011764-502-M 30 July 1973 MEMO TO: File FROM: Soon K. Cho SUBJECT: RAM1D 1. Introduction When an E- or H-polarized plane wave is incident normally upon a convex cylindrical body of general surface impedance, the electric current induced on the surface for each polarization can be described by an integral equation of the second kind. When the plane wave incidence is not normal but oblique, there then results a pair of coupled integral equations (Andreasen, 1965). A computer program, RAM1D, has been written for an E- or H-polarized normal plane wave Incidence, to compute the bistatic or monostatic scattering cross section of a convex cylindrical body via an approximate solution of the integral equation of the surface current. A brief discussion is given in Section 2 for the integral equations involved. RAM1B deals with a pair of coupled integral equations for an oblique plane wave incidence, involving second derivatives of the Green's functions as kernels, even though the obliqueness in incidence is not carried through but abandoned in the end. It is worth noting that a second derivative of the Green's function involved in RAM1B gives rise to a singularity which is not integrable in the Riemann sense. A recognition of these factors in RAM1B motivated RAM1D. For a special case where an E-polarized normal plane incident wave illuminates a perfectly conducting convex cylindrical body, the integral equation DISTRIBUTION Knott Laxpati Hiatt/File Liepa Cho Senior 11764-502-M = RL-2240

011764-502-M of the surface current is reduced to that of the first kind with a Green's function as its kernel. As a consequence of the symmetry possessed by the Green's function with respect to the source and the observation coordinates, the kernel matrix derived from the integral equation through algebratization is symmetric. Although this fact is not exploited in RAM1D for this special case, a user could easily implement it with a slight modification to SUBROUTINE MATRIX, if so desired. For a perfectly conducting body, the computation of the surface impedance of the body can be dispensed with altogether and this is done in RAM1D. The computation of the Hankelrs functions involved is based on the polynomial approximation (Abramowitz, Segun, 1968). In Section 3, a scheme used in computing the geometric factors of interest in solving the algebraic equation is presented in some detail. Although the computation of the set of end points of the cells of the body contour is not needed for our present case, the formula for the computation of the cell end points is included in the analysis for a future reference. Finally, in Section 4, RAM1D listing is given, including the time tally of each SUBROUTINE for a sample calculation. 2. Integral Equations of Surface Current Density Functions For a normal plane incident wave of the E- or H-polarization, the surface electric current induced on a convex cylindrical body of general surface impedance is given by an integral equation of the second kind (Senior, Knott, 1973)'. Thus, for s, s' E C, the body contour (see Fig. 1), we have H-polarization: -H s (ss) () +K 4 n(s ') (s (kr1 1 K(s)d(ks') + z 2 (s) s c + 4 ) (St) H(1)(kl1) K(s)d(ks'), (2.1) C Note typographic sign errors in Eqs. (3-1) and (3-2) of this reference. 2

011764-502-M E-polarization: i 1 (2) t (2A, Y Ei(s)a ( r)(l)s)K (s)+ 4f )(s)~. n(s')H(l)(k|rl) K(s')d(ks )+ 0z 2 z C + H(1)C(kr)K(s)d(ks') C (2.2) where A r (s)-(s') t(st): the outward unit normal vector at s X sv, principal value integral. y '2 X C Fig. 1: Geometry of a convex cylindrical body contour. (n, T, t) is the local right-hand cartesian coordinate system. 3

011764-502-M Following the usual procedure, Eqs. (2. 1) and (2.2) are algebratized in the forms M -H(S) A.. K(s.) i 1, 2, 3,..., M (2.3) j=l where M denotes the number of the cells of C and s. the midpoint of the ith cell, A, and 1 Aij [i 4 rij n(sj ) (1 -6ij kA + (1) (L1+2 kAi + (1)s.) o ij 4 ' + H(1)(k|V|Xl-6.} L ' (2.4) i, j= 1, 2, 3,..., M; and M Y Ei(s) B.. K(s.), i, 2, 3,..., M. (2.5) 0o z I = J j 3] j where 32 1kA + 1+i in(Ln - 0.4228 56.. + kA. +H() (k|l|) (1-6.), (2.6) j ly 4 i, j l, 2, 3,..., M; 4

011764-502-M for i j j, iJ for ix j, i, j z 1, 2, 3,..., M. (2) (2) Equations (2.4) and (2.5) show that, as long as n ((sj) 0, by replacing r )(s.) by 1/( 1(sj), r( (s.)B.. A.. (2.7) J 13 1ij demonstrating that, as pointed out by Knott and Senior (Knott, Senior, 1973), the use of the duality relations between E, H, /u and e enables one to infer the solution of one integral equation from that of the other, for an imperfectly conducting body. 3. Geometry of the Contour of a Convex Cylindrical Body Let the z-axis of the (x, y, z) Cartesian coordinate system be parallel to the axis of a convex cylinder and let C denote the contour of the cylinder in any fixed z-plane. Next, let C be decomposed into a set of segments of circular arcs of various radii of curvature. In the special limiting case where the segment is a straight line, we treat it separately. Thus, assuming that the contour of the convex cylinder, C, consists of a set of segments of circular arcs of finite radii of curvature and straight lines, in general, we wish to derive the formulas for the geometric quantities involved in solving the integral equations (2.1) and (2. 2). If a segment is simulated by a circular arc, we assume that the Cartesian coordinates of the end points of the arc and the angle subtended by the arc are known; if the segment is a straight line we only need to specify the coordinates of the end points of the line. We further assume that the contour C is symmetric with respect to the x-axis, say. This last assumption is made solely so as to enable us to specify the surface impedance along the contour in a convenient manner. Then it is sufficient to consider only the part of the contour in the upper half-plane. 5

