AD-AO786 860 NAVAL SURFACE WEAPONS CENTER WHITE OAK LAS SILVER SP--ETC F/6 20/14 mSINGULARITY EXTRACTION OF EMP RESPONSE WAVEFORMS BY PRONY'S RET--ETCIU)MAT RI A S LEVECKIS

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1. REPORT NUMBFI 12. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER

NSh WLR77-99 _gft~'7y ~ __________

4. TtTL4tm ......) . S. TYPE OF REPORT & PERIOD COVERED

-INGULARITY ;XTRACTION OF EMF RESPONSE 'iaWAVEFORMS BY PRONY'S METHOD. N

7. AUTHOR(#) 6. CONTRACT OR GRANT NUMBER(s)

Alg is S.4eveckis/

9. PERFORMING ORGANIZATION NAME AND ADDR . 10. PROGRAM ELEMENT. PROJECT, TASK

Naval Surface Weapons Center' d47

Silver Spring, MD 2C910 62734NtZF3± 8 1

11. CONTROLLING OFFICE NAME AND ADDRESS - 12. REPORT DATE'

Naval Surface Weapons Center , i I-May-98PISilver Spring, MD 20910 6NUMBEA PAGES

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IS. SUPPLEMENTARY NOTES

19 KEY WORDS (Continue on reverse aide if neceeeery md Identify by block number)

PronySingularity Expansion MethodNyquist Criterion

EMP

Sempex20. ABSTIACT (Continue on revere, aide if neceeary and identify by block number)* The Singularity Expansion Method has grown from a theoretical idea to a

numerical technique that calculates the poles of a given scattering object.It has spawned the study and growth of Prony's method, which calculates the

1: poles of the scattering object, from the scatter field that is in response

to an incident electromagnetic wave. Here is presented the results of anattempt to use Prony's method on EMP response waveforms on ships. The tech-

nique to acquire the experimental data is detailed; the numerical problem is

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20. ABSTRACT (Continued)

Uposed, and the goals are quickly simplified to attempting to find the bestcurve-fit to the waveform, using complex exponentials, and then hoping toeliminate the curve-fit poles. A method is presented that aids in thelphysical pole# selection process, and explained also is the significance ofthe right half plane poles in-the results.

DTICSELECTEJUL 2 1 1980 j

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NSWC/WOL TR 77-99

PREFACE

The study of EMP responses, has as of late, become a numericalprocedure, and recently in the technique called Singularity Expansion,a new method has been proposed which evaluates the poles and residuesof any response waveform. The goal is that these calculated poles beeasily related to the physical structure of the scattering/receivingstructure. This report explores an area of use for one of thesemethods - Prony's method. It examines the possibility of using theProny technique on EMP response waveforms, where the scattering objectis a complicated structure.

Chapter 1 discusses the reasons that SEM may be of interest inship EMP response analysis. Chapter 2 provides background in SEM andthe Prony technique. Chapter 3 presents the computer code that isbased on the Prony method and describes its use. Chapter 4 describesthe EMPRESS EMP simulation facility, and presents a summary of theexperimental work conducted. Chapter 5 introduces a technique thataids in the elimination of the "curve fit" poles from the calculatedresults. In addition, the process of the data anlaysis is described.

I would especially like to thank the following people for theirhelp and valuable comments during the course of the work. ProfessorJin A. Kong, MIT Advisor; Dr. M. L. VanBlaricum of Mission ResearchCorporation, Santa Barbara, California; Mary Laanisto, Neal Stetson,George Brackett, and Robert Baran, all of the Naval Surface WeaponsICenter, White Oak Laboratory, Silver Spring, Maryland. The resultsof this research were submitted to the Department of ElectricalEngineering and Computer Science, Massachusetts Institute ofTechnology, Cambridge, Massachusetts in partial fulfillment of the

requirements for the degree of Master of Science.

Released by:

D. B. COLBY, Hea

Electronics Systems Department

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NSWC/WOL TR 77-99

CONTENTS

Page

CHAPTER 1 EMP AND SHIP COUPLING .... ................ 11.1 EMP ........ ....................... 11.2 EMP Coupling .................. 11.3 Background for SEM Analysis for Ships ....... 3

CHAPTER 2 METHODS OF POLE CALCULATION .... ........... 72.1 Previous Methods .................. 72.2 Tesche's Method and SEM .............. ... 82.3 The Prony Method for Singularity Extra ction. . 122.4 Frequency Domain Prony .... ............ . 16

CHAPTER 3 THE COMPUTER CODE: SEMPEX ... ........... .. 173.1 Simplications for Analysis ... .......... . 173.2 SEMPEX Code Methods .... .............. . 183.3 SEMPEX Input Parameters .... ........... ... 21

CHAPTER 4 EXPERIMENTAL FIELD WORK .... ............. ... 244.1 The EMPRESS Facility .... ............. . 244.2 Test Equipment ..... ................ . 294.3 Preliminary Shipboard Work .. .......... . 304.4 Field Work on Shore .... .............. ... 35

CHAPTER 5 NUMERICAL ANALYSIS . . .... .......... 375.1 Digitization and the Test Waveform ...... . 375.2 Formulation of the Nyquist Criterion

Constraint ..... ................. . 43

CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS ........... ... 576.1 Conclusions ...... .................. . 576.2 Recommendations ..... ................ . 57

REFERENCES ...... ................... . 59

APPENDIX A COMPARISON OF THE TIME AND FREQUENCY DOMAIN

PRONY TECHNIQUES .... .............. .. 63

APPENDIX B CHARACTERISTICS OF THE EMPRESS ELECTRIC FIELD. 65

APPENDIX C RESULTS OF THE FOUR TEST CASES .......... ... 67

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ILLUSTRATIONS

Figure Page

1.1 EMP Collector Flowpaths in a Ship ..... .......... 21.2 High Amplitude Current Carrying Mast Cables .. ...... 43.1 SEMPEX Flowchart ........ ................... .. 193.2 Defining a Waveform for SEMPEX .... ............ .233.3 Defining S-Plane for Physical Pole Location ...... .234.1 EMPRESS ...................................... 254.2 NSWC Field Test Facility, Solomons, MD ........... .. 264.3 EMPRESS Electric Field Waveform .... ........... .274.4 EMPRESS Electric Field Fourier Transform ......... ... 284.5 Data Acquisition Setup ...... ................ .324.6 A 20 NS/CM Waveform ...... ................. .344.7 Waveform Merging ........ ................... .. 344.8 Four Field Experiments ...... ................ .365.1 Test Waveform ........ .................... .385.2 Windowing Techniques ....... ................. .. 445.3 Limits of Pole Calculation Analysis ... ......... .465.4 Nyquist Criterion Method Flowchart .... .......... .485.5 Using the Nyquist Criterion Method .... .......... .. 495.6 Waveform Used For Runs (1-4) .... ............. ... 505.7 Reconstructed Waveform Error Versus Number of Poles

Requested ......... ..................... .56B.1 Field Measurement Points Downriver From EMPRESS . 66C.1 Test No. 1 Waveform ....... ................. .. 68C.2 Pole Plot of Test No I Waveform. . ......... 6PC.3 Fourier Transform of Test No. 1 Waveform ......... ... 70C.4 Pole Plot of Test No. 2 Waveform ... ........... .. 71C.5 Test No. 3 Waveform ....... ................. .. 72C.6 Test No. 3 Waveform, RMS Error Versus NPOLES ........ 73C.7 Pole Plot of Test No. 3 Waveform ... ........... .. 74

, C.8 'rest No. 4 Waveform..................75C.9 Pole Plot of Test No. 4 Waveform ... ........... .. 76C.1O Test No. 4 Waveform, RMS Error Versus NPOLES ........ 77

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NSWCNYOL TR 77-ge

TABLES

Table Page

3.1 SEMPEX Input Data ......................224.1 The Dash 2 Probe ..................... 315.1 Test Waveform Without Filter. ............. 405.2 Test Waveform Requesting 34 Poles ..............415.3 Shifting Pole Locations.................425.4 Waveform with Many Curve Fit Poles. .......... 515.5 RunI..........................525.6 Run2..........................535.7 Run 3.........................545.8 Run4..........................55

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Chapter 1

EMP AND SHIP COUPLING

1.1 EMP

Nuclear weapons effects have for a long time been a majorconsideration in military systems research and development. It wasonly in the 1960's that the nuclear weapons effect known as theelectromagnetic pulse (EMP) was identified as a potentially harmfulphenomenon, especially in the case of a high altitude burst (HAB).After an HAB, when a high yield nuclear weapon is detonated abovethe ionosphere, the only significant effect on the earth's surfaceis this EMP. All other effects, such as the shock wave, the heatwave, and the radiation are absorbed by the atmosphere. Today thestudy of EMP effects is very important, since most electronicequipment utilizes semiconductor components. State-of-the-artintegrated circuit equipment, unlike vacuum tube devices, can beextremely vulnera le to any type of transients. Transients cancause upset, or even destruction, if the transients Pxceed the normaluse or breakdown specifications. The increasing sophistication innuclear strategy and armament, puts much more emphasis now on thehardening of defense systems against nuclear effectf, including EMP.It has been theorized by the Defense Nuclear Agency , that an HABcould give an attacking force as much as a thirty minute advantage,by upsetting or destroying electronic devices in parts of thedefense system.

1.2 EMP COUPLING

Ways in which EMP can couple into electronic equipment on aship are shown in Figure 1.1. Pathways I are the primary means ofEMP coupling to the ship's systems. The pulse itself is notdetectable on most waveforms taken on a ship; it may appear only

in making the first peak of a waveform a little higher than wouldbe expected. On a ship, the effect of pseudo-focusing is commondue to the large structures wit high amplitude resonant currents.The presence of the original EMP would be unascertainable in such awaveform. All pathways down the II section are almost insignificant.The amount of energy coupled down these paths depends on theamplitudes of the resonances of the skin current in the vicinity ofthe aperture, the size of the aperture, and the size of the room.The apertures can be modeled by infinitesimal or large dipole current

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NSWC/WOL TR 77-99

INCIDENT PULSE

ETRA-a- RESONANT APE RTURESCABLESSHIP'S

--flow STRUCTURES

la STUFFINGIa TUBE

INTERNAL

FIELDS

CCABLES

I lid

IcEN 0EQUIPMENT E

FIG. 1.1 EMP COLLECTOR FLOWPATHS IN A SHIP

RNSWCiWOL TR 77-99

sources depending on the size of the window. In a test that wasconducted in the largest room with the largest apertures on a ship,it was found that the internal fields caused by aperture couplingwere insignificant except very close to the aperture. Thus theenergy in pathways IIb and Id is small and only in IIc could itbecome significant if the equipment is very near large windows. Inpathway I the resonant ship structures have high amplitude currentsand by radiation couple to other ship's structures and externalcables. The worst case can be found in cables that are routed upthe mast. The external cables then pass through the deck of theship and radiate fields into the internal compartments, Figure 1.2.Good grounding techniques between the external cable and thestuffing tube through the deck (Ia of Figure 1.1) have been shownto attenuate the internal fields, but do not eliminate the problem.The energy in the internal cables couples to other internal cablesvia the Il loop. The internal fields that are part of the loop,could affect equipment via path Ic. Field tests have shown that theinternal cables that originate on the mast still radiate high amplitudeelectromagnetic fields even several decks below, before finallyreaching the equipment by path Ib. Thisis the most serious case inEMP penetration into ship's equipment.

1.3 BACKGROUND FOR SEM ANALYSIS FOR SHIPS

The resonant currents on the ship's structures and cables showdamped oscillations. These oscillations correspond to the naturalmodes of the current on the structure that are the solutions to theintegral equations describing the current for the structure. Theycan be represented as a sum of complex exponentials, with theexponent, (s = o + jw) termed a pole, specifying a pole location onthe complex s-plane. The location of the poles is only a function ofthe physical properties of the structure involved. The coefficientsof the complex exponentials are the complex residues at each polelocation in the s-plane or the "coupling coefficients" times thenatural odes, since they depend on the form of the excitingwaveform . The Prony method of Singularity Expansion Method (SEM)determines the poles and the coupling coefficients for the EMPtransient response waveform, which should be the poles of theresonating structure(s). Thus, it is actually extracting poles froma waveform, the reverse of the expansion process. Interest in thismethod was brought about by the presence and dominance of the shipstructures' resonating currents in most data taken on a ship.

