011136-1-F 11136-1 -F = RL-2209 THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING Radiation Laboratory TRANIENT RADIATION AND RECEPTION OF ELECTROMAGNETIC ENERGY Final Report (15 December 1972 - 31 May 1975) Grant GK 36867 by Professor Chen-To Tal 21 May 1975 Prepared for: National Science Foundation Washington, D.C. 20550 Ann Arbor, Michigan

011136-1-F THE UNIVERSITY OF MICHIGAN Sum.mary The research completed under this Grant covered the following problems: 1) Transient Radiation from resistlvely loaded transmission lines and thin biconical antennas, 2) Radiation and reception of transients by lear antennas, 3) Eigenfunction expansion of dyadic Green's ftotions. Problem 1) was the Ph.D. thesis topic of Mr. Harold E. Foster, who completed his work in the Spring of 1973. Problem 2) was the subject matter of a chapter, bearing the same title, written jointly by the principal investigator, C. T. Tai and Dr. Dipak Sengupta for a forthcoming book, Transient Electrom netic Fields edited by L. B. Felsen of the Polytechnic Institute of New York. The book is soeduled to be published in 1975. Problem 3) was not a main topic of the original grant. However, as a result of an indaequate treatment of the subject matter in the principl investigator's book, some effort was spent to improve that treatment. The complete work was too long to be published as a journal article, but a technical report was submitted to the National Science Foundation in July 1973. Subsequently, the same report was published as one of the Mathematical Notes by the Weapons System Laboratory of the Kirtland Air Force Base, as requested by Dr. Carl Baum of that laboratory. The elgenfunction expansion of dyadic Green's functions pertaining to cavities have also been found. Functions pertaining to cylindrical and spherical cavities were part of a Ph. D. thesis written by Pawel Rosenfeld, which was completed in March 1974. Different representations of the dyadic Green's function for a rectangular cavity have been investigated very thoroughly by the principal investigator, and will be presented as an oral paper at the forthcoming meeting of the AP-S/URSI International Symposium in Urbana, Illinois, June 2-5, 1975. Discussion of Research For convenience, a copy of the Abstract of Harold E. Foster's Ph.D. thesis is attached to this report. The Abstract gives a detailed synopsis of the work. Of particular significance is the treatment of a resistive loading that varies linearly with 1

011136-1-F THE UNIVERSITY OF MICHIGAN position along a transmission line. The problem was solved exactly in terms of Airy functions of complex argument. This model is used to check the numerical analysis for other problems which could not be solved analytically in closed form. For the problem of radiation and reception of transients by linear antennas, we emphasize the importance of the concept of the transfer function between the transmitting antenna and the receiving antenna. According to our view, the description of the transient field of a transmitting antenna alone is of limited use, because the detection of any electromagnetic sial inevitably requires a receiving antenna. Thus, the characteristics of the receiving antenna play an Impurtant role in determining the ultimate response of a transient signal. In our work, we apply the method of Fourier transform to the formulation of the problem and carry out the analysis for some simple interaction problems such as the transfer function analysis between dipole and dipole, dipole and loop. For resistively loaded antennas, we depend heavily on the numerical method to obtain the final result. By properly loading the antenna, it is possible to create transient fields which exhibit very little oscillatory response to step-voltage excitation. In the work on dyadic Green's functions, we have now amended the missing singular term in all the eigenfunction expansions of the dyadic Green's functions previously studied in the principal investigator's book. This unusual error was not detected for several years in spite of the fact that the manuscript of the book had been reviewed by many experts in this field. The missing singular term only affects the field in a source region. This is presumably the reason why it was not discovered until the whole completeness problem was critically examined by Professor Per-olof Brundell of the University of Lund, Sweden. Since then we have reconstructed all the functions based on a new expansion technique using the inhomogeneous equation for the dyadic Green's function pertaining to the H-field as the starting point, namely, VX YVXG-k r G xHi [ a( R-u' H k H The solenoidal nature of the function GH, i.e., V GH = 0, enables us to construct n. aH 2