011764-502-M A. Circular Arc Segments Consider a circular arc segment, S. Let its end points w a (xz, y ), Wb= (xb, yb) and the angle subtended by S, 0, be given. Let w t (x, y ) denote the center of curvature of S. We divide S into N equal subarcs (or cells) and consider a j th subarc, As.. Let (x., y.) denote the midpoint of j j j As. and (X., Y.) and (Xj+1, Yj+ ) its left end and the right end points. Let n. denote the outward unit normal vector at (x., y.). Given w a wb and 0, we wish to derive the formulas for a set of the a b unit outward normal vectors, the midpoints, the arc lengths and the end points of N subarcs. See Fig. 2. A A njy i/ (8/2+T- q) j wa w c Fig. 2: Geometry of a segment of circular arc. Let d denote the chord length connecting wa, wb. Then, a b 6

011764-502-M d /(x x)2+(y y)2 Db- a + 'Yb- Ya 2 r sin, (3.1) 2 where r is the radius of curvature of S. Hence e r d/2 sin- (3.2) Let AxXb-x Ay =y b-Ya. Define by sin TrAy/d, cos T X Ax/d. Then, w (xc, yc) is given by c c c x x + rsin( +T), c a 2 ycx ya- rcos( 2+ T); or X X +(Ax+Aycot) e c a2 2 e0 0. (3.3) 1 0 Y =a+ 2 (Ay- Ax cot) c 2 2 The jth subarc, As., is bounded by (X., Yj) and (Xj+, Yj+ ). Its midpoint is (x., yj). Let qj denote the angle subtended by the arc extending to the point (xj, yj) from w and let AO denote the angle subtended by As. Then, from Fig. 2, it is seen that qu= (2j-1) 2 Qj.(j-1)A; jl1, 2, 3,..., N (3.4) 7

011764-502-M where Qj is the angle subtended by the arc extending to the point (X., Y.) from w. We now have a e x,= x -rsin(O+T-q.) I c 2 1. Y +r cos( +T-q), j 1, 2, 3,..., N. Eliminating xC, y by use of Eq. (3.3) from the above relations, we obtain the coordinates of the midpoint of Aj: 1 x +1 fi x x. x +2- (A + Ay cot) (1 - cosqj)-(Ay- Ax cot- ) sin q. (3.5) y.' Y + (Ax+ Aycot ) sin q + (y -Ax cot ) ( - cosqj. 21n e o0, q. (2j-1); j =1, 2, 3,..., N. Replacing q. in Eq. (3.5) by Qj, we obtain the coordinates of the left end point of Aj: Xj xa+ (Ax+ Aycot) (1-cosQ) -(Ay-Axcot-) sin Q j a 2 2 2 (3.6) Y. Ya + ( Ax+Aycot )sinQj +(Ay-Ax cot -) (1 cosQ Q 9 0, Q l)A; 1, 2, 3,..., N. We note that XN+1 b YN+1 Yb. Let us now consider the outward unit normal vector at (x., y.) of As.. See Fig. 3. 8

011764-502-M y A Y ^ /2+T-q. A (X Y x.yj1 (Xj. Y.) 0 Fig. 3: The outward unit normal vector at the midpoint of the jth subarc (or cell), AS.. J A From Fig. 3, we get the x, y components of n. as 3 n x - sin ( 0 + - q), jx 2 1 e n. = cos (- + T-qj) or sin- ~ nj. - Ax + Ay cot ) cos q+(Ay-Axcot ) sin (3.7) sin.- r Q n. s d( Ax+ Ay cot ) sinq - (Ay-Ax cot ) cos q Y d L # 0, Ae qj= (2j-1) 2 j= 1, 2, 3,..., N. The length of the arc extending to (x., y.) from wa, which we denote by Sj, is 3 3 9

011764-502-M Sj rq r(2j-) 2 (3.8) j = 1, 2, 3,..., N, and the length of each subarc (or cell) is rA0. B. Linear Segments When the segment is a straight line, we assume that the end points w, Wb are given. Note that 0 as defined in 3. A is zero in this case. Referring to Fig. 4, it is seen that Sda (ax + (A)y)2 p (3.9) (3.10) s N j N' j = 1, 2, 3,..., N. y A n. i.)T b Y.) 1iy x 0 Fig. 4: Geometry of a straight line segment. Since d S. Ax j a cosT - X - d S. 3 10

011764-502-M we get Noting that jth cell, Asj, as s. x. x + -Ax, j a d S. Y.= y d j a d d. s. (2j -1) 2N; j = 1, 2, 3,..., N. (3.11) - x, we can write the coordinates of the midpoint of the d 2N j=1l, 2, 3,..., N Yj = ya+ 2N (2j -1) 0 X 0. For the outward unit normal vector at (x., y.) of Ax], we have Ax n. cos T JX d j = 1, 2, 3,..., N. n. =-sinT-x d jy d The coordinates of the left end point of the j th cell are X.x Ax j j 2N j=1, 2, 3,..., N. y = v - j -j 2N ' e 0. (3.12) (3.13) (3.14) We note that N+l' Xb YN+1 Yb ' 11