Interest in SEM at the Naval Surface Weapons Center (NSWC) grewout of several ongoing projects. One was a study of antennaequivalent circuits. This was based on a Illinois Institute ofTechnology Research Institute (IITRI) feasibility study , using thereciprocity of antennas to predict the full-threat level EMP response.The technique is to calculate the equivalent circuit of the antenna,

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in its environment, by exciting the antenna at the base with apulse injection source and simultaneously measuring the radiatedelectric field. Using the equivalent circuit obtained by thismethod it would then be a simple matter to compute the response ofthe antenna to any excitation. The method is cumbersome and couldbe simplified by representing the antenna response by a set of poles,extracted by SEM. Another project for which SEM seemed to have ananswer was the design of injection sources for full-threat leveltesting. Here the EMP response of the antenna system, the antennain its environment, the cable routing, all intercoupling between thecables, would be measured at the input to the equipment. Aftercalculating the poles of the system from the response, it would bepossible to design a damped sinewave injection source that couldreproduce the waveform, but at full threat level. A new projectwhich provided the motivation for this thesis, was to examine thepole locations of antennas, with respect to their locations on theship. It thereby would be possible to identify the structures thatcontribute the most to a specific antenna's response.

The results of an SEM analysis of a response waveform wouldprovide much information on the antenna, and the antenna system'senvironment. Concrete answers might make it possible to redesignthe ship's superstructure, or simply place antennas in a more oPtimumlocation to reduce the coupling from structures with high resonantcurrents. With the identification of the structures, they could alsobe included in theoretical models to provide more realisticpredictions for ship systems' EMP responses.

I

Chapter 2

METHODS OF POLE CALCULATION

2.1 PREVIOUS METHODS

Previous to today's quick numerical methods of SEM muchanalysis had been done to find the natural frequencies of man-objects. The first attempt seems to be attributed to Pocklington

5

who constructed the integro-differential equation for current, andsolved it for the first mode of a small thin ring. He also was ableto derive the first mode coupling 6coefficient for a wave incidentupon the ring on its axis. Oseen and ifallen derived naturalfrequencies for thin wires. Oseen applied the method of retardedpotentials to thin wires, and derived asymtotic formulas for thenatural frequencies. [fallen derived a pure integral equation forcurrent op a wire, and was able to solve for the weakly damped modes.K. K. Mel extended the liallen integral equation to include thin wireantennas of arbitrary geometry. In the past years, much of theresearch effort has been geared to the formulation of numericalmethods of solution to the integral equations, though t e analyticaltechnique was recently revived and expanded by L. Marin". fie adaptedthe [fallen integral equation for electric field formulation andanalytically determined the approximate fields for scattering fromseveral thin wire structures. fie concedes that the analytical |

approach is feasible for the location of the natural modes torepresent the late and possibly the intermediate times of an impulseresponse, due to the small number of poles needed to accomplish this,and because poles in the weakly damped part of the spectrum areeasier to find. lie leaves the calculation of poles to reT 6 esent theearly times of a waveform to numerical techniques. Marinconstructed a numerical technique to solve the magnetic fieldintegral equation to find some of the poles of the prolate spheroid.lie reduced the integraol equation to two sets of coupled onedimensional integral -,ua1ions and then solved these numerically.Miller, Poggio, and Burke developed a numerical method to solvethe electric field integral equation for thin wires. In addition,they suggested that the magnetic field integral equation is bestused for the analysis of solid surfaces, while the electric fieldformulation is better for the solution of thin wire problems.

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NSWC/WOL TR 77-99

2.2 TESCHEI'S METHOD AND SEM

Carl Bau12 presented the Singularity Expansion technique forthe general problem of electromagnetic scattering from finiteconducting bodies, in defining th1 3natural frequencies, modes, andcoupling coefficients. F. Tesche developed a technique tonumerically calculate the poles and coupling coefficients for thinwire structures. Knowing these parameters makes it much simpler tocalculate the response due to any excitation at any incident angle;just as in circuit theory, knowing the impulse response, the polesof the circuit, it is easy to calculate the response to an arbitraryexcitation. Previously, using the time harmonic analysis with Fourierinversion, it was necessary to recalculate almost the entire problemto find the response when jyt the angle of incidence changed. Here,using the method of moments , an electric field integro-differentialequation is transformed into a matrix equation, where the zeroes ofthe determinant of the system "impedance" matrix are the naturalfrequencies (s = a + jw) of the system. For each of the naturalfrequencies, there is a residue matrix, which in the electric fieldcase can be defined by two vectors, which are identical.

Consider a wire of length L and diameter a, whose axis iscoincident with the z axis in (z,,e) space. A step function wave-front is incident from the negative z direction and encounters the

wire at z = 0 and t = 0. An integro-differential equation can bewritten for the z-directed current I(z,s). Starting with a generalintegral equation

-- .J -soEt (2.1)

with the integral operator defined by

- J)(r) VV - ) f G(r,r',s) . J(r') ds' (2.2)

where G(r,r',s) is the free space Green's function

-s r-r'I

G(r,r',s) - e c (2.3)Tr ( ,, r-r'l)

and s is the complex natural frequency

c is the speed of light

". t(z,s) is the incident tangential electric field

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sz cose

Ft(z,s) = 1_'i(s) sine e C (2.4)

where for a step wave, in the s-domain

.(s) -s (2.5)

By assuming that there is no 0 variation, equation (1.1) can berewritten

L

(-sE°Et (zs)) =f I(zs) z- K(z,z',s) dz' (2.6)

0

where K(z,z',s) is the kernel of the equation defined by

K(z,z',s) = C 2i e 1 (2.7)

0

and the point of observation R is

Rn 2 2 0 (2.8)

Previous methods solved eouation (2.6) for many jw and then Fourierinverted for the time-domain response. In Tesche's analysis, thetechnique is to find the poles of the current function I(z,s) inthe s-plare in a manner similar to circuit analysis, i.e., let theincident electric field be Lero and find the non trivial solutionsto the homogeneous integro-differential equation. The equation canbe transformed by the method of moments to a matrix equation

(Z(s)) (I(s)) = (V(s)) (2.9)

(Z(s)) the impedance matrix is a square n x n matrix, (I(s)) is thecurrent response vector, and (V(s.) the voltage or source vector,both cf timer.sion n. Now (V(s)) is set to zero and the non trivialsolutions will he those s. such that

dt fV(s)) = (2.10)

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NSWC/WOL TR 77-99

Solving equation (2.9) for the response vector

(I(s)) = (Y(s)) (V(s)) (2.11)det (Z(s))

and then by Fourier inversion one obtains the current time response

1 0 (Y s) ) et(it)) 2- t (Z(s)) (V(s)) es ds (2.12)

o - j W

now let

(Y(s)) (.13

det (Z(si)) s-s(

This is the sum over all the poles in the complex s-plane, and (Ri)is the residue matrix. This matrix is a dyadic and can berepresented by the outer product ?f two vectors of dimension n: (M.)the natural mode vector, and (C.) where (Ci) is the couplingvector, where the vectors are the solutions of the followingequations respectively

(Z(si)) (Mi) : 0 (2.14)

(Z(s i))T (Ci) = 0 (2.15)

In the electrical field formulation these vectors are identical andtherefore

(Ri) (Ci) (Ci)T (2.16)

now with (V(s)) represented as a step function in the s-domain(V0 (s))/s)

1 o + j w (Ci)(Ci)T Vo(s) st1 0 +we ds( 7i(t) = s-si sd2

Go-j)

which can be rewritten into

10

"ww am 0&, 0' 0.. 0

NSWCNWOL TR 77-99

(t) 1 (Ci)(C) V(S) est2it) J si sssi) -

(Ci) (Ci) T St

( si V (S) es t ds (2.18)

Interchanging the orders of summation and integration, and closingthe contour on s = a at infinifg, the integrals can be evaluatedby the Cauchy Intergal formula

i(t) = (C i)(Ci)T U(t) (V0 (Si))

E (Ci)(Ci)T U(t) (V (0)) (2.19)

i si -

where U(t) is a diagonal matrix of unit step functions which becomesthe identity matrix after the wavefront of the incident wave hascompletely passed the structure at z = L, i.e., for t > L/c. Thelast sum of the equation (2.19) must be zero because the responsefor the late times must decay to zero, and this term is timedependent. Now

i(t > L/c) = X (Ci)(Ci)T (V(si)) 1 esi (2.20)i

This is just a sum of damped sinusoids. Baum has defined a couplingcoefficient n

ri = (Ci)T (V0 (si)) (2.21)

such that v. = C., the mode vector, is the only term dependent uponthe inciden i fie d

s.it/s.

i(t) = V i e 1 e (2.22)

In the numerical search procedure to find the pole locations,i.e., the solutions to equation (2.10), the determinant of thesystem impedance matrix is regarded as a complex function. It isexpanded in a Taylor series and only the first two terms are keptto obtain

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NSWC/WOL TR 77-99

(i- s 0) ddet (Z(s0 )) (2.23)

a - det (Z(so))

An initial guess must be made for the pole, then by an iterativeprocess similar to the Newton-Raphson method for real functions,the pole locations are calculated. The residue at each pole skis found by evaluating

(Rk) = lim E [z(s+E)J - 1 (2.24)CE*O

where s = s + c for small enough E such that the value of (Rk) doesnot change, and

w ss2 (s-s k ) (z(.2)

-11

where (Z(s))l becomes undefined at s sk Tesche calculated thepoles for thin wire and a sphere, compareh'them with previousresults and showed the simplicity of recalculating the time-domainresponse for different :ngles of the incident wavefront.

2.3 THF PRONY NETHOD FOR SINGULAITY EXTRACTION

In 1974, Van Plaricum and Mittra 16 presented a new techniquefor locating the poles of objects. Their method was based on thepremise that if the transient response can be calculated from thepoles, then e poles can be determined from the transient response.Van Blaricum forma1zl. and expanded the procedure that is basedon rrony's algorithm which represents a signal in terms ofcomplex exponentials. The derivation of the Prony technique ispresented here.

Giver that it is possible to represent a waveform by a sumof damped sinusoids, an impulse response for the electromagneticscattering of an object can be defined

ff(r,s) = £ ni(s,P)vi[s-si]-l (2.26)i=l1

where 1 is the polarization of the incident wave

12

NSWC/WOL TR 77-99

and in the time domain

~sitHt(r,t) = U(t-to 0 n i (sq)vi e (2.27)

i-l

where t > 0, due to the presence of the impulse exciter at t -0since tfe impulse function cannot be represented by a sum of complexexponentials. Thereby, knowing the impulse response, the responseR(s), due to an arbitrary function F(s) is then simply found by

R(s) = F(s) H(s) (2.28)

Then the time domain response R(t) is retrieved by an inverseLaplace transform, exactly as in circuit theory. R(t) can be anyfunction that can be acquired experimentally, and will have theform

si t

R(t) = 1 Ai e (2.29)

where A. is defined as the residue at the pole si . Here as opposedto equaions (2.26) and (2.27), the summation is truncated at N,since realistically any waveform that can be generated is bandlimited,and can contain only a finite number or poles. The waveform will bedigitized and then can consist of M discrete equally spaced samples.

N simAtR(,) = m = Ai e m = 0,1,2.. .M- (2.30)

i=l1

where Tm mAt is the time of the R th data sample, and At is thespacing between the samples. EquatTon (2.30) is actually representa-tive of K nonlinear equations in 2N unknowns, and can be rewrittenin the expanded form

R= AIXI 0 + A2X2 0 + A3X3 + ........... + ANXN 0 (2.31.0)RI =AX

0 + 20 +A 0 ........ . A X (2.31.1)

1 1 2X 2 A 3X 3 N N

RMI=A1X1 M+A 2X2M- +AX 3 M1 ........ . ANXNM 1 (2.31.M-1)

13

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wheresiAt

X= e t (2.3-)

It is necessary to solve the equation for the N pole locations andthe N values of the residues at the pole locations; therefore, Mmust be at least 2N to be able to solve the equations. Now letXis be the solution to the following polynomial

a0 + alXl + a .2x .2 ...... + aNXN = 0 (2.33)

where the solution to the polynomial can be expressed as

(X-XI)(X-X2 ) ...... (X-XN) = 0 (2.34)

and more simply

N NL aXm = (X-Xi) = 0 (2.35)m=0 i=l

To determine the coefficients a., equation (2.31.0) is multipliedby a ,and equation (2.31.1) by'a and so on until (2.31.N) ismultiplied by a. Then the results are summed, and since the X.are solutions to the polynomial, the sum is also equal to zero.

a 0R 0 + aIR 1 + .... + aNRN = 0 (2.36)

The same procedure is used again, except that this time ao multiplies(2.31.1), and a (2.31.2) and so on, and this process is repeateduntil M-N lineal difference equations are obtained

a0 R0 + a R1 + ...... + aNRN = 0 (2.37)

a0 R1 + a1R2 +. ..... + aNRN+ 1 = 0

a0R2 + .

a0RMN_ 1 + alRMN + ..... + aNRM 1 = 0

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I7NSWC/WOL TR 77-99

rewritten as

NF0 apR = 0 p+k = m = 0,1,2. .... M-1 (2.38)0= p +k

This is called PiRny's difference equation where the sampled valuesR satisfy the N order linear difference equation. Since R. arekhown, the equations can be solved for a.. When M = 2N then theequations are solvable exactly, and if M1 > 2N, the equations can be

solved approximately by the least squares method. By normalizingaN to unity, equation (2.38) can be rewritten in matrix form

A B = C (2.39)

where

Ro R .......... R0 1 N-i

R1 R2 .......... RNA =(2.40.A)

R2 ..