011136-1-F THE UNIVERSITY OF MICHIGAN the eigenfunction expansion of GH using two sets of solenoidal vector wave functions. The corresponding GE function is then found by E2 = GE V GH Z: The discontinuous behavior of GH across a current source region is responsible for ii the singular term of GE as a result of the differential operation on a discontinuous function. As a sequel to this work, we have completed the derivation of the dyadic Green's functions pertaining to cavities. In the thesis of Pawel Rosenfeld, the function pertaining to rectangular, cylindrical and spherical cavities were derived. The Abstract of his thesis attached. Alternative representations of the rectangular cavity functions have since been thoroughly examined. This work can be considered as a supplement to the work of Morse and Feshbach Methodsof Theoretical Physic, Vol. II, McGraw-Hill Book Company, 1953] who derived one complete expression of GA, the function pertaining to the vector potential, but gave one incomplete expression in another place. We have now found all the alternative expressions for GE The relations between GE, G and G are also discussed in detail. This work will be presented in the forthcoming meeting of AP-S/URSI in Illinois. Publications under Grant GK-36867 1. C.T. Tai, "On the eigenfunction expansion of dyadic Green's functions", Proc. IEEE, 61 p. 480, April 1973. 2. C.T. Tai, "Eenfuntion Expansion of Dyadic Green's Functions", technical report submitted to NSF in July, 1973; the same report was reissued by the Weapons System Laboratory, Kirtland Air Force Base, as Mathematics Note No. 28, dated September 1973. 3. P. Rosenfeld, "The Electromagnetic Theory of Three-dimensional Inhomogeneous Lenses and the Dyadic Green's Functions for Cavities", Ph.D. Thesis, Department of Electrical and Computer Engineering, The University of Michigan, 1974. 3

011136-1-F THE UNIVERSITY OF MICHIGAN 4. H. Z. Foster, "Transient Radiation from Resistively Loaded Transmission Lines and Thin Bicouloal Anasll", Ph. D. Thesis, Department of Electrical and Computer E tlaeerig, The University of Michigan, 1973. 5. H. E. Foster mad C.T. Tai, "Inhomogeneous Resistive Loading of Transmission Medla", paper presented at Fall URSI Meeting, Boulder, Colorado, October 14-17, 1974 Persomel NSF Grant 36867 was awarded to The University of Michigan and was administered in the Radiation Laboratory under the direction of Professor Chen-To Tai. He was assisted by Harold E. Foster and Pawel Rosenfeld, both graduate students in The University of Michigan before they completed their work in 1973 and 1974 respectively. Yu-Ping Lu, antbr graduate student, assisted the project mainly in computer programming and computation. 4

480 PROCEEDINGS OF TIlE IEEE, APRIl[ 1)973 frequency shift is proportional to the cosine of the angle of arrival of the incoming wave referred to the direction of vehicle travel, the frequenIcy axis can also be scaled in angle of arrival. The angle speciti s a cone only so it is impossible to differentiate between paths:I11viig from the right or left (or up or down). The figures vividly ill1 urate that most of the scattered radio energy arriving at the moilile ehicle at the bottom of the 10-m-wide canyon formed by the surro ilding buildings was traveling generally up and down the street (nearly naximuim I)oppler shift plus or minus). Only a small part of the scatt red energy arrives from a path nearly perpendicular to the street. If he buildings were not present, the path between transmitter tand receiver would be at an angle of 420 to the direction of vehicle tr:rvel. This 2~ angle corresponds to a Doppler shift of +2.2 cycies-/m. For sa ll excess delays there is some energy arriving at this utn]e. Referenc [3 ] includes some description of the scattering env'irnmaenct that i applicable to the major features on this scattering iic tion. Briefly, lor excess delays up to about 1.5;Ls, the scattered ciieriv comnes froi buildings in the vicinity of the vehicle. Up to,i!.i;..3 i s, scatteriTi is from other buildings on lower *Manhattan I-,::tid. 'hle general Ia k of scattered power at excess delays of from about 2 to 8 us is due to the East River and Hudson River open areas surrounding the isl nd. From about 8 us to 10 Us., the relatively strong peaks that are D)pler shifted by the negative maximum;iana;unlt 1/X are reflection. from Brooklyn across the East River. 