011764-502-M 4. List of RAM1D The list of RAMID, along with the time tally of the program, is given here. RAMiD took 23.8 seconds to compile in IBM 360 and for a sample case of a perfectly conducting ogive the contour of which was divided into 48 cells, the CPU time for evaluation of bistatic cross section at 37 points was 31.7 seconds for the case of H-polarization. A. RAM1D C..... MAIN4 P[RGRAM.*0 C Th CE ENTI IE A N AL YS IS I I i4ADE IN TilE CA RTI ES IAN J9- 0 R DII NA 3rY36 r - C NOR A LIZED W IT'I RESPECT AO TH'IEr, WAVELENGTH. CO O I/i0,1 A/AK,AK2,A K4,, RA:[ A AN, RA N)IV, M, NSEG, lAr CCMMONl/BOB/X(50),Y(50 ),YN(50),N(5'), DS )(50)yS(50),ZS (50) COM ON/LOC/PIN2 (50),P HI (50) $,AA( 5051),A IAG(50),PHASE(50 COMON/BOD/IPRINTFIOPTIPPIPSC, ALPHA COMPLEX PINC, PlIAAZS IAT 1 RADIAN=57.23573 RADIINV=0.017453 AK4=1. 570796 AK23. 1415927 AK-=6. 2831846 READ (5,100) MNSEGALPHA R EAD (5, 200) IO PT, I PPIP I PNT,IRS IF(IOPT.EQ.0) GO TO 30 IF(IPP. GE.2) Go TO 20 DO 10 I=1,M 10 READ(5,300)X(() #Y (T) XN( ),YN(I),DSQ(I),S((),ZS(I) GO TO 30 20 CONTINUE DO 15 I=i,ti1 15 R E AD (5,3f35 0) X (I),Y (T),XN (I),YN (I),DSQ (I) 30 CONTINUE CALL GEOM CALL MATRIX CALL SCATT 1 00 F Oi M AT (2 I 5, F 1 O. 5) 200 FORi".%AT ( 41I 3) 300 FORMAT(77. 2) 350 FORMAT(5F7.2) STOP END 12

Ull'ib4.-:3uz-m *....6THIS SUBiROUTINE COMPUTES 6OT11 3EOMETRICr- FAC.'TORS OF INTEREST C AiND THE SUP."&ACE IMPEDANZE WHElN TIlE BODY' IS lilPERFlE.C-TLY OD2PN G. C THE BODY CONTOUR, ASSUMED r o H3E SYK.IETRIC ABOUI! THEi-l X-AXIsfSAYflS C COINSIDERED AS ZCO0MPOSED) OF A SET OF CIRC.ULAR S')EG7MEN r S.NJP E H AFr C A STRAiG"J'HT LINE IS A SPE"CIAL CASE DF A CIRCULAR ARC'.*COM.MON/I3OA/AKAK2,AK4,rRADL AN, RADINVMiNSEGIATT COMMON/BOB/X(50), Y(50),XN(SO),YN (50),DSQ(53) ra3 (5 0), Z3- (5 0) COrIMON/BOC/PINC(50) PHI(50) IAA(5O,r5l),AMAG(50),rPHASE(50) C"-Ot'MON/BO00/IPRINTIOPTLPP,IPSC%,rALPHA D It E-NS IO0N L L (5) CI'PE P2 LNC,PHIrAA COMPLEX ZAZBZFACZS ISTAR'A=l DO 99 N-U=lNSEG LL(NU) =NU C NOTE THAT XAvYA,,vXBvYB ARE NOR~MALIZED QUANrirIES C WITH RESPECT TO THE WAVELENGTH* R EAD (5 r1 00) MM, XAvYA, XB, YBvA NG,MMD= 2*MM I END= l MM IF(NU.GT.1) GO TO 5 GO TO 6 I EliDEI+ I END 6 I A-,(IPP.GE.2.) GO TO 10 h EAD (5,f 1 50) -LZFD RMlZA ZB3,fZF AC, ZE X WiRIrT E (6 r 4 00) 10' DX=XB-XA DY=Yb-YA CIIO)RD=SQRT (DX*DX+DY*DY) IF(ANo-.EQ.0.0) GO TO 20 THlETA2=O. 5*ANG*RADINV A LFA=TdETA2l/MM DTH'lETA=2 * 0* ALFA SIN BSIN (THETA2) TANB=TAN (THETA2Z) IF(AIIG.EQ.180.0)GO TO 25 TrR'EX=DX DY/TA NB T RAY=DY- DX/TANB GO TO 26 25 TiE'X=DX T'RAY=DY 26 DA-RC:=ALFA*CilORD/SINB DARC20. 5*DARkC 20 DARC=CHOPRD/MM DAt.<C2=0. 5*DARC 30 IF(IPP.GE.21)) GO TO 33 WRITE(6,L450) GO TO 34~ 33 WRITE(6,500) 34~ CONTINUE DO 75 I=ISTART,IEND F AC'T=2*I- 1 13