RMN 1 RM- N ........... RM 2

a0 RN

aI RN+l

a2 RN+2 (2.40.B)

• (2.40.C)

LN- 1 RM-1

Then the ai can be determined by

B = (A) - I C for M- =2N (2.41)

B = (ATA)-l ATC for M > 2N (2.42)

1s

NSWC/WOL TR 77-99

Twhere matrix A is rectangular, but (A A) is square and invertible.The X.s are calculated by finding the roots of equjtion (2.33), bya polynomial roo 0 finding technique. Van Blaricum suggestsMuller's method , as the "quickest and most accurate". Fromequation (2.32)

si = ln Xi/At (2.43)

The last step is to determine the values of the residues A.. Thisinvolves the solution of equation (2.31) now in matrix forA

D E = F (2.44)

Now as with equation (2.41) the A i are found by11

E + (D)- F (2.45)

Blaricu 17X'an Blaricum further shows the relationshiP 2 etween the Pronymethod and the 2 orrington difference equation , and the Padeapproximation

Thus the poles of a system are extracted from the time-domainresponse waveform making it possible to locate the poles andresidues of very complicated structures, wjhout even trying to

write the integral equation. Van Blaricum presented his resultsof trial runs with the Prony method on the response of a 1.0 meterdipole with a half-ligth to radius ratio of 100, comparing resultswith those of Tesche . The Prony method poles matched those ofTesche very well and in addition provided more higher frequenciespoles and extracted some extra poles that "have been attributed" tothe current exciting source.

2.4 FREQUENCY DOMAIN PRONY

Brittingham, Miller, and Willows2 3'2 4'2 5 derived a procedurefor the extraction of poles from real frequency information, whichessentially is the Prony method in the frequency domain. Appendix Areproduces a table in reference 23 that shows the similarities of theproperties between the time-domain and the frequency-domain Pronytechniques. Results on computer generated electromagnetic data provedto be accurate (in relation to the time-domain Prony) though therewas a distinct decrease in accuracy in attempting to find the polesin a waveform that contained more than twenty pole pairs.

The experimental data that are acquired in ship field tests arein the time-domain. The method chosen for the work in this report,is the time-domain Prony method as presented by Van Blaricum andMittra. The other methods, are morlSsuited for such areas as radartarget detection and discrimination , and the EMP study ofstructures, such as missiles and airplanes.

16

NSWC/WOL TR 77-99

CHAPTER 3

THE COMPUTER CODE: SEMPEX

3.1 SIMPLICATIONS FOR ANALYSIS

Using the Prony technique as it stands today, involves severalsimplifications of the real problem. For one, equation (2.26) shouldactually contain an entire function in s.

H(r,s) = li ri(s,) vi(s-si)- I + W(r,s) (3.1)i=l1 1

This in the general case is required by the Mittag-Leffler Theorem27

According to the theorem, to guarantee the convergence of theinfinite series in equation (3.01), each of the pole terms must havean entire function associated with it. The coupling coefficientswhen defined to be functions of s, as in equation (2.21), have theprovision to include an entire function, this is coupled to the factof whether the coupling coefficient is Class I or Class I, as willbe discussed later. The entire function appears from the Laplacetransform, and having no singularities in the finite s-plane, itmust rise and fall faster than an exponential. The entire functionthen is limited in influence only to the early-time of a waveform,since the later times are dominated by the complex exponentials.In ggeral the entire function has been neglected in all previouswork , due to the disinterest in the early-time response and thecomplexity of the solution.

There have now been defined two types of coupling coefficients,n.(s). The Class I couplij coefficients are the type that are usedi Van Blaricum's analyses This assumes that all the resonancesare excited at the same time, i.e., all the damped sinusoids of thecomplete object start at the same time t=O, exactly when the initialwavefront hits the object. The Class II coefficients are much moredifficult to calculate, but give a much better early-time representa-tion. The coefficients in the time-domain are staggered, such thatthe damped resonances due to structures that have not yet beenilluminated by the incident wave do not contribute to the waveform.Hence, the early time response will be much smoother, or lessabrupt, when these coefficients are used. For the case of this work,with a ship broadside to the incident field, it would take about

17

NSWC/WOL TR 77-99

40 nanoseconds for an electromagnetic wave to cross a ship withrailing to railing width of about 13 meters. The error that thiscould produce using Class I coefficients corresponds to stretchinga 10 MHz wave, that is 1,000 microseconds long, by 0.040 microseconds,and observing the frequency shift. The worst case would have abouta 4% maximum error. For a ship, the major resonant structures areat the center, and the other structures, if any, at the sides of theship contribute very little for most applications of ship analysis.Class I coefficients seem to be the rule then, in addition to thefact that there is not, as of yet a program to extract the poles andtheir Class II coupling coefficients from the transient response.Class II coefficients should be used, or some amendments to theresults must be added when the problem is a very large structureilluminated such that the transit time is long compared to fractionsof a period of the dominant resonances. Such a case is the illuminationof a ship when the incident wave is propagating from the stern to thebow, or vice versa. Then in the presence of two masts on the ship,there would be two distinct sets of poles and their residues, thatare being excited at different times.

In equation (3.01) the exponential of (s-si) should, for thegener ? case be -m, as opposed to -1, denoting'the multiplicity oftile i pole. It has been determined though, that in scatteringproblems for spheres, there are only simple poles, and it isconjectured that the same is true for all other finite sized oNectsas well, since there has not been any evidence to the contraryMultiple poles have been shown to exij in some problems, such aspoles in the exciting source - a ramp with a double pole at tneorigin, and with resistively loaded an}nnas, such that they.arel7critically damped, as found by Tesche . Even so, Van Blaricumextended Prony's method to include the calculation of any multiplepoles.

3.2 SEMPEX CODE METHODS

Lager, Hudson, and Poggio developed a computer code, SEMPEX,based on Prony's method for the Air Force Weapons Laboratory'sCDC 7600. This program neglects the entire function, assumes simplepoles and class I coupling coefficients. The program was acquiredand implemented on NSWC's CDC 6500. Implementation required a bitof debugging to eliminate some errors such as complex multiplicationof very large numbers and division by zero. In addition a change wasmade in the calculation of At which took into account the beginningtime of the digitized waveform. Figure 3.1 presents the flowchart ofthe program with appropriate numbers denoting the equations in theChapter 2 derivation of Prony's method.

The input data for the program are 512 equally spaced datasamples of a response waveform. This is presumably for compatibilitywith a Tektronix 7912 digitizer. The mean of the waveform iscalculated and subtracted from each sampled value. This apparentlyis not a required step, and in fact causes some confusing results,

18

NSWC/WOL TR 77-99

STARTA

IREAD INPUT MULLER'S METHOD (2.33)DATA FRX

FREAD SAMPLED 1 2.30) COTRDCIN (.4WAVEFORM FOR A

ISUBTRACT MEAN WAVEFORM RECONSTRUCTIONFROM WAVEFORM FROM CALCULATED POLES

FOURIER TRANSFORM 1PLOT POLESBANDPASS FILTER

CHOOSE POINTS FOR UNPHYSICAL POLESPRONY'S METHOD

PR'S AALYI7PLE

CROUT REDUCTION (2.39)EN

FIG. 3.1 SEMPEX FLOWCHART

19

J.r

NSWCIWOL TR 77-99

(errors), if one is not aware of its presence. The original intentof this operation was to correct for drift in someoscilloscopes on a transient range. The waveform is Fouriertransformed and bandpass filtered at the user's option, betweenchosen frequencies. The bandpass filter is a perfect one, such thatenergy in frequencies outside the pass band is identically zero.It must be remembered at this point that the poles of the filter are

* introduced here. With an inverse transformation, a waveform isretrieved, and of the original 512 points only a few are used forthe Prony analysis subject to several constraints to be explainedlater. In the case that M > 2N in equation (2.30), then the leastsquare method is used to transform the matrix A of equations (2.39and 2.40A) to a square one. Thus in equation (2.39)

[A] B]= C (3.2)

it is necessary to solve for the coefficients ai in vector B tominimize

Sr.- (3.3)= 1

where

r F C B (3.4)

the vector of residuals. The minimum occurs at that point when allthe partial derivatives with respect to the coefficients ai are zero.Thus, we are looking for the vector B such that

V = 0 (3.5)

whc re

(3.6)V a 1 2 ' 3 . .ct auo, a ,t 2 3 N

Then these are the forms of the "normal equations of least square",where now

A A B = A C

is solved for vector B, and the A A matrix is s9 yare. This matrixequation is solved by the Crout reduction method ' The roots ofthe po0 lomlial equa1tion (2.33) are found by MIuller's method. The

residtes are calculated b- equation (2.44), by again employing theCrout reduction technique. The calculated poles and the residuesare now plotted and are used to construct a waveform for thecomlTparison with the original response. Then, if selected, the userhas the option to eliminate poles which are obviously not physical,and a 'ain recalculate the response for the comparison.

20

NSWC/WOL 77-99

3.3 SEMPEX INPUT PARAMETERS

The user must provide the following data, as shown in Table 3.1.

RESPIN the array that contains the original 512 sampled datapoints.

NPOLES is the number of poles that are to be extracted fromthe function. If the number of poles is known, NPOLES should bechosen to be two to four more, to allow a better fit of the data,and to provide a check on the actual number of poles. The extraneouspoles should have very small residues and, therefore, be easilyidentifiable. When the number of poles is unknown, the procedure asstated in the instructions is to ask for as many poles as possibleand then to observe the results. When far too many poles have beenrequested the matrix A in equation (3.7) will become singular andthe program will exit. In such a case the number of requested polesshould be reduced for each run, and the poles with the large residues,due to the resonating structures, should remain stationary, while the"curve fit" poles will experience a great deal of movement in thes-plane.

NBEGN - is the index number (1 - 512) of the first data pointto be used in the analysis. NBEGN should be chosen such that it isafter the time that the driving function has decayed or gone to zero.This especially must be the case if the driying function cannot beexpressed as a sum of complex exponentials li At times, the drivingfunction such as the Guassian pulse, which cannot be expressed asa sum of damped exponentials and does not decay to zero untilinfinity, the excitation function must be deconvolved from theresponse function. When the exciting function can be expressed as asum, and does not fall to zero quickly, then the poles of theexciting function will also appear in the output poles in the s-plane.

NDECI - determines the sampling interval for the points that

are chosen for the Prony analysis. Figure 3.2 illustrates a typicallysampled waveform. NDECI must be small enough to avoid pole foldover,analogous in aliasing in the discrete Fourier transform. It mustsatisfy the Nyquist criterion.

L tNDFCI 2 -- (3.8)

wheremax

tt - the spacing between the original data points

max the highest frequency component in the waveform

21

I,9

,, . .. - . . .; x " ' ' ': ' " .. . .

NSWC/WOL TR 77-99

Table 3.1. SEMPEX Input Data

Parameters for CalculatedParameters for Prony's Method Pole Manipulation

RESPIN RESNPOLES RHPNBEGN PREALNPTS PIMAGNDECIFLOWFHIGH

NTPS - is the number of points that Prony's method will workon. One constraint is that

NPTS 2 NPOLUS (3.9)

this is equivalent to the statement M > 2 NPOLES gives an exactsystem, such that matrix A in equation (3.2) is square. ForM > 2 N, the least square method is employed. Another constrainton NPTS is that the index of the last point must be less than 512.