'I he ne-ativelv shifted sma peak at 6 and the positively shifted smat;!l!,peak between 8 and 9 s do not seem to correspond to any simnle reflecting location and t us are probably due to multiple reilectiorn. Par;ameter: that describe the nultipath delay spread of this path are defined and tabulated in [3]. ACKNOWLE GMENT The author,wishes to thank NW.:Legg, A. J. Rustako, R. R. NMurra., and C. 0. Stevens for their he) in taking the data, \V. E. Iec f;or his help in the data reduction, a.ld D. Vitello and \V. Mammei for mnaking the three-dimensional comititer plots. REFE RENCES 1 R.E. Kt i..nedy, d 'i:;:g Di.;crsi:'c Ccmn:;.unicalion C. c;;Nns. NC-.X Ycr: V.'2ey ntori~-^nr1, i960, ch..3 [2] I). i. C(ox, O'Delay Dolppler characteristics of multipat propagation at 910 MHz:m a,:turban mobile radio environment," IEEIE 7"r ns. Antennas Propagal.. vol. AP-20, pi). 625-635, Sept. 1972. [3 —."91) _MHIz urtban mobile radio propagation: Mul th characteristics in Newv \'ork Cit-," submitted to IEEE Trans. Commun. I] \V' R. R. \'oung, Jr., and L. Y. Lacy, "Echoes in transmissiA at 450 megacycles roin land-to-car radio units," Proc. IRE, vol. 38. pp. 255- 8. Mar. 1950. 5] (;. I.. Turin, F. D. Clapp, T. L. Johnston. S. B. Fine, and D. La-vry. A statistical iodiel of urban multipath propagation," IEEE Trans. VIt. Technol., vol. T-2 i, pp. 1-9, Feb. 1972. 16] D. C. Cox, 'Doppler spectrum measurements at 910 MHitz ox a suburban imobrile radio pathl," Proc. IEEE. vol. 59. pp. 1017-1018, June 197 [7j Fl. B. Barrow, L. G. Abraham, \V. M. Cowan, Jr., and R. M. Galla t.'Indirect arnoolieric measurements utilizing Rake tropospheric scatter t ehniques-!::t I: The Rake tropospheric scatter technique." Proc. IEEE, vtU 57. pp. 537-551, Apr. 1969. [!S I). C. Cox, "Time and frequency domain characterizations of multipat propa-ation at 910 MlHz in a suburban mobile radio environment,' Radio Sci., vol. 7, Dec. 1972. Professor Per-Olof Brundell of the University of Lund i.,l. Sweden, has kindly called my attention to an error in the trt,,(It,.: t of the eigenfunction expansion of the dyadic Green's functii,.,! - cussed in my book [1 ]. In that work, only two sets of solenoMiil (, tor wave functions are used in the expansion of the dyadic deltr, fri,, tion 16(R-R'). These two sets are indeed complete within the,, - of solenoidal vector fields but not within the space of all vector fidt As Ib(R-R') is not solenoidai, having the divergence equal t,, Va(R-R'), the two sets of solenoidal vector wave functions,m. ployed are not sufficient to represent such a quantity. As a result of the error, the singular behavior of the eigenfiincti,, expansion of the dyadic Green's function is not correctly forintulatcl in my bxook. Of the residue series derived in the book, all of them halpen to be valid for R$R' except the one dealing with a recta.ngrlar waveg.ide with moving medium which is incorrect. However, the method used to dterive these residue series is rmisleading and unsatisfactory. For the slpherical case based on the algebraic method, the singular behavior of the dyadic Green's function can be preserved if so desired although only the result for R/-R' is given in the book. In this letter we outline a method whereby only two sets of solenoidal vector wave functions are sufficient to derive the correct solution which is valid everywhere including the source region. We consider, for example, the dy;dic Green's function of the first kind ~pertaining to a rectangular waveguide which satisfies the equation: V X V X G(R I R') - k2G(R I R') = 16(R - R'). (I) By taking the curl of (1) we obtain V X V X V X Gi(R I R') - k:V X Gi(R I R') = V X [75(R - R')]. (2) The function V X 1b(R -R') is solenoidal so it can be expanded in terms of the vector eigenfunctions M and N defined by (h = X sin sin n eihl (3) mvrx nr-ry ih[ N,( h) = V V X Cos - cos cos_ e'zi (4) K a b Th ize rer th4 Gi wl ta R se 11 hz gi d( d( d, b, w al ti li Crl where 2 = 7)h + k_2 These vector wave functions satisfy the boundary condition at the walls of the waveguide corresponding to x=0, x =a, y=0, and y- h: i X V x M,,,(h) = 0 fi X V X Nmn(h) 0. They are compatible with the requirement that iXG1(RIR') =) on the boundary of the waveguide. Ve now let VX13(R-R')-R fI Z [Mo.n(h) Aon(h) +NV,n)B(h)Bhldh (5) where A and B are two unknown sets of vector coefficients to be (determined. The integral in (5) is to be interpreted in the generalized sense as the Fourier integral representation of the distribution V X [Ils(R —R'). By means of the orthogonal relations of the fiun:ctions MoL,,(h) and N,,(h) we find 2 - So Ao~.