011764-502'-M D Sd (I) =PARUL IF (I PP. GE. 2.) GO IO 35 S (I) =FACT*DARC2 TF(IZFOlR1) 40,f45,j50 4jQ P OAE R= (1.-RE A L(Z3))/Z EX PO'lR?=0. 5*2OWEzR*POW"R Z-'S () ZAj%*EXI? (P OWEI'R) GO T3'1 3 5 45ZS (T) =7ZAC* (ZA+ZB*S (I) **ZEX) GO TO0 35 S.C 7 S (I) = ZlAC* (ZA+ZB3*EXP (-S (I) *ZEX)) 3 5 IA —F(ANG.EQ.O.O)GO TO 60 cSIN=cSN (ARG1) X (I) =XN+O.5* (-REX*COSQ1ITRAY*Si-NQ) Y (I) =YA+0. 5* (TREX*SINQ+~iMA Y*C-OSQ1 ) XN () =-STN3* (2REX*C0SQ+TIAY*SINQ) /CWDRD YN (1) =SL-iNB* (TIEX*SINQ-TiRAY*COSQ) CIE 60C Y. (I) =XA+FACT*DX/IriMD Y (I) =YA+FACT*DY/IIMD XN (I) =-DY/CHORD YN (I) =DX/CEIltRD 7C JF(T,?P.G-E.2)GO TO 714 WRITE (6,42100) L"(NIJ),X (I),Y (I),XN (L)fYN (I) pDSQ(I),S (I) Zs (I) GO TO 75 74 WFITE (b,3 00) LL(NU)rIpX (T),rY(I),rXN (I),YN (I) D,DS2(I) 75 %- CONT I NUI I F (1 P?. GE.2) GO TO 80 GO TO 3.5 80 WPT TE(65 10) 85 CONTINUE D0 99 I=ISTrARTIEND EF(I?,P.GE.2)GO TO 86 s (IS) =S (AI) Zs (IS) =Zs (I) 8 6 D SQ (T:) = DSQUE ) X (IS) =X (I) Y (IS) -Y(I) X N (TS N (I) fN, (:S) =YN (I) IF(IP7?.GE.2)GO T0 88 WRlTTE(6,250)LL(U)IXIS,(I)X N(IS) YN (I S)DSQ (I S) 1S (I S) Zs (IS) 9 9 co0N:TI Nr Es lOC FOR~v'AT(I-r5,5Fl0.5) 1 50 FPORl.AT (I 3, 7F7.2) 200 FORM ATr(5X,1I5,2X, I3, 8F7. 3) 2 50 FO-IMAT (6 Xr I5,r2X v13, 2F7. 3, 1 X aF7. 3 r1XF7. 3, 2X F7 3,1 XF7. 3 2F7. 3) 14

011764-502-M 300 FORMA T(5X,5, 2X,13,5F7.3) 350 FORNIAT (obX,I5,2X,I3,2F7.3,1X,F7. 3,1 X,F7. 3,2X,F7.3) 400 FORMAT(1H1, 7HIZFORM=T3,2X, 3HZA=2F7.2,2X,3HZB3=2F7.2, 1 2X,5IZI- AC=2F7. 2,2X,4HZEX=F7.2/1 450 FORMAT(3X,7IISLlMENT,4X,1li,3X,41HX (I),3X,14lY (I),2X, 5fiBN(I), 1 2X,5A:YN(T),1X,6[IDSQ (I),3X,41S(I),9X,5fIZS () //) 500 FOIR\AT(3X,7HS;GMENT,4X,1[I,3X,4LX(I),3X,4HY(I),2X,5HXN(I, 1 2X, 5HYN (I), 1X,6HDSQ(I)//) 460 FORMAT (1H1,3X,7HSEG IENT,3X,2HIS,2X,5HIX(IS),2X,5HY(IS},2X,6HXN (IS), 1 2X,6HYN (IS),2X,7HiDSQ (IS),3X,5HS (IS),7X, 7HZS (IS)//) 510 FORMAT (1 11 1, 3X,7 HS EGMENT, 3X,2 2IIS,2 X, 5HX (IS), 2X, 5HY (IS), 2 K, 6rKN(I S) 1 2X, 6 iYN (IS) 2X,7HDSQ (IS) //) RETURN END SUBROUTINE MATRIX C.....TI!IS SUlRBOUTINE COMPUTES TWO MATRICES DERIVED FROM A SJRFACE C CURRENT INTEGRAL EQUATION FOR A CONVEX CYLINDRICAL SCATTERER F:)R C EITHER L-,3PR,H-POLARIZATIDN:AN M BY 1 COLUMN MATRIX FOR THE C NRCEMAL PLAN. WAVE INCIDENCE AND AN M BY M COEFFICIENr?lArRIX C OF Tl{E CURRENT MATRIX. COMMON/BOA/AK,AK2,AK4,RADIAN, RADINV, M,NSEG, IAr CCYMO/BOB/X(50), Y(50), X N (50),YN (50), DSQ(53), (50), Z3 (5 3) COM:0CN/BOC/PINC (50),PHI (50),AA(50,51) 5 AMAG(50) PIlPHASE(50 COruh.OR!/BOD/IPRINT,IOPT,IPP, IRS-, ALPHA CC:PLEX PINC, PHI, AA COMPLEX ZS, BMA1,BMA2,CK4 CK4 =CMPLX (0.,AKi4) ARG1=CCS (ALPIHA*RADINV) APG2=SIN (ALPHA*RADINV) WRITE (6,500) DO 100 I=1^,r AFG3=AK* (X (I)*ARGI+Y(I)*ARG2) PINC (I) =CMPLX (COS (ARG3),-SIN(AR33)) T1=REAL (PINC (I)) T2=AIMAG (PINC (I)) PH=ATAN2 (T2,T 1) *RADIAN WRITE (6,510) IT1,T2,PH DO 100 J=I,M IF(I.EQ. J)GO TO 10 DX=X (I) - X (J) DY=Y (I) -Y (J) RD=SQRT (DX*DX+DY*DY) CNRi= (DX*XN (J) +DY*YN (J)) /RD R ANS= AK* R D CALL HANK (RANS,0,BJ0OBYO) BMA2=AK4*DSQ (J) *CMPLX (BJO, BYO) CALL HANK (RANS, 1,BJ1,BY1) BVA1=CK4*CNR*DSQ (J) *CMPLX(BJ1,BY1) G0 TO 20 15