NPTS < 512 - NBEGN + 1 (3.10)NDECI

FLOW - specifies the low frequency cutoff of the perfectbandpass filter.

FHIGH - specifies the high frequency cutoff of the perfectbandpass filter. To avoid distortion of the retrieved waveform,the inverse Fourier transformed filtered spectrum, due to windowing,the bandpass filter must be chosen carefully such that FLOW andF-HIG11 occur in deep notches of the original 2ectrum, that are atleast two orders of magnitude below the peak

The following parameters are only used to eliminate thepossible nonphysical poles from the set that are extracted by theProny technique. Figure 3.3 shows the location of PREAL, PIMAG,and RIIP. RES is used to eliminate poles that decay too fast.

22

I u . + . " . . : + ,, & '

. + . . . . . . . . • . . . . ..+ . .. . . " ' + | . . . .. . . . . . . . . . . . . .. . . . . . . ... . . .

NSWC/WOL TR 77-99

NBEGN=6

NDECI=3

x x S .

CL •. x x TIME-Sx

"" x I. x jx • x w x

X. %

x %x~ . ORIGINAL 512 POINTSX xxX POINTS TO BE USED

FOR PRONY'S METHOD

FIG. 3.2 DEFINING A WAVEFORM FOR SEMPEX

PIMAG

PREAL UPPER LEFT HALF PLANE RHP

LOWER LEFT HALF PLANE

-PIMAG

FIG. 3.3 DEFINING S-PLANE FOR PHYSICAL POLE LOCATION

i 23

NSWC/WOL TR 77-99

CHAPTER 4

EXPERIMENTAL FIELD WORK

-. 1 11W EMPRESS FACILITY

The Naval Surface Weapons Center operates a test facility forI'MP rclated experiments in Solomons, Maryland, designed and builtby I I'rt'. The facility, EMPRESS (EMP Radiation Environment Simulatorfor Ships) is located near the mout-F-oT the Patuxent River, such thatt:. iest site could be acccssible by destroyer class ships. TheE MPRiSS Facility provides a subthreat level environment for analysisof shipboard EMP response and coupling. The facility consists of apulse generator, and a top loaded conical antenna.

The pulse generator is a 2.5 MW peaking capacitor source(PULSPAK 8000), de-ioged and built by Pulsar Associates, Inc., andit operates in the vertical mode as a monoconic pulser. It uses a50 stage high inductance Mar,: .1enerator and each stage can be chargedto as much as 50 kV, and has a total output capacitance of 4 nF and amaximum stored energy of 12 kJ. Because of the "high" Marx inductancedistributed peaking capacitors are used to obtain a risetime of10 nanoseconds.

ligure 4.1 shows EMPRESS; the Marx generator, peaking capacitors,uniilical cord for remote control and the bottom of the 300 conicalan tenna.

W'ith the closing of the main spark gap in the pulser, a pulsei. ftred into the 28 meter tall conical antenna, which, in the earlytime looks like an impedance of 75Q, to radiate the verticallypoiari:cd fields. To prevent and/or delay reflections that woulddegrade the EMP simulation, the antenna is top loaded by aLyIindrical transmission line that is 380 meters long and is

terminated in 300Q to ground.

The layout of EMPRESS with respect to the "interaction area" ispictured in Figure 4.2. The . eld in and around the interactionarea has been well documented' and some relevant excerpts appearin Appendix B. Figures 4.3 and 4.4 show a sample EMP electric fieldand its Fourier transform. rhe spectrum is fairly constant to 1 MHz,then drops off at about 18 dB/decade to 20 M111z and then 26 dB/decadethrough 100 MHz. Thus, frequencies above 70 Mltz are down by at least

24

'ri.

NSWC/WOL TR 77-99

CONIC ANTENNA

50 STAGE MARX GENERATOR

PEKNI CAPACITORS

UMBI LICALWETR

FIG. 4.1 EMPRESS

25

NSWC/WOL TR 77-99

VERTICAL CONIC

ANTENNA & PULSER

HORIZONTALTRANSMISSION.4 LINE

INTERACTION AREAAND EX-USS LAF FEY

30012TERMINATION 40MBU

0 50M loomSCALE

FIG. 4.2 NSWC FIELD TEST FACILITY, SOLOMONS, MD

26

j NSWCIWOL TR 77-99

1%

0,0

UUw(

UU

anILcc

W

U.

ci-

3aindW iNUu

27N

NSWCIWOL TR 77-99

Ijc

0(a

'U z

00

'U UJ

AU

U.

00

3anilldwv> -j~VWO

28 '

NSWCIWOL TR 77-99

two orders of magnitude. The amplitude at the anchorage point is2.7 kV/meter providing an environment for subthreat level experimen-tation. The EMP simulating field is a propagating plane wave at theship and the peak magnetic field is 2,700/377 A/meter, or 7.2 A/m.

The test vessel, on which all experiments were conducted, is therecently decommissioned USS LAFFEY. The LAFFEY is a DD class destroyer,with three five inch guns and MK 46 torpedoes. Though decommissioned,all electronic gear is still on board for EMP coupling tests. Sincethe ship is only recently inactive, all electrical connections arestill good, and have not yet been affected by adverse weather andsalt.

4.2 TEST EQUIPMENT

During the EMP coupling experiments at NSWC, a system for dataacquisition has been developed. It consists of isolated test equip-ment with which it is possible to get hard copy pictures of anytransient response. Data that have been recorded in the course ofEMP work are: transient electric and magnetic fields, skin currents,bulk cable currents, and a variety of current and voltage measurementsat antenna and electronic equipment terminals. For this experiment,the measurements are transient short circuit currents at the basesof antennas and ends of cables. The main instrument in such a test"setup" is a Tektronix 454A oscilloscope operating in the singlesweep mode. The 454A's bandwidth of 150 MHz well exceeds the 70 MHzbandwidth of the EMPRESS Facility. Data is recorded from the face ofthe oscilloscope using a Polaroid Type 32 oscilloscope camera. Thetransient response is input to the oscilloscope via a test cable whichcan be connected to a variety of sensors and probes. The waveformsrecorded on Polaroid film are brought back 3ho NSWC laboratories fordifferent types of analyses and cataloging . Previously the maininterest was the peak amplitudes of waveforms for determination ofthreshold levels for upsets, but now with Prony's method the completewaveform becomes very important and the pictures are fully utilized.

The scope naturally is subject to EMP upset and many precautionshave been taken to reduce this possibility. With the ship at itsanchorage point there are two types of power available - shorepower and a diesel generator. Since transients can be induced onall cables, power lines included, neither of these power sourcescan be utilized. Power, therefore, is obtained from rechargeable12V lead/acid automotive batteries. A fully charged battery canoperate a 454A for almost 12 hours. 12 VDC is converted with a300 W inverter and upsets due to power line transients are thuseliminated. Protection against electromagnetic fields in the testingenvironment is also required. It has been found that the operationof an oscilloscope in a high amplitude electromagnetic transientfield will seriously affect the resulting waveform. For this purposeeach oscilloscope is shielded by a metal box - "scope box". Thescope box is made from aluminum, for ease of handling, and is just

29

NSWC/WOL TR 77-99

large enough to contain the oscilloscope, the power inverter, and

tile camera. The battery is situated outside the scope box in anotheraluminum box, and battery power is transmitted to the inverter via ashielded umbilical cable.

Tle scope box is equipped with feed-through connectors forshielded connections between incoming signals on RG 214 or RG 58/Ucoaxial cables and short cables that connect to the scope. The testcable usually used, is double shielded coaxial RG 214. It has beenfound that in high field areas, such as near the mast, even with asmuch as twenty feet of cable no appreciable coupling can be detectedinto the test cable itself.

In this test of antenna currents, the test cable was connectedto a current probe tSingcr 91530-2), i.e. "Dash-2". Data on thecurrent probe are sho%,n in Iable -1.1. The Dash 2 probe can beclamped around a cable to Imeasure current, oi a shorted loop tomeasure magnetic field, tith appropriate calibration factors. Thebasic data acquisition etni1 sho%,n in Figure 4.5.

In actual 011' explrimentat ion, problems with triggering thescope have shown up, and tibe process of obtaining a complete sweepon the oscilloscope, including a short presweep to guarantee theeginning, of tile w'aveform i qJuite an art. Ith sb e fon tat"

external triggering using a short piece of RG 38/IJ attached over aground plane, the scope box, works well for setups on deck. Withinthe ship though, another Dash 2 can he connected to a cable, whichis known to carry high amplitude current, to externally trigger the454A.

The measurement of short circuit current, I was accomplishedby making a small adapter that would short the i er conductor totile outer in a loop just big enough to accommodate the Dash 2 probe.The I adapter was made small so that there would be little or no;agnet c f;eld coupling through the loop. The adapter fits onto aconnector at the base of the antenna, and is also used at the cableend measurements inside the ship, at the radio equipment.

I .3 RIlL IMINARY SIIIPBOARI) WORK

As mentioned in the introduction, one motivation for the studyof pole extraction at NSWC, was to identify and analyze shipboardresonant structures with respect to antenna location. For thisexperiment, two identical antennas on the ex-USS LAFFFY were chosen,as each was located in radically different environments omi the ship.In addition there was interest in them, since in a previous shiptest it was discovered that the short circuit current at the basecontained relatively high amplitude currents in the 2-15 MHz range,much below the pass band of the antenna, 250-400 Mliz. One of theantennas is on top of a gun mount in the forward part of the deck,and the other is on the top of the mast. The two antennas also

30

" , ,¢a' . . ... _: * .I, . __,_... .... .. ... .

NSWC/WOL TR 77-99

Table 4.1. The Dash 2 Probe

Parameter Singer 91550-2

Frequency Range 10 KHz - 100 MHz

Transfer Impedance 1.0 Q

RF Current Range 0 - 100 amps

Max Power Current 350 amps (60 Hz)

Max Power Voltage No limit; subject toadequate conductorinsulation

Output Load Impedance 50 + jO 0

Sensitivity 10 ma = 10 my, across 50Q

Output Connector Type N

Window Size 32 mm diameter

Dimensions 90 mm longx 74 mmwide x36 mm thick

Signal Cable Length No limit; subject to cablefrequency response

31

NSWC/WOL TR 77-99

I VP

I ro-

low,

-'4 w

le g ' A

NSWC/WOL TR 77-99

have radically different cable routings. For the mast antenna, thecable drops directly down alongside the mast and then down two decksto the Combat Information Center, the CIC. The cable from the gun1mount antenna runs down through the gun mount, up and down several

decks in the interior of the ship, then along a wall on the outsideand finally endinil up in the Radio Transmitter Room.

The exp(ci' rlitt ,as to provide for Prony analysis, open circuitvoltaic jk I .-<irt circuit current (I waveforms, from bothat the has4 itt nnas, anj the ends the cables at theequipment ii t ., ,cctive rooms. Poles from this analysis wouldbe usetil in t, , in of full-threat injection sources at theequiflllt:s ni'1t !i. imiial, provide knowledge of the effect of cablerouting on ti.. IL-10 wire aveform, and identify the resonatingStructures aud to , hat degree they contribute to the response of thetwo antennas. re I ininr" experimentation at the NSWC EMP Facility,showed extremely coniplicated waveforms, which weredifficult to digitize. At this point it was determinedthat to increase the accuracy in digitization and to not overlookthe finer aspects of the waveform, it would be necessary to use thedelay time multiplier feature on the Tektronix 454A. The delay-timemultiplier makes it possible to get a 2 microsecond waveform on ahorizontal scale of 20 ns/cm. Figure 4.6 is a picture of a 20 ns/cmwaveform and shows the first 200 ns of the waveform. The delay-timemultiplier makes the screen of the oscilloscope look like a 200 nswindow. This window is moved from one end of the waveform to theother, with pictures taken at each window position which overlapsthe previous picture by a half, so that there is a complete set ofpictures, which when merged together will show a 2 microsecondwaveform. Figure 4.7 shows part of the merged waveform and theseparate pictures through the 200 ns window, that combine for thecomplete waveform. Each picture of the waveform was literally cutup and pieced together, by hand and then by computer for the Pronyanalvsis.