(h) = ~ V' x Mod(- -h) vabkc, 2 - Bo B~h)....'X N,,,(-h) -rabk,2 where the primed functions are defined with respect to the pnrintl' variables (x', y', z') pertaining to R'. Applying now the Ohm- a1;t'!1'i method to (2), we can express V XtG(Rj R'), which represents at:olnoidal vector field, in a simnilar form. The expression for V X G (KR K?' is found to be V x G,(RIR') = 5 2-S m n iaibk,'(h1 + kr - Kt2) -[IM.(h)V' X M.'( —) + N,.(h)V' X N,,(- h) \" [i i I t i On the Eigenfunction Expansion of Dyadic Green's Functions CHEN-TO TAI Abstract-As a result of an erroe, the singular behavior of the eigenfunction expansion of the dyadic Green's function is not correctly formulated in my book (C. T. Tai, Dyadic Green's Functions in Electromagnetic Theory, Scranton, Pa.: International Textbook, 1971). The correct expressions are given here and an improved method for deriving the residue series is presented. Manuscript received November 21, 1972. TIe- author is with thie Radiation Laboratory. University of Michigan, Ann Artior, Mlicih. 48105, and with the Institute of Electromagnetic Theory. University of L,und. Lund. Sweden.

PROCEEDINGS LETTERS 481 In view of (1), we have G,(R { R') - [V X V X G(R |) -b(R - R') _- - ^lfIy^^ o __ = - Fs + abk 2(h, + k ' - k2) -IV X Mn(h)V X MN,'(-h) + V X Ne,(hI)V' X N,(-h) dh.. (7) The integral in (7), representing V XVXG,(RI R'), contains a generalized singular function which can be extracted from the integral. The remaining part can be evaluated in a clos-cd form by the residue theorem. The final expression for Gi(Rj R') is given by k2 ab m kgk2 -(R | R' =^(R - - R') i 2 6 {M,,,(k)Mn,'(-k,)+ N,,,^n(k)Noml(-k) M n.( —k)M n '(k ) + N,(-kg)No(k) (8) where ka denotes the guided wave numbler (k - k) 12)l. The series contained in (8) converges to a distribution as R approaches R'. For Rf R', the top series applies to z>z' and the bot tom one to z <z'. The series contained in (8) is the sarne as that founld in the author's book [1, p. 79, eq. (8) ]. However, the expression for GI(Rj R') derived here has an extra term — z^a(R-R')/kl which together with the series gives the complete representation of Gi(R]R') in the entire spatial domain of the waveguide includin, the source region. The method described here is applicable to.il eicenfunction expansions of the dyadic functions which satisfy an eriluation of the form represented by (1). The details of evalritinrt the integral representing VXVXG (RI R') are rather intricate. They will be described elsewhere together with other cases. ACKNOWIE DGM E NT The author wishes to thank Prof. P. 0. Brundeil for many valuable discussions, particularly for his analysis of the problem based on the theory of distributions, which ultimately lead to the method outlined here. The technical hellp which the a;Jljori received from Prof. 0. Einarsson is also very mnlh??prcciate. REFERENCES [1] C. T. Tai. Dyadic Green's Functions in Electromagnetic Theory. Scranton. Pa.: International Textbook, 1971. A Note on the Modified Kalman Filter r Channel Equalization JON W. IMARK Abstract-An applicati'x of the Kalman filter to equalization of a ligital communication chanrll is described. The resultant modified Caiman equalizer (KE) is a inlinear system in which the channel ap gains are estimated via a ecision feedback aUproach and the litial state variable is estimated a prediction process. I. INTRODIUTION In the design of a digital ccmmunkation system a tapped-delay ne model has often been used to repreient the dispersive channel. the digital message is encoded such hat it is characteristically milar to white Gaussian noise, the chanMl model may be formuted using a state variable representation\similar to the filtering odel advanced by Kalman [I. Here the \iate variables are the ccessive rnessage symbols. Under a known hiannel condition the alman. filter, which is a dual to the channel rdclel, represents the timum linear equalizer in that the nlmber of ahunlizer taps needs ly be the same as the number of channel taps. PXor a conventional ear equalizer (CLE) the performance is a direct function of the xree of freedom associated with the CLE [61, [7 I.