e j 8 e j8 8 e j9 8 e j5 8 e j0 8 e j6 8 e j9 8 e j5 8 (4) Butler ad Lowe [], Moody [] ad Macamara [6] have used a teratve process for dervg the S-parameters of Butler matrces The recurrece relatos assocated wth ths process are ow establshed ad Secto 3 These relatos are solved, ad the close-form expressos for the S-parameters of [S r ] case of a Butler matrx are reported By aalyzg the systematc desg of a Butler matrx establshed by Moody [], t ca be oted that the term S (M) = S ( ) of a lossless Butler matrx ca be wrtte as: ( ) ( ) S ( ) = exp jψ () 0 (5) wth Ψ () ( 0 = φ ) where φ = (6) =0 Moreover, t ca be observed that the term S ( )m for m =, 3,, depeds o the term S ( )(m ) as follows: If m s eve, the: If m s odd, the: S ( )(m) = S ( )(m ) exp ( j/) (7) S ( )(m) = S ( )(m ) exp [ j ( χ m m = m )] { [ = S ( )(m ) exp j (χ m ) m (8) The value of m Eq (8) s reported Table for ay (odd) dex m

5 Progress I Electromagetcs Research C, Vol 6, 0 7 Table Value of m versus the odd dex m of the term S ( )m m N m N m m 3 4 N m Table Values of χ m versus the odd dex m of the term S ( )m whe N m = 5 N m 5 m x (m) x (m) x (m) χ m O the other had, χ m Eq (8) s gve by Eq (9): N m χ m = x (m) (9) where the N m values of x (m) are determed for all raks m havg the same N m For a gve N m, x (m) s deduced from the sequece of Nm, followed by a sequece of Nm 0 Ths sequece of ad 0 s the repeated tmes For dervg χ m from Eq (9) the followg steps ca be followed: = Step : Fd the value of N m assocated wth m by usg Table ; Step : Derve the values of the x (m) ( =, 3,, N m ) for rak N m ad detfy the correspodg values x (m) ; Step 3: Calculate χ m usg Eq (9) Table summarzes the results of these steps whe N m = 5 It ca be deduced, for example, that χ 7 = 3 4 = 9 ad χ = = The teratve approach descrbed by Moody [] for desgg a lossless Butler matrx allows dervg the term S ( l)(m) wth l =, 3,, as follows: S ( l)(m) = S ( l )(m) exp [j φ m ] (0) where the determato of the phase gradet φ m s llustrated wth Fgure [, 6]

6 7 Leclerc et al Fgure φm [, 6] Illustrato of the determato of the phase gradet Equatos (7) (8) ad (0) are the recurrece relatoshps that gover the computato of the scatterg parameter S ( l)(m) The value of the phase gradet φ s such that φ = M whle the other gradets φ m are determed by recursve expressos combg the Eq () reported [6] wth Eqs () (4), that s: φ p φ p = = p [; M], m N () M φm (p ) = φp p [; M], p N () φ M φ φ M M = 3 [, ], N (3) M φ φ φ m M = m M m M (4) m [; ], N, m [0; M ], m N Note that the phase gradet φ m takes alteratvely egatve ad postve values 3 CLOSE-FORM EXPRESSIONS OF THE S-PARAMETERS FOR LOSSLESS AND SYMMETRIC BUTLER MATRICES If we combe Eqs (7) ad (8) ad use Eq (5), the S-parameter S ( )(m) ca be derved as follows: If m s eve, the: S ( )(m) = exp j ( m ) ( ) Ψ () 0 m 4 m =3,5, ( χ ) (5)

7 Progress I Electromagetcs Research C, Vol 6, 0 73 If m s odd, the: S ( )(m) = exp j Ψ () 0 (m ) 4 m ( ) m =3,5, ( χ ) (6) Eqs (5) (6) are the close-form expressos of the S ( )(m) of ay Butler matrx They are derved from the aalyss prevously descrbed Secto of the teratve process proposed [] They ca be establshed by mathematcal ducto as show Appedx A From the kowledge of the phase gradet φ m, the S-parameter S ( l)(m) gve Eq (0) ca the be deduced: S ( l)(m) = S ( )(m) exp [j (l ) φ m ] (7) Whe m s eve, S ( l)(m) ca be determed by combg Eq (5) ad Eq (7); whe m s odd, Eq (6) ad Eq (7) are combed The resultg S-parameter S ( l)(m) s the gve by: whe m s eve, the: S ( l)(m) = exp j ( m ) ( ) whe m s odd, the: S ( l)(m) = exp j m ( ) Ψ () 0 (l ) φ m m 4 m =3,5, ( χ ) (8) Ψ () 0 (l ) φ m (m ) 4 m =3,5, ( χ ) (9) Eqs (8) ad (9) are the close-form expressos of S ( l)(m) (wth l > ) of ay Butler matrx These expressos ca be establshed by mathematcal ducto as show Appedx A 4 CONCLUSION For the frst tme to our kowledge, geeral close-form expressos have bee derved to determe the scatterg matrx of a lossless ad