The complexity of the waveform required the windowing of thewaveform and the piecing together of the results. This proved to bea difficult task. First of all, the waveforms are notidentical from one picture to another, in the same window. Thiscould be due to a number of reasons. One is that thelocation of the ship can change by as much as thirty degrees due towind and tides, etc. In piecing together the waveform, it was

usually almost impossible to match peaks, and the matching processdegenerated into a guessing game providing a mostly useless waveform.Initially it was hoped that the *discontinuities at the match pointsand the differeaces especially in the higher frequency components,would only introduce a small perturbation into the Problem, butsince Pronv's method is very noise sensitive1 7, this was notthe case. To make an experimental procedure like this work, it isabsolutely necessary for the waveform to be identical from EMP shotto 011t shot, especially when dealing with such a touchy problem asscattered fields to hegin with. The complexity of the waveform

33

L"-'..

NSWC VVUL TH 77 99

ii~.....l.EEE.dmwmmmmm N

VERTICAL - 20 MV/CM

HOR IZONTAL -20 NS/CM

FIRST PART OF THE WAVEFORM

FIG. 4.6 A 20NS'CMWAVEFORM

1ST PICTURE 2ND PICTURE 3RD PICTURE

VERTICAL 20 MV CM HORIZONTAL 20NS/CM

FIG. 4. WVAVF I O11M MERG.ING

1 •.. # ,.minmmmmmmmmmi.

NSWC/WOL TR 77-99

quickly brought to the surface another problem. How many poles couldmake up such a waveform, and how many poles are there in thescattered field from such a complex object as the superstructure ofa ship anyway? The Prony's method program, in its present state,occupying almost 150 K of core for the extraction of 35 poles couldnot be expanded any more. And it is a fact that the scatter from aship would include more than 17 pole pairs. In addition, because theuse of the Prony method had not yet withstood a rigorous test ofanalyzing real experimental waveforms, it became clear that theprogram could not handle a waveform this noisy or complex.

4.4 FIELD WORK ON SHORE

The resulting decision was to put aside the problem of scatterfrom a ship, and begin to concentrate on the use of Prony's methodoa experimental waveforms. One of the antennas was removed from thegun mount and brought ashore, where it was erected on top of a shack50 meters from the pulser. Here various scattering tests could beperformed with the antenna as a receiver, and could hopefully providecleaner and simpler waveforms. The antenna was retainedfor this experiment, because of the remaining interest in th2 analvsisof its EMP response.

Metal objects in and around the area of the pulser were removedas much as pessible such that the scattered field from an extraneousobject would only reach the site of the shack about 200 ns after theEMi was first incident on the antenna. In addition, this extraneousscattered field would also be much lower in amplitude, due to thqsizes of the extraneous objects, and the inverse dependence on r- ofthe scattered field. The only large structure available for use inthis test was a 5 m by 1.5 m (dia) sewer pipe. Unfortunately thelowest resonance of the pipe was a bit too high, on the order of30 Mz, a bit above the 2-20 Mflz frequencies that are usually recordedin field tests. Testing included the recording of pictures of theshort circuit current waveforms of the antenna, without the cylinderin the field. The experiment required the gradual moving in of thecylinder toward the antenna to see the change in waveform as thescattering object was brought closer and closer. The cylinder wasfinally put on top of the shack next to the antenna and then, evensuch that the antenna was on the shadow side of the cylinder. Fromthe waveforms it seemed that the scattered field from the cylinderias having no effect on the response of the antenna, until it wasplaced very near the antenna. This is due to the low amount ofenergy in the dominant frequency range of the antenna response inthe scattered field. Four sets of data were chosen for the Pronyanalysis. 1ibe first set was the FMP electric field waveform,recorded i;i the clear field, in the exact location where the antennawas placed. The second set was the AS-1018 response to YMP, wheretile antenna %,as mounted on top of the shack, and the short circuitcurrent ias. recorded at the base. The third set was the Isc response

35

Jf

NSWC/WOL TR 77-99

of the antenna with the cylinder next to it on top of the shack; andthe fourth was the 1 response with the cylinder placed such thatthe antenna was on tfs shadow side. Figure 4.8 illustrates thevarious conditions of each test. The results of the analysis of thefour test waveforms are listed in Appendix C.

PULSER

O SEWER PIPE

TEST NO. 1 B-DOT SENSOR ON SHACKIN PLACE OF ANTENNA

TEST NO. 2 ANTENNA IN PLACE ON SHACKf NO SEWER PIPE

TEST NO. 3 ANTENNA IN PLACE ON SHACKSEWER PIPE AS SHOWN

TEST NO. 4 ANTENNA IN PLACE ON SHACKSEWER PIPE AS SHOWN

SHACKTEST NO. 4

TEST NO. 3

ANTENNA-'-" I

FIG. 4.8 FOUR FIELD EXPERIMENTS

36)

NSWC/WOL TR 77-99

CHAPTER 5

NUMERICAL ANALYSIS

5.1 DIGITIZATION AND THE TEST WAVEFORM

The analysis process begins with the digitization of thewaveform. A Tektronix Graph Tablet was used in conjunction %ith aTektronix storage graphics terminal. The terminal is part of theNSWC CDC 6500 timesharing system. The software for the terminalallows for the manipulation of the waveform in almost any possibleway: merging waveforms, deleting sections, adding sections, Fouriertransform, etc. SEMPEX though, currently demands that the waveformbe in the form of 512 equally spaced data samples. A program waswritten to perform this transformation, also allowing for the optionof "fitting" the original digitized waveform with line segments, ora cubic spline. The new "fit" curve was sampled at the 512 equallyspaced points of time ranging over the entire waveform. The cubicspline was mainly used, because it provided smoother waveforms -possibly less noisy ones, concurrently. The line fit routine wasmore useful on the more complicated waveforms, when the cubicroutine sometimes had the tendency to let the fit curve wander awayfrom the real curve.

A test of the SIMPEX routine was performed by generating afunction from five pole pairs, as shown in Figure 5.1, and thenattempting to extract them.

J +6

f(t) = .5 e - 1 0 tsin(2ff7xlO6t) +

.S e-l'3xl+6tsin(2710 7t) +

.74 e-4Xl0+ 6 tsin(2fl.5xl07t) +.5e 1 +6t 7t

.65 e 1 0 tsin( 2 2.4xl0 t) +

.44 e-8xl0+6tsin(272x107 t) (5.1)

ihe waveform was bandpass filtered between two typical frequencies(2-35 MIz), as would be done for an experimental waveform. Theresults were surprising, in that the runs requesting, ten and twelve

37

2

NSWC/WOL TR 77-99

4.0-

2.0-

-;0 .

-2.0

0

SECONDS X 10 7

FIG. 5.1 TEST WAVEFORM

38

NSWC/WOL TR 77-99

poles (NPOLES = 10, 12), the extracted poles were not even close tothe input ones. When requesting 14 poles, the input poles wereretrieved within a few percent error. Asking for more poles simplyshowed a lot of shifting in the residue value of each pole pair,and some shifting in the pole locations. It seemed clear that theProny technique would be useless, if for a perfect waveform from tenpoles input, the exact ten poles could not be extracted. The resultsfrom experimental waveforms could then be expected to be incompre-hensible. It was at this point that the "mean subtraction" problemwas realized, and the bandpass filter was thrown out. Subtractionof the mean from the waveform (see 3.2) caused very low frequencypoles to appear, that originally had nothing to do with the waveform.

The bandpass filter was removed for several reasons. Thefilter itself has poles, and these are introduced into the waveformat the filtering stage; this complication of the problem is not

needed. Further, the Fourier transformation introduces noise, andthe inverse transformation does not remove it, but only increasesit. The effect of noise in a waveform, is well documented byprevious authors and, has been determined to be catastrophic.Noise must be on the order of 1-5% of peak or 35 dB signal to noise,in order for the Prony technique to extract quickly recognizablepoles. Some have tried changing the filter, by giving it differentroll-offs, but to no avail.

The test waveform was run again after the elimination of themean and the filter. The result is shown in Table 5.1. Ten poleswere requested, and the same ten poles were returned, exactly, withvery small error in the reconstructed waveform. A run was maderequesting the full 35 poles to be extracted; the result is shownin Table 5.2. The ten real poles were extracted, and 25 curve fitpoles are all at least four orders of magnitude below the largest.

The least square method was adopted for all waveforms. Thoughguaranteeing the presence of curve fit poles, it provides Prony'smethod a little "more room to work with." In this stroke, theproblem now becomes the process of finding the best curve fit polesand then hoping to identify the physical poles, as opposed tosearching for the physical poles of which there is no conception ofnumber, location, or magnitude of residue.

The waveform that was anlyzed first was that of the I responseof the antenna in the clear field. An initial set of runs 5 howedthat all the poles, not only the ones with the small residuesshowed considerable movement in the s-plane from run to run, suchthat the hope of identifying the real poles of the system werelost (see Table 5.3). Upon.examining the input parameters andthe resulting poles, a method that not only finds the best fit polesfor the waveform but also eliminates many pole groupings that hadbeen considered correct in previous analyses was constructed.

39

L L *4

j NSWCfINOL TR 77-99

Table 5.1. Test Waveform Without Filter

Magnitude Real Part Imaginary Partof Residue of Pole Location of Pole Location(ampl) (nepers) (Hz)

1. . 49902E+00 - .99805E+06 .69863E+07

2. .49902E+00 - .99805E+06 - .69863E+07

3. .49873E+00 - .12975E+07 .99805E+07I4. .49873E+00 - .12975E+07 - .99805E+07

5. . 74416E-01 - .39922E+07 .14971E+08

6. .74416E-01 - .39922E3+07 - .14971E+08

7. .43810E+00 - .79844E+s07 .19961Ei-08

8. .43810E+00 - .79844E+07 - .19961E+'08

9. .64873E+00 - .99804E+06 .24951E+08

10. .64873E3+00 - .99804E3+06 - .24951E+08

NPOLES = 10 NDECI = 10NBEGN = 2 F2NYQ = 25.55 MlizNPTS = 40 RMS ERROR =7.57 X 10-6%

40

NSWCIWOL TR 77-99

Table 5.2. Test Waveform Requesting 34 Poles

Magnitude Real Part Imaginary Partof Residue of Pole Location of Pole Location(ampl) (nepers) (Hz)

1. .52265E-06 - .11192E+08 .13740E-102. .10337E-19 .29781E+08 - .32506E-093. .30214E-06 - .73795E+07 .74302E-074. .16201E-06 - .74207E+07 .67890E+075. . 16177E-06 - .74207E+07 - .67890E+076. .49902E+00 - .99805E+06 .69863E+077. .49902E+00 - .99805E+06 - .69863Ei-078. .49873E+00 - .12975E+07 .9980SE+079. . 49873E+00 - .12975E+07 - .99805E+07

10. . 12305E-05 - .83308E+08 - .11403E+0811. . 12305E-05 - .83308E+08 .11403E+0812. . 74416E-01 - .39922E+07 - .14971E+08013. . 74416E-01 - .39922E+ ' .14971E+08

)14. . 28055E-06 - .19563E+08 .16723E+08is1. . 28263E-06 - .19563E+08 - .16723E+08

16. .43810E+00 - .79844E+07 .19961E+0817. . 43810E+00 - .79844E+07 - .19961E+08

18. .91430E-11 .76394E+07 .20344E+0819. .91523E-ll .76394E+07 - .20344E+0820. .64873E+00 - .99805E+06 .249SlE+0821. .64873E+00 - .99805E+06 - .24951E+0822. . 23851E-07 - .19129E+07 - .25585E+0823. .23924E-07 -.191293+07 .25S85E+0824. .14472E-09 .34404E+07 - .29714E+0825. .14491E3-09 .34404E+07 .29714E3+0826. .47873E3-09 .29847E3+07 .33494E+40827. .47891E3-09 .29847E3+07 - .33494E+0828. .10298E3-07 - .70296E3+07 .34713E3+0829. .10348E3-07 - .70296E3+07 - .34713E+0830. .2315413-07 - .89421E3+07 - .38494E+0831. 29030 -,.89421E3+07 .38494E+0832. .14870E3-11 .89367E+07 - .38911E3+0833. .14867E-11 .89367E+07 .38911E+08

34. .47071E3-14 .12482E+08 - .4258313+08

NPOLLS = 34 NDECI = 6NB17GN = 2 F2NYQ = 42.58 MHz 6NPTS = 84 RMS ERROR = 2.302 x 10 6

41

NSWC/WOL TR 77-99

Table 5.3. Shifting Pole Locations

Imaginary Part Imaginary Partof Pole Location of Pole Location

Run #1 Run #2(MHz) (MHz)