\ptinum linear Manuscript received October 10, 1972; revi:s 'd November 9, 197This work was ported by the National Research Couoncil of Canada. rhe author is with the Department of Electrical Engineering, niversity of:erloo, Waterloo. Ont., Canada. equalization is attained by the CLE only when the number of taps is infinlte. Also, the tap gains further away from the main or central tap ar smaller and are difficult to adjust. Thus a Kalman equalizer (KE) which needs only a finite number of taps to attain optimality is definiely superior to the CLE from the implementation point of view. In.e KE the estimates of the state variables constitute the estimates f the transmitted message symbols. A utilization of the Kalman filttr for channel equalization under a known channel condition has bee described recently [2. Applicatioh of the Kalman filter to channel equalization requires 1) a knowledgt of the initial state variable estimate and the initial covariance mat ix, and 2) a knowledge of the channel impulse response. If the iniial estimates are outside the feasible region to admit convergence, the alman filter will diverge. In the equalization context, a divergent situation will not necessarily mean an unstable condition. Rather, th system error will run away. Unless the initial state variable and tbe initial covariance estimates are chosen judiciously, the stringentrequirement of initial conditions may obviate the optimalitv of the lFalman filter as an equalizer. Also, for unknown channels, the gains of twe tapped-delay line model (which correspond to the components of tie output vector in the Kalman filter model) need to be measured or 4stimated. Taking into account the optin-ality of the Kalmani filter, ~his letter is concerned with the estimation of an initial state variableestimate and the adaptive computation of the channel tap gains {cj.\ II. PRSBLEM STATEMENT ILet the discrete messagelsequence {a(n)} be sample-to-sample independent. I.et the digital Vommunication channel be characterized by a tapped-delay line nmdel with a finite number of nonzero tap gains i;->rcf -r,, where cn i the main tap with I precursors and M delay echoes. The channel nray be represented by the following state vector equation: m(n) -- Fm(n\- 1) + ga(n). (1) with the channel output given by y(n) = ct(n)m( ) -+ 'n) = x(n) + (n) (2) where F is an (MAL+ l)X(~L+~L+ )) system matrix which characterizes the tapped-delay line model, is an (M —L+4I)X1 input vector, and c(n) is an (M-,-L-+1) X1 butput vector (channel tap gain vector). F and g are given by I 0 ~ F- 0 1 j ~ and\g i E. (3) In (1) and (2), m'(n ( (m (n),, mo(n), * \, nv(n)) is a state vector, x(n) is the channel output scalar, and,7(') is a sample of an additive white (aussian noise sequence. The sluerscript t denotes matrix tranlspose. In the KE the final state varinble estimate represents the estimate of the message symbol, i.e., d(n- -) = -ih(l(n n). Our problem is to devise an adaptive technique fot computing tlhe conditional estimate \ a(n - M) - E{a(n - M) | y(n), y(n - 1), ~* * (1) E'a(n - M) y(n)}, n > 11 (4) where the last equality holds due to the implicit Gaiss-Markov properties of the {y(n)} sequence. Also, it is assumed Vlat at the start, the precursors of the KE have been filled with data.\ II. ADAPTIVE MODIFIED KALMAN EQUALIZER (KE) If the transmitted sequence {a(n)} is assumed to be whit Gaussian noise, the state vector m(n) is a Gauss-MTarkov vector. Aiplication of tle conditional mean (4) while taking into accour the implicit Gauss-Markov property of -v(n)} leads to the Kayman filtering equations m(n I n) = m(n | n - 1) + k(n)[v(n) - x(n I n - 1)] 5) m(nn n- 1) = FrM(n - 1 I n - 1) + gm_.(n in - 1) 6) T (n |n - 1) = cp(nam ( n n - 1). (7) The unknown parameters in (5) to (7) are m-_tL(n n —1), k(n), and II I I I

ABSTRACT THE ELECTROMAGNETIC THEORY OF THREE-DIMENSIONAL INHOMOGENEOUS LENSES AND THtE DYADIC GREEN'S FUNCTIONS FOR CAVITIES by Pawel Rozenfeld Co-Chairmen: Chen-To Tai, Chiao-MIin Chu In this thesis the dyadic Green's functions of a number of cavities have been derived and the characteristics of some inhomogeneous lenses have been investigated. To facilitate the treatment of problems involving cavities we have found the expressions for the dyadic Green's functions pertaining to rectangular, cylindrical and spherical cavities. Expressions for the electric and magnetic field involvin'g the Green's functions are, presented. An example of the application of the dyadic Green's function technilique to the computations of the input admittance of the rectangular cavity is given. The lenses covered in our work include: the Luneburg, Eaton-Lippmann and Eaton. The dyadic Green's functions for electric and magnetic dipoles in the presence of these lenses are found. The expressions for the electric field of an Huygens source in the presence of an inhomogeneous lens are constructed. Radiation patterns and the bistatic scattering cross sections for the small-diameter lenses and the directivity and the distribution of the energy around the geometrical focus of the Luneburg lenses are examined In detail.