8 74 Leclerc et al large symmetrcal Butler matrx These close-form expressos have bee establshed by mathematcal ductos They allow a drect computato of the Butler scatterg matrx from ay value of ACKNOWLEDGMENT Ths work s supported by COBHAM ad the Frech Space Agecy (CNES) APPENDIX A THE CLOSE-FORM EXPRESSIONS OF BUTLER SCATTERING MATRIX DERIVED FROM MATHEMATICAL INDUCTION The close-form expressos reported ths paper are establshed by mathematcal ducto ths appedx A Let us show by mathematcal ducto that the scatterg coeffcet S ( )(m) s gve by: whe m s eve, the: ) { [ exp ( S ( )(m) = j ( m ) ( Ψ () 0 m 4 ) whe m s odd, the: S ( )(m) = exp j m ( ) m =3,5, ( χ ) Ψ () 0 (m ) 4 m =3,5, ( χ ) (A) (A) Step : It s straghtforward to show that these expressos are true for m =, ad 3 As a matter of fact: Applyg m = Eq (A), the followg expresso s obtaed: ( ) ( ) S ( )() = exp jψ () 0 (A3) where Ψ () 0 s gve by: Ψ () ( 0 = ) = ( ) (A4) =0

9 Progress I Electromagetcs Research C, Vol 6, 0 75 The same expresso s obtaed from the Moody s teratve approach [] Applyg m = Eq (A) t remas the followg expresso: ( ) [ ( S ( )() = exp j Ψ () 0 )] (A5) Aga, the same expresso s obtaed from the Moody s teratve approach [] whe m = 3, Eq (A) s used: ( S ( )(3) = ( = ) [ ( exp j Ψ () 0 ) [ ( exp j Ψ () 0 )] )] ( χ 3) 3 (A6) O the other had, aalyzg Moody s systematc desg [], Eq (A7) s obtaed: ( ) [ ( S ( )(3) = exp j ( ) )] (A7) The by dvdg Eq (A6) by Eq (A7), the result s Therefore, these two expressos are equal Cosequetly for the frst values of m, the Eqs (A) (A) are true Step : Assumg that Eqs (A) ad (A) are true at rak m, let us show that they are true at rak m : If m s eve, usg the recurrece relato Eq (7) ad Eq (A) (sce m s odd), we fd: S ( )(m) = S ( )(m) exp ( j/) S ( )(m) = exp j = ( )( ) m Ψ () 0 (m ) 4 m =3,5, (A8) (χ ) (A9) Eq (A9) s detcal to Eq (A) whch m s replaced by m

10 76 Leclerc et al If m s odd, usg the recurrece relato Eq (8) ad Eq (A) (sce m s eve), we obta: { [ S ( )(m) = S ( )(m) exp j (χ m) m (A0) S ( )(m) = exp j Ψ () 0 m 4 m ( ) m (χ ) (A) =3,5, Eq (A) s detcal to Eq (A) whch m s replaced by m Cosequetly, to deduce the step, f Eqs (A) ad (A) are assumed to be true at rak m, t s show that they are also true at rak m Step 3: From step ad step, we coclude that Eqs (A) ad (A) are true for ay rak m A Let us show by mathematcal ducto that the scatterg coeffcet S ( l)(m) s gve by: whe m s eve, the: S ( l)(m) = exp j ( m ) ( ) whe m s odd, the: S ( l)(m) = exp j m ( ) Ψ () 0 (l ) φ m m 4 m =3,5, ( χ ) (A) Ψ () 0 (l ) φ m (m ) 4 m =3,5, ( χ ) (A3) Step : Eqs (A) (A3) are tested for Butler matrces whe =, ad 3 The resultg matrx [S r ] s the compared wth the matrx [S r ] derved from the teratve process reported by Moody []: For =, the Butler matrx s reduced to a 90 coupler From [] the [S r ] matrx s the gve as follows: ( ) [ ] e [S r ] = j/ e j/ (A4)