1. 6.5663 4.5368

2. 12.946 7.7514

3. 13.071 12.751

4. 19.620 21.382

5. 28.379 28.671

6. 39.886 39.123

7. 43.463 43.972

8. 55.396 S5.654

9. 67.526 66.651

10. 67.825 68.990

11. 79.491 83.343

12. 91.279 85.235

NPOLES = 25 NPOLES = 24NBEGN 1 NBEGN = 1NPTS = 84 NPTS = 84NDECI = 6 NDECI 6F2NYQ = 91.2 MHz F2NYQ = 91.2 MHzRMS ERROR = 4.48% RMS ERROR = 8.08%

42

J " ± . ... ..| .. . . . I 11 II .. . .. . . .... .. .. .. .. . " . . . . . ..:

NSWC/WOL TR 77-99

5.2 FORMULATION OF THE NYQUIST CRITERION CONSTRAINT

The key to this constraint method is to sample the original 512points in such a way that Prony's method is constrained to look forpoles whose imaginary part (w) will be below twice the Nyquist rate.This in general brings about two results. The first is psuedo-filtering. Instead of the noisy Fourier transformations and theperfect bandpass filter, by specifying the Nyquist frequency, thepole locations calculated will have to be below twice that frequency,thus providing a type of filtering. The other result is polefoldover. Ideally (computer generated cases) there can be aninfinite number of poles for any structure in the impulse response.In an actual field case, the incident field's frequency spectrum isknown, such that the effects of pole foldover, while existing, canbe minimized by proper choice of an upper frequency. This frequencyis defined such that, it is substantially above the dominantfrequencies of the waveform, the imaginary part of the pole locationsof the structure. Called F2NYQ, it is calculated by

F2NYQ NDECI 2TX (5.2)

where NDECI is the sampling interval for the points, and TMAX is thetotal duration of the sampled waveform. Of four input parameters(NDECI, NPOLES, NPTS, NBEGN), two (NPTS, NBEGN) are used to specifythe size and position of the window, through which only part of thewaveform is observed and analyzed. Using windowing makes itpossible to look at several sections of a waveform and then toaverage the results, in an effort to try to find the real poles.The process of windowing is gaining popularity as a way ofeliminating the curve-fit poles from the results. The basis forthis is that, through windowing, the real poles of the system shouldnot be subject to great deviations in location, from window towindow, while the curve fit poles will, or at least should. Severaltypes of windowing techniques are illustrated in Figure 5.2.

Specifying the number of poles is the final step in theconstraint method. Prony's method is thus forced to look for NPOLESbelow a certain frequency, where there may be n poles in the realsystem. By observing the resulting calculated poles for a certainF2NYQ, for many NP E 2e will soon notice a phenomenom alreadyrecorded by others - I . This is the appearance of the righthalf plane (R11P) poles. These poles are known to be nonphysical,and therefore should be eliminated. The other poles calculated inthat run should also be discarded since their residues and theirpole locations are affected by the RHP poles. It is not correct toeliminate them and then recalculate the residues for the remainingpoles.

A simple explanation of the phenomenon of the appearance of theRHP poles is as follows. Given: there are n poles in the physical

43

NSWC/WOL TR 77-99

WINDOW NO. 1 WINDOW NO. 2 WINDOW NO. 3

WINDOWNO. 3 INDOW NO.A WINDOWNO.

jWINDOW NO.1 WINDOW NO. 2 WINDOW NO.3

FOR LATEYTIME CHARACTERZATION

FIG. 5.2 WINDOWING TECHNIQUES

44

2 A-

NSWC/WOL TR 77-99

system below F2NYQ. If NPOLES < n is requested, one can expectlarge errors in the waveform that is reconstructed using thesepoles, since with fewer than n poles the waveform cannot beadequately described. As NPOLES - n, the errors should quicklybecome small, and when NPOLES gets bigger than n more curve fitpoles are added to the results such that the fine structures of thedigitized waveform can be described, thus slowly lowering the errorof the reconstructed waveform with increasing NPOLES. Finally,when NPOLES > n, there is so mucn pressure on the Prony method thatit begins to seek poles in the RHP to cancel the effects of theexcess curve fit poles in the left half s-plane. Thus when righthalf plane poles appear, it is clear that there is too much pressureon the method for the given F2NYQ. This provides an upper limit forthe possible number of poles in the bandwidth of F2NYQ. It may thenbe observed that if F2NYQ is raised sufficiently, the RHP poles willdisappear. There are then three things to consider in the analysisof a waveform: the size and location of the windows, NPOLES, andF2NYQ.

Figure 5.3 illustrates the physical limits of the analysisprocess. The figure is drawn for a typical waveform encounteredin iMP response work for ships. These are pictures of waveformsrecorded on a scale of 50 ns/cm for a total size of .5 microseconds,for use with the currently available Prony's method program SEMPEX.The graph shows the possible F2NYQs plotted against the NPOLES thatcan be requested, and defines the boundaries, within which, the polescan be calculated. The rightmost boundary NPOLES = 35 is defined bythe computer program. For a system containing more than 17 polepairs it would be necessary to expand the program, thus moving thevertical boundary further to the right. This would require muchmore computer core than is currently used, which is 150K, half thecapacity of the NSWC CDC 6500, and the full capacity for daytime

use. The top boundary is not of interest, but it is subject to aconstraint set of F2NYQ and NPTS (> 2 NPOLES) as discussed below.The left-most boundary is also immaterial for one can always specifyNPOLtS = 2, and there are few cases that contain only one pole pairin the data. The location of the lower boundary is of most interest.It is near this boundary line that most of the results are obtained.The boundary can be calculated by

min NPTS = 2 NPOLES + 2 (5.3)

requiring a least square fit, and such that

max NDICI = 512 / min NPTS (5.4)

which provides for the analysis of the computer waveform of 512points, and the boundary is then defined by

512rain F2NYQ - max NDECI 2 TMAX (5.5)

45

Lt

NSWC/WOL TR 77-99

cn

LU-J0

0.

N0

-J0

In z

8 0

4o4

NSWCNWOL TR 77-99

It is not advisable to make runs using min NPTS, because the leastsquare method is not given enough points to work with. It isgenerally better to have more points for practicality and a betterfit to the data. Therefore, the boundary is only a theoretical one,and though results can be obtained on it, the practical boundarywould be at a slightly higher F2NYQ for the same NPOLES. In theanalysis proposed here, the method is to look at the results ofspecific F2NYQs by fixing NDECI and then moving left and right inthe plane by varying NPOLES. There is also another limit aspreviously stated, that of pole foldover. For the purposes of shipEiMP response the lowest F2NYQ that can be requested is on the orderof 40 M z: (Figure 4.4) , thus the dotted boundary in Figure .7 .

The comput ation method is illustrated in Figure 5. 1, in flow-chart form. Given a wa 'eform the min F2NYQ is set by the practicalphysical characteristics of the system. The first run is made in theright hand bottom corner of the "allowable" section, Figure 5.5,point ;\. If no RiIl' poles appear then this means that not enoughpoles were forced into the band below F2NYQ. Then, it is necessaryto lower F2NYQ, or to increase the number of poles to increase thepressure on the method. Since tile run was made at the point with themaximum possible pressure, all that can be done is to make more runsfollowing path I, Figure 5.5, down the bottom border left and down,to see if there were many high frequency curve fit poles, such thatRill' poles may then appear. Such a case is illustrated in Table 5.4.Notice the large number of "high frequency" poles. If at point Bstill no R11P poles appear, then there are too many poles, includingthe curve fit poles, in the system for SMlPlFX to extract. Manycurve fit poles implies that the waveform is noisy either due to theoriginal data, or the digitization process. If RI1P poles do appearthen path If, up and left, may- be foliowed to lessen the pressure,until the point is reached where the Rill' poles disappear again. Ifat point A, RilP poles appear, then too many poles for the F2NYQ wererequested and either F .NYQ is raised o- NPoIES is lowered, or both.This procedure of course can be started and used in any part of theplane, and eventuallv one will converge to a point where the Rill' polesappear.

\s can be seen there are many wiVs to find that border at whichthe Rill, poles appear and it is known that the real pole locations canhe foeud, Table 5.4. Th,: Use of this Method will be illustrated intile following example O54 One of thie wavetod1s ta1e il tile field.Reslts fOl tile oter waveforms can be found in Appendix C.

Figure -,.(, shIows the waveformi that i,as analyzed. Runl I,Table 5.5, produced no RIIP poles and therefore more pressure wasrequ ired on the method. F2NYQ = 45 Mt: was chosen, due to tiefrequency s pectrumi of the incident wave, and tile largest possiblenumber of poles was chosen for Run 2, Table 5.o. Here three Rill'poles appeared, one almost a DC pole, and two at S Mh1z ; therefore,less pressure must now be exerted at the same [2NYO. In Run 3,

47

: .... . .,,•-...

NSWC/WOL TR 77-99

START

FIND PRACTICAL ]MIN F2NYQ

MAKE RUN ATMAX NPOLESMIN F2NVO

CONTINUE THIS LOOP UNTIL THE MAX NPOLES HAVEBEEN FOUND THAT WILL NOT RESULT IN RHP POLES

POE CONSTRAINT

Y N_ MAKE RUNSFOLLOWINGLOWERBOUNDARYDOWN AND

LOWER PRESSURE LEFTMAKE RUN AT FIG. (5.05)LOWER NPOLES

OR INCREASE PRESSUREHIGHER F2 NYO MAKE RUN AT

HIGHER NPOLES

LOWER F2 NYQ

END-TOO MANY

POLES IN

WAVEFORM

FIG. 5.4 NYOUIST CRITERION METHOD FLOWCHART

I.. .

- -- - - - - -- - - PA T H I

RHP POLES

FIG. 5.5 USING THE NYQUIST CRITERION METHOD

49

NSWC/WOL TR 77-99

80.0

40.0-

E 0.0

-40.0 SECONDS X 10'7

-80.0-

FIG. 5.6 WAVEFORM USED FOR RUNS (1-4)

so

NSWCANOL TR 77-99

Table 5.4. Waveform with Many Curve Fit Poles

Magnitude Real Part Imaginary Partof Residue of Pole Lccation of Pole Location

(amp 1) (nepers) (Hz)

1. .1842513+01 -.11616E3+08 .37664E3+07

2..821:l-11616E3+08 -.:37664E3+074. .868321>01 - .690821E+06 S .88961-+075. .26391:+01 -.511452E+07 .5286E+085. .2359)71:+01 - .51145E3+07 -. 12867E3+087. .479451:+00 -. 232115+07 -. 15017F3+087. .4794513+00 - .23231E3+087 .15017E3+088. .479871'+00 -.624261F+07 -.19041E+0810. .2988713+00 - .68261'1+07 -.19042E3+08

101. .298256400 -.131106E+08 .190219+0812. .82564E3+00) - .13110E3+08 -.22739E3+0813. .495414-02 -. 123110E+08 -. 284029E+0814. .494391:-02 - .129311-408 -.28402E3+0814. .19491-.+00 -. 18511-+08 -. 321402E+0816. .1997613 '+00 - .1851513+08 -.35210E3+0817. .937213-02 - .5262313+07 .43S1713+0818. .953721-0 - .5262313+07-.417+0

19 3701-01-.051o+0 .483217+08

20. .3770013-01 - .90551E3+07 -. 4832213+0821. .7933113-01 -.306431F+08 -. 485342E+0821. .793311' '-01 - .306431'+08 -.485341'+08

23. .6)89(6913-02 --. 7972'713+07 .57972F-+0824. .08969f."-02" - .797271:+07/ - .57972E3+082.7. .7446213-01 - .5790513+08 .599511,+0820. .744621-01-o - .579051-'+08 - .5995113+082-17. .4662-913-02 - .785631:+07 .690141-3+0828. .400291:-0 2 - . 8 5631:+07 - .6901413+08

29. 999621'- 02 - 1369214+08 .7570813+0830. .996213-02 - .1 369213+08 - .757081i+08.-1 . .6805613-02 - .1499713+08 .78163E-+0832. .6865613- 02 .1499714+08 - .7816311+08SS. .2491013 '-02 -. 7887F+07 .83649EiI+08

34.2 4 9 11 -021 -. 7880713+07 -. 83649E3+08

Note the great number of poles with small residues whose imaginarypai-t is above 35 M111z. More than half of the poles are thus

NSWCIWOL TR 77-992

Table 5.5. Run 1

Magnitude Real Part Imaginary Partof Residue of Pole Location of Pole Location

(ma) (neperes) (Hz)