ABSTRACT TRANSIENT P ADIATION FROM. RESISTIVELY LOADED TRANSMISSION LINES AND THINT BICONICAL ANTENNAS by Harold Edwin Foster Co-Chairmen: Chen-To Tai, Ralph E. Hiatt This dissertation presents a theoretical analysis of the radiation and reception of transient electromagnetic signals by resistively loaded transmission lines and thin biconical antennas. The resistively loaded transmission line analysis, in addition to providing an advance in its own right, supplies a basis for the study of transients in antennas. Transmission line theory and mechanisms apply to the modeling of a variety of antennas and to the detailed understanding of their performance. The transient analysis is attacked by the Fourier transform approach to make use of establislhedU concept' such as Lilat of impedance. Investigations are performed first in the frequency domain and then transformed into the time domain for inspection of transient results. The Fast Fourier Transform technique of truncating series of sinusoids provides some economy where numerical computations are needed for the transformations. General transmission line equations are developed to account for time-dependent and position-dependent transmission line parameters. These equations are then specialized to accommodalte the case under investigation which is that of an open-circuited transmission line loaded with series resistance. Several functional distributions of resistance alongr the transmission line are considered. For a resistive loadling that varies linearly with position along the line, a closed form expression for the resulting current on the line is found in terms of Airy functions with complex arguments. For resistance distributions other than linear, t'he transmission line equation does

not in general possess a closed form functional solution. These problems are solved by a numerical analysis which is implemented digitally on a computer. An examination is made of the control which the different resistance distributions exert over the transmission line's input impedance, current distribution, radiated transient waveform and received transient waveform. It is shown that an inverse functional form of resistance loading is optimum, based on criteria of nmaximizing current on the transmission line while minimizing reflections. Discretely lumped resistance loadingos as well as continuous resistance distributions are analyzed. Results of the discrete and continuous analyses are in excellent agreement when the discrete resistances are sufficiently close together. The concept of a position-dependent characteristic impedance is developed for the resistively loaded transmission line. In addition to varying with position along the line, this quantity also differs in the forward and backward directions. Such characteristic impedances are formulated in general and for the several resistance loading functions that are treated in this dissertation. An approximate step function voltage source is considered to energize the loaded transmission lines. The resulting current waveforms at positions along the line and the resulting transient radiated fields are computed. The different shapes of the transient waveforms that are radiated in different directions from the loaded transmission line are shown. The radiation patterns which change shape with time are also computed and shown. The maximum amplitude of radiation, over all time, is shown to be in an off-broadside direction that is consistQnt with a predominantly traveling wave. This occurs for the optimally loaded transmission line in which reflected waves are minimized. Reception of transient signals by a resistively loaded transmission line is formulated in terms of the vector effective height function. Transient radiation coupling between different loaded transmission lines is also formulated

in this way. The conical transmission line fields associated with a thin biconical antenna are used in an analysis of the transient behavior of this antenna. Determination of transient currents on the antenna takes account of internal complementary fields. Radiation of transient waveforms is analyzed. Transient reception is analyzed via the vector effective height function of the antenna..