11 Progress I Electromagetcs Research C, Vol 6, 0 77 The same scatterg matrx s obtaed from the close-forms Eqs (A) (A3) by takg = ad m = Eq (A), m = Eq (A3) ad l = ad both equatos For =, aalyzg a 4 4 Butler matrx, the followg [S r ] matrx s derved from []: e j/4 e j3/4 e j/4 e j4/4 ( ) e j/4 e j0/4 e j5/4 e j3/4 [S r ] = e j3/4 e j5/4 e j0/4 e j/4 e j4/4 e j/4 e j3/4 e j/4 (A5) By substtutg by Eqs (A) (A3), the same matrx s obtaed For = 3, the [S r ] matrx s gve Eq (4) The same matrx s obtaed from Eqs (A) (A3) Cosequetly for the frst values of m, the Eqs (A) (A3) are true Step : Assumg that Eqs (A) (A3) are true at rak, let us show that they are true at rak : Followg [], t ca be observed that scatterg parameters correspodg to a odd rak the frst le of a Butler matrx allows dervg the overall [S r ] matrx As a matter of fact: () the frst le of the [S r ] matrx the scatterg parameters assocated wth a eve rak ca be deduced from odd rak parameters by subtractg a phase of / duced by the frst coupler; () the other les of the [S r ] matrx ca be derved from the frst oe usg Eq (7) Cosequetly, we cosder oly the scatterg parameters havg a odd rak the frst le of the [S r ] matrx Moody graph [] allows dervg the Butler matrx ad the correspodg [S r ] matrx Eq (A3) gves the same matrces As a matter of fact, cocerg the phase of the scatterg parameters (the comparsos of the magtudes are straghtforward): arg ( S ( )()) : ( from Moody []: φ ) =0 (A6) from Eq (A3): Ψ () 0 = ( ) ( ) (A7)

12 78 Leclerc et al arg ( S ( )(3)) : from Moody []: ( φ ) = from eq (A3): Ψ () 0 3 ( χ 3) ( ) (A8) (A9) arg ( S ( )(5)) : ( ) from Moody []: φ 5 ( φ ) = from eq (A3): Ψ () 0 (A0) 5 ( χ ) =3,5 3 3 (A) 4 Cosequetly, to coclude the above-metoed step, f Eqs (A) ad (A3) are assumed to be true at rak, they are also true at rak Step 3: From step ad step, we coclude that Eqs (A) ad (A3) are true for ay rak REFERENCES Butler, J ad R Lowe, Beam-formg matrx smplfes desg of electrocally scaed ateas, Electroc Desg, Vol 9, 96 Moody, H J, The systematc desg of the Butler matrx, IEEE Trasactos o Ateas ad Propagato, Vol, No 6, , Shelto, J P ad K S Kelleher, Multple beams from lear arrays, IRE Trasactos o Ateas ad Propagato, Vol 9, No, 54 6, 96 4 Alle, J L, A theoretcal lmtato o the formato of lossless multple beams lear arrays, IRE Trasactos o Ateas ad Propagato, Vol 9, No 4, , 96 5 Jaeckle, W G, Systematc desg of a matrx etwork used for atea beam steerg, IEEE Trasactos o Ateas ad Propagato, Vol 5, No, 34 36, 967

13 Progress I Electromagetcs Research C, Vol 6, Macamara, T M, Smplfed desg procedures for Butler matrces corporatg 90 hybrds or 80 hybrds, IEE Proceedgs, Vol 34, No, 50 54, Maddah-Al, M ad K Forooragh, A compact Butler matrx for WLAN applcato, Mcrowave ad Optcal Techology Letters, Vol 5, 94 98, 00 8 Dall Omo, C, T Moedere, B Jecko, F Lamour, I Wolk, ad M Elkael, Desg ad realzato of a 4 4 mcrostrp Butler matrx wthout ay crossg mllmeter waves, Mcrowave ad Optcal Techology Letters, Vol 38, , Djeraf, T, N J G Foseca, ad K Wu, Desg ad mplemetato of a plaar 4 4 Butler matrx SIW techology for wde bad hgh power applcatos, Progress I Electromagetcs Research B, Vol 35, 9 5, 0 0 Mohamed Al, A A, N J G Foseca, F Coccett, ad H Aubert, Desg ad mplemetato of two-layer compact wdebad Butler matrces SIW techology for Ku-bad applcatos, IEEE Trasactos o Ateas ad Propagato, Vol 59, No, 503 5, 0