01 1764 -bUZ-M 10 i3~Al=0CMPLX(0.5,0.O) BMA2=DSQ(J)*CMPLX(.AK4,,ALOG(AK4*DSQ(J)hO0,223) 22'0 IF(IPP.EQ.,3)GO TO 25 IF(IPP.EQ.0.Oa1"i.IPP.EQ.2)GO TO 30 AA (I,J) =3MA1+BMA2*ZS (3) GO TO 100 25 AA(I,J)=BMA1 GO TO 100 30 IF(IP?.EQ.2)GO TO 35 A A(I,, J) =i3M-A 1*ZS (J) +E$NA2 GO TO 100 35-1 A A (IT,3)8=MA2 100 CONT'INUE 50C FOPMAT(lHi,1O0X,1HI,5Xj2OHPLANE WAVE INCIDENCE/) 510 FOP. MAT (9 X, 13, 3X, 3F7.2) RETUR~N END SUBRiOUTINEL' SCATT.~..HISSUBROUTINE COMPUET1S BOTIN THlE SURFACL 2[IRRENT DI STRIi3UTION AND 2CITHER A B3ISTATICOR BACK SCATIEI&RING~ CROSS' sECrioN CO:~tON/DOA/AKAK2,rAK4#,,RAL)IAN#,FADINV,1,NSE,,.(;G,iAr C'~OMON/,BOB/X (50),Y (50),XN(5Q),YN(50),DSQ(50),S(50),ZS(50) COrMMON/BOC/PINC0(0) PilL(50),AA (50,51),AM~AG (5-0), PlAS3E(5-)) CCOMMtIN/BOD/lPRINTT-OPT,.IPP, IPSC,r ALPHA DIIIENSION PrS (100o,6),A (6) DATA A/lH-,F1H-,1H-,1HR,1HiX,lH*/ COMPLEX PINCvPHIAA %O. "EX SUMvZSAMwP IF (IiSC.EQ. 1)G0 TO 10 READ (5, 200) FIRST, FINAL,,STE P NBIT=1+IFlX (A3S (F-INAL-FIRST) /STEP) WEITE(6,2l0)lRSCNBITFIRSTFINALo,STEP GO TO 20 16

011764-502-M i C A'RTT1E,((:6, 2 6O0) I IiS C, A L PHA 2C CALL ZVO8 (A As, MPINC,,PHIllAT) I F'I PE0. 0) W RI TE (6,r3 00) IF(IPD.EQ.1) WRITE (6, 3 01) I F (ILP?.EQ. 2) WRHIT E (6-, 302) IF (IPP.EQ. 3) WRITE (6,303) 3C iWRI T E (6,2 80) DOl,,A 3S =CAS1Pl I(I PH.ASE! (I) = ATAN2 (AI MAG (PHIl (I)) REAL (PHI (I) ) 35 YPIT E"(6,13 05) I, AMA G (I) PH AS E (I) IVF(I P PINTI. EQ. 1)GO TO. 4Q0 W PIT E (b,,12) h 4 FOlM AT (H1) 1 DO 1 3 I= 1,6 DO 13 J=1,il lF(I.EQ.1)PTS(J,I)=0.5 IF(I.EQ.2)PA.TS (JrI)=. 0 IF(I PP. G1.2)GO TO 14 IV (I. Q. 5) PTS (JI) =0. 5*A IM A G(ZS (J))+0.5 (110 TO0 1,5 1 4 EF(L.E 4) P'TS (3, 1) = 0. 5 F (I. Q5) PTS (J,I) =0. 5 15 CONT-ANUEv 1CON T -NUEP CALL Gk-PM(S,,PT-3o,5,10,rM,6,Mli,51,fA) 4 C F (I -SC.. —Q. 1)GO TO 80.-iRITAE (6,r 3 10) THETA =FIRST DO 72) I=11NBIT -RTI EJT A=TH-E T A *,rA D IN V 3 T=SLIN (R THETA) CT= CO 0S (R T HTA) S U MC MPL X (0. 0.) DO 60 J31,M?DOTl=CT* XI (J) + ST*YN (J) AI<G-K * (CT *X (3) + ST *Y (3) AMP=PII1l(J) *DJ5 (3) *Ct P LX (CO0S (A R),SI N (A RG)) iF(rpP.FQ.3)G~o TO 45 I(I L). Ev.0.R.I P P.EQ.2)G TO 50 S UM= S11N+ A P* (RDOT N-zS (3)) GO0 TO 60 45 SUM=SUM+AMP*RDOTN UO TO 60 -SC If(IlP.ElQ.2)GO TO 55 S UM=SJ l+ AM P* ( DOT N*ZS (J) -1. GO TO0 60 17