1. .34985E3+01 - .19344E3+08 . 43716E+072. .34985E3+01 - .19344E+08 - .43716E+073. . 15853E+01 - .17377E+08 .69290E+074. . 15853E+01 - .17377E+08 - .69290Ei-075. .33907E+01 - .69217E+07 .14317E+086. .33907E3+01 - .69217E+07 - .14317E+087. .48341E3+00 - .5388013+07 .17596E+088. .48341E3+00 -.53880E+07 - .17596E+089. .85324E+00 - .11644E+08 .18579E+0810. .8532413+00 - .11644E3+08 - .18579E+0811. .16288E3+01 - .32862E+08 .2S082E+0812. .16288E3+01 - .32862E+08 - .25082E+08

13. .13408E3+00 - .19282E4-08 .38361E+0814. .13408F3+00 - .19282E+08 -.38361E+0815. .98865L-02 -.62617E3+07 .47158E+081b. .9886513-02 - .62617E3+07 - .47158E+0817. .2921111-01 -.76612E3+07 .52787E3+0818. .29,111 , () - .76612E3+07 -.5278713+08

1..2251.LU(+o0 - .66557E+08 .58033E3+08* 225 181 +00 - .6655713+08 - .58033E+08

21. .7133310 -.89836E+07 .64131E+0822., .71333E>02 - .89836E3+07 - .64131E3+0823. .192781>-01 -.13249E+08 .75961E+0824. .192781>01l - .13249E+08 - .75961E+08

25. .2193413-01 - .16992E+08 .81596E3+082o . .219341L-01 - .16992E+08 - .81596E3+0827. .3882513-01 - .24262E+08 .87869E+0828. .3882513-01 -.24262E3+08 -.87869E3+0829. .88954f7-02 - 81530E+07 .92629E3+0830. .889541:-02 - .81530E3+07 - .92629E+0831. .34823E3-02 - .46219E3+07 .10393E3+09

32..32823F-,-02 - .46219E3+07 - .10393E3+09.33. .731911:-02 - .10197E3+08 .1144313+0934. .731911E-02 - .1019713+08 - .11443E3+09

NPOLI3S = 34 NDECI = 5NBEGN = 2 F2NYQ =117.5 MHzNPTS = 100 RMS ERROR =8.72%

52

NSWCIWOL TR 77-99

Table 5.6. Run 2

Magnitude Real Part Imaginary Partof Residue Of Pole Location of Pole Location

(ma) (neperes) (Hz)

1. .15006E-02 .12459E3+08 .6994513-092. .23807E+01 - .14536E+08 -.45008E+073. .23807E3+01 - .14536E3+08 .45008E3+074. .32854E-01 .41202E+07 - .87541E+075. .32854E-01 .41202E+07 .87541E+076. .25443E3+01 - .57049E+07 - .14098E3+087. . 25443E+01 - .57049E3+07 .14098E3+088. .46477E+00 - .30601E3+07 - .16997E3+089. .46477E+00 - .30601E+07 -16997E+08

10. .48234E3+00 - .84151E+07 - .21333E+0811. .48234E3+00 - .84151E+07 .21333E+0812. . 12089E+01 - .19126E+08 - .25246E+0813. .12089E3+01 - .19126E+08 .2524613+0814. .3127813-01 - .71038E+07 - .29891E+08

15. .31278E3-01 - .71038E+07 .29891E+0816. .69807E3-01 - .12013E+08 - .39126E3+0817. .69807E3-01 -.12013E3+08 .39126E3+0818. .6209013-02 - .2449213+07 - .47464E+0819. .6209013-02 - .24494E3+07 .47464E+0820. .43172F3+00 - .43748E3+08 - .48954E3+08

NPOLES = 20 NDECI =12NBEGN = 2 F2NYQ = 48.95 MHzNPTS = 42 RMS ERROR =0.286%

53

NSWC/WOL TR 77-99

Table 5.7, note that the RHP poles have disappeared, and a pole onthe order of 5.7 MHz reappeared. This pole can be seen in Run 1,but it is missing in Run 2, thus showing the unreliability of resultsthat contain RHP poles. It is also interesting to see that theinclusion of the RHP poles in Run 2 lowered the error of thereconstructed waveform by an order of magnitude. With even lesspressure in Run 4, Table 5.8, the error is up a bit with all polelocations slightly shifted. As a check, many runs were made atF2NYQ with 4 NPOLES 20 to see if the error at any other pointdropped to a low level, indicating that the waveform might beadequately described by even fewer poles. The resulting curve isshown in Figure 5.7. Notice how the error drops significantlyin the beginning up to NPOLES = 10, then levels out somewhat. Itis here that more curve fit poles are being calculated than realpoles.

Table 5.7. Run 3

Magnitude Real Part Imaginary Partof Residue of Pole Location of Pole Location

(ma) (nepers) (Hz)

1. .21493E+01 -.13607E+08 .44044E+072. .21493E+01 -. 13607E+08 -.44044E+073. .83460E-01 -. 18361E+07 .62077E+074. .83460E-01 -.18361E+07 -.62077E+075. .26701E+01 -.66503E+07 .13923E+086. .26701E+01 -.66503E+07 -. 13923E+087. .95226E+00 -. 52590E+07 .16525E+088. .95226E+00 -.52509E+07 -.16525E+089. .68501E+00 - .1421E+08 .20655E+0810. .68501E+00 -. 11421E+08 -.20655E+0811. .11797E+01 -.17158E+08 .25151E+0812. .11797E+01 -.17158E+08 -.25151E+0813. .91204E-01 -.17377E+08 .28695E+0814. .91204E3-01 -.17377E+08 -.28695E+0815. .82218E-01 -.13772E+08 .39290E+0816. .822181E-01 -.13772E+08 -.39290E+0817. .46269+00 -.29978+08 .48594E+0818. .0687E+00 -.13574E+08 .48594E+08

NPOIES = 18 NDECI = 12NBEGN = 2 F2NYQ = 48.95 MHzNPTS = 42 RMS ERROR = 7.23%

54

/I

NSWC/WOL TR 77-99

Table 5.8. Run 4

Magnitude Real Part Imaginary Partof Residue of Pole Location of Pole Location

(ma) (nepers) (Hz)

1. .27729E+01 -.17377E+08 .40765E+072. .27729E+01 -. 17377E+08 -.40765E+073. .15631E+01 -.19830E+08 .71158E+074. .15631E+01 -. 19830E+08 -.71158E+075. .40284E+01 -.79235E+07 .14295E+086. .40284E+01 -.79235E+07 -.14295E+087. .13182E+01 -.97781E+07 .17612E+088. .13182E+01 -.97781E+07 -.17612E+089. .36972E+01 -.33842E+08 .21366E+08

10. .36972E+01 -.33842E+08 -.21366E+0811. .13981E+01 -.14187E+08 .23880E+0812. .13981E+01 -.14187E+08 -. 23880E+0813. .11507E+00 -.16169E+08 .38732E+0814. .11507E+00 -.16169E+08 -.38732E+0815. .12225E+00 -.18730E+08 .46431E+0816. .12225E+00 -.18730E+08 -.46431E+08

NPOLES = 16 NDECI = 12NBEGN = 2 F2NYQ = 48.95 MHzNPTS = 42 RMS ERROR = 8.79%

55

L o . . - . .. .. .. ...... ... - - e ' . .

NSWC/WOL TR 77-99

aUcoD-

-

0

ta

z

CC

j- 0

z 0Cc

0

I-.

z

U7 U-

IIO&13 SWftU IN3:)U3d

56

Sam "M

NSWC/WOL TR 77-99

CHAPTER 6

CONCLUSIONS AND RECOMMENDATIONS

6.1 CONCLUSIONS

This work has shown that Prony's method is a possibleexperimental data reduction technique. The Nyquist CriterionConstraint method proposed, provides a limit on the possible numberof real poles in a system within a given bandwidth. The lower limitis provided by the length of the waveform recorded, and the upperlimit of the bandwidth is set by the dominant frequencies in thewaveform with respect to a Nyquist rate. The process of windowingcan be used with some success to try to eliminate the curve fitpoles but in general all poles shift about from run to run. Theseshifts can be observed mostly in the real part of the pole location.Van Blaricum1 7 showed that the real part is more susceptible to noise,by making many runs with several waveforms all corrupted by variouslevels of "random" noise, from 2 dB to 49 dB signal to noise ratios.Using Prony's method on response waveforms taken aboard a ship istherefore a questionable proposition at this time. It would bevery useful for the cleaner, simpler waveforms that can berecorded on or near the mast and its cables; but for the analysisof scattered fields as done here, the waveforms are too noisy andcause great inaccuracy and concern. Identification of some of the realpoles can be accomplished by simple calculations and the locationsof the others must for now be simply accepted as given in the runs.There still is no good way to eliminate the curve fit poles fromthe results. In its present form this method provides usefulpoles of transient responses and certainly provides an excellentway to store data in about 10% to 20% of space required for a Kdigitized waveform.

6.2 RECOMMENDATIONS

Noise reduction should be the primary concern. Some aspects ofit could include meticulous digitization practices, a study of thenumerical procedures, their contributions to and effects upon noisein the waveform. Examples are, deconvolution, types of filtering,and transformations. Some preliminary work has already been done.

57

- '"' i - ' ... .. .. " l ' :-- II II~ [ II .. . . .. ... .. .

NSWC/WOL TR 77-99

but none as yet seems to be a good technique 1 '' 3 5 The SEMPEXcode should be trimmed down; it should be possible to cut down thesize of needed core by tens of K. This would provide enough space

*for the calculation of more poles as was seen necessary from theexperiments dealing with the scattering from the sewer pipe

*(Appendix C). The possibility of averaging many runs togethershould be done numerically on the computer. In this way it would bepossible to specify the size and shape of the area in the s-planewithin which the poles will be averaged. For instance, ellipsesshould be used instead of circles, since it is known that the realpart of the pole is subject to more shifting.

There is certainly great room for improvement in this area sincenot much work has been accomplished in methods for the identificationof the real poles of a system. First the Nyquist Criterion should befurther examined for usefulness, especially in cases where theresponse is clearly bandlimited in frequency. Also there may be some

*answers in observing the effects of sub-Nyquist rate sampling in the*pole locatis. The idea of forcing Prony's method to S8ntain some

known poles has already been proposed by Van Blaricum ,by forcingthe response function to contain the poles of the driver. The Pronymethod should also be examined for use with Class II coefficients,and perhaps in the description of the ignored entire function; thesetwo elements should also be examined in depth for use in betterwaveform representation. A study of pole movement should beconsidered for the case where finite objects are brought into theproximity of each other.

Singularity extraction from the time domain response waveformhas great promise, for its applications are very powerful, but theroot of all evil - the method's sensitivity to noise, must be dealtwith before significant progress can be made in the process ofcorrectly indentifying the real poles of the excited system.

58

NSWC/WOL TR 77-99

REFERENCES

1. IT Research Institute, Defense Nuclear Agency EMP AwarenessCourse Notes, I)NA 2772 T, Aug 1971.

2. A. S. Leveckis, "EMP Coupling Through Apertures," Bachelor'sThesis, MIT, May 1976.

3. C. E. Baum, "The Singularity Expansion Method," in TransientElectromagnetic Fields, L. B. Felsen, Ed. Springer, New York,1976, Chapter 3.

4. W. C. Emberson and R. H. Hueneman, NOL431-69001, "PreliminaryStudy of Time-Domain Measurement of Antenna Parameters,"Jun 1970.

5. 11. C. Pocklington, "Electrical Oscillations in Wires," Proc.Cambridge Phil. Soc., vol. 9, pp. 324-332, 1897.

6. C. W. Oseen, "Uber die elektischen Schwingungen an dunnenStaben," Arkiv Mat. Astr. Fysik. Vol. 9, no. 30, pp. 1-27, 1914.

7. E. LHallen, "Theoretical Investigations into the Transmittingand Receiving Antennae," Nova Acta Reg. Soc. Sci. Upsaliensis,vol. 11, no. 4, pp. 1-44, 1938.

8. K. K. Mei, "On the Integral Equations of Thin Wire Antennas,"I1I11 Trans. on Ant. and Prop., pp. 374-378, May 1965.

9. L. Marin, "Natural Modes of Certain Thin Wire Structures,"

Interaction Note No. 186, AFWL, Albuerque, NM, Aug 1974.

1o. L. Marin, "Natural Mode Representation of Transient Scatteringfrom Rotationally Symmetric Bodies and Numerical Results forthe Prolate Spheroid," Interaction Note No. 119, AFWL,Al huquerque , NM, Sep 1972.