011764-502-M 60 CONTINUE D'Is'RSC=20. 0*ALOG 10 (CAB3S(SUM) )+1.987 ii ITE I6,r350) I,?HETAr DBRSC 70 Ti'iET A=TH ETA +S TI P GO TO 999 80C W RITIE(6 L4 00) R ALP HAl'AL PHA* RA DI NV ST=S1EN (RALHA) =C0CO (RALPIIA) S UM= C"!IPLX (0.,r 0.) DO 110 JblM R )OT1%NCT*XL1 (J) +ST *YN (J) Ai.RG=-AK* (CT*X (J) +ST*Y (J)) A[,1P=?dI(J)*DSQ IJ) *CM~PLX (COS (AR"34),SIN (ARG)) ILF(IPP.EQ.3)GO TO 85 PF(I-.PP.Q. 0.OR.IP P. EQ. 2) GO TO 9 0 SUMt SUIJ~+AMP* (i1tDOTN-ZS (J)) GO To 1 10 d5 SUM=SUM+AM?*RDOT.N 90v IF(IPP.EQ.2)GO TO 95 S UM=SUMi+AMP* (RDOT N*ZS (J) -1.0) 95 S UJI= SU11- AM 2 110 C CNIrT NU E DLHS C=2O0. 0* ALOG 1O (CAB S (S UM) )+ 1. 987 W'PTFE"(6,'450) AL1'HADB3RSC 9990 CON T:1"U E 2C.00C FIOR!A:'(3F1O. 5) 2 10 FGUP'1A-' (1F1, 5Xf5HIRSCT13,2X,5HNBIT13, 2X, 6HFIRST=FIO.5v I 2X,6HiFINAL=F1O0. 5,2X, 5HSTE"P=F1O0.5) 2>,-0 F OR MwA 7 (1h1 1 5 Xa, 5HIIIS C= 3, 2.X,6 HAL P 1A=F 10. 5) -AO F-oRM4A7 (2 X 5 1HCUiRENT DISTR I BJTI:JN FDR EP-LOLAR IZATION, NJN-ZERO ZS/) 3 0"1 F.7CR %IA7(2 X 5 1H2:URR1E NT. DILS'TRIBUTIDN FOR [I-LOLARIZArION, NJN-ZERD ZS/) 33J"2 F Ot~i Ar (5XK,,4Lf1iCU;RRENT DISTR IBUTION, FOR " —POLA.RIZATIDN, ZS=0f) 30u"3 FfC AT (5 X,Li 4HC U R R,NT D ISTR IB3U T1O.N. FOR H-POLARIZATION,ZO/ 2130- F OR-" AX7(2X 1 1 HCELL NIJMBEii,2 X,711A MAG7 f)r2 X,8dPiiAS E(I)) 35 FO3M AT (7 X el3,SX, F7. 2,A X rF7.2) 3 10 F Oaki A " 1H1,L4Xl1H[REC. POINTS,,5X,5c-HriETA,,3X,,23HJBLrAriz; ROiss SScri 1 C NJ/) 3 50 F CRE tiT (4X4r13 11X, F7.2.,lO X rE12. ) 4)j0 F QIAI h-(1H1,5Xf1OHINC. ANGLI~ 2X,28HBA-CKSCATrEHING CR033 3EC2IIOf) 40 F ORM ArT (b X,F"10. 5, 1S5X E 12. 4) i 'NDtI 18

U.Ll (t oqP3Uz~l S;1J3PoUTIiNE- 7ZVO8 (AtNXs, YIIAT) 1)H1EN-SION A (50, 51 )I X (50), Y (50), L(50) M,M(50) COM1P LE X AI.DOB I GA,HOLD Y, X TANTLGE1 LIM~ If (IAT- 1) 200,200,300 200 CONTINUE DC=. — xPL X ( 1.0,r0.0) D0 80 K=ION L (K) = K MiT -Ti = BIGA=A (K, K) 1)0 20 J=KN DO)0 1 I=K, N 10 I1F (C Aii S (BI GA)- AIJS (A (IJ)) 1 5, 20, 2() 15 BIGA=A(IJ) L (K) =1I M (K) =J 20 CO'0N TI N UE J3 L (K) I-rF (LI-K) 35,35,25 -25 Do 30 I11,N f!CLD=-A (K,I) A (K, I)=A (J, I) 3 0 A(3,I)I=HOLD 3 5 ~M(K) IF (I-K) 45,r45,38 38 DO 40 3=1,N HOLD=-A (J,,K) A (3, K) = A (3, I) 140 A (JJ ) =HOLD 145 IF (C AB3S (B IGA)) 48,46,148 446 D=C1PL.X (0.0,0.0) RETU I~N 148 DO 55 I1l1,N IF (I-K) 50, 55,5 0 50 A (ItK)=A(I, K)/(-BIGA) 55 CONTINUE C REDUCE M~ATRIX DO 65 1=1IN DO 605 J=1,IN IF (1I-K) 6-0,65,60 600 IF (J-K) 62,65,62 62 A(IpJ)=A(IvK)*'A(KJ) +A(IJ) 65 C"-ON TINU E C DIVIDE ROW BY PIVOT DO 75 J=1,N IF (3-K) 70,75,,70 70 A (KJ) =A (K J)/BIG A 75 CONTINUE C PRODUCIT OF PIVOTS D=D* BIGA A(K,!K-)=1.O00,r0.OOO) /BIGA 80 CONTINUE 3 N=N DMAG=C ABS (D) *(2. **BN) 19

Ul l' b4-b UZ-M K=N 100 K=K-1 IF (K) 150,150,105 105 I=L (K) IF (I-K) 120, 120,108 10b DO 110 J=1,N HOLD=A (J,K) A (J,K) =-A (J,I) 110 A(J,I)=IIOLD 120 J=M (K) IF (J-K) 100,100,125 125 DO 130 I=1,N HOLD=A(K,I) A (K,I) —A (J,I) 13C A (o, I) H J L. GO TO 100 150 CONTINUE 300 CONTINUE DO 210 I=1,N Y (I) =CMPLX(0.0,0.0) DO 210 J=1,N 210 Y (I)=A (I,J) *X (J)+Y(I) RE'URN END 20

011764-502-M S i iROIJPrINE HAMlK (Ii, N, 3JIB Y) D I:'IENStION A (7),t(7),C(7),rD(7), 7), ( 7),G (7),[ ( 7) DATAADC,,,FG /.0 e& -2. 2 11999 97O, 1. 26 56 208,0 I-03 1 63 866Or &0.04444790,-O.0,03941440,r 0.00C2 1000, 0.36746691s, 0.6055936t')-0O.7143503841 ES 0.253001 17,-0.01426 121'4, 0.001427961l,-0.0002!4345,3 &0.50000000,-0.50'2L49985, 0.21093573, &- 0. 0395 42 89, 0. 0044331 9, -0 0003 176 1, -0. 0000 1 1 0 &-0. 636 6 1 980, 0. 22 1209 10, 2.16 82 7090,;,- 1.316148270, 0.312_39510,-0.014C09760, 0.00273730, &, C.797d88456,-0.00O000,77,-0.005527140, &- 0. 0000951 2, 0. 00 1 37237,-0. 000728053, 0.0001141476, &'-0.78539816o,r-0.014166397,f-O.0003o39145, &S 0.002'o257.3,-0.800054125,-0.00029333, 0.00013558, & 0.797381456,- 0.00000156, 0.01659667, & 0. 00017105,-0. 00249511,r 0.001130553,-0. 00023033, &-2.356191449, 0.121499612, 0.00005650,,- 0.00637879, 0.00074348, 0.00C7982'4,-0.000291b5f IF(R.LE.0.0)GO TO 50 IF(P.GT.3.0)GO TO 20 X=R/3. 0 X=X*~'X IF (N. N-L.0) GO TO 1 0 CALL ADAM(AXY) L3J=y CALL ADAM (BXY) 3Y=0.6366198*ALOG (0.5*R) *j3J+Y 1 0 I F(N.NE.1 )GO TO 6 0 CALL ADAM(CX,Y) BJ=R*Y CALL ADAM(D,X,Y) BY=0.6366198*ALOG (0.5*R) *BJ4'Y/R R ET UR~N 20 X =3.0/ R TF (N.NE.0) GO TO 3 0 CALL ADAM (E,XY) G'OOD=Y/SQRT- (II) CALL ADAM(F,X,Y) GO TO 140 30 I(N.N'E. 1 )GO TO 6 0 CALL ADAM (GXY) GOOD=Y/SQRTA (R) CALL ADAM (H,XY) '40 Tir=+ Y i3J=GOOD*COS (T) BY=GOOD*SIN (T) R ET U RN 50 MP=MP+1 60 P=PM P+ 1 RETURN ENXD 2 1

011764-502-M S UB{O UT I NE A DA ~1l(Z, vX fY) DIMEN3-ION e0-(7) Y= X*C (7) DO 1 0 I= 1,j5 10 Y=X * (C (7 - I) + Y) Y=Y*C ( 1) ii ETU RN E N D 22

011764~-502-M SUB PO IV:IN E GPI (Xi Ye Le S, ml N, W, LN, A) C THIS SrJBROUTEtNE ~1AS BEEN ESPECIAL.LY MODIFIEF FOR USE WITH 1,AM1B C* CONTROL C C** CALL GPM(X, Y, L, S, M, N, Wo, LN, A) C** WHERE C * * X = ARRAY OF INDEP.ENDENT VALUJRS, DIIENS0IONED K (M) C** ~ Y = ARRAY OF' SETS OF DEPENDE'NT VALUES, DIMSNSIONED Y (Mt '). C** L = NUMbER OF LINES TO BE SKIPPED BEFORE 0ISPLAf. C**S = NUMBER OF SPACES FPIM LEFT SIDE DP PAGE TO C** BE SKIPPED BEFORE DISPLAY. C** M = NUMBER POINTS IN EACH SET. N= NUMBER OF SETS OF POINTS. C** W = WIDTH OF DISPLAY IIN PRINT SPACES. C** LN = LENOGTH OF DISPLAY IN PRINT LINES. C** ~A= ARRAY OF SINGLE CHARACTERS, DIMENSIO31NED A(N),,rQ C** REPRESENT THE TREND FOR EACH sEr (EX.- DATA &/1HA, C** 1Hi3,...ETC.) DI.MENSION X(M), A(N) DIMENSION PLOT(51,100),Y(100,r6j C INTEGER 5, W, W1 C DATA BLANK/1H -/rVERT/1HI/,HlORZ,17IlH=/ C C** CHECK MAXIMUM WIDTH AND LIENGTH RL-QUESTED AND C ** E~XIT IF NOT CORRECT IArF (S+W.GT. 131) GO TO 900 IF (L+LN.GT. 58) GO TO 800 C C** FIND MINIMUM AND MAXIMUM OF X AND Y XMAX=X(1) XMIN=X (1) C DO 10 I=2,M IF (X(L).G-T. XMAX) XMAX=X(I) 10 IF (X(I).LT. XMIN) XMIN=X(I) C YMIN=0. YMAX=.2. 5 C C** COMPUTE SCALE FACTOR P FOR K, FOR Y P=FLOAT (W-1) / (XMAX~-XMIN) Q=FLOAT (LN- 1)1 (YMAX-YMIN) C C** BLANK PLOT ARRkY DO 30 L=1,W 2

011764-502-M L)(' 30 JJ=1,LN 3 i? PLOT (J,I) = BLANK C X' ('CNSTRICT BORDER OF DISPLAY DO) 40 J=1,LN L~= 1.'L AOT {.J, I) =V ERT 40 L)T (J, I) =VERT C W 1=W-1 DO 50 1-2,41 - = 1 0 =1 P LOT (J, I) =HORIZ JLN O?-LOT (J, I) =HORIZ C C*~, CO:PUTE SUBSCRIPTS AND INSERT TREND CHARACTER IN C. PLO'T ARRAY C. DO 60 I= 1,M DO t0 J=1,N I=1 +INT(0. 3+?*(K (I)-XMIN)) J1=LN-iNT (0.5+Q* (Y (I, J) -YMIN)) Ub PLOT (J1,1) =A (J) C*' SKIP L LINES BEFORE BEGINNINO DISPLAY PRINTING C DO 70 K=1,L 70 PRINT 600 t j3 FORMAT (1H ) C C* WRITE OUT PLOT ARRAY, SKIPPING S SPACES BEFORE PRINrIN] Cv* EACH LINE 0F DISPLAY C ' DO 80 J=1,LN P, PidI, 1 601, (BLANK, K=1,S), (PLOT (JI), I=IW) '51 FCORMAT (132A1) PRINTr 602, XMIN, XMAX, YMI N, YMAX o02 FORMAT (1H0,5,Xt6HXM.IN =E16. 8,10X,6HXMAX =E16.8, l0X, X 6HY.1IN =E16.8,10X,6HYMIAX =E16.8) RETURN C C*' ERROR MESSAGES BEFORE TERMINATION C' X,)0D PRINT 603, L, LN J3 FORMAT (30!{AL+LN IS GREATER THAN 58 L =I3,5X,4HLN =I3) CALL SYSTEM 930 PRINT 604, S, W 604 FORMAT (30HiAS+W IS GREATER THAN 131 S =I3,5X,3HW =I3i CALL SYSTEM R iETU N ND 24

011764~-5U'2-M B. TimeTal SPF N T# 00000-00000 1.33 -8 PLO0G C0000-00000 0. 314 C, ~ B. —; 0OCO-GOOGO 2 3. 101 CD V D 00000-00000 1.55 C!VIY# 00000-00000 1d2.76 1 sy00000-00000. 1.24 s IV 00000-00000 1.55 1 4TIINTALY 00000-01 BFF ' 0.00 *tFAKLEMTS OOCCO-OOI2FF 3.0 10 00F00-00FF? 3. 521 01000-OlOFF 0.d 16 01100-012FIF 0.001 01300-013 FF 0. 17 011400-OI14FF 183.62 01500-0s2FF 0.001 M1A IN 00000-003FiT- 0. 00 ADAM 00000-COOFF 1. 551 00100-OC1iFF 7.071 GEOMi 00000-OOCCF 0.00 OODOO-GODFF 0.17 OOEOO-t0C-:FF 0.001 IIATRIX OOCOO-OO3FF 00 00L400-OO4FF 0341 00500-OO5FF 0. 521 006"00-OO6FF 0.341 C3700-OC7FT 0.00 HANK 0 00 00-0 (0 WF,& 0.521 00100-001F~ 0.03 00200-002F',172 00300-303FF 2.4 1 00400-004F2' 1. 55 00500-OOSFF- 0.171 SC AT? 00000-019FF 0.00 ZVOB 000 00 -OC4FF 0.00 00500-o30u5FFA 2.93 30600-OC6'FF 0.86 00700-'00'7vF 3*145 C3800-00dFf o.901) 00900-OO9FFVA 0.3141,OOAOO-OCAFF 0.521 OOBCO-OCJ?3-F 0.52 GIU OGOOQ-057FF 0.001 IBCOV )0 00 0 O2.FF 0.00 &c300OQ3FF 0.341 00400-0O4FF 0.34 00500'0051'F 0.171 ~060-"06FF 0.00 A~tDCONI# O0 00-0 0 0FE 0.001 F-C VZO 0(0100-OcliFF 0.00 FCVLO 00200-004F1 0.00 I FCVIO 00500-OO5FF 0.00 I 00600-0O6FF 0. 17 25

011764-502'-M C0700-OO07FF 1.0 3 0""0800-00dFF 0.00 00900-OOG9FFT. 0.69 FCVEO COAOO-OOB-FF 0.00 F cYvC O0CO0-0CCYF? 0.001 FCVTHB'- 00D00-O1OFFv 0.001 FIOC~i OOOOC-'QO1FF 0.001 00200>002FFV 0. 171 00300~-004FA. 0.00 O0500~-005FF 0. 17 00600-OOEFI' 0.001 26