11. !. K. Miller, A. J. Poggio, and G. J. Burke, "An IntegroDifferential Technique for the Time-Domain Analysis of ThinWire Structures, I. Numerical Results," in J. of Comp. Phy.,.,vol. 12, pp. 24-48, 1973.

59

NSWYC/WOL TR 77-99

12. C. F Bauim, ":Elec t roma gnet ic T ran s ienit I nt eract ionl w ith1 Obect swith Emtphas is Oin Finite Size Objects, and Some Aspects ofFrans jent Plulse Production,'' Spring 1972 IIRSI Meeting,Washington, 1). C. , 1972.

13. F. Tesche , ''Oi tile Ana irs is of Scat tering and Antenna Probl emsUsigthe Singu.lar itv Fxpans ion Trechnique," 11113 Trans. APS,

No. I1, 1973

14 R . arrigton , Fielid Comput at ion br Momeont Methods , TheMac~lillan (C0., New York, 1968.

15 R. V. Chur11chill, Complex Variables and Applications, McGraw-HIill (Co., New York , 1900

10. NM. L. Van Blaricum and R. Mi ttra , ''A Technique for Fxtractinithle Poles and Res idues of a System lDirectlyv from Its TransijentResponse,'' Interact ion Note 245, AFIV , Albuiquerque, NMI, Feb 1975 .

17 M . L. Van Bia ricum , ''Techniques for Extract ing thle ComplexResonances of a Sys ter Direct ly from'its Trans jent e oe,PhDl Dissertation, Uinivers itv of Illinois, 1970.

18. R. Prony, ''Issai Experimeontal et Aayique sur les lois de laIDilzitail ite do Eluides Elastiques et stir Cels del la ForceE Xpan S iV ye e 1,a VapueI01 1.1e I' al1kool1, a Di f ferent es Temperatures,"

.1 . ' Ecol e Polytech, vol. 1, no. 2, pp. 2-T7 , Pari x, 1795

19 . 1F. B. Iidebrand , Introduct ion to Numerical Analvs is , McGraw-IHIill Co. , N ew YorIk, 197 4, pp., 45S7 - 4 o

20. 1). . lawrenice, ''Polynomial Root Finder,'' ELI., ILivermore , CA,V IC Rep). (7212- 001 , Dec 1 90o.

I1. M. S. Corriiigton,''ip i fied Calculation of* Transient Responlse,P'roc . 11:11, vol. 53, Mar 19o5 , pp. 287-292.

2.K. I_. Su, Tiine-Doma in Svnthies is of- Linear Networks, PrenticeHaii nliF 1TTs7,, 19-1 , pp. -'3-2o-.

N3 .N. Br i tt i ng ham , V . K. M1iller I,, and .J . L. Will1ows , ''The[i.,r iva t ionl o f Sim11ple Pole0S inl a T ranls fer 1un1c t ionl from Real1 -V reQqke uc \v In f'ormlat io Oi '' 11CRI, - ;52050o , ,ll. , 1, irve r-mo e , CA , Apr 19 760

- 1. . \. INr i tt i lig I, . K. M ille0r , and J1. L.. W ifIIows ''"TheOc r i va t ionl o f s impleNI Pole 10;n a T ran S fecr 171unct ionl I- om1 Real1 -IrequenlCIcv InI for '1mat ion11, Par It 2': Re'su It s from071 Rea" 1 l M Da t a ,'C iR.I - 5 'I18 , 1 , , 1. i v re l)r or , (CA, .\u: 109o.

- 1.I. N . Br itV ch m , 1. K . \1il H e r, and J. l'I.. W ilson , "''TheDcri i ati en otfSim Po I e in1 a 1 ran s1 fe r v nnc t io f)l 'rom Rea I -

I rce nec it InO Co ia t i onl Part I , t CR 321 v I, LI ,I ve re LA

'Tjll

NSWC/WOL TR 77-99

2o. D. L. Moffatt and R. K. Mains, "Detection and Discrimination ofRadar Targets," IEEE Trans. APS, vol. 23, no. 3,- 1975, pp. 358-367.

-. CG. Carrier, M. Krook, and C. Pearson, Functions of a ComplexVariable, Mc(raw-lill, New York, 1966.

S. V. I . Uaurn, "The Singularity Expansion Method," in TransientI 1cctro na-netic Fields, 1.. B. Felsen, ed., Springer, New York,1 , halter 3.

I LscheC, ''Appiication of the Singularity Expansion Methodto \nalvs is of Impedance Loaded Linear Antennas," Sensor and" :uIlt ioni .Note 177, AFI'L, Albuquerque, NM, May, 1973.

3). K. Iage r, II. C. Hudson, and A. 3. Poggio, "User's Manualfor SI MI\ .'" I.LL,, livermore, CA, 1970.

31. F. B. lildebrand, Introduction to Numerical A.,alvsis, McGraw-lill CO., New York, 1974, IrP. 545-549.

32. 1). L. lager, A. J. Poggio, and HI. G. Hudson, "Suipressing SomeNoise Difficulties in Prony "rocessing," presented at USNC/URSIConference, Amherst, MA , 197(,.

33. R. J. Ilaislmaier, "The llectromagntic Pulse Radiation Envi ronmentSimulator for Ships (EMPRIESS)", presented at NATO EMC Workshop,Naval .\ii Test Center, Patuxent, MD, 190.

34. F. J. Nicosia, "Requirements, Tasks, and Attributes of tEMIP DataMIanagement,' NSW C/;OL/TR 5- 25, Fe 10-5.

35. 1. G. Dudley, "Random errors in Prony's MIethod," presented attUSNC/tJRSI Conference, Amherst, NMA, 1) 7.

.. 1. L. Van Blaricum, "A Method for the Analysis of EMNP SimulatorData,' MRC , Santa Barbara, CA, Dec 1970.

{W(A}

a.l

NSWC/WOL TR 77-99

APPENDIX A

COMPARISON OF THE T[ME AND FREQUENCY DOMAIN PRONY TECHNIQUES

Time Domain Frequency Domain

N N

s -jjr q_ ,Go .t ~' R

1 S s i

ear . equation N Ni

(suoerscript de- F(s) =I s-s. - (s-s.)notes the absence 1..3jI

of the i~j tern)

N N

Introduce poly- Z. ;,ai ; = aI i s i

notal! eynm -sllon ;= I:=-i--O 1

N-i

X= e 3 bi b T' jO

Samnle data fp F. F(S Kko,k

(note: equally s'-ced (not,: sv.7ples can 1esnples) arbitraxily spaced)

DE-velop linaar N-1 N-ir 0-'sya .. for 'ki~ai r (e...a.-N. .b.) = N?Slste.nOT fl. a 1polynomia b ir i - kcoe- fic lents

N Iiki" fki . k-

i = 0 ..... ,-N N!<i- s; i 0,...,

(b

b. -- R.b

r-1-01 un- (system has 2Nj real un-

kno;ns and K+! 2") kno-,is and M+! N)

:ilin1;s. N Nron "'a. 0 a =

X N

fro I F" V ki- i. i :

03- i

63

NSWC/WOL TR 77-99

APPENDIX B

CHARACTERISTICS OF THE EMPRESS ELECTRIC FIELD

Given a cylindrical coordinate system (r,e,z) with r = z = 0at the pulser, the components of the vertically polarized electricfield, as measured with the B-dot sensor, at the anchor point in theinteraction area are:

Field Strength Percentage ofMeasurement (kV/m) Principal Component

2.7 100%

E .174 6.4%

L .27 10%r

The peak electric field versus distance from EMPRESS as measuredwith the B-dot sensor are: (see Figure B.1)

Peak Field Strength DistanceLocation (V/m) (m)

Pl Anchor Point 2700 350

02 Near Bridge 1000 1100

43 Off Solomons Pier 420 2200

l4 Buoy - 8 270 3300

05 Buoy - 6 130 4100

65

• "" -" I II III I . . . . . . I1| - - I -- - II .. .

NSWCIWOL TR 77-99

z0

z <-

z <JDl 4n

Lux c

u

22Lui

x 02 -~cc

o r.

0) .1 0.0CL

CLu

0 0Lu

cn-

L-U-

66

NSWC/WOL TR 77-99

APPENDIX C

RESULTS OF THE FOUR TEST CASES

Many runs were made of the four test waveforms. Here, arepresented the results. Pictured are: the waveforms, the rms errorof the reconstructed waveform and the pole plot arrived at byaveraging the results of the runs. As can be seen, some poles canbe confirmed initially by a Fourier transform and later in theirrecurrent appearances from test case to test case. RHP poles couldnot be forced out in the runs of Test #3 and #4. This is due to theextra poles that are introduced into the system when the sewer pipeis presented as a scatterer in the field. An interesting note isthat the 4.5 iH: pole of the pulser vanished when the AS 1018receiving antenna was on the shadow side of the cylinder, as inTest #4.

The rest 42 waveform has already been analyzed in Tables 5.5through 5.8 and :iures 5.( and 5.7 here in addition is presentedthe pole plot of- the waveform.

67

NSWC/WOL TR 77-99

30.0-

20.0

E> 1t0.0

0.07.004 14.009 21.013 2\8 .0 17 35.021

-00SECONDS X 10-8

-10.01

FIG. C.1 TEST NO. 1 WAVEFORM

68

NSWC/WOL TR 77-99

- 100.0

x

0 (MHz)

-10.0

I I I 1.0-8.0 -7.5 -7.0 -6.5 -6.0 -5.5

REAL PART OF POLE LOCATION (LOG(NEPERS))

X POLE IDENTIFIED BY AVERAGING SEVERAL RUNS SIZE OF MARK RELATESTO THE UNCERTAINTY OF LOCATION

SREAL POLES CONFIRMED BY INDEPENDENT FOURIER TRANSFORM

FIG. C.2 POLE PLOT OF TEST NO. 1 WAVEFORM

69

" 4 . . .. . . .

NSWC/WOL TR 77-99

100

Lu

0

I~10.1

10-17 0

FRQECY(z

07

NSWC/WOL TR 77-99

100.0

IX

xX (MHz)

10.0

I I I I I 1.0-7.5 -7.0 -6.5 -6.0 -5.5

REAL PART OF POLE LOCATION (LOG(NEPERS))

> POLE IDENTIFIED BY AVERAGING SEVERAL RUNS

POLE CONFIRMED BY EITHER FOURIER TRANSFORMOR APPEARANCE IN OTHER POLE PLOTS

FIG. C.4 POLE PLOT OF TEST NO. 2 WAVEFORM

71

NSWC/WOL TR 77-99

80.0

40.0

0.0 "

-40.0 SECONDS X 10'8

-40.0

F6%. C.5 TEST NO. 3 WAVEFORM

72

NSWC/WOL TR 77-99

0

C--0

0

0

'a

CR

00c;~~ c;L

UOUH3SWU I33Ua

73a

NSWC/WOL TR 77-99

-100.0

X

7<K

(MHz)

10.0

| | [ I1.0-7.7 -7.3 -6.8 -6.4

REAL PART OF POLE LOCATION (LOG(NEPERS))

X POLE IDENTIFIED BY AVERAGING SEVERAL RUNS

I& POLE CONFIRMED BY EITHER FOURIER TRANSFORMEDOR APPEARANCE IN OTHER POLE PLOTS

FIG. C.7 POLE PLOT OF TEST NO. 3 WAVEFORM

74

/ II II I .. . . ll "1"111 I . .. ... .. ..

NSWCANOL TR 77-99

80.0-

40.0-

)4. SECONDS X 10-7

-80.0L

FIG. C.8 TEST NO. 4 WAVEFORM

75

NSWC/WOL TR 77-99

- 100.0

X

AK (MHz)

x - 10.0

L I I I I , 1.0-7.5 -7.0 -6.5 -6.0 -5.5

REAL PART OF POLE LOCATION (LOG(NEPERS))

X POLE IDENTIFIED BY AVERAGING SEVERAL RUNS

POLE CONFIRMED BY EITHER FOURIER TRANSFORM

OR APPEARANCE IN OTHER POLE PLOTS

FIG. C.9 POLE PLOT OF TEST NO. 4 WAVEFORM

76

NSWCIWOL TR 77-99

Lac.,

0

LU-j0

LO zNl

ca

0

N A wLU

0LU

Lo >

CdLU

o I

LA.

HOHUi3 SIAJ I N33U~d

77

NSWC/WOL/TR 77-99

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NSWC/WOL/TR 